#  Time-Frequency Analysis
Lecture Notes

Markus Faulhuber  NuHAG, Faculty of Mathematics, University of Vienna

(Summer Term 2021
Version: )

These lecture notes accompanied the course Time-Frequency Analysis given at
the Faculty of Mathematics of the University of Vienna in the summer term
2021. The material is suitable for an advanced undergraduate course in
mathematics or a mathematics class for PhD students. Besides standard linear
algebra and calculus only some basics from functional analysis are needed. A
course in Fourier analysis may be of advantage, but is not needed. The course
contained 4 academic units per week. The appendices and Section 11 were not
presented in class.

I gratefully acknowledge the feedback from my students throughout the course
and for pointing out several typos in the lecture notes. In particular, the
comments of L. Köhldorfer, F. Moscatelli, I. Shafkulovska, and E. Stefanescu
helped to improve these lecture notes.

###### Contents

  1. Some Engineering Terminology
  2. Notation and Recap
    1. Notation
    2. The Fourier Transform
    3. Fourier Series
  3. 1 Basic Concepts in Time-Frequency Analysis
  4. 2 The Short-Time Fourier Transform
  5. 3 Quadratic Time-Frequency Representations
    1. 3.1 The Spectrogram
    2. 3.2 The Rihaczek Distribution
    3. 3.3 The Ambiguity Function
    4. 3.4 The Wigner Distribution
  6. 4 The Poisson Summation Formula
    1. 4.1 The Whittaker-Nyquist-Kotelnikov-Shannon Sampling Theorem
  7. 5 Uncertainty Principles
  8. 6 Gabor Systems and Frames
    1. 6.1 Frames
    2. 6.2 Gabor Systems
    3. 6.3 Gabor Systems over Lattices and Point Sets
  9. 7 The Symplectic and the Metaplectic Group
    1. 7.1 The Symplectic Group
    2. 7.2 The Heisenberg Group
    3. 7.3 Representation Theory
      1. 7.3.1 The Stone-von Neumann Theorem
    4. 7.4 The Metaplectic Group
  10. 8 Function Spaces
    1. 8.1 Modulation Spaces
      1. 8.1.1 Time-Frequency Analysis of Distributions
      2. 8.1.2 Properties of Modulation Spaces
    2. 8.2 Feichtinger’s Algebra
    3. 8.3 Wiener Amalgam Spaces
    4. 8.4 The WKNS Sampling Theorem Re-Visited
  11. 9 The Zak Transform and Density Principles
    1. 9.1 The Zak Transform
    2. 9.2 The Balian-Low Theorem
    3. 9.3 Density of Gabor Frames
  12. 10 The Bargmann Transform
  13. 11 The Frame Set of a Window Function
    1. 11.1 The Frame Set of a Gaussian
      1. 11.1.1 The Theorem of Lyubarskii and Seip and Wallstén
    2. 11.2 The Frame Set in Higher Dimensions
  14. A Appendices
    1. A.1 More Uncertainty Principles
    2. A.2 Hilbert-Schmidt Operators
    3. A.3 Neumann Series
    4. A.4 Toeplitz Matrices and Laurent Operators
    5. A.5 Proof of Hudson’s Theorem
    6. A.6 Caratheodory’s Inequality
    7. A.7 Tensor Hilbert Spaces

Hitherto communication theory was based on two alternative methods of signal
analysis. One is the description of the signal as a function of time; the
other is Fourier analysis. Both are idealizations, as the first method
operates with sharply defined instants of time, the second with infinite wave-
trains of rigorously defined frequencies. But our everyday experiences —
especially our auditory sensations — insist on a description in terms of both
time and frequency.

– [14] D. Gabor. Theory of Communication, Journal of the Institution of the
Electrical Engineers, 93(26):429–457 (1946).

Short biography. Dennis Gabor (originally Dénes Gábor, June 5, 1900 – February
8, 1979) was born in Budapest, Austria-Hungary at that time. He was an
electrical engineer and physicist. After gaining his doctorate from the
Charlottenburg Technical University in Berlin, now known as the Technical
University of Berlin, he worked for Siemens AG. In 1933 he left Nazi Germany,
as he was considered to be Jewish. Britain invited him to work at the
development department of the British Thomson-Houston company (BTH) and he
also became a British citizen. In 1947 he invented holography, for which he
was awarded the Nobel Prize in Physics in 1971:

“For his invention and development of the holographic method.”

https://www.nobelprize.org/prizes/physics/1971/gabor/biographical/

Information. The main sources for these lecture notes are the textbooks by
Folland [13], de Gosson [16] and Gröchenig [19]. Whenever a lack of references
occurs, one of these sources may be consulted.

## Some Engineering Terminology

In this section we are going to introduce some important notions from sampling
theory. We mainly sum up the introduction from [55] and quote some passages
verbatim. One aim of this section is to introduce some engineering terms and
‘translate’ them into mathematical terms.

The most important term in communication engineering and signal analysis is
the term signal. Actually, there are several types of signals which we need to
distinguish, namely continuous (analog) signals and discrete (digital)
signals. Whenever we only speak of a signal, we mean the first type.
Mathematically, a signal is nothing but a (continuous) function, which is why
we may use the words signal and function interchangeably. From a physical
point of view, a signal $f(t)$ may represent the voltage difference at time
$t$ between two points in an electrical circuit or the pressure of a sound
wave. A digital signal, on the other hand, is just a sequence of numbers
$(c_{k})_{k\in\mathbb{Z}}$.

A prototype task in communication might be to convert an analog signal into a
digital signal, transmit the digital signal to a receiver and convert it back
to an analog signal (maybe even the original one). By processing a signal $f$
we mean operating on it in some fashion and usually also require that this
operation is reversible. Mathematically, this means we apply some
transformation (linear or not) to the function $f$ which we require to be
invertible.

Lastly, we want to transmit information in a “cheap” fashion, meaning that we
want to extract as little as possible data from a signal in order to
reconstruct it at the receiver. This is where sampling theory begins.

By sampling a signal $f$, i.e., evaluating $f$ at discrete points, we apply a
(linear) transformation to convert an analog signal into a digital signal.
Ways of converting the discrete signal back to the continuous signal usually
go under the name sampling theorem. We will study a specific sampling theorem
(actually the sampling theorem par excellence) later on. There, it will be of
importance that the signal is band-limited. This simply means that the so-
called spectrum of the signal has finite support. On the other hand, a signal
which only lasts for a finite duration is called time-limited. The classical
uncertainty principle tells us that it is impossible for a signal to be both,
time- and band-limited. This would pose great problems for digital
communication, but (luckily) signals can be almost time- and band-limited at
the same time.

Throughout the manuscript, we will also assign more physical and engineering
terminology to certain mathematical concepts.

## Notation and Recap

One of the most essential tools in time-frequency analysis is the Fourier
transform of a function $f$. The values $f(x)$ represent the temporal behavior
of a signal and its Fourier transform $\widehat{f}(\omega)$ describes the
amplitudes of the occurring frequencies.

The variables $x$ and $\omega$ will usually be in $\mathbb{R}^{d}$. For the
special case $d=1$, $x$ is a point in time and $\omega$ is a specific
frequency. More generally, $x$ can be chosen from a locally compact Abelian
group $G$ and $\omega$ is then a character from the dual group $\widehat{G}$.
In the case of $\mathbb{R}^{d}$, the dual group $\widehat{\mathbb{R}^{d}}$ is
isomorphic to $\mathbb{R}^{d}$ and we will not distinguish between them.
Another prominent example of a locally compact Abelian group is the
$d$-dimensional torus $\mathbb{T}^{d}$ which has dual group $\mathbb{Z}^{d}$
(and vice versa).

We start with introducing some notation and a collection of essential results.

### Notation

Vectors in $\mathbb{R}^{d}$ are always considered to be column vectors;

$x=\begin{pmatrix}x_{1}\\\ \vdots\\\ x_{d}\end{pmatrix}$ (0.1)

However, for convenience reasons we will usually write

$x=(x_{1},\ \ldots,\ x_{d}).$ (0.2)

Another notation we will use is the following

$x^{2}=x\cdot x=x^{T}x.$ (0.3)

Hence, $\cdot$ denotes the standard Euclidean inner product on
$\mathbb{R}^{d}$ and $x^{T}$ is, of course, the transpose of $x$. The
Euclidean norm is denoted by $|.|$ and given by

$|x|=\sqrt{x\cdot x}\ .$ (0.4)

If $M\in\mathbb{R}^{d\times d}$ is a square matrix, we will also use the
notation

$Mx^{2}=x\cdot Mx\quad\text{ and }\quad M^{-T}=(M^{-1})^{T}=(M^{T})^{-1}.$
(0.5)

In the latter case, $M$ needs to be invertible (of course).

The complex vector $z\in\mathbb{C}^{d}$ consists of a real and imaginary part
denoted by

$z=\Re(z)+i\ \Im(z).$ (0.6)

We will use the same notation in $\mathbb{C}^{d}$ as in $\mathbb{R}^{d}$;

$z^{2}=z\cdot z=z^{T}z,\quad z\in\mathbb{C}^{d}.$ (0.7)

However, note that this is not the inner product on $\mathbb{C}^{d}$. The norm
of an element in $\mathbb{C}^{d}$ is given by

$|z|^{2}=\overline{z}\cdot z=z^{*}z,\quad z^{*}=\overline{z}^{T}.$ (0.8)

The integral

$\int_{\mathbb{R}^{d}}f(x)\,dx=\int_{\mathbb{R}}\dots\int_{\mathbb{R}}f(x_{1},\
\ldots,\,x_{d})\,dx_{1}\dots dx_{d}$ (0.9)

is the Lebesgue integral of $f$ on $\mathbb{R}^{d}$ and the measure of a
measurable set $\Omega\subset\mathbb{R}^{d}$ is given by

$|\Omega|=\int_{\mathbb{R}^{d}}\raisebox{2.0pt}{$\chi$}_{\Omega}(x)\,dx,$
(0.10)

where $\raisebox{2.0pt}{$\chi$}_{\Omega}$ is the indicator function on the set
$\Omega$. For a number $1\leq p<\infty$, the $L^{p}$-norm, or simply $p$-norm,
of a function $f$ is denoted by

$\lVert f\rVert_{p}=\left(\int_{\mathbb{R}^{d}}|f(x)|^{p}dx\right)^{1/p}.$
(0.11)

The Banach space of all functions with finite $p$-norm is denoted by
$L^{p}(\mathbb{R}^{d})$. Of course, a function is then only defined up to an
equivalence class and $f$ is actually a representative of the class. The usual
adjustment for the case $p=\infty$ is

$\lVert f\rVert_{\infty}=\operatorname*{ess\,sup}_{x\in\mathbb{R}^{d}}|f(x)|.$
(0.12)

At this point we re-call Hölder’s inequality. For $f\in L^{p}(\mathbb{R}^{d})$
and $g\in L^{q}(\mathbb{R}^{d})$, with $\frac{1}{p}+\frac{1}{q}=1$, we have

$\lVert f\,g\rVert_{1}\leq\lVert f\rVert_{p}\lVert g\rVert_{q}.$ (0.13)

For $p=2$, we actually get a Hilbert space with inner product

$\langle
f,g\rangle=\int_{\mathbb{R}^{d}}f(x)\,\overline{g(x)}\,dx,\qquad\langle
f,f\rangle=\lVert f\rVert_{2}^{2}.$ (0.14)

In particular, Hölder’s inequality for the case $p=q=2$ now implies the
Cauchy-Schwarz inequality by using the triangle inequality.

$|\langle
f,g\rangle|=|\int_{\mathbb{R}^{d}}f(t)\overline{g(t)}\,dt|\leq\int_{\mathbb{R}^{d}}|f(t)g(t)|\,dt=\lVert
fg\rVert_{1}\leq\lVert f\rVert_{2}\lVert g\rVert_{2}.$ (0.15)

A signal $f\in L^{2}(\mathbb{R}^{d})$ is said to have finite energy, as
$\lVert f\rVert_{2}^{2}$ is considered as the energy of a signal. Thus, the
Hilbert space $L^{2}(\mathbb{R}^{d})$ is also called the space of finite-
energy signals.

### The Fourier Transform

The Fourier transform is usually first defined for functions in
$L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ and then extended to all of
$L^{2}(\mathbb{R}^{d})$ by a density argument. We will skip the technical
details here and define the Fourier transform right away for functions in
$L^{2}(\mathbb{R}^{d})$. We stress the fact that there are several different
normalizations for the Fourier transform and that the choice of normalization
affects all further normalizations which we need to make in the sequel.

######  Definition 0.1.

The Fourier transform of $f\in L^{2}(\mathbb{R}^{d})$ is given by

$\mathcal{F}f(\omega)=\widehat{f}(\omega)=\int_{\mathbb{R}^{d}}f(x)\,e^{-2\pi
i\omega\cdot x}\,dx.$ (0.16)

In engineering terms, the function $\widehat{f}$ is called the amplitude
spectrum of the signal $f$. The notation $\mathcal{F}f$ reflects the fact that
the Fourier transform is a linear operator acting on a function space, whereas
the notation $\widehat{f}$ refers to the fact that the Fourier transform of an
$L^{2}(\mathbb{R}^{d})$ function is again a function (in
$L^{2}(\mathbb{R}^{d})$).

A signal $f$ is completely described by its amplitude spectrum $\widehat{f}$
(and vice versa) and can be written as a continuous superposition of its
spectral values;

$f(x)=\int_{\mathbb{R}^{d}}\widehat{f}(\omega)e^{2\pi ix\cdot\omega}\,d\omega$
(0.17)

We identify the above formula as the inverse Fourier transform
$\mathcal{F}^{-1}$. For the formula to hold point-wise, $\widehat{f}$ needs to
be integrable and $f$ needs to be continuous and integrable (continuity is
necessary for the point-wise statement and integrability to assure that we can
define $\widehat{f}$). The inversion formula holds on $L^{2}(\mathbb{R}^{d})$
(a.e.) by using the inversion formula on the dense subspace
$L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ and a density argument.

For an integrable function $f\in L^{1}(\mathbb{R}^{d})$, we note the fact

$\lVert\widehat{f}\rVert_{\infty}\leq\lVert f\rVert_{1},$ (0.18)

which is refined in the following lemma.

######  Lemma 0.2 (Riemann-Lebesgue).

If $f\in L^{1}(\mathbb{R}^{d})$, then $\widehat{f}$ is uniformly continuous
and

$\lim_{|\omega|\to\infty}|\widehat{f}(\omega)|\to 0.$ (0.19)

The next result shows that the Fourier transform is actually a unitary
operator on $L^{2}(\mathbb{R}^{d})$.

######  Theorem 0.3 (Plancherel).

For $f\in L^{2}(\mathbb{R}^{d})$ we have

$\lVert f\rVert_{2}=\lVert\widehat{f}\rVert_{2}.$ (0.20)

More generally, we have Parseval’s formula.

######  Theorem 0.4 (Parseval).

For $f,\,g\in L^{2}(\mathbb{R}^{d})$, we have

$\langle f,g\rangle=\langle\widehat{f},\widehat{g}\rangle.$ (0.21)

As a next result, we will give the extension of equation (0.18).

######  Theorem 0.5 (Hausdorff-Young).

Let $1\leq p\leq 2$ and let $q$ be such that $\frac{1}{p}+\frac{1}{q}=1$. Then

$\mathcal{F}\colon L^{p}(\mathbb{R}^{d})\to
L^{q}(\mathbb{R}^{d})\qquad\textnormal{ and
}\qquad\lVert\widehat{f}\rVert_{q}\leq\lVert f\rVert_{p}.$ (0.22)

######  Definition 0.6.

The convolution of two functions $f$ and $g$ is defined as

$f*g\,(x)=\int_{\mathbb{R}^{d}}f(y)g(x-y)\,dy$ (0.23)

We recall that convolution is taken to point-wise multiplication under the
Fourier transform.

$\displaystyle\mathcal{F}(f*g)(\omega)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}f(y)g(x-y)\,dy\right)e^{-2\pi
i\omega\cdot x}\,dx$ (0.24)
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}f(y)g(x-y)\,dy\right)e^{-2\pi
i\omega\cdot y}e^{-2\pi i\omega\cdot(x-y)}\,dx$ (0.25)
$\displaystyle\stackrel{{\scriptstyle\textnormal{Fubini}}}{{=}}\int_{\mathbb{R}^{d}}f(y)e^{-2\pi
i\omega\cdot y}\left(\int_{\mathbb{R}^{d}}g(x-y)e^{-2\pi
i\omega\cdot(x-y)}\,dx\right)\,dy$ (0.26)
$\displaystyle=(\mathcal{F}f\,\mathcal{F}g)(\omega).$ (0.27)

Also, we will use Young’s convolution inequality.

######  Theorem 0.7 (Young’s Inequality).

Let $1\leq p,q\leq r\leq\infty$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1$
and suppose $f\in L^{p}(\mathbb{R}^{d})$ and $g\in L^{q}(\mathbb{R}^{d})$.
Then

$\lVert f*g\rVert_{r}\leq\lVert f\rVert_{p}\lVert g\rVert_{q}.$ (0.28)

In particular, if $f,\ g\in L^{1}(\mathbb{R}^{d})$, then the convolution
satisfies

$\lVert f*g\rVert_{1}\leq\lVert f\rVert_{1}\lVert g\rVert_{1}\ ,$ (0.29)

We remark that (0.29) shows that $L^{1}(\mathbb{R}^{d})$ is a Banach algebra
under convolution and, by employing the Lemma of Riemann-Lebesgue, that the
Fourier transform maps $L^{1}(\mathbb{R}^{d})$ into a (dense) subalgebra of
$C_{0}(\mathbb{R}^{d})$.

The following theorem and its proof are taken from [13, App. A] and states
that the Fourier transform of a Gaussian is another Gaussian.

######  Theorem 0.8 (Fourier Transform of a Gaussian).

Let $M$ be a $d\times d$ matrix with real entries and the properties

$M^{T}=M\quad\text{ and }\quad M>0,$ (0.30)

where the second assertion means that $M$ is positive definite. Then, for any
$\omega\in\mathbb{R}^{d}$ we have

$\int_{\mathbb{R}^{d}}e^{-\pi Mx^{2}}e^{-2\pi i\omega\cdot
x}\,dx=\det(M)^{-1/2}e^{-\pi M^{-1}\omega^{2}}$ (0.31)

###### Proof.

The proof proceeds in 3 steps. First, we prove the case for $d=1$, then for
the case that $M$ is a diagonal matrix and, lastly, the general case stated in
the theorem.

Step 1. Let $d=1$ so $M$ is a real, positive scalar. Let

$I(\omega)=\int_{\mathbb{R}}e^{-\pi Mx^{2}}e^{-2\pi i\omega x}\,dx.$ (0.32)

Then, we are allowed to differentiate under the integral and we have

$\displaystyle I^{\prime}(\omega)$ $\displaystyle=\int_{\mathbb{R}}(-2\pi
ix)e^{-\pi Mx^{2}}e^{-2\pi i\omega
x}\,dx=\frac{i}{M}\int_{\mathbb{R}}\dfrac{d}{dx}(e^{-\pi Mx^{2}})e^{-2\pi
i\omega x}\,dx$ (0.33) $\displaystyle\stackrel{{\scriptstyle\text{Int.~{}by
parts}}}{{=}}-\frac{i}{M}\int_{\mathbb{R}}e^{-\pi
Mx^{2}}\dfrac{d}{dx}(e^{-2\pi i\omega x})\,dx=-\frac{2\pi\omega}{M}I(\omega).$
(0.34)

Therefore, we have that
$\frac{I^{\prime}(\omega)}{I(\omega)}=-\frac{2\pi\omega}{M}$, or
$\frac{d}{d\omega}\log(I(\omega))=-\frac{2\pi\omega}{M}$. Therefore,

$\log(I(\omega))=-\frac{\pi\omega^{2}}{M}+C,$ (0.35)

or, equivalently

$I(\omega)=e^{-\frac{\pi\omega^{2}}{M}+C}=C_{1}\,e^{-\frac{\pi\omega^{2}}{M}}.$
(0.36)

Now,

$C_{1}=I(0)=\int_{\mathbb{R}}e^{-\pi Mx^{2}}\,dx=M^{-1/2},$ (0.37)

which shows the result for the first case.

Step 2. Suppose $M$ is a $d\times d$ diagonal matrix. Then the integral under
consideration factors;

$\int_{\mathbb{R}^{d}}e^{-\pi Mx^{2}}e^{-2\pi i\omega\cdot
x}\,dx=\left(\int_{\mathbb{R}}e^{-\pi M_{1,1}x_{1}^{2}}e^{-2\pi
i\omega_{1}x_{1}}\,dx_{1}\right)\ldots\left(\int_{\mathbb{R}}e^{-\pi
M_{d,d}x_{d}^{2}}e^{-2\pi i\omega_{d}x_{d}},\,dx_{d}\right).$ (0.38)

Hence, the result follows by employing the result from the first step.

Step 3. Now, suppose $M$ is not a diagonal matrix. Since $M^{T}=M$ there is a
rotation $R$ such that $R^{T}MR=D$ is diagonal. Setting $x=Ry$ and using the
fact that $R^{T}=R^{-1}$, by step 2 we obtain

$\displaystyle\int_{\mathbb{R}^{d}}e^{-\pi Mx^{2}}e^{-2\pi i\omega\cdot
x}\,dx$
$\displaystyle=\underbrace{|\det(R)|}_{=1}\int_{\mathbb{R}^{d}}e^{-\pi(Ry)\cdot
M(Ry)}e^{-2\pi i\omega\cdot Ry}\,dy=\int_{\mathbb{R}^{d}}e^{-\pi
Dy^{2}}e^{-2\pi i(R^{T}\omega)\cdot y}\,dy$ (0.39)
$\displaystyle=\det(D)^{-1/2}e^{-\pi(R^{-1}\omega)\cdot
D^{-1}(R^{-1}\omega)}=\det(M)^{-1/2}e^{-\pi M^{-1}\omega^{2}}.$ (0.40)

∎

Another prominent function in this manuscript is the indicator function of the
interval $[-\frac{1}{2},\frac{1}{2}]$. Its Fourier transform is the
$\operatorname*{sinc}$ function, also called cardinal sine.

$\displaystyle\mathcal{F}\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega)$
$\displaystyle=\int_{\mathbb{R}}\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(x)e^{-2\pi
i\omega x}\,dx=\int_{-1/2}^{1/2}e^{-2\pi i\omega x}\,dx$ (0.41)
$\displaystyle=\frac{e^{-2\pi i\omega x}}{-2\pi
i\omega}\Big{|}_{x=-1/2}^{1/2}=-\frac{e^{-\pi i\omega}-e^{\pi
i\omega}}{2i\pi\omega}=-\frac{\sin(-\pi\omega)}{\pi\omega}=\frac{\sin(\pi\omega)}{\pi\omega}=\operatorname*{sinc}(\omega).$
(0.42)

We note that the value at 0 equals 1 (by using l’Hospital’s rule) and that the
$\operatorname*{sinc}$-function is not Lebesgue-integrable. However, it is a
finite energy signal (since
$\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}\in L^{2}(\mathbb{R})$).
Thus, $\operatorname*{sinc}$ possesses a Fourier transform as an
$L^{2}(\mathbb{R})$-function. Since it is an even function, its Fourier
transform and its inverse Fourier transform 111Note that the Fourier transform
of the $\operatorname*{sinc}$-function cannot be computed by real integration
methods, but that it can be computed using complex methods (using the homotopy
version of Cauchy’s integral theorem and the residue theorem). The advantage
of the complex method is that the singularity at $x=0$ (which is a removable
singularity) can be avoided by choosing a “roundabout” contour near 0.
coincide;

$\mathcal{F}\operatorname*{sinc}(\omega)=\mathcal{F}^{-1}\operatorname*{sinc}(\omega)=\mathcal{F}^{-1}(\mathcal{F}\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]})(\omega)=\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega).$
(0.43)

### Fourier Series

By

$e_{k}(t)=e^{2\pi ik\cdot t},\quad t\in\mathbb{T}^{d}$ (0.44)

we denote the complex exponential with frequency $k\in\mathbb{Z}^{d}$. We say
that $e_{k}$222More generally, if $\omega\in\mathbb{R}^{d}$, we say that
$e_{\omega}(t)=e^{2\pi i\omega\cdot t}$ is a pure frequency of frequency
$\omega$. However, note that, in general, $e_{\omega}$ need not be periodic
with period 1. is a pure (integer) frequency. Recall that the set of pure
integer frequencies, i.e.,

$\\{e^{2\pi ik\cdot t}\mid k\in\mathbb{Z}^{d}\\}$ (0.45)

is an orthonormal basis for $L^{2}(\mathbb{T}^{d})$ or $L^{2}([0,1]^{d})$. It
is easy to see that the system is orthonormal by a simple calculation (which
we perform only for $d=1$ for notational convenience, the $d$-dimensional case
follows by iterated integration.);

$\langle e_{k},e_{l}\rangle_{L^{2}(\mathbb{T})}=\int_{\mathbb{T}}e^{2\pi
ikt}e^{-2\pi ilt}\,dt=\int_{\mathbb{T}}e^{2\pi
i(k-l)t}\,dt=\begin{cases}1,&k=l\\\ 0,&\text{ else}\end{cases}$ (0.46)

By the Weierstrass approximation theorem, the trigonometric polynomials
$\sum_{k=-N}^{N}c_{k}e_{k}(t)$ are dense in $C(\mathbb{T}^{d})$ and,
therefore, in $L^{2}(\mathbb{T}^{d})$. Therefore, the set of pure integer
frequencies is an orthonormal basis for $L^{2}(\mathbb{T}^{d})$. As a
consequence, a function $f\in L^{2}(\mathbb{T}^{d})$ can be expanded as

$f(t)=\sum_{k\in\mathbb{Z}^{d}}c_{k}e^{2\pi ik\cdot t},$ (0.47)

with coefficients

$c_{k}=\langle f,e_{k}\rangle_{L^{2}(\mathbb{T}^{d})}.$ (0.48)

Plancherel’s theorem holds in this case as well;

$\lVert(c_{k})\rVert_{\ell^{2}(\mathbb{Z}^{d})}=\lVert
f\rVert_{L^{2}(\mathbb{T}^{d})}.$ (0.49)

Also, for $f\in L^{1}(\mathbb{T}^{d})\supset L^{2}(\mathbb{T}^{d})$333Note
that we do not have such an inclusion for $L^{p}(\mathbb{R}^{d})$-spaces
because the Lebesgue measure on $\mathbb{R}^{d}$ is not finite. the Lemma of
Riemann-Lebesgue holds;

$\langle f,e_{k}\rangle_{L^{2}(\mathbb{T}^{d})}\to 0,\quad|k|\to\infty.$
(0.50)

Furthermore, using the multi-index convention $|k|=k_{1}+\ldots+k_{d}$ now, if
we denote the space of trigonometric polynomials of order not larger than $N$
by

$T_{N}=\\{p_{N}(t)=\sum_{|k|=-N}^{N}c_{k}e^{2\pi ik\cdot t}\mid
t\in\mathbb{T}^{d},\,c_{k}\in\mathbb{C},\,N\in\mathbb{N}\\},$ (0.51)

then the truncated Fourier series is the best approximating element in $T_{N}$
for any $f\in L^{2}(\mathbb{T}^{d})$;

$\lVert f-\sum_{|k|=-N}^{N}\langle f,e_{k}\rangle
e_{k}\rVert_{L^{2}(\mathbb{T}^{d})}\leq\lVert
f-p_{N}\rVert_{L^{2}(\mathbb{T}^{d})},\quad\forall p_{N}\in T_{N}.$ (0.52)

Equation (0.47) admits the following interpretation. A periodic signal $f\in
L^{2}(\mathbb{T}^{d})$ can be decomposed into pure waves of integer
frequencies with amplitudes $(c_{k})_{k\in\mathbb{Z}^{d}}$. The amplitudes
$(c_{k})_{k\in\mathbb{Z}^{d}}$ are derived from taking linear measurements of
the signal $f$ with respect to the orthonormal system $\\{e_{k}\\}$.

A goal of time-frequency analysis, following the ideas in [14], is the
decomposition of non-periodic signals into “simple building blocks” similar to
(0.47), such that the occurring coefficients can be interpreted as the
amplitudes of the building blocks. In particular, the coefficients should
contain some information on how “strong” a certain frequency is at a certain
point in time.

##  1 Basic Concepts in Time-Frequency Analysis

We consider functions from the Hilbert space $L^{2}(\mathbb{R}^{d})$ and
define the following operators;

$T_{x}f(t)=f(t-x)\qquad\textnormal{ and }\qquad M_{\omega}f(t)=f(t)\,e^{2\pi
i\omega\cdot t}.$ (1.1)

These operators (illustrated in Figure 1) are usually called translation
operator and modulation operator, respectively. However, in the field of time-
frequency analysis we say $T_{x}$ is a time-shift by $x$ and $M_{\omega}$ is
called a frequency-shift by $\omega$. These operators are unitary on
$L^{2}(\mathbb{R}^{d})$, i.e.,

$\langle T_{x}f,T_{x}g\rangle=\langle f,g\rangle\qquad\textnormal{ and
}\qquad\langle M_{\omega}f,M_{\omega}g\rangle=\langle f,g\rangle.$ (1.2)

(a) A Gauss function. (b) A time-shifted Gauss function.

(c) A frequency shifted Gauss function (real part) with its Gaussian envelop.
(d) A time-frequency shifted Gauss function (real part) with its Gaussian
envelop.

Figure 1: The standard Gauss function $g_{0}(t)=2^{1/4}e^{-\pi t^{2}}$ and its
time-, frequency- and time-frequency-shifted copies (real part) $T_{2}g_{0}$,
$M_{2}g_{0}$, $M_{2}T_{2}g_{0}$, respectively.

In general, they do not commute. We have the canonical commutation relation

$M_{\omega}T_{x}=e^{2\pi i\omega\cdot x}\,T_{x}M_{\omega},$ (1.3)

which is at the heart of time-frequency analysis. Equation (1.3) shows that
time-shifts and frequency-shifts do not commute in general. However, they
commute up to a phase factor. A phase factor is a complex number of modulus 1
($|c|=1$, $c\in\mathbb{C}$). The commutation relation for time- and frequency-
shifts (1.3) will accompany us throughout the course. Sometimes the non-
commutativity may simply be ignored (e.g., because we are interested in
absolute values), other times the bookkeeping of the exponential factors might
be annoying but important and again other times the non-commutativity will be
crucial to obtain (beautiful) results right at the heart of time-frequency
analysis.

Equation (1.3) follows from a direct computation;

$\displaystyle M_{\omega}T_{x}f(t)=f(t-x)\,e^{2\pi i\omega\cdot
t}=f(t-x)\,e^{2\pi i\omega\cdot(t-x)}e^{2\pi i\omega\cdot x}=e^{2\pi
i\omega\cdot x}\ T_{x}M_{\omega}f(t).$ (1.4)

It is immediate from (1.3) that time-shifts and frequency shifts commute if
and only if $\omega\cdot x\in\mathbb{Z}$.

Furthermore, for any $1\leq p\leq\infty$, time- and frequency-shifts (and
hence time-frequency shifts) are isometries on $L^{p}(\mathbb{R}^{d})$, i.e.,

$\lVert M_{\omega}T_{x}f\rVert_{p}=\lVert f\rVert_{p}\ .$ (1.5)

Next, we study the behavior of $T_{x}$ and $M_{\omega}$ under the Fourier
transform. By a direct calculation, we will see that

$\mathcal{F}T_{x}=M_{-x}\mathcal{F}\qquad\textnormal{ and
}\qquad\mathcal{F}M_{\omega}=T_{\omega}\mathcal{F}.$ (1.6)

We start with the first assertion.

$\displaystyle\mathcal{F}(T_{x}f)(\xi)=\int_{\mathbb{R}^{d}}f(t-x)\,e^{-2\pi
i\xi\cdot t}\,dt=\int_{\mathbb{R}^{d}}f(t)\,e^{-2\pi
i\xi\cdot(t+x)}\,dt=M_{-x}(\mathcal{F}f)(\xi).$ (1.7)

The second formula follows analogously.

$\displaystyle\mathcal{F}(M_{\omega}f)(\xi)=\int_{\mathbb{R}^{d}}f(t)\,e^{2\pi
i\omega\cdot t}e^{-2\pi i\xi\cdot t}\,dt=\int_{\mathbb{R}^{d}}f(t)\,e^{-2\pi
i(\xi-\omega)\cdot t}\,dt=T_{\omega}(\mathcal{F}f)(\xi).$ (1.8)

This shows that the use of the word frequency-shift is justified for the
operator $M_{\omega}$.

By combining (1.3) and (1.6) we get

$\mathcal{F}M_{\omega}T_{x}=T_{\omega}M_{-x}\mathcal{F}=e^{2\pi i\omega\cdot
x}M_{-x}T_{\omega}\mathcal{F}$ (1.9)

The composition of $T_{x}$ and $M_{\omega}$ is called a time-frequency shift,
which we denote by

$\pi(\lambda)=M_{\omega}T_{x},\qquad\lambda=(x,\omega)\in\mathbb{R}^{2d}\,.$
(1.10)

In the sequel, we will also make use of other, familiar operators. We start
with the isotropic dilation operator, which we denote by $D_{a}$. It acts on a
function in the following way;

$D_{a}f(t)=a^{-d/2}f(a^{-1}t),\qquad a>0,\ t\in\mathbb{R}^{d}.$ (1.11)

The factor $a^{-d/2}$ makes the operator unitary. We are also interested in
the operator’s behavior under the Fourier transform. This will be helpful to
get an intuitive understanding for uncertainty principles, introduced later
on.

$\mathcal{F}D_{a}=D_{\frac{1}{a}}\mathcal{F}\ .$ (1.12)

This can easily be seen by the following calculation. First, note that
$\Phi_{a}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $t\mapsto a^{-1}t$ is a linear
diffeomorphism on $\mathbb{R}^{d}$ for $a>0$. Hence,

$\displaystyle\mathcal{F}(D_{a}f)(\omega)$
$\displaystyle=\int_{\mathbb{R}^{d}}a^{-d/2}f(a^{-1}t)e^{-2\pi i\omega\cdot
t}\,dt$ (1.13)
$\displaystyle=\int_{\mathbb{R}^{d}}a^{-d/2}f\left(\Phi_{a}^{-1}(a^{-1}t)\right)\underbrace{\left|\det\left(D\Phi_{a}^{-1}\right)\right|}_{=a^{d}}e^{-2\pi
i\omega\cdot\Phi_{a}^{-1}(t)}\,dt$ (1.14)
$\displaystyle=\int_{\mathbb{R}^{d}}a^{d/2}f(t)e^{-2\pi ia\omega\cdot
t}\,dt=D_{\frac{1}{a}}(\mathcal{F}f)(\omega).$ (1.15)

We will also encounter a more general (anisotropic) dilation operator, which
acts on functions in a similar way.

Furthermore, we note that the Fourier transform takes differentiation to
multiplication with a monomial and vice versa. By $\partial_{k}$, we denote
the partial derivative in the $k$-th component. Then

$\mathcal{F}\left(\partial_{k}^{n}f\right)(\omega)=(2\pi
i\omega_{k})^{n}\,\mathcal{F}f(\omega)\qquad\textnormal{ and
}\qquad\mathcal{F}\Big{(}(-2\pi
it_{k})^{n}f\Big{)}(\omega)=\partial_{k}^{n}(\mathcal{F}f)(\omega).$ (1.16)

It suffices to carry out the proof for $n=1$. First, assume that
$f\in\mathcal{S}(\mathbb{R}^{d})$ (the Schwartz space of rapidly decreasing
functions). Then, by using integration by parts, we see that

$\displaystyle\mathcal{F}\left(\partial_{k}f(t)\right)(\omega)$
$\displaystyle=\int_{\mathbb{R}^{d}}\partial_{k}f(t)e^{-2\pi i\omega\cdot
t}\,dt=\underbrace{f(t)e^{-2\pi i\omega\cdot
t}\Big{|}_{t=-\infty}^{\infty}}_{=0}-\int_{\mathbb{R}^{d}}f(t)(-2\pi
i\omega_{k})e^{-2\pi i\omega\cdot t}\,dt$ (1.17) $\displaystyle=(2\pi
i\omega_{k})\mathcal{F}f(\omega).$ (1.18)

The result for higher order derivatives follows by induction and the result
for $f\in L^{2}(\mathbb{R}^{d})$ follows by a density argument. For the second
assertion we use the differential quotient and denote the unit vector in the
$k$-th direction by $\mathbf{e}_{k}$.

$\displaystyle\lim_{\delta\to
0}\frac{\widehat{f}(\omega+\delta\mathbf{e}_{k})-\widehat{f}(\omega)}{\delta}$
$\displaystyle=\int_{\mathbb{R}^{d}}f(t)e^{-2\pi i\omega\cdot
t}\lim_{\delta\to 0}\frac{e^{-2\pi i\delta\mathbf{e}_{k}\cdot
t}-1}{\delta}\,dt$ (1.19) $\displaystyle=\int_{\mathbb{R}^{d}}f(t)e^{-2\pi
i\omega\cdot t}(-2\pi it_{k})\,dt.$ (1.20)

The exchange of integration and taking the limit is justified by the dominated
convergence theorem and for taking the limit we used l’Hospital’s rule. Again,
the result for $n>1$ follows by induction and the result for
$L^{2}(\mathbb{R}^{d})$ by a density argument.

##  2 The Short-Time Fourier Transform

The short-time Fourier transform is the fundamental tool in time-frequency
analysis. In a distributional sense, it can be seen as a generalization of the
Fourier transform, as we will see later. It aims to overcome the drawback that
$f(t)$ only gives temporal information of a signal $f$ while under the Fourier
transform we obtain only information about its frequency distribution
$\widehat{f}(\omega)$. We will start with an illustrative example of a multi-
component signal.

######  Example 2.1.

Let $f$ be the following multi-component signal.

$\displaystyle f(t)=\begin{cases}e^{4\pi it}+e^{8\pi it},&-3\leq t\leq-1\\\
e^{2\pi it},&-1\leq t\leq 1\\\ e^{2\pi it^{2}},&1\leq t\leq 4\\\
0,&\text{else}\end{cases}.$ (2.1)

(a) Real part of the multi-component signal. (b) Imaginary part of the multi-
component signal.

Figure 2: A multi-component signal consisting of a superposition of two pure
frequencies with 2 and 4 Hz, followed by a pure frequency of 1 Hz and a linear
chirp, i.e., the frequency increases linearly. (a) Absolute value of the
positive frequencies of $\widehat{f}(\omega)$. Figure 3: The positive part of
the Fourier transform of the multi-component signal. There are peeks around
the major occurring frequencies at 1,2 and 4 Hz. Also, the contributions of
the chirp are apparent throughout the frequency band $[2,8]$ Hz.

In order to obtain local information of the signal $f$, we can use a window
function $g$. We will use the standard box function $b_{0}$, which is the
generator of all $b$-splines, and the standard Gaussian window $g_{0}$.

$\displaystyle b_{0}(t)=\begin{cases}1,&-\frac{1}{2}\leq t\leq\frac{1}{2}\\\
0,&\text{else}\end{cases}\qquad\text{ and }\qquad g_{0}(t)=2^{1/4}e^{-\pi
t^{2}}.$ (2.2)

We note that both windows are normalized $\lVert b_{0}\rVert_{2}=1$ and
$\lVert g_{0}\rVert_{2}=1$.

(a) Real part of the signal localized around 0 with the box function $b_{0}$.
(b) Real part of the signal localized around 0 with the standard Gaussian
$g_{0}$.

Figure 4: Localization of the multi-component signal with the box function and
the standard Gaussian.

(a) Absolute value of the Fourier transform of the localized signal using
$b_{0}$. (b) Absolute value of the Fourier transform of the localized signal
using $g_{0}$.

Figure 5: The Fourier transform of the windowed signals reveals a local
behavior as well.

However, a joint time-frequency distribution should yield a picture in the
time-frequency plane which reveals the local behavior of the signal. We
illustrate such a representation with 2 spectrograms. The precise definition
of a spectrogram will be given in Section 3.1. As can already be seen from
Figure 4 and Figure 5, the choice of the window crucially affects the
localization properties of the signal. This fact will also become apparent in
Figure 6.

(a) A spectrogram using the window $b_{0}$. (b) A spectrogram using the window
$g_{0}$.

Figure 6: The spectrograms of the multi-component signal with different
localization windows. The $x$-direction indicates the position to which the
window is shifted and the $\omega$-direction shows the values of the energy
distribution of the local Fourier transfroms.

We note that the use of the box function $b_{0}$ does not yield a satisfying
localization of the signal in the Fourier domain, as smoothness of a function
affects the decay of its Fourier transform. Therefore, the localized Fourier
transform with window $b_{0}$ decays only like $\frac{1}{\omega}$. This can
also be seen in Figure 6 (a).

After this motivating example, we come to the definition of the short-time
Fourier transform (STFT) and discuss some of its properties. We mainly follow
the textbook by Gröchenig [19, Chap. 3].

######  Definition 2.2.

For a (non-zero) function $g$, called a window function, the short-time
Fourier transform of a function $f$ with respect to the window $g$ is given by

$V_{g}f(x,\omega)=\int_{\mathbb{R}^{d}}f(t)\overline{g(t-x)}e^{-2\pi
i\omega\cdot t}\,dt=\langle
f,\pi(\lambda)g\rangle,\qquad\lambda=(x,\omega)\in\mathbb{R}^{2d}.$ (2.3)

For the moment, we only give this formal definition without further
specification of $f$ and $g$. The STFT is also called sliding-window Fourier
transform or voice transform. The notation $V_{g}f$ comes from the latter
name. The name sliding window Fourier transform comes from the interpretation
of $V_{g}f(x,.)$ ($x$ fixed) as a local Fourier transform of $f$ around the
point $x$. As $x$ varies, the window $g$ slides along the $x$-axis to all
possible positions. Written as an inner product, the STFT is often interpreted
as a linear measurement, taken at the point $\lambda\in\mathbb{R}^{2d}$.

######  Example 2.3.

We will compute the STFT of the standard Gaussian

$g_{0}(t)=2^{d/4}e^{-\pi t^{2}},$ (2.4)

so $\lVert g_{0}\rVert_{2}=1$, with itself as a window.

$\displaystyle V_{g_{0}}g_{0}(x,\omega)$
$\displaystyle=2^{d/2}\int_{\mathbb{R}^{d}}e^{-\pi
t^{2}}e^{-\pi(t-x)^{2}}e^{-2\pi i\omega\cdot t}\,dt$ (2.5)
$\displaystyle=2^{d/2}\int_{\mathbb{R}^{d}}e^{-\pi(t+\frac{x}{2})^{2}}e^{-\pi(t-\frac{x}{2})^{2}}e^{-2\pi
i\omega\cdot(t+\frac{x}{2})}\,dt$ (2.6) $\displaystyle=2^{d/2}e^{-\pi
i\omega\cdot x}\int_{\mathbb{R}^{d}}e^{-2\pi(t^{2}+\frac{x^{2}}{4})}e^{-2\pi
i\omega\cdot t}\,dt$ (2.7) $\displaystyle=e^{-\pi i\omega\cdot
x}e^{-\frac{\pi}{2}x^{2}}2^{d/2}\int_{\mathbb{R}^{d}}e^{-2\pi t^{2}}e^{-2\pi
i\omega\cdot t}\,dt$ (2.8) $\displaystyle=e^{-\pi i\omega\cdot
x}e^{-\frac{\pi}{2}x^{2}}\underbrace{\int_{\mathbb{R}^{d}}e^{-\pi
t^{2}}e^{-2\pi i\frac{\omega}{2^{1/2}}\cdot
t}\,dt}_{\mathcal{F}g_{0}\left(\frac{\omega}{2^{1/2}}\right)}$ (2.9)
$\displaystyle=e^{-\pi i\omega\cdot x}e^{-\frac{\pi}{2}(x^{2}+\omega^{2})}.$
(2.10)

Thus, the STFT of a Gaussian is again a Gaussian (in the time-frequency plane
with a precisely determined phase factor).

We will now study the basic properties of the STFT. Besides writing the STFT
as an inner product of $f$ and $M_{\omega}T_{x}g$, we can also write it in
various other ways, which can be useful at times.

$\displaystyle V_{g}f(x,\omega)$
$\displaystyle=\mathcal{F}(f\,T_{x}\overline{g})(\omega)$ (2.11)
$\displaystyle=\langle\widehat{f},T_{\omega}M_{-x}\widehat{g}\rangle$ (2.12)
$\displaystyle=e^{-2\pi
ix\cdot\omega}\mathcal{F}(\widehat{f}\,T_{\omega}\overline{\widehat{g}})(-x)$
(2.13) $\displaystyle=e^{-2\pi
ix\cdot\omega}(f*M_{\omega}\overline{g}^{\vee})(x)$ (2.14)
$\displaystyle=(\widehat{f}*M_{-x}\widehat{\overline{g}}^{\vee})(\omega),$
(2.15)

where $g^{\vee}(x)=g(-x)$ is the flip operation.

So far, we have not discussed when the STFT is well-defined. Clearly, all of
the above formulas make sense for $L^{2}(\mathbb{R}^{d})$ functions because if
$f,g\in L^{2}(\mathbb{R}^{d})$, then the product $f\,T_{x}\overline{g}\in
L^{1}(\mathbb{R}^{d})$ by the Cauchy-Schwarz inequality and
$V_{g}f(x,\omega)=\mathcal{F}(f\,T_{x}\overline{g})(\omega)$ is defined point-
wise. For most of the time, we will be happy with the assumption that $f,g\in
L^{2}(\mathbb{R}^{d})$ (because then we may use Parseval’s theorem and
Plancherel’s theorem). However, by Hölder’s inequality (0.13)
$f\,T_{x}\overline{g}\in L^{1}(\mathbb{R}^{d})$ whenever $f\in
L^{p}(\mathbb{R}^{d})$ and $g\in L^{q}(\mathbb{R}^{d})$ with
$\frac{1}{p}+\frac{1}{q}=1$, so the STFT is again defined point-wise in this
case.

The following result can partially be seen as a “Riemann-Lebesgue-like”
statement for the STFT.

######  Lemma 2.4.

Let $f,g\in L^{2}(\mathbb{R}^{d})$, then $V_{g}f$ is uniformly continuous on
$\mathbb{R}^{2d}$.

###### Proof.

This follows mainly from the continuity of the operator groups $\\{T_{x}\\}$
and $\\{M_{\omega}\\}$, i.e.,

$\lim_{x\to 0}\lVert T_{x}f-f\rVert_{2}=0$ (2.16)

and

$\lim_{\omega\to 0}\lVert M_{\omega}f-f\rVert_{2}=\lim_{\omega\to 0}\lVert
T_{\omega}\widehat{f}-\widehat{f}\rVert_{2}=0.$ (2.17)

The statement follows by combining the two results. ∎

Using the Lemma of Riemann-Lebesgue, we also conclude that
$\lim_{|\omega|\to\infty}V_{g}f(x,\omega)=0$ for any $x$ (because
$f\,T_{x}\overline{g}\in L^{1}(\mathbb{R}^{d})$ for $f,g\in
L^{2}(\mathbb{R}^{d})$).

The following property of the STFT shows a fundamental concept in time-
frequency analysis. The usual interpretation is that, up to a phase factor,
the Fourier transform rotates the time-frequency plane (by $90^{\circ}$ in the
case $d=1$). Also, it shows that if we localize a signal $f$ with a window
$g$, we localize its spectrum $\widehat{f}$ with the Fourier transform
$\widehat{g}$ of the original window.

######  Proposition 2.5 (Fundamental Identity of Time-Frequency Analysis).

For $f,g\in L^{2}(\mathbb{R}^{d})$, the following holds;

$V_{g}f(x,\omega)=e^{-2\pi
ix\cdot\omega}V_{\widehat{g}}\widehat{f}(\omega,-x).$ (2.18)

###### Proof.

The proof is a straight-forward computation.

$\displaystyle V_{g}f(x,\omega)$ $\displaystyle=\langle
f,M_{\omega}T_{x}g\rangle=\langle f,e^{2\pi i\omega\cdot
x}T_{x}M_{\omega}g\rangle$ (2.19) $\displaystyle=e^{-2\pi i\omega\cdot
x}\langle f,T_{x}M_{\omega}g\rangle=e^{-2\pi i\omega\cdot
x}\langle\mathcal{F}f,\mathcal{F}T_{x}M_{\omega}g\rangle$ (2.20)
$\displaystyle=e^{-2\pi i\omega\cdot
x}\langle\mathcal{F}f,M_{-x}T_{\omega}\mathcal{F}g\rangle=e^{-2\pi
i\omega\cdot x}V_{\widehat{g}}\widehat{f}(\omega,-x).$ (2.21)

∎

The identity is sometimes also referred to as the basic identity of time-
frequency analysis. Also, we conclude that
$\lim_{|x|\to\infty}V_{g}f(x,\omega)=0$ for any $\omega$ because
$|V_{g}f(x,\omega)|=|V_{\widehat{g}}\widehat{f}(\omega,-x)|$ and we may apply
the Lemma of Riemann-Lebesgue to $\widehat{f}\,T_{\omega}\widehat{g}$, for
$\widehat{f},\widehat{g}\in L^{2}(\mathbb{R}^{d})$.

The next formula is known as the covariance principle. Its interpretation is
that a time-frequency shift by $(\xi,\eta)\in\mathbb{R}^{2d}$ simply
translates the STFT in the time-frequency plane (again, up to a phase factor).

######  Proposition 2.6 (Covariance Principle).

For $f,g\in L^{2}(\mathbb{R}^{d})$, we have

$V_{g}(M_{\eta}T_{\xi}f)(x,\omega)=e^{-2\pi
i\xi\cdot(\omega-\eta)}V_{g}f(x-\xi,\omega-\eta),\qquad
x,\xi,\omega,\eta\in\mathbb{R}^{d}.$ (2.22)

###### Proof.

The proof is a straight-forward computation.

$\displaystyle V_{g}(M_{\eta}T_{\xi}f)(x,\omega)$ $\displaystyle=\langle
M_{\eta}T_{\xi}f,M_{\omega}T_{x}g\rangle$ (2.23) $\displaystyle=\langle
f,T_{-\xi}M_{-\eta}M_{\omega}T_{x}g\rangle$ (2.24) $\displaystyle=\langle
f,e^{2\pi i\xi\cdot(-\eta+\omega)}M_{\omega-\eta}T_{x-\xi}g\rangle$ (2.25)
$\displaystyle=e^{-2\pi i\xi\cdot(\omega-\eta)}V_{g}f(x-\xi,\omega-\eta)$
(2.26)

∎

We remark that the above proof can be adapted and that the result stays true
whenever $V_{g}f$ is defined. The only $L^{2}(\mathbb{R}^{d})$ specific
property we used this time was the inner product $\langle.,.\rangle$, so
whenever this bracket is well-defined (e.g., by duality) the proof holds
(e.g., for $\mathcal{S}(\mathbb{R}^{d})$ and
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ or $L^{p}(\mathbb{R}^{d})$ and
$L^{q}(\mathbb{R}^{d})$ with $\frac{1}{p}+\frac{1}{q}=1$).

The STFT also enjoys a property which is quite similar to Parseval’s identity
(0.21).

######  Theorem 2.7 (Orthogonality Relations).

Let $f_{1},f_{2},g_{1},g_{2}\in L^{2}(\mathbb{R}^{d})$, then
$V_{g_{k}}f_{k}\in L^{2}(\mathbb{R}^{2d})$ for $k\in\\{1,2\\}$ and

$\langle V_{g_{1}}f_{1},V_{g_{2}}f_{2}\rangle_{L^{2}(\mathbb{R}^{2d})}=\langle
f_{1},f_{2}\rangle\overline{\langle g_{1},g_{2}\rangle}.$ (2.27)

###### Proof.

For technical reasons, we first assume that $g_{k}\in
L^{1}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})\subset
L^{2}(\mathbb{R}^{d})$ (dense). Therefore, $f_{k}\,T_{x}g_{k}\in
L^{2}(\mathbb{R}^{d})$ for all $x\in\mathbb{R}^{d}$ and Parseval’s formula
(0.21) applies to the integral in $\omega$.

$\displaystyle\langle
V_{g_{1}}f_{1},V_{g_{2}}f_{2}\rangle_{L^{2}(\mathbb{R}^{2d})}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}V_{g_{1}}f_{1}(x,\omega)\overline{V_{g_{2}}f_{2}(x,\omega)}\,d\omega\,dx$
(2.28)
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\mathcal{F}(f_{1}\,T_{x}\overline{g_{1}})(\omega)\,\overline{\mathcal{F}(f_{2}\,T_{x}\overline{g_{2}})(\omega)}\,d\omega\right)\,dx$
(2.29)
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}f_{1}(t)\overline{f_{2}(t)}\;\overline{g_{1}(t-x)}g_{2}(t-x)\,dt\right)\,dx\,.$
(2.30)

As $f_{1},f_{2}\in L^{2}(\mathbb{R}^{d},dt)$, the product
$f_{1}\,\overline{f_{2}}\in L^{1}(\mathbb{R}^{d},dt)$ and, by assumption, the
product $\overline{g_{1}}g_{2}\in L^{1}(\mathbb{R}^{d},dx)$. We may therefore
exchange the order of integration. We have

$\displaystyle\langle
V_{g_{1}}f_{1},V_{g_{2}}f_{2}\rangle_{L^{2}(\mathbb{R}^{2d})}$
$\displaystyle=\int_{\mathbb{R}^{d}}f_{1}(t)\overline{f_{2}(t)}\left(\int_{\mathbb{R}^{d}}\overline{g_{1}(t-x)}g_{2}(t-x)\,dx\right)\,dt$
(2.31) $\displaystyle=\langle f_{1},f_{2}\rangle\overline{\langle
g_{1},g_{2}\rangle}.$ (2.32)

By a density argument the result extends to $g_{k}\in L^{2}(\mathbb{R}^{d})$,
$k=1,2$. ∎

A shorter proof can be obtained by introducing the following unitary operators
for functions of $2d$ variables and the tensor product of two functions;

$\mathcal{T}_{a}F(x,t)=F(t,t-x)\quad\text{ and
}\quad\mathcal{F}_{2}F(x,\omega)=\int_{\mathbb{R}^{d}}F(x,t)e^{-2\pi
it\cdot\omega}\,dt.$ (2.33)

The operators $\mathcal{T}_{a}$ is an asymmetric coordinate change and the
operator $\mathcal{F}_{2}$ is the partial Fourier transform in the second
($d$) variable(s). The tensor product of two functions is denoted by

$(f\otimes g)(x,\omega)=f(x)g(\omega)$ (2.34)

Second Proof of the Orthogonality Relations. If $f,g\in
L^{2}(\mathbb{R}^{d})$, then

$\displaystyle\langle
V_{g_{1}}{f_{1}},V_{g_{2}}f_{2}\rangle_{L^{2}(\mathbb{R}^{2d})}$
$\displaystyle=\langle\mathcal{F}_{2}\mathcal{T}_{a}(f_{1}\otimes\overline{g_{1}}),\mathcal{F}_{2}\mathcal{T}_{a}(f_{2}\otimes\overline{g_{2}})\rangle_{L^{2}(\mathbb{R}^{2d})}$
(2.35) $\displaystyle=\langle
f_{1}\otimes\overline{g_{1}},f_{2}\otimes\overline{g_{2}}\rangle_{L^{2}(\mathbb{R}^{2d})}$
(2.36) $\displaystyle=\langle f_{1},f_{2}\rangle\overline{\langle
g_{1},g_{2}\rangle}.$ (2.37)

$\square$

As a corollary, we obtain a Plancherel-like result for the STFT.

######  Corollary 2.8.

For $f,g\in L^{2}(\mathbb{R}^{d})$, we have

$\lVert V_{g}f\rVert_{2}=\lVert f\rVert_{2}\lVert g\rVert_{2}\,.$ (2.38)

In particular, if $\lVert g\rVert_{2}=1$ then

$\lVert f\rVert_{2}=\lVert V_{g}f\rVert_{2},\qquad\forall f\in
L^{2}(\mathbb{R}^{d}).$ (2.39)

Hence, in this case the STFT is an isometry from $L^{2}(\mathbb{R}^{d})$ into
$L^{2}(\mathbb{R}^{2d})$.

We note that (2.39) tells us that $f$ is completely determined by $V_{g}f$.
Furthermore, we have the following implication

$\langle
f,M_{\omega}T_{x}g\rangle=0,\,\forall(x,\omega)\in\mathbb{R}^{2d}\qquad\Longrightarrow\qquad
f=0.$ (2.40)

This means that the set of time-frequency shifts of the window $g$ is complete
in $L^{2}(\mathbb{R}^{d})$ and, hence, spans a dense subspace of
$L^{2}(\mathbb{R}^{d})$.

######  Remark.

In particular, the last result tells us that

$X=\text{span }\\{M_{\omega}T_{x}g_{0}\mid(x,\omega)\in\mathbb{R}^{2d}\\}$
(2.41)

is dense in $L^{2}(\mathbb{R}^{d})$, where $g_{0}(t)=2^{d/4}e^{-\pi t^{2}}$ is
the $d$-dimensional standard Gaussian. We used Plancherel’s theorem in order
to prove this result (for general $g\in L^{2}(\mathbb{R}^{d})$), but there are
other proofs as well (e.g., using Fourier series [19, Chap. 1.5]). Now, we
will prove that Plancherel’s theorem is also a consequence of the fact that
$X$ is dense in $L^{2}(\mathbb{R}^{d})$.

For $f\in X$ we have

$f(t)=\sum_{k=1}^{n}c_{k}M_{\omega_{k}}T_{x_{k}}g_{0}.$ (2.42)

When computing $\lVert f\rVert_{2}=\langle f,f\rangle$, we need to compute
inner products of time-frequency shifted Gaussians. We recall that

$\displaystyle\langle g_{0},M_{\omega}T_{x}g_{0}\rangle=e^{-\pi
ix\cdot\omega}e^{-\frac{\pi}{2}(x^{2}+\omega^{2})}$ (2.43)

We compute

$\displaystyle\langle M_{\eta}T_{\xi}g_{0},M_{\omega}T_{x}g_{0}\rangle$
$\displaystyle=\langle g_{0},T_{-\xi}M_{\omega-\eta}T_{x}g_{0}\rangle$ (2.44)
$\displaystyle=\langle g_{0},e^{2\pi
i\xi\cdot(\omega-\eta)}M_{\omega-\eta}T_{x-\xi}g_{0}\rangle$ (2.45)
$\displaystyle=e^{-2\pi i\xi\cdot(\omega-\eta)}e^{-\pi
i(x-\xi)\cdot(\omega-\eta)}e^{-\frac{\pi}{2}((x-\xi)^{2}+(\omega-\eta)^{2})}$
(2.46) $\displaystyle=e^{-\pi
i(x+\xi)\cdot(\omega-\eta)}e^{-\frac{\pi}{2}((x-\xi)^{2}+(\omega-\eta)^{2})}$
(2.47)

Now, we perform the same computation for the Fourier transforms of the time-
frequency shifted Gaussians.

$\displaystyle\langle\mathcal{F}(M_{\eta}T_{\xi}g_{0}),\mathcal{F}(M_{\omega}T_{x}g_{0})\rangle$
$\displaystyle=\langle e^{2\pi
i\eta\cdot\xi}M_{-\xi}T_{\eta}\mathcal{F}g_{0},e^{2\pi i\omega\cdot
x}M_{-x}T_{\omega}\mathcal{F}g_{0}\rangle$ (2.48) $\displaystyle=e^{2\pi
i\eta\cdot\xi}e^{-2\pi i\omega\cdot x}\langle
g_{0},T_{-\eta}M_{\xi-x}T_{\omega}g_{0}\rangle$ (2.49) $\displaystyle=e^{2\pi
i\eta\cdot\xi}e^{-2\pi i\omega\cdot x}\langle g_{0},e^{2\pi
i\eta\cdot(\xi-x)}M_{\xi-x}T_{\omega-\eta}g_{0}\rangle$ (2.50)
$\displaystyle=e^{2\pi i\eta\cdot\xi}e^{-2\pi i\omega\cdot x}e^{-2\pi
i\eta\cdot(\xi-x)}e^{-\pi
i(\xi-x)\cdot(\omega-\eta)}e^{-\frac{\pi}{2}((\xi-x)^{2}+(\omega-\eta)^{2})}$
(2.51) $\displaystyle=e^{-\pi
i(\omega-\eta)\cdot(x+\xi)}e^{-\frac{\pi}{2}((x-\xi)^{2}+(\omega-\eta)^{2})}.$
(2.52)

In the last step we used the fact that $(x-\xi)^{2}=(\xi-x)^{2}$. Hence, these
computations show that

$\langle
M_{\eta}T_{\xi}g_{0},M_{\omega}T_{x}g_{0}\rangle=\langle\mathcal{F}(M_{\eta}T_{\xi}g_{0}),\mathcal{F}(M_{\omega}T_{x}g_{0})\rangle$
(2.53)

Therefore

$\displaystyle\langle f,f\rangle$
$\displaystyle=\langle\sum_{k=1}^{n}c_{k}M_{\omega_{k}}T_{x_{k}}g_{0},\sum_{l=1}^{n}c_{l}M_{\omega_{l}}T_{x_{l}}g_{0}\rangle$
(2.54)
$\displaystyle=\langle\sum_{k=1}^{n}c_{k}\mathcal{F}(M_{\omega_{k}}T_{x_{k}}g_{0}),\sum_{l=1}^{n}c_{l}\mathcal{F}(M_{\omega_{l}}T_{x_{l}}g_{0})\rangle$
(2.55) $\displaystyle=\langle\widehat{f},\widehat{f}\rangle,$ (2.56)

by linearity of the inner product. Summing up, this shows that Plancherel’s
theorem is equivalent to the completeness of the set of time-frequency shifted
standard Gaussians.

After knowing that the information of $f$ is completely contained in $V_{g}f$,
the question that pops up is: How can we recover $f$ from $V_{g}f$?

Before we can answer this question, we need a quick excursion in vector-valued
integrals. In time-frequency analysis, superpositions of time-frequency
shifts, such as

$f=\iint_{\mathbb{R}^{2d}}F(x,\omega)(M_{\omega}T_{x}g)\,d(x,\omega).$ (2.57)

are of utmost importance. Note that $g$ is time-frequency shifted by
$(x,\omega)$, but it also depends on its argument. Hence, the integral gives
us back a vector (i.e., a function) in the Hilbert space
$L^{2}(\mathbb{R}^{d})$, which depends on the same argument as $g$. Therefore,
we may also speak of a function-valued integral. More generally, integrals may
also be matrix- or operator-valued. Later on, we will use a weak formulation
for such integrals. Furthermore, consider the following example. If $F\in
L^{2}(\mathbb{R}^{2d})$, then the conjugate-linear functional

$\ell(h)=\iint_{\mathbb{R}^{2d}}F(x,\omega)\overline{\langle
h,M_{\omega}T_{x}g\rangle}\,d(x,\omega)$ (2.58)

is a bounded functional on $L^{2}(\mathbb{R}^{d})$. This can be seen by
applying the Cauchy-Schwarz inequality and the Plancherel-like result from
Corollary 2.8;

$|\ell(h)|\leq\lVert F\rVert_{L^{2}(\mathbb{R}^{2d})}\lVert
V_{g}h\rVert_{2}=\lVert F\rVert_{L^{2}(\mathbb{R}^{2d})}\lVert
g\rVert_{2}\lVert h\rVert_{2}.$ (2.59)

This means that $\ell$ defines a unique function
$f=\iint_{\mathbb{R}^{2d}}F(x,\omega)M_{\omega}T_{x}g\,d(x,\omega)\in
L^{2}(\mathbb{R}^{d})$ with norm $\lVert f\rVert_{2}\leq\lVert
F\rVert_{L^{2}(\mathbb{R}^{2d})}\lVert g\rVert_{2}$ and satisfying
$\ell(h)=\langle f,h\rangle$.

With this notion at hands, we will be able to prove the inversion formula for
the STFT.

######  Theorem 2.9 (Inversion formula for the STFT).

Let $g\in L^{2}(\mathbb{R}^{d})$ and choose $\widetilde{g}\in
L^{2}(\mathbb{R}^{d})$ such that $\langle\widetilde{g},g\rangle\neq 0$. Then
we have for all $f\in L^{2}(\mathbb{R}^{d})$

$f=\frac{1}{\langle\widetilde{g},g\rangle}\iint_{\mathbb{R}^{2d}}V_{g}f(x,\omega)M_{\omega}T_{x}\widetilde{g}\,d(x,\omega).$
(2.60)

###### Proof.

By Corollary 2.8 $V_{g}f\in L^{2}(\mathbb{R}^{2d})$. Therefore, the following
vector-valued integral is a well defined function $\widetilde{f}\in
L^{2}(\mathbb{R}^{d})$;

$\widetilde{f}=\frac{1}{\langle\widetilde{g},g\rangle}\iint_{\mathbb{R}^{2d}}V_{g}f(x,\omega)M_{\omega}T_{x}\widetilde{g}\,d(x,\omega).$
(2.61)

By using the orthogonality relations, we see that

$\displaystyle\langle\widetilde{f},h\rangle$
$\displaystyle=\frac{1}{\langle\widetilde{g},g\rangle}\iint_{\mathbb{R}^{2d}}V_{g}f(x,\omega)\underbrace{\langle
M_{\omega}T_{x}\widetilde{g},h\rangle}_{\overline{V_{\widetilde{g}}h}}\,d(x,\omega)$
(2.62) $\displaystyle=\frac{1}{\langle\widetilde{g},g\rangle}\langle
V_{g}f,V_{\widetilde{g}}h\rangle_{L^{2}(\mathbb{R}^{2d})}=\frac{1}{\langle\widetilde{g},g\rangle}\langle
f,h\rangle\overline{\langle g,\widetilde{g}\rangle}$ (2.63)
$\displaystyle=\langle f,h\rangle.$ (2.64)

This formula holds for any $h\in L^{2}(\mathbb{R}^{d})$ and we conclude that
$\widetilde{f}=f$. ∎

Note that, as $g\neq 0$, we may actually choose $\widetilde{g}=g$ in the
inversion formula. Hence, in this case the inversion formula reads

$f=\frac{1}{\lVert
g\rVert_{2}^{2}}\iint_{\mathbb{R}^{2d}}V_{g}f(x,\omega)M_{\omega}T_{x}g\,d(x,\omega)=\frac{1}{\lVert
g\rVert_{2}^{2}}\iint_{\mathbb{R}^{2d}}V_{g}f(\lambda)\,\pi(\lambda)g\,d\lambda,\qquad\lambda=(x,\omega)\in\mathbb{R}^{2d}.$
(2.65)

Of course, this formula simplifies a bit if we assume that $\lVert
g\rVert_{2}=1$,

$f=\iint_{\mathbb{R}^{2d}}V_{g}f(\lambda)\,\pi(\lambda)g\,d\lambda.$ (2.66)

Hence, a function $f\in L^{2}(\mathbb{R}^{d})$ can be written as a continuous
superposition of a time-frequency shifted window with weights obtained from
its STFT. In this sense, (2.60) is similar to the inversion formula of the
Fourier transform. The main difference is that the elementary building blocks,
i.e., the complex exponential $e^{2\pi ix\cdot\omega}$, are not in
$L^{2}(\mathbb{R}^{d})$ whereas the elementary functions
$M_{\omega}T_{x}\widetilde{g}$ are usually picked to be particularly nice
$L^{2}(\mathbb{R}^{d})$-functions.

The time-frequency analysis of a signal now usually consists of three steps.

Analysis. Given a signal $f$, its STFT $V_{g}f$ is computed. We will interpret
it as a joint time-frequency distribution of $f$. The window $g$ crucially
influences the analysis and one usually demands that $g$ and $\widehat{g}$
decay sufficiently fast.

Processing. The STFT $V_{g}f(x,\omega)$ is then transformed into a new
function $F(x,\omega)$. Some typical processing steps are truncation of
$V_{g}f$ to a region where something interesting seems to happen or where
$|V_{g}f|$ is above a given threshold. In an engineering language such
processing steps are called feature extraction, signal segmentation and signal
compression, depending on the purpose. Furthermore, the STFT is also used as a
pre-processing tool in the area of machine learning.

Synthesis. The processed signal is then reconstructed by using the modified
inversion formula

$\widetilde{f}=\frac{1}{\langle\widetilde{g},g\rangle}\iint_{\mathbb{R}^{2d}}F(x,\omega)M_{\omega}T_{x}\widetilde{g}\,d(x,\omega).$
(2.67)

Note that signal $\widetilde{f}$ is reconstructed from $F(x,\omega)$ and not
from $V_{g}f(x,\omega)$. Hence, $\widetilde{f}$ may naturally differ from $f$.
Also we remark again that distinct windows might be used for the analysis and
the synthesis.

It is also customary to write the inversion formula as a superposition of
rank-one operators. Consider a Hilbert space $\mathcal{H}$ and let
$u\otimes\overline{v}$ denote the rank-one operator defined by

$(u\otimes\overline{v})(h)=\langle h,v\rangle_{\mathcal{H}}u,\quad
u,v,h\in\mathcal{H}.$ (2.68)

Then, the inversion formula (2.60) is the following continuous resolution of
the identity operator

$I_{L^{2}(\mathbb{R}^{d})}=\frac{1}{\langle\widetilde{g},g\rangle}\iint_{\mathbb{R}^{2d}}M_{\omega}T_{x}\widetilde{g}\otimes\overline{M_{\omega}T_{x}g}\,d(x,\omega).$
(2.69)

##  3 Quadratic Time-Frequency Representations

Again, for most of this chapter we will follow the textbook by Gröchenig [19].

Quadratic representations are often interpreted as joint (probability)
densities of a function $f\in L^{2}(\mathbb{R}^{d})$ and its Fourier transform
$\widehat{f}\in L^{2}(\mathbb{R}^{d})$ on $\mathbb{R}^{2d}$. The space
$\mathbb{R}^{2d}$ might be called time-frequency plane or phase space,
depending on the specific situation.

Such joint time-frequency representations were investigated by E. Wigner [53]
in the context of quantum mechanics with the goal of finding a joint
probability distribution for the position and momentum variables. Now,
quadratic representations are also popular in engineering and we will call
them (quadratic) time-frequency representations. Mathematically, we are
looking for a sesquilinear form $G(f,g)(x,\omega)$; that is $G$ is linear in
the first argument $f$ and conjugate linear in the second argument $g$. Then,
there are two ways to make $G$ “quadratic” in $f$. We either consider
$Cf=|G(f,g)|^{2}$ with $g$ fixed or $Cf=G(f,f)$. In both cases the quadratic
form satisfies

$C(c_{1}f+c_{2}h)=|c_{1}|^{2}\,Cf+|c_{2}|^{2}\,Ch+c_{1}\overline{c_{2}}\,G(f,h)+\overline{c_{1}}c_{2}\,G(h,f),\quad
c_{1},c_{2}\in\mathbb{C}.$ (3.1)

The non-linearity in (3.1) causes problems because the superposition of two
signals $f$ and $h$ introduces the cross-terms $G(f,h)$ and $G(h,f)$. These
are often hard to separate from the main components of interest $Cf$ and $Ch$.
Also, the analysis and interpretation of these cross-terms is difficult.

On the positive side, a quadratic time-frequency representation of the form
$G(f,f)$ does not depend on a window $g$ and should display the time-frequency
content of $f$ in a pure, unobstructed form.

###  3.1 The Spectrogram

In this section we are going to briefly discuss the spectrogram and some of
its properties. We have seen examples already at the beginning of Section 2.

######  Definition 3.1.

Let $g\in L^{2}(\mathbb{R}^{d})$ be a window with the property $\lVert
g\rVert_{2}=1$. Then, the spectrogram of $f$ with respect to the window $g$ is
given by

$\text{SPEC}_{g}f(x,\omega)=|V_{g}f(x,\omega)|^{2}.$ (3.2)

As already apparent from its definition, the spectrogram $\text{SPEC}_{g}f$
inherits properties from the STFT. In particular, it is covariant and
preserves the energy and, furthermore, is non-negative (by definition);

  * •

$\text{SPEC}_{g}f(x,\omega)\geq 0,\qquad\forall(x,\omega)\in\mathbb{R}^{2d}$,

  * •

$\text{SPEC}_{g}\left(M_{\eta}T_{\xi}f\right)(x,\omega)=\text{SPEC}_{g}f(x-\xi,\omega-\eta)$
by Proposition 2.6, which we also called the Covariance Principle,

  * •

$\lVert\text{SPEC}_{g}f\rVert_{L^{1}(\mathbb{R}^{2d})}=\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{2}$ by Corollary 2.8.

The last point demonstrates the fact that, up to normalization of $f$, the
spectrogram may serve as a probability density on $\mathbb{R}^{2d}$ of the
joint time-frequency content of a signal $f$.

###  3.2 The Rihaczek Distribution

We will now study the Rihaczek distribution as described in [20], which is a
rather simple time-frequency representation. It is more or less simply the
tensor product of $f$ and its Fourier transform $\widehat{f}$.

$Rf(x,\omega)=f(x)\overline{\widehat{f}(\omega)}e^{-2\pi ix\cdot\omega}.$
(3.3)

The Rihaczek distribution is intimately connected to the STFT by the Fourier
transform (on $\mathbb{R}^{2d}$)

$Rf(x,\omega)=\widehat{V_{f}f}(\omega,-x).$ (3.4)

This is confirmed by a direct computation.

$\displaystyle\widehat{V_{f}f}(\omega,-x)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}V_{f}f(\xi,\eta)e^{-2\pi
i(\xi\cdot\omega-x\cdot\eta)}\,d\eta\right)\,d\xi$ (3.5)
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}f(t)\overline{f(t-\xi)}e^{-2\pi
i\eta\cdot t}\,dt\right)e^{-2\pi
i(\xi\cdot\omega-x\cdot\eta)}\,d\eta\right)\,d\xi$ (3.6)
$\displaystyle=\int_{\mathbb{R}^{d}}\underbrace{\left(\int_{\mathbb{R}^{d}}\left(\mathcal{F}\left(f\,T_{\xi}\overline{f}\right)(\eta)\right)\,e^{2\pi
ix\cdot\eta}\,d\eta\right)}_{=\mathcal{F}^{-1}\left(\mathcal{F}\left(f\,T_{\xi}\overline{f}\right)\right)(x)}e^{-2\pi
i\xi\cdot\omega}\,d\xi$ (3.7)
$\displaystyle=\int_{\mathbb{R}^{d}}f(x)\overline{f(x-\xi)}e^{-2\pi
i\xi\cdot\omega}\,d\xi=f(x)\int_{\mathbb{R}^{d}}\overline{f(\xi)}e^{-2\pi
i(-\xi+x)\cdot\omega}\,d\xi$ (3.8)
$\displaystyle=f(x)\overline{\int_{\mathbb{R}^{d}}f(\xi)e^{2\pi
i(-\xi+x)\cdot\omega}\,d\xi}=f(x)\overline{\widehat{f}(\omega)}e^{-2\pi
ix\cdot\omega}$ (3.9) $\displaystyle=Rf(x,\omega).$ (3.10)

We can also state (3.4) in a more general way.

$R(f,g)(x,\omega)=f(x)\overline{\widehat{g}(\omega)}e^{-2\pi
ix\cdot\omega}=\widehat{V_{g}f}(\omega,-x).$ (3.11)

We call $R(f,g)$ the cross-Rihaczek distribution of $f$ and $g$. In this
context, we should $Rf=R(f,f)$ actually call the auto-Rihaczek distribution of
$f$. We note that, up to the complex exponential, the STFT factors under the
Fourier transform. We will mainly use $Rf$ when presenting the classical
uncertainty principle of Heisenberg, Pauli and Weyl.

###  3.3 The Ambiguity Function

Another time-frequency representation, which is up to a complex exponential
the same as the STFT, is the cross-ambiguity function of two functions $f$ and
$g$.

######  Definition 3.2.

For $f$ and $g$ in $L^{2}(\mathbb{R}^{d})$, their cross-ambiguity function is
defined as

$A(f,g)(x,\omega)=\int_{\mathbb{R}^{d}}f(t+\tfrac{x}{2})\overline{g(t-\tfrac{x}{2})}e^{-2\pi
i\omega\cdot t}\,dt=e^{\pi ix\cdot\omega}V_{g}f(x,\omega).$ (3.12)

For the case $f=g$, we write $Af=A(f,f)$ and call it the (auto-)ambiguity
function of $f$.

By definition, it follows immediately that

$\overline{Af(-x,-\omega)}=Af(x,\omega).$ (3.13)

Particularly, $Af(0,0)$ is a real value. Of course, most properties of the
STFT carry over to the cross-ambiguity function. In particular the
orthogonality relations in Theorem 2.7 imply

$\lVert Af\rVert_{L^{2}(\mathbb{R}^{2d})}=\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{2}.$ (3.14)

However, the ambiguity function $Af$ is a quadratic representation of a
function $f$, whereas the STFT with window $g$, i.e., $V_{g}f$, is a linear
transformation of $f$. Therefore, $f$ can only be recovered up to a phase
factor by $Af$. In particular we have $A(cf)=Af$ for any $|c|=1$. Such a $c$
is called a phase factor. If $f(0)\neq 0$, we have the following inversion
formula

$f(x)=\frac{1}{\,\overline{f(0)}\,}\int_{\mathbb{R}^{d}}Af(x,\omega)e^{\pi
ix\cdot\omega}\,d\omega.$ (3.15)

This can be obtained as follows. First, we note that for fixed
$x\in\mathbb{R}^{d}$, $Af(x,.)$ is the Fourier transform of $t\mapsto
f(t+\tfrac{x}{2})\overline{f(t-\tfrac{x}{2})}$. Hence, by the Fourier
inversion formula we have

$f(t+\tfrac{x}{2})\overline{f(t-\tfrac{x}{2})}=\int_{\mathbb{R}^{d}}Af(x,\omega)e^{2\pi
it\cdot\omega}\,d\omega.$ (3.16)

By setting $t=\tfrac{x}{2}$, we obtain the desired result. Furthermore, it is
easy to check that for a phase factor $|c|=1$, equation (3.15) holds for $cf$
as well.

Of course, it is not quite “fair” to compare the inversion formula for the
auto-ambiguity function with the inversion formula for the STFT. If we
introduce an “auto-STFT”, i.e., $V_{f}f$, then we can also only recover $f$
from $V_{f}f$ up to a phase factor. The key difference is that for the STFT,
we completely know the window function $g$. Therefore, for the cross-ambiguity
function, we may completely recover $f$ from $A(f,g)$ by knowing $g$.

The ambiguity function occurs naturally in radar applications and, therefore,
is often called the radar ambiguity function. We will briefly discuss a
simplified model as described in [19, Chap. 4.2]. Suppose that the distance
and speed of an unidentified flying object (UFO) are to be determined. For
this purpose a transmitter/receiver station sends a test signal $f$ towards
the UFO. The signal $f$ is reflected and a part of the reflected signal is
received as an echo $e$. The test signal might be a “pulse” of short duration
of the form $f(x)=\sigma(x)e^{2\pi i\omega_{0}\cdot x}$., where $\sigma$ is a
nice envelope signal of slow variation. This means that
$\operatorname*{supp}(\widehat{\sigma})\subset[-W,W]$ with $W$ small compared
to the carrier frequency $\omega_{0}$, so
$\operatorname*{supp}(\widehat{f})\subset[\omega_{0}-W,\omega_{0}+W]$. Let $r$
be the distance between the transmitter/receiver station and the UFO, $v$ the
relative speed between the station and th UFO and $c$ the speed of light.
Then, the echo $e$ is received with a time lag $\Delta t=\frac{2r}{c}$.
Moreover, each frequency $\omega$ in the band $[\omega_{0}-W,\omega_{0}+W]$ of
$f$ undergoes a Doppler shift $\Delta\omega=-\frac{2\omega v}{c}$. Since the
bandwidth $2W$ is small compared to $\omega_{0}$, the frequency shifts can be
approximated by $\Delta\omega\approx-\frac{2\omega_{0}v}{c}$, independently of
the exact form of $f$. We ignore any further distortion of the signal $f$ in
this example, therefore the echo $e$ has the form

$M_{\Delta\omega}T_{\Delta t}f.$ (3.17)

At the receiver, the echo is then compared to time-frequency shifts of the
original signal $f$ by taking the correlation

$|\langle e,M_{\omega}T_{x}f\rangle|=|V_{f}f(x-\Delta
t,\omega-\Delta\omega)|=|Af(x-\Delta t,\omega-\Delta\omega)|.$ (3.18)

The values of $\Delta t$ and $\Delta\omega$, and hence the distance $r$ and
the velocity $v$ of the UFO, can be determined by means of the following
lemma.

######  Lemma 3.3.

Let $f\in L^{2}(\mathbb{R}^{d})$, $f\neq 0$, then

$|Af(x,\omega)|<Af(0,0)=\lVert f\rVert_{2}^{2},$ (3.19)

for all $(x,\omega)\neq(0,0)$.

###### Proof.

The inequality

$|Af(x,\omega)|=|\langle f,M_{\omega}T_{x}f\rangle|\leq\lVert f\rVert_{2}^{2}$
(3.20)

is just the Cauchy-Schwarz inequality. Equality holds if and only if
$M_{\omega}T_{x}f=cf$ (i.e., we have linear dependence) for some
$(x,\omega)\in\mathbb{R}^{2d}$ and $|c|=1$. Now, assume $x\neq 0$ and further
assume that $T_{x}|f|=|cM_{\omega}T_{x}f|=|cf|=|f|$, then $|f|$ would be
periodic with period $x$. However, as $f\in L^{2}(\mathbb{R}^{d})$, it cannot
be periodic unless $f=0$. If $x=0$ and $\omega\neq 0$, then, analogously,
$T_{\omega}|\widehat{f}|=|\widehat{f}|$ would be $\omega$-periodic. Therefore,
$|Af(x,\omega)|$ is maximal if and only if $(x,\omega)=(0,0)$.

The value of $Af(0,0)$ yields the energy of $f$, i.e.,

$Af(0,0)=\int_{\mathbb{R}^{d}}f(t)\overline{f(t)}\,dt=\langle
f,f\rangle=\lVert f\rVert_{2}^{2}.$ (3.21)

∎

Now, to find the lag $(\Delta t,\Delta\omega)$, one determines experimentally
the values $(x_{0},\omega_{0})$ where the correlation function $|\langle
e,M_{\omega}T_{x}f\rangle|$ takes its maximum. According to the lemma above,
the position of this maximum is the desired lag $(\Delta t,\Delta\omega)$.

We give now two examples of auto-ambiguity functions, which are used to
measure time-frequency concentrations.

######  Example 3.4.

  1. (a)

The ambiguity function of the box function is given by

$Ab_{0}(x,\omega)=\begin{cases}\frac{\sin(\pi\omega(1-|x|))}{\pi\omega},&|x|\leq
1\\\ 0,&\text{else}\end{cases}$ (3.22)

Figure 7: Illustration of the function
$b_{0}(t+\tfrac{x}{2})b_{0}(t-\tfrac{x}{2})$ for some $x$ with $|x|\leq 1$.

We show this by a directly computing

$Ab_{0}(x,\omega)=\int_{\mathbb{R}}b_{0}(t+\tfrac{x}{2})b_{0}(t-\tfrac{x}{2})e^{-2\pi
i\omega\cdot t}\,dt.$ (3.23)

We distinct the cases $x\in[0,1]$ and $x\in[-1,0]$. We start with the case
$x\in[0,1]$.

$\displaystyle\int_{\mathbb{R}}b_{0}(t+\tfrac{x}{2})b_{0}(t-\tfrac{x}{2})e^{-2\pi
i\omega t}\,dt$ $\displaystyle=\int_{\tfrac{-1+x}{2}}^{\tfrac{1-x}{2}}e^{-2\pi
i\omega t}\,dt$ (3.24)
$\displaystyle=\int_{\tfrac{-1+x}{2}}^{\tfrac{1-x}{2}}\cos(2\pi\omega
t)\,dt=\frac{\sin(2\pi\omega
t)}{2\pi\omega}\Bigg{|}_{t=\tfrac{-1+x}{2}}^{\tfrac{1-x}{2}}$ (3.25)
$\displaystyle=\frac{\sin(2\pi\omega\frac{1-x}{2})}{2\pi\omega}-\frac{\sin(2\pi\omega\frac{-1+x}{2})}{2\pi\omega}$
(3.26) $\displaystyle=\frac{\sin(\pi\omega(1-x))}{\pi\omega}.$ (3.27)

The case $x\in[-1,0]$ follows easily by the symmetry
$Ab_{0}(x,\omega)=Ab_{0}(-x,\omega)$ and the result follows.

  2. (b)

Next, we compute the ambiguity function of the 1-dimensional standard Gaussian
$g_{0}(t)=2^{1/4}e^{-\pi t^{2}}$, $t\in\mathbb{R}$ (the case for of the
$d$-dimensional standard Gaussian follows just as easily.) As an in-between
step, we will use the dilation operator $D_{a}g(t)=a^{-1/2}g(a^{-1}t)$ and its
behavior under the Fourier transform and Theorem 0.8, which shows that
$\mathcal{F}g_{0}=g_{0}$.

$\displaystyle Ag_{0}(x,\omega)$
$\displaystyle=2^{1/2}\int_{\mathbb{R}}e^{-\pi(t+\tfrac{x}{2})^{2}}e^{-\pi(t-\tfrac{x}{2})^{2}}e^{-2\pi
i\omega\cdot t}\,dt$ (3.28)
$\displaystyle=e^{-\pi\tfrac{x^{2}}{2}}\int_{\mathbb{R}}\underbrace{2^{1/2}e^{-2\pi
t^{2}}}_{D_{1/\sqrt{2}}g_{0}}e^{-2\pi i\omega\cdot t}\,dt$ (3.29)
$\displaystyle=e^{-\pi\tfrac{x^{2}}{2}}\mathcal{F}\left(D_{1/\sqrt{2}}g_{0}\right)(\omega)=e^{-\pi\tfrac{x^{2}}{2}}D_{\sqrt{2}}\mathcal{F}g_{0}(\omega)=e^{-\pi\tfrac{x^{2}}{2}}D_{\sqrt{2}}g_{0}(\omega)$
(3.30) $\displaystyle=e^{-\tfrac{\pi}{2}(x^{2}+\omega^{2})}.$ (3.31)

Note, that we also have the relation $A(f,g)(x,\omega)=e^{\pi
ix\cdot\omega}V_{g}f(x,\omega)$. Hence, using the result from Example 2.3 that
$V_{g_{0}}g_{0}(x,\omega)=e^{-\pi
ix\cdot\omega}e^{-\frac{\pi}{2}(x^{2}+\omega^{2})}$ we would have obtained the
result without any further computations.

(a) Density-plots of the absolute values of the auto-ambiguity functions of
the box function $b_{0}$ and the standard Gaussian $g_{0}$.

(b) 3d-plots of the absolute values of the auto-ambiguity functions of the box
function $b_{0}$ and the standard Gaussian $g_{0}$.

Figure 8: Time-frequency concentration of the box-function $b_{0}$ and the
standard Gaussian $g_{0}$.

######  Remark.

We see that, for $x=0$, we obtain the Fourier transform of the box function

$Ab_{0}(0,\omega)=\mathcal{F}b_{0}(\omega)=\operatorname*{sinc}(\omega),$
(3.32)

which demonstrates, again, that the frequency resolution of the box function
is not suitable for measuring time-frequency content. On the time-side, the
ambiguity function measures the auto-correlation of a signal. Since the box
function is an even function (i.e., $g^{\vee}=g$), for $\omega=0$, we obtain
the convolution of the box function with itself by using l’Hospital’s rule

$Ab_{0}(x,0)=\lim_{\omega\to
0}\frac{\sin(\pi\omega)(1-|x|)}{\pi\omega}=1-|x|.$ (3.33)

For the second example, we see that for $x=0$, we obtain the auto-correlation
of the Gaussian spectrum

$Ag_{0}(0,\omega)=e^{-\frac{\pi}{2}\omega^{2}},$ (3.34)

which is a dilated Gaussian, and not the standard Gaussian. The same is true
in the case $\omega=0$;

$Ag_{0}(x,0)=e^{-\frac{\pi}{2}x^{2}}.$ (3.35)

This is, as in the case of the box function, the convolution of the Gaussian
with itself.

What causes the difference in the spectral behavior of the box function
compared to the Gaussian, is the fact that $Ab_{0}(0,\omega)$ does actually
not give the Fourier transform of $b_{0}$, but the auto-correlation of the
spectrum $\mathcal{F}b_{0}$. But, the auto-correlation corresponds to a
convolution in this case. Now, the $\operatorname*{sinc}$-function is
invariant under convolution, as can easily be seen by the following
computation;

$(\operatorname*{sinc}*\operatorname*{sinc})(\omega)=(\mathcal{F}b_{0}*\mathcal{F}b_{0})(\omega)=\mathcal{F}(\underbrace{b_{0}\,b_{0}}_{=b_{0}})(\omega)=\mathcal{F}b_{0}(\omega)=\operatorname*{sinc}(\omega).$
(3.36)

Note that, in general, we have

$Af(0,\omega)=\mathcal{F}(f\,\overline{f})(\omega)=(\mathcal{F}f*\mathcal{F}\overline{f})(\omega)\quad\text{
and }\quad Af(x,0)=(f*\overline{f}^{\vee})(x).$ (3.37)

$\diamond$

Since, we have the relation $A(f,g)(x,\omega)=e^{\pi
ix\cdot\omega}V_{g}f(x,\omega)$, it is clear that the ambiguity functions
enjoys similar properties as the STFT. In particular, the orthogonality
relations and the Parseval-like identity hold;

$\displaystyle\langle
A(f_{1},g_{1}),A(f_{2},g_{2})\rangle_{L^{2}(\mathbb{R}^{2d})}$
$\displaystyle=\langle e^{\pi ix\cdot\omega}V_{g_{1}}f_{1},e^{\pi
ix\cdot\omega}V_{g_{2}}f_{2}\rangle_{L^{2}(\mathbb{R}^{2d})}$ (3.38)
$\displaystyle=\langle
V_{g_{1}}f_{1},V_{g_{2}}f_{2}\rangle_{L^{2}(\mathbb{R}^{2d})}=\langle
f_{1},f_{2}\rangle\overline{\langle g_{1},g_{2}\rangle}.$ (3.39)

Therefore, we also have

$\lVert A(f,g)\rVert_{L^{2}(\mathbb{R}^{2d})}=\lVert f\rVert_{2}\lVert
g\rVert_{2}.$ (3.40)

As the ambiguity function has a symmetry in the time-frequency shifts,
$A(f,g)(x,\omega)=\langle\pi(-\frac{\lambda}{2})f,\pi(\frac{\lambda}{2})g\rangle$,
the analogue to Proposition 2.5 (Fundamental identity of TFA) becomes more
symmetric as well;

$\displaystyle A(f,g)(x,\omega)$ $\displaystyle=e^{\pi
ix\cdot\omega}V_{g}f(x,\omega)=e^{\pi ix\cdot\omega}e^{-2\pi
ix\cdot\omega}V_{\widehat{g}}\widehat{f}(\omega,-x)$ (3.41)
$\displaystyle=e^{-\pi
ix\cdot\omega}V_{\widehat{g}}\widehat{f}(\omega,-x)=A(\widehat{f},\widehat{g})(\omega,-x).$
(3.42)

Lastly, we discuss a covariance result for the ambiguity function. For $f,g\in
L^{2}(\mathbb{R}^{d})$, $x,\omega,t\in\mathbb{R}^{d}$ and
$\xi_{1},\xi_{2},\eta_{1},\eta_{2}\in\mathbb{R}^{d}$, we have

$\displaystyle\,A(M_{\eta_{1}}T_{\xi_{1}}f,M_{\eta_{2}}T_{\xi_{2}}g)(x,\omega)$
(3.43)
$\displaystyle=\int_{\mathbb{R}^{d}}(M_{\eta_{1}}T_{\xi_{1}}f)(t+\tfrac{x}{2})(M_{\eta_{2}}T_{\xi_{2}}f)(t-\tfrac{x}{2})e^{-2\pi
i\omega\cdot t}\,dt$ (3.44)
$\displaystyle=\int_{\mathbb{R}^{d}}f(t+\tfrac{x}{2}-\xi_{1})\overline{g(t-\tfrac{x}{2}-\xi_{2})}\,e^{2\pi
i\eta_{1}\cdot(t+\frac{x}{2})}e^{-2\pi i\eta_{2}\cdot(t-\frac{x}{2})}e^{-2\pi
i\omega\cdot
t}\,dt\quad\begin{subarray}{c}\phantom{=}\end{subarray}_{(t\mapsto
t+\tfrac{\xi_{1}}{2}+\tfrac{\xi_{2}}{2})}$ (3.45) $\displaystyle=e^{\pi
ix\cdot(\eta_{1}+\eta_{2})}\int_{\mathbb{R}^{d}}f(t+\tfrac{x-\xi_{1}+\xi_{2}}{2})\overline{g(t-\tfrac{x-\xi_{1}+\xi_{2}}{2})}\,e^{-2\pi
i(\omega-\eta_{1}+\eta_{2})\cdot(t+\frac{\xi_{1}+\xi_{2}}{2})}\,dt$ (3.46)
$\displaystyle=e^{\pi ix\cdot(\eta_{1}+\eta_{2})}e^{-\pi
i\omega\cdot(\xi_{1}+\xi_{2})}e^{\pi
i(\eta_{1}-\eta_{2})\cdot(\xi_{1}+\xi_{2})}\int_{\mathbb{R}^{d}}f(t+\tfrac{x-\xi_{1}+\xi_{2}}{2})\overline{g(t-\tfrac{x-\xi_{1}+\xi_{2}}{2})}\,e^{-2\pi
i(\omega-\eta_{1}+\eta_{2})\cdot t}\,dt$ (3.47) $\displaystyle=e^{\pi
ix\cdot(\eta_{1}+\eta_{2})}e^{-\pi i\omega\cdot(\xi_{1}+\xi_{2})}e^{\pi
i(\eta_{1}-\eta_{2})\cdot(\xi_{1}+\xi_{2})}A(f,g)(x-\xi_{1}+\xi_{2},\omega-\eta_{1}+\eta_{2}).$
(3.48)

In particular, for $\xi_{1}=\xi_{2}=\xi$ and $\eta_{1}=\eta_{2}=\eta$ we
obtain

$A(M_{\eta}T_{\xi}f,M_{\eta}T_{\xi}g)(x,\omega)=e^{2\pi
i(x\cdot\eta-\omega\cdot\xi)}A(f,g)(x,\omega).$ (3.49)

This shows that shifting, both, $f$ and $g$ in the time-frequency plane, only
yields a phase factor.

###  3.4 The Wigner Distribution

The Wigner distribution was “invented” by Eugene Wigner in 1932 [53] in the
context of quantum mechanics. We quote from the introduction in [17]:

In this paper [[53]; author’s note], Wigner introduced a probability quasi-
distribution that allowed him to express quantum mechanical expectation values
in the same form as the averages of classical statistical mechanics. There
have been many speculations about how Wigner got his idea; his eponymous
transform seems to be pulled out of thin air.

We will now study some of the properties of this almost mysterious transform,
including Hudson’s theorem, which will essentially take us all the way to very
recent mathematical research. Besides the textbook of Gröchenig [19], the book
by de Gosson [17] may be consulted as a further reference and for more
information on the Wigner transform.

######  Definition 3.5.

The Wigner distribution of a function $f\in L^{2}(\mathbb{R}^{d})$ is given by

$Wf(x,\omega)=\int_{\mathbb{R}^{d}}f(x+\tfrac{t}{2})\overline{f(x-\tfrac{t}{2})}e^{-2\pi
i\omega\cdot t}\,dt.$ (3.50)

The cross-Wigner transform of two functions $f,g\in L^{2}(\mathbb{R}^{d})$ is
given by

$W(f,g)(x,\omega)=\int_{\mathbb{R}^{d}}f(x+\tfrac{t}{2})\overline{g(x-\tfrac{t}{2})}e^{-2\pi
i\omega\cdot t}\,dt.$ (3.51)

We quote from the introduction in [17]:

Truly, Wigner’s definition, in modern notation did not remind of anything one
had seen before; a rapid glance suggests it is something of a mixture of a
Fourier transform and a convolution. And, yet it worked! For instance, under
some mild assumptions on the function $\psi$ [which may be a quantum state or
wave function; author’s note] we recover the probability amplitudes
$|\psi(x)|^{2}$ and $|\phi(p)|^{2}$ [$\phi=\widehat{\psi}$, author’s note] of
quantum mechanics:

$\displaystyle\int_{-\infty}^{\infty}W\psi(x,p)\,dp=|\psi(x)|^{2},$ (3.52)
$\displaystyle\int_{-\infty}^{\infty}W\psi(x,p)\,dx=|\phi(p)|^{2};$ (3.53)

assuming $\psi$ normalized to unity and integrating any of these equalities,
we get

$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W\psi(x,p)\,dp\,dx=1,$ (3.54)

so that the Wigner transform $W\psi$ can be used as a mock probability
distribution.

The fact that we (can) only call the Wigner transform a quasi-distribution
stems from the fact that it fails to be non-negative in general. The only
exceptions come from Gaussian functions, which is Hudson’s Theorem. Before we
prove that $Wf$ is a quasi-distribution, we need to describe the connections
to the other time-frequency representations.

We recall that the reflection of a function $g$ is denoted by $g^{\vee}$;

$g^{\vee}(t)=g(-t),\qquad t\in\mathbb{R}^{d}.$ (3.55)

The following lemma shows the algebraic connection between $V_{g}f$ (and hence
$A(f,g)$) and $W(f,g)$.

######  Lemma 3.6 (Algebraic Relations).

For $f,g\in L^{2}(\mathbb{R}^{d})$ we have

$W(f,g)(x,\omega)=2^{d}e^{4\pi
ix\cdot\omega}V_{g^{\vee}}f(2x,2\omega)=2^{d}A(f,g^{\vee})(2x,2\omega).$
(3.56)

###### Proof.

We will use the substitution $u=x+\tfrac{t}{2}$ in the definition of $W(f,g)$
and compute

$\displaystyle W(f,g)(x,\omega)$
$\displaystyle=\int_{\mathbb{R}^{d}}f(x+\tfrac{t}{2})\overline{g(x-\tfrac{t}{2})}e^{-2\pi
i\omega\cdot t}\,dt$ (3.57)
$\displaystyle=2^{d}\int_{\mathbb{R}^{d}}f(u)\overline{g(-(u-2x)}e^{-2\pi
i\omega\cdot(2u-2x)}\,du$ (3.58) $\displaystyle=2^{d}e^{4\pi
ix\cdot\omega}\langle f,M_{2\omega}T_{2x}g^{\vee}\rangle$ (3.59)
$\displaystyle=2^{d}e^{4\pi ix\cdot\omega}V_{g^{\vee}}f(2x,2\omega).$ (3.60)

∎

We will now collect properties of the Wigner distribution from the STFT. Some
of these properties readily follow by using Lemma 3.6.

######  Proposition 3.7.

For $f,g\in L^{2}(\mathbb{R}^{d})$ the cross-Wigner distribution has the
following properties.

  1. (i)

$W(f,g)\in L^{\infty}(\mathbb{R}^{2d})\cap C_{0}(\mathbb{R}^{2d})$, in
particular $W(f,g)$ is uniformly continuous on $\mathbb{R}^{2d}$ and is
bounded by

$\lVert W(f,g)\rVert_{\infty}\leq 2^{d}\lVert f\rVert_{2}\lVert g\rVert_{2}.$
(3.61)

  2. (ii)

$W(f,g)=\overline{W(g,f)}$. In particular, $Wf$ is real-valued.

  3. (iii)

For $\xi_{1},\xi_{2},\eta_{1},\eta_{2}\in\mathbb{R}^{d}$ we have

$\displaystyle\,W(M_{\eta_{1}}T_{\xi_{1}}f,\,M_{\eta_{2}}T_{\xi_{2}}g)$ (3.62)
$\displaystyle=$ $\displaystyle\,e^{\pi
i(\xi_{1}-\xi_{2})\cdot(\eta_{1}+\eta_{2})}e^{2\pi
ix\cdot(\eta_{1}-\eta_{2})}e^{-2\pi
i\omega\cdot(\xi_{1}-\xi_{2})}\,W(f,g)(x-\tfrac{\xi_{1}+\xi_{2}}{2},\omega-\tfrac{\eta_{1}+\eta_{2}}{2}).$
(3.63)

In particular, $Wf$ is covariant, i.e.,

$W(M_{\eta}T_{\xi}f)(x,\omega)=Wf(x-\xi,\omega-\eta).$ (3.64)

  4. (iv)

$W(\widehat{f},\widehat{g})(x,\omega)=W(f,g)(-\omega,x)$

  5. (v)

Also, we have the Parseval-like identity, which goes under the name Moyal’s
formula;

$\langle W(f_{1},g_{1}),W(f_{2},g_{2})\rangle_{L^{2}(\mathbb{R}^{2d})}=\langle
f_{1},f_{2}\rangle\overline{\langle g_{1},g_{2}\rangle}.$ (3.65)

###### Proof.

The proofs follow mainly by direct computation.

  1. (i)

The assertion that $W(f,g)$ is uniformly continuous and decaying to 0 follows
from the respective properties of $V_{g}f$ by using Lemma 3.6. To show the
$L^{\infty}$-bound we compute

$\displaystyle|W(g,f)(x,\omega)|$
$\displaystyle=2^{d}|V_{g^{\vee}}f(2x,2\omega)|=2^{d}|\langle
f,M_{2\omega}T_{2x}g^{\vee}\rangle|$ (3.66) $\displaystyle\leq 2^{d}\lVert
f\rVert_{2}\lVert M_{2\omega}T_{2x}g^{\vee}\rVert_{2}=2^{d}\lVert
f\rVert_{2}\lVert g\rVert_{2}.$ (3.67)

In the above calculations we used, again, Lemma 3.6 and the Cauchy-Schwarz
inequality. Also we used that $M_{\omega}$ and $T_{x}$ are unitary and that
flipping $g$, i.e., $g\mapsto g^{\vee}$, is unitary as well.

  2. (ii)

By using the substitution $t\mapsto-t$, we compute

$\displaystyle W(f,g)(x,\omega)$
$\displaystyle=\int_{\mathbb{R}^{d}}f(x+\tfrac{t}{2})\overline{g(x-\tfrac{t}{2})}e^{-2\pi
i\omega\cdot t}\,dt$ (3.68)
$\displaystyle=\overline{\int_{\mathbb{R}^{d}}\overline{f(x-\tfrac{t}{2})}g(x+\tfrac{t}{2})e^{-2\pi
i\omega\cdot t}\,dt}$ (3.69) $\displaystyle=\overline{W(g,f)(x,\omega)}.$
(3.70)

  3. (iii)

We compute

$\displaystyle\,W(M_{\eta_{1}}T_{\xi_{1}}f,M_{\eta_{2}}T_{\xi_{2}}g)(x,\omega)$
(3.71)
$\displaystyle=\int_{\mathbb{R}^{d}}(M_{\eta_{1}}T_{\xi_{1}}f)(x+\tfrac{t}{2})\overline{(M_{\eta_{2}}T_{\xi_{2}}g)(x-\tfrac{t}{2})}e^{-2\pi
i\omega\cdot t}\,dt$ (3.72)
$\displaystyle=\int_{\mathbb{R}^{d}}f(x-\xi_{1}+\tfrac{t}{2})\overline{g(x-\xi_{2}-\tfrac{t}{2})}e^{2\pi
i\eta_{1}\cdot(x+\frac{t}{2})}e^{-2\pi i\eta_{2}\cdot(x-\frac{t}{2})}e^{-2\pi
i\omega\cdot
t}\,dt\quad\begin{subarray}{c}\phantom{=}\end{subarray}_{(t\mapsto
t+\xi_{1}-\xi_{2})}$ (3.73) $\displaystyle=e^{2\pi
ix\cdot(\eta_{1}-\eta_{2})}\int_{\mathbb{R}^{d}}f(x-\tfrac{\xi_{1}+\xi_{2}}{2}+\tfrac{t}{2})\overline{g(x-\tfrac{\xi_{1}+\xi_{2}}{2}-\tfrac{t}{2})}e^{-2\pi
i(\omega-\frac{\eta_{1}+\eta_{2}}{2})\cdot(t+\xi_{1}-\xi_{2})}\,dt$ (3.74)
$\displaystyle=e^{2\pi ix\cdot(\eta_{1}-\eta_{2})}e^{-2\pi
i\omega\cdot(\xi_{1}-\xi_{2})}\int_{\mathbb{R}^{d}}f(x-\tfrac{\xi_{1}+\xi_{2}}{2}+\tfrac{t}{2})\overline{g(x-\tfrac{\xi_{1}+\xi_{2}}{2}-\tfrac{t}{2})}e^{-2\pi
i(\omega-\frac{\eta_{1}+\eta_{2}}{2})\cdot t}\,dt$ (3.75)
$\displaystyle=e^{2\pi ix\cdot(\eta_{1}-\eta_{2})}e^{-2\pi
i\omega\cdot(\xi_{1}-\xi_{2})}e^{\pi
i(\xi_{1}-\xi_{2})(\eta_{1}+\eta_{2})}W(f,g)(x-\tfrac{\xi_{1}+\xi_{2}}{2},\omega-\tfrac{\eta_{1}+\eta_{2}}{2}).$
(3.76)

  4. (iv)

We note that $\widehat{g^{\vee}}=\widehat{g}^{\vee}$. We compute

$\displaystyle W(\widehat{f},\widehat{g})(x,\omega)$
$\displaystyle=2^{d}e^{4\pi
ix\cdot\omega}\langle\widehat{f},M_{2\omega}T_{2x}\widehat{g}^{\vee}\rangle$
(3.77) $\displaystyle=2^{d}e^{4\pi ix\cdot\omega}\langle
f,T_{-2\omega}M_{2x}g^{\vee}\rangle$ (3.78) $\displaystyle=2^{d}e^{-4\pi
ix\cdot\omega}\langle f,M_{2x}T_{-2\omega}g^{\vee}\rangle$ (3.79)
$\displaystyle=W(f,g)(-\omega,x).$ (3.80)

In the above computations we used Lemma 3.6, Parseval’s formula (0.21) and the
commutation relations (1.3).

  5. (v)

We compute

$\displaystyle\,\iint_{\mathbb{R}^{2d}}W(f_{1},g_{1})(x,\omega)\overline{W(f_{2},g_{2})(x,\omega)}\,dxd\omega$
(3.81) $\displaystyle=$
$\displaystyle\,2^{2d}\iint_{\mathbb{R}^{2d}}V_{g_{1}^{\vee}}f_{1}(2x,2\omega)\overline{V_{g_{2}^{\vee}}f_{2}(2x,2\omega)}\,dxd\omega$
(3.82) $\displaystyle=$ $\displaystyle\,\langle
f_{1},f_{2}\rangle\overline{\langle g^{\vee}_{1},g^{\vee}_{2}\rangle}$ (3.83)
$\displaystyle=$ $\displaystyle\,\langle f_{1},f_{2}\rangle\overline{\langle
g_{1},g_{2}\rangle}$ (3.84)

When changing from the integral formulas to the inner products, we made the
substitution $(2x,2\omega)\mapsto(x,\omega)$, which is why the factor $2^{2d}$
disappeared. Also, we used the orthogonality relations (2.27) of the STFT.

∎

Before we proceed, we introduce some more notation. By $\mathcal{T}_{s}$ we
denote the symmetric coordinate change defined by

$\mathcal{T}_{s}F(x,t)=F(x+\tfrac{t}{2},x-\tfrac{t}{2}).$ (3.85)

We also note that its inverse is given by

$\mathcal{T}_{s}^{-1}F(x,t)=F(\tfrac{x+t}{2},x-t).$ (3.86)

Furthermore, we re-call the notion of the partial Fourier transforms
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, which transform a function of two
arguments only in the first and the second argument, respectively.

$\mathcal{F}_{1}F(x,\omega)=\int_{\mathbb{R}^{d}}F(t,\omega)e^{-2\pi ix\cdot
t}\,dt.\qquad\text{ and
}\qquad\mathcal{F}_{2}F(x,\omega)=\int_{\mathbb{R}^{d}}F(x,t)e^{-2\pi
i\omega\cdot t}\,dt.$ (3.87)

Using these (unitary) operators, the cross-Wigner distribution can be written
as

$W(f,g)(x,\omega)=\mathcal{F}_{2}\left(\mathcal{T}_{s}(f\otimes\overline{g})(x,t)\right)(x,\omega),$
(3.88)

where $(f\otimes g)(x,\omega)=f(x)g(\omega)$ is the already familiar tensor
product. From this factorization, we easily obtain an inversion formula for
the (auto-)Wigner distribution.

$f(x)\overline{f(y)}=\left(\mathcal{T}_{s}^{-1}\mathcal{F}_{2}^{-1}Wf\right)(x,y)=\int_{\mathbb{R}^{d}}Wf\left(\tfrac{x+y}{2},\omega\right)e^{2\pi
i(x-y)\cdot\omega}\,d\omega.$ (3.89)

Now, if $f(0)\neq 0$, then

$f(x)=\frac{1}{\overline{f(0)}}\int_{\mathbb{R}^{d}}Wf\left(\tfrac{x}{2},\omega\right)e^{2\pi
ix\cdot\omega}\,d\omega.$ (3.90)

Of course, this inversion formula is true only up to a global phase factor
$|c|=1$.

The next lemma is highly important and reveals a deep connection between the
ambiguity function and the Wigner distribution via the Fourier transform.

######  Lemma 3.8.

For $f,g\in L^{2}(\mathbb{R}^{d})$, we have

$W(f,g)(x,\omega)=\mathcal{F}A(f,g)(\omega,-x)\qquad\text{ and }\qquad
A(f,g)(x,\omega)=\mathcal{F}W(f,g)(\omega,-x)$ (3.91)

###### Proof.

We note that the Fourier transform on $\mathbb{R}^{2d}$ factors into the
partial Fourier transforms on $\mathbb{R}^{d}$, i.e.,
$\mathcal{F}=\mathcal{F}_{1}\mathcal{F}_{2}=\mathcal{F}_{2}\mathcal{F}_{1}$.
By using (3.88) we have

$\displaystyle\mathcal{F}^{-1}W(f,g)(x,\omega)$
$\displaystyle=\mathcal{F}_{1}^{-1}\mathcal{F}_{2}^{-1}\mathcal{F}_{2}\mathcal{T}_{s}(f\otimes\overline{g})(x,\omega)$
(3.92)
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{T}_{s}(f\otimes\overline{g})(t,\omega)e^{2\pi
it\cdot x}\,dt$ (3.93)
$\displaystyle=\int_{\mathbb{R}^{d}}f(t+\tfrac{\omega}{2})\overline{g(t-\tfrac{\omega}{2})}e^{2\pi
ix\cdot t}\,dt$ (3.94) $\displaystyle=A(f,g)(\omega,-x).$ (3.95)

This proves the first assertion. The second one follows analogously.

$\displaystyle\mathcal{F}W(f,g)(\omega,-x)$
$\displaystyle=\mathcal{F}_{1}\mathcal{F}_{2}\mathcal{F}_{2}^{-1}\mathcal{T}_{s}(f\otimes\overline{g})(\omega,x)$
(3.96)
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{T}_{s}(f\otimes\overline{g})(t,x)e^{-2\pi
i\omega\cdot t}\,dt$ (3.97)
$\displaystyle=\int_{\mathbb{R}^{d}}f(t+\tfrac{x}{2})\overline{f(t-\tfrac{x}{2})}e^{-2\pi
i\omega\cdot t}\,dt$ (3.98) $\displaystyle=A(f,g)(x,\omega).$ (3.99)

The computations certainly hold for Schwartz functions (or on the dense
subspace $L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$) and the formulas
extend to all $f,g\in L^{2}(\mathbb{R}^{d})$ by continuity of $\mathcal{F}$
and $\mathcal{T}_{s}$. ∎

After this discussion on the connection with other time-frequency
representations, we will discuss the support properties of the Wigner
distribution. To simplify notation, we are only going to discuss this in the
case $d=1$ (hence, for intervals and not for $d$-dimensional hyper cubes).

######  Lemma 3.9.

For $f\in L^{2}(\mathbb{R})$, if
$\operatorname*{supp}(f)\subset[a,b]=\mathcal{Q}_{[a,b]}$, then
$Wf(x,\omega)=0$ for $x\notin\mathcal{Q}_{[a,b]}$. Likewise, if
$\operatorname*{supp}(\widehat{f})\subset\mathcal{Q}_{[c,d]}$, then
$Wf(x,\omega)=0$ for $\omega\notin\mathcal{Q}_{[c,d]}$.

###### Proof.

$Wf(x,\omega)\neq 0$ is possible only if
$x+\frac{t}{2}\in\operatorname*{supp}(f)$ and
$x-\frac{t}{2}\in\operatorname*{supp}(f)$. Then
$x=\frac{1}{2}(x+\frac{t}{2})+\frac{1}{2}(x-\frac{t}{2})\in\mathcal{Q}_{[a,b]}$.
Thus $Wf(x,\omega)=0$ for $x\notin\mathcal{Q}_{[a,b]}$.

The second statement follows from the relation
$Wf(x,\omega)=W\widehat{f}(\omega,-x)$. ∎

So, in contrast to the ambiguity function (and the STFT), the Wigner transform
preserves the support properties of $f$ and $\widehat{f}$ exactly (so does the
Rihaczek distribution). Also, note that the auto-Wigner distribution is taken
as a joint (quasi-) probability distribution in phase space. The
interpretation of the STFT usually depends on the auxiliary window and the
“sliding” of the window $g$ spreads out the support of $f$.

The next result is at the origin of interest in quantum mechanics and shows
that the Wigner distribution yields the correct marginal densities, which then
allow to interpret the Wigner distribution as a joint (quasi-) probability
density function.

######  Lemma 3.10.

Let $f,\widehat{f}\in L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$. Then

$\int_{\mathbb{R}^{d}}Wf(x,\omega)\,d\omega=|f(x)|^{2},$ (3.100)
$\int_{\mathbb{R}^{d}}Wf(x,\omega)\,dx=|\widehat{f}(\omega)|^{2}.$ (3.101)

In particular, if $\lVert f\rVert_{2}=\lVert\widehat{f}\rVert_{2}=1$, then

$\iint_{\mathbb{R}^{2d}}Wf(x,\omega)\,d(x,\omega)=1.$ (3.102)

###### Proof.

We use the algebraic relation 3.6 between the Wigner transform and the STFT
and write the STFT as a convolution of Fourier transforms (see (2.11));

$|Wf(x,\omega)|=2^{d}|V_{f^{\vee}}f(2x,2\omega)|=2^{d}|\widehat{f}*M_{-2x}\widehat{\overline{f}}^{\vee}(2\omega)|.$
(3.103)

By Young’s Theorem 0.7 we have the convolution inequality (0.29) which tells
us that the convolution of two $L^{1}(\mathbb{R}^{d})$-functions is again an
$L^{1}(\mathbb{R}^{d})$-function. Hence, under the Assumption that
$\widehat{f}\in L^{1}(\mathbb{R}^{d})$ and (3.103) we see that, for any fixed
$x\in\mathbb{R}^{d}$, $Wf(x,\omega)\in L^{1}(\mathbb{R}^{d},d\omega)$.
Therefore, the Fourier inversion formula for the partial Fourier transform
$\mathcal{F}_{2}$ is applicable and yields

$\displaystyle\int_{\mathbb{R}^{d}}Wf(x,\omega)\,d\omega$
$\displaystyle=\left(\mathcal{F}_{2}^{-1}\left(\mathcal{F}_{2}\mathcal{T}_{s}(f\otimes\overline{f})\right)\right)(x,0)=\mathcal{T}_{s}(f\otimes\widehat{f})(x,0)=|f(x)|^{2}.$
(3.104)

Now, using Proposition 3.7(iv) we compute

$\int_{\mathbb{R}^{d}}Wf(x,\omega)\,dx=\int_{\mathbb{R}^{d}}W\widehat{f}(\omega,-x)\,dx=|\widehat{f}(\omega)|^{2}.$
(3.105)

∎

In the language of quantum mechanics, this means that
$\int_{\mathbb{R}^{d}}Wf(x,\omega)\,d\omega$ yields the probability density
$|f(x)|^{2}$ for the position variable and
$\int_{\mathbb{R}^{d}}WF(x,\omega)\,dx$ yields the density $|\widehat{f}|^{2}$
for the momentum variable.

Also, we remark that, for a suitable subspace of $L^{2}(\mathbb{R}^{d})$ such
that $Wf\in L^{1}(\mathbb{R}^{2d})$ (which allows us to use Fubini’s theorem),
we have

$\lVert
f\rVert_{2}^{2}=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}Wf(x,\omega)\,dx\,d\omega=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}Wf(x,\omega)\,d\omega\,dx=\lVert\widehat{f}\rVert_{2}^{2}.$
(3.106)

Thus, Lemma 3.10 implies Plancherel’s theorem (on this subspace, which is
actually dense).

We will now discuss the positivity of the Wigner distribution and, hence,
whether it can serve as an energy density or joint probability density. We
mention right away that, in general the Wigner distribution fails to be
positive. For example, let $f$ be an odd function, i.e.,
$f(t)=-f^{\vee}(t)=-f(-t)$, then

$Wf(0,0)=\int_{\mathbb{R}^{d}}f(\tfrac{t}{2})\overline{f(-\tfrac{t}{2})}\,dt=-2^{d}\int_{\mathbb{R}^{d}}f(t)\overline{f^{\vee}(-t)}\,dt=-2^{d}\int_{\mathbb{R}^{d}}|f(t)|^{2}\,dt=-2^{d}\lVert
f\rVert_{2}^{2}<0.$ (3.107)

Negative values cannot be interpreted as a probability or as an energy
density. However, a single point in the time-frequency plane or phase space
also has no physical meaning. Now, the hope is that, maybe there is a “nice”
subspace of “physically realizable” functions for which the Wigner
distribution is positive? The answer to this question is given by Hudson’s
theorem [29].

######  Theorem 3.11 (Hudson).

Assume that $f\in L^{2}(\mathbb{R}^{d})$. Then $Wf(x,\omega)>0$ for all
$(x,\omega)\in\mathbb{R}^{2d}$ if and only if $f$ is a generalized Gaussian,
i.e.,

$f(t)=e^{-\pi t\cdot At+2\pi b\cdot t+c},$ (3.108)

where $A\in GL(\mathbb{C},d)$ is an invertible $d\times d$ matrix with entries
from $\mathbb{C}$ and positive definite real part ($\Re(A)>0$) and with
$\overline{A}^{T}=A^{*}=A$, $b\in\mathbb{C}^{d}$ and $c\in\mathbb{C}$.

We will postpone the proof to a later point as we need more preparation for
the proof.

A bit more generally, Theorem 3.11 can also be stated as follows [24], [40].

######  Theorem 3.12 (Hudson-Lieb).

Let $f,g\in L^{2}(\mathbb{R}^{d})$ (non-zero). Then $W(f,g)\geq 0$ for all
$(x,\omega)\in\mathbb{R}^{2d}$ if and only if $f=K\,g$, $K>0$ and $g$ is a
generalized Gaussian of the form (3.108). In this case, $W(f,g)(x,\omega)>0$
for all $(x,\omega)\in\mathbb{R}^{2d}$.

We note that $W(f,g)$ is not necessarily real-valued. However, if it is non-
negative and real-valued, then $f$ and $g$ are the same generalized Gaussian
up to a positive constant and the cross-Wigner distribution is essentially an
auto-Wigner distribution and actually positive everywhere. Hudson’s original
theorem dates back to 1974 and the extension of Lieb was established in 1990.

A related and hard problem, which still remains open in full generality, is to
characterize pairs of functions $f$ and $g$ for which $W(f,g)(x,\omega)\neq 0$
for all $(x,\omega)\in\mathbb{R}^{2d}$. Only very recently examples of pairs
$(f,g)$, which are not Gaussians and yield zero-free Wigner distributions, or
likewise zero-free STFTs and ambiguity functions, have been found and
described by Gröchenig, Jaming and Malinnikova in their article [24] published
in 2020. The prototype example they give is obtained from the one-sided
exponential.

######  Example 3.13.

Let $f$ be a one-sided exponential, i.e.,

$\displaystyle f(t)=\raisebox{2.0pt}{$\chi$}_{\mathbb{R}_{+}}(t)e^{-\pi
t}=\begin{cases}e^{-\pi t},&t>0\\\ 0,&else\end{cases}.$ (3.109)

Figure 9: The one-sided exponential function.

Assume $x\geq 0$, then the ambiguity function is given by

$\displaystyle Af(x,\omega)$
$\displaystyle=\int_{\mathbb{R}}\raisebox{2.0pt}{$\chi$}_{\mathbb{R}_{+}}(t-\tfrac{x}{2})\raisebox{2.0pt}{$\chi$}_{\mathbb{R}_{+}}(t+\tfrac{x}{2})e^{-\pi(-\frac{x}{2})}e^{-\pi(t+\frac{x}{2})}e^{-2\pi
i\omega t}\,dt$ (3.110) $\displaystyle=\int_{\frac{x}{2}}^{\infty}e^{-2\pi
t}e^{-2\pi i\omega t}\,dt=\int_{\frac{x}{2}}^{\infty}e^{-2\pi
i(\omega-i)t}\,dt=\frac{e^{-2\pi i(\omega-i)t}}{-2\pi
i(\omega-i)}\Bigg{|}_{t=\frac{x}{2}}^{\infty}$ (3.111)
$\displaystyle=\frac{e^{-\pi i(\omega-i)x}}{2\pi i(\omega-i)}.$ (3.112)

The part for $x<0$ follows analogously and we get

$Af(x,\omega)=\frac{e^{-\pi i(\omega-i)|x|}}{2\pi i(\omega-i)}.$ (3.113)

In particular, $Af(x,\omega)\neq 0$ for all $(x,\omega)\in\mathbb{R}^{2}$. It
follows immediately that $W(f,f^{\vee})\neq 0$ by the algebraic relation in
Lemma 3.6.

Figure 10: 3d plot and density plot of the absolute values of the ambiguity
function of the 1-sided exponential. While it is well-concentrated in the time
domain, the localization in the frequency domain is not satisfying, as the
decay is only $\mathcal{O}(\frac{1}{\omega})$. This is caused by the
discontinuity of the complex exponential (as smoothness on the time-side
corresponds to fast decay in the frequency domain).

On the other hand, it follows from the Hudson-Lieb Theorem 3.12 that if $f=\pm
f^{\vee}$, i.e., $f$ is either an odd or even function, then
$A(f,f^{\vee})=\pm Af$ can only be positive if $f$ is a generalized Gaussian.

  * •

If $f=f^{\vee}$, i.e., $f$ is even, then $W(f,f^{\vee})=Wf$. Also, $Wf$ is
real-valued and $Wf(0,0)=2^{d}\lVert f\rVert_{2}^{2}$. By Hudson’s theorem the
Wigner transform is only positive for Gaussians. Therefore, in this case the
Wigner distribution must possess zeros if $f$ is not a generalized Gaussian.
Now, by the algebraic relation given by Lemma 3.6 the same holds for the
ambiguity function $Af$.

  * •

If $f=-f^{\vee}$, i.e., $f$ is odd, then $W(f,f^{\vee})=-Wf$ and,
$W(f,f^{\vee})(0,0)=2^{d}\lVert f\rVert_{2}^{2}$. Therefore, as $Wf$ is real-
valued, the Wigner distribution $W(f,f^{\vee})$ is real-valued and by the
Hudson-Lieb theorem either possesses zeros or $f$ is a generalized Gaussian.
Again, by Lemma 3.6 the same is true for the ambiguity function $Af$.

This shows that if the ambiguity function $Af$ and cross Wigner distribution
$W(f,f^{\vee})$ are zero-free, then $f$ does naturally not possess “simple”
symmetries, meaning that it cannot be an even or an odd function, unless it is
a Gaussian.

Curiously, this prototype example was (almost) already given by Augustus
Janssen more than 20 years earlier in 1996. In [33], Janssen already computed
the ambiguity function of the one-sided exponential (evaluated only at integer
points, i.e., $(x,\omega)\in\mathbb{Z}^{2}$). However, it was not noticed that
a zero-free ambiguity function was already given in [33] until the appearance
of [24]. The authors re-discovered this example independently as the one-sided
exponential is also a prototype example of so-called totally positive
functions of finite type. These functions have attracted considerable
attraction in recent years in Gabor analysis and we will encounter them later
on as well.

######  Example 3.14.

Next, we will compute the Wigner distribution of the point measure or delta
distribution $\delta_{0}$, where $\langle\delta_{x},f\rangle=\overline{f(x)}$.
For $f\in\mathcal{S}(\mathbb{R}^{d})$, by using (3.88), we have

$\displaystyle\langle W(\delta_{0},\delta_{0}),f\rangle$
$\displaystyle=\langle\mathcal{F}_{2}\mathcal{T}_{s}(\delta_{0}\otimes\overline{\delta_{0}}),f\rangle$
(3.114)
$\displaystyle=\langle\delta_{0}\otimes\overline{\delta_{0}},\mathcal{T}_{s}^{-1}\mathcal{F}_{2}^{-1}f\rangle$
(3.115)
$\displaystyle=\overline{\mathcal{T}_{s}^{-1}\mathcal{F}_{2}^{-1}f(0,0)}$
(3.116)

Using the formula (3.86) for $\mathcal{T}_{s}^{-1}$, we obtain

$\displaystyle\overline{\mathcal{T}_{s}^{-1}\mathcal{F}_{2}^{-1}f(0,0)}$
$\displaystyle=\overline{\int_{\mathbb{R}^{d}}f\left(\tfrac{x+y}{2},\omega\right)e^{2\pi
i(x-y)\cdot\omega}\,d\omega}\ \Bigg{|}_{x=y=0}$ (3.117)
$\displaystyle=\int_{\mathbb{R}^{d}}\overline{f(0,\omega)}\,d\omega$ (3.118)
$\displaystyle=\langle\delta_{0}\otimes\mathbf{1},f\rangle,$ (3.119)

where $\mathbf{1}$ is the constant 1-function. Therefore,
$W\delta_{0}=\delta_{0}\otimes\mathbf{1}\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$.

##  4 The Poisson Summation Formula

The Poisson summation formula is a highly useful tool in time-frequency
analysis, but also in many other branches of mathematics. It relates the
periodization of a function $f$ to a Fourier series on the torus
$\mathbb{T}^{d}$ with Fourier coefficients obtained from the Fourier transform
on $L^{2}(\mathbb{R}^{d})$.

######  Definition 4.1.

Given a function $f$, its $\alpha$-periodization is defined by

$\mathcal{P}_{\alpha}f(t)=\sum_{k\in\mathbb{Z}^{d}}f(t+\alpha k),\qquad
t\in\mathbb{R}^{d},\alpha>0.$ (4.1)

We note that if $f\in L^{1}(\mathbb{R}^{d})$, then $\mathcal{P}_{\alpha}f\in
L^{1}(\mathbb{T}^{d})$. If $\alpha=1$, we will omit the index and write

$\mathcal{P}f(t)=\mathcal{P}_{1}f(t).$ (4.2)

Also, we have the following property.

######  Lemma 4.2.

If $f\in L^{1}(\mathbb{R}^{d})$, then for all $\alpha>0$ we have

$\int_{\mathbb{R}^{d}}f(t)\,dt=\int_{[0,\alpha]^{d}}\left(\mathcal{P}_{\alpha}f(t)\right)\,dt.$
(4.3)

###### Proof.

We note that the cubes $\alpha k+[0,\alpha]^{d}$ form a (disjoint) partition
of $\mathbb{R}^{d}$ (up to an overlap of measure 0). We compute

$\displaystyle\int_{\mathbb{R}^{d}}f(t)\,dt$
$\displaystyle=\sum_{k\in\mathbb{Z}^{d}}\int_{\alpha
k+[0,\alpha]^{d}}f(t)\,dt$ (4.4)
$\displaystyle=\int_{[0,\alpha]^{d}}\left(\sum_{k\in\mathbb{Z}^{d}}f(t+\alpha
k)\right)\,dt.$ (4.5)

As $f\in L^{1}(\mathbb{R}^{d})$, the exchange of summation and integration is
justified by Fubini’s theorem. ∎

We are now ready to prove the Poisson summation formula in its standard
version.

######  Proposition 4.3 (Poisson Summation Formula).

Let $f$ be continuous and $\varepsilon>0$, $C>0$. Assume that $|f(t)|\leq
C(1+|t|)^{-d-\varepsilon}$ and $|\widehat{f}(\omega)|\leq
C(1+|\omega|)^{-d-\varepsilon}$. Then

$\sum_{k\in\mathbb{Z}^{d}}f(t+k)=\sum_{l\in\mathbb{Z}^{d}}\widehat{f}(l)e^{2\pi
il\cdot t}.$ (4.6)

The above identity holds point-wise for all $t\in\mathbb{R}^{d}$ and both sums
converge absolutely for all $t\in\mathbb{R}^{d}$.

###### Proof.

The decay conditions imply that $f$ as well as $\widehat{f}$ are in
$L^{1}(\mathbb{R}^{d})$ and by assumption (and the Lemma of Riemann-Lebesgue)
both are continuous. As $f\in L^{1}(\mathbb{R}^{d})$ we have that
$\sum_{k\in\mathbb{Z}^{d}}f(t+k)=\mathcal{P}f(t)\in L^{1}(\mathbb{T}^{d})$. We
want to check the Fourier coefficients of $\mathcal{P}f$.

$\displaystyle\mathcal{F}_{\mathbb{T}^{d}}(\mathcal{P}f)(l)$
$\displaystyle=\int_{\mathbb{T}^{d}}\mathcal{P}f(t)e^{-2\pi il\cdot t}\,dt$
(4.7)
$\displaystyle=\int_{\mathbb{T}^{d}}\left(\sum_{k\in\mathbb{Z}^{d}}f(t+k)e^{-2\pi
il\cdot(t+k)}\right)\,dt$ (4.8)
$\displaystyle=\int_{\mathbb{R}^{d}}f(t)e^{-2\pi il\cdot
t}\,dt=\mathcal{F}f(l)=\widehat{f}(l).$ (4.9)

By assumption $\sum_{l\in\mathbb{Z}^{d}}|\widehat{f}(l)|<\infty$, we see that
$\mathcal{P}f$ has an absolutely converging Fourier series

$\mathcal{P}f(t)=\sum_{k\in\mathbb{Z}^{d}}f(t+k)=\sum_{l\in\mathbb{Z}^{d}}\widehat{f}(l)e^{2\pi
il\cdot t}.$ (4.10)

∎

######  Remark.

  1. (i)

The conditions in Proposition 4.3 are stronger than needed for the point-wise
equality to hold, but they ensure absolute convergence of both series as well
as the point-wise equality. The conditions on $f$ and $\widehat{f}$ under
which the formula holds point-wise have been worked out, e.g., by Gröchenig in
[18].

  2. (ii)

Note that we assume that $f\in L^{1}(\mathbb{R}^{d})$, which, by the Lemma of
Riemann-Lebesgue, implies that $\widehat{f}$ is continuous. On the Fourier
side, we also assume that $\widehat{f}\in L^{1}(\mathbb{R}^{d})$, which also
implies that $\mathcal{F}^{-1}\widehat{f}$ is continuous. Many authors hence
drop the condition on $f$ being continuous as they implicitly assume that
$f(t)=\mathcal{F}^{-1}\widehat{f}(t)$.

  3. (iii)

If we replace the absolute convergence in Proposition 4.3 by convergence in
$L^{2}(\mathbb{R}^{d})$ and pointwise equality by equality almost everywhere,
we obtain a weaker version of the Poisson summation formula.

If $\sum_{k\in\mathbb{Z}^{d}}f(t+k)\in L^{2}(\mathbb{T}^{d})$ and
$\sum_{k\in\mathbb{Z}^{d}}|\widehat{f}(k)|^{2}<\infty$, then (4.6) holds
almost everywhere.

  4. (iv)

The Poisson summation formula can also be written for arbitrary lattices
$\Lambda\subset\mathbb{R}^{d}$ and their duals. For a lattice
$\Lambda=A\mathbb{Z}^{d}$, $A\in GL(\mathbb{R},d)$, the dual lattice is given
by $\Lambda^{\perp}=A^{-T}\mathbb{Z}^{d}$ and the Poisson summation formula
reads

$\sum_{\lambda\in\Lambda}f(t+\lambda)=\textnormal{vol}(\Lambda)^{-1}\sum_{\lambda^{\perp}\in\Lambda^{\perp}}\widehat{f}(\lambda^{\perp})e^{2\pi
i\lambda^{\perp}\cdot t}$ (4.11)

  5. (v)

In analytic number theory the Poisson summation formula yields the following
functional equation for the Jacobi theta function
$\theta(\tau)=\sum_{k\in\mathbb{Z}}e^{\pi i\tau k^{2}}$, $\tau\in\mathbb{H}$;

$\theta(\tau)=\sqrt{\tfrac{i}{\tau}}\ \theta(-\tfrac{1}{\tau})$ (4.12)

This functional equation was already used by Bernhard Riemann to come up with
the functional equation for the Riemann zeta function $\zeta(s)$.

Using Proposition 4.3 for the function $M_{\omega}T_{x}f(t)$ at $t=0$, we
obtain

$\displaystyle\sum_{k\in\mathbb{Z}^{d}}f(k-x)e^{2\pi i\omega\cdot k}$
$\displaystyle=\sum_{k\in\mathbb{Z}^{d}}M_{\omega}T_{x}f(k)=\sum_{l\in\mathbb{Z}^{d}}\mathcal{F}(M_{\omega}T_{x}f)(l)$
(4.13) $\displaystyle=\sum_{l\in\mathbb{Z}^{d}}e^{2\pi i\omega\cdot
x}M_{-x}T_{\omega}\mathcal{F}f(l)=\sum_{l\in\mathbb{Z}^{d}}\widehat{f}(l-\omega)e^{2\pi
ix\cdot(\omega-l)},$ (4.14)

or, likewise, by exchanging the order of translation and modulation we obtain

$\sum_{k\in\mathbb{Z}^{d}}f(k-x)e^{2\pi
i\omega(k-x)}=\sum_{l\in\mathbb{Z}^{d}}\widehat{f}(l-\omega)e^{-2\pi ix\cdot
l}.$ (4.15)

###  4.1 The Whittaker-Nyquist-Kotelnikov-Shannon Sampling Theorem

We are now going to state and prove a sampling theorem for band-limited
functions. The information theoretical problem is as follows. Assuming that a
continuous signal is band-limited, how many discrete measurements does one
need in order to be able to completely determine the signal by its samples?

The solution to this question is often (only) referred to as the Nyquist-
Shannon Sampling Theorem. However, it is also known as the Whittaker-Nyquist-
Kotelnikov-Shannon (WNKS) Sampling Theorem 444Often, only the contributions of
the information theorists H. Nyquist (1928) and C. Shannon (1949) are
mentioned, but the mathematical foundations were known already decades before
to E. Whittaker (around 1915). Probably the problem had been treated
mathematically already even earlier. For a long time the work of V. Kotelnikov
from 1933 was not known outside Russia and it became more prominent only in
the 1950s, after Shannon’s work. Kotelinkov’s work was probably the first work
which treated the problem of sampling a continuous function in an information
theoretical context [39] (see also [55, Chap. 2]).

######  Definition 4.4.

A function $f\in L^{2}(\mathbb{R})$ is called band-limited with band limit
$\frac{\mathsf{B}}{2}$ and band width $\mathsf{B}$ if its Fourier transform
$\widehat{f}$ is supported on the interval
$[-\tfrac{\mathsf{B}}{2},\tfrac{\mathsf{B}}{2}]$, i.e.,
$\widehat{f}(\omega)=0$ for all $|\omega|>\frac{\mathsf{B}}{2}$.

We will now state and prove the sampling theorem for the case $\mathsf{B}=1$,
but the general case follows in the same manner.

######  Theorem 4.5 (WNKS Sampling Theorem).

Let $f$ be a band-limited function with band limit $\frac{1}{2}$. In
particular

$f(t)=\int_{-1/2}^{1/2}\widehat{f}(\omega)e^{2\pi it\cdot\omega}\,d\omega,$
(4.16)

with $\widehat{f}\in L^{2}([-\frac{1}{2},\frac{1}{2}])\left(\subset
L^{1}([-\frac{1}{2},\frac{1}{2}]\right)$ and, hence, $f$ is also continuous.
Then $f$ can be reconstructed from its samples on $\mathbb{Z}$ by the formula

$f(t)=\sum_{k\in\mathbb{Z}}f(k)\frac{\sin(\pi(t-k))}{\pi(t-k)},\quad
t\in\mathbb{R}.$ (4.17)

The series converges absolutely and uniformly on compact subsets of
$\mathbb{R}$.

###### Proof.

Note that $\widehat{f}$ coincides with its periodization on
$[-\frac{1}{2},\frac{1}{2}]$, i.e.,

$\widehat{f}(\omega)=(\mathcal{P}\widehat{f}(\omega))\,\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega)=\sum_{l\in\mathbb{Z}}\widehat{f}(\omega+l)\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega).$
(4.18)

We apply the Poisson summation formula (in reverse) to obtain

$\sum_{l\in\mathbb{Z}}\widehat{f}(\omega+l)=\sum_{k\in\mathbb{Z}}f(k)e^{-2\pi
ik\omega}.$ (4.19)

Next, we use the fact that $f$ is the inverse Fourier transform of
$\widehat{f}$ to apply the Poisson summation formula;

$\displaystyle f(t)$
$\displaystyle=\mathcal{F}^{-1}(\widehat{f})(t)=\int_{\mathbb{R}}\widehat{f}(\omega)e^{2\pi
i\omega
t}\,d\omega=\int_{\mathbb{R}}\sum_{l\in\mathbb{Z}}\widehat{f}(\omega+l)\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega)e^{2\pi
i\omega t}\,dt$ (4.20)
$\displaystyle=\int_{\mathbb{R}}\sum_{k\in\mathbb{Z}}f(k)e^{-2\pi
ik\omega}\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega)e^{2\pi
it\omega}\,d\omega=\sum_{k\in\mathbb{Z}}f(k)\underbrace{\int_{\mathbb{R}}\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}(\omega)e^{2\pi
i(t-k)\omega}\,d\omega}_{=\mathcal{F}^{-1}\left(\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]}\right)(t-k)}$
(4.21)
$\displaystyle=\sum_{k\in\mathbb{Z}}f(k)\frac{\sin(\pi(t-k))}{\pi(t-k)}$
(4.22)

Our arguments all work in the $L^{2}(\mathbb{R})$-sense, however, by applying
the Cauchy-Schwarz inequality (for the space $\ell^{2}(\mathbb{Z})$) to our
statement, we actually see that the convergence is absolutely and uniformly on
compact subsets of $\mathbb{R}$. ∎

The value $\frac{1}{2}$, i.e., $\mathsf{B}=1$ was of course chosen on purpose
to obtain a statement and a proof in a simple form. More generally, if
$\widehat{f}$ is supported on the interval
$[-\frac{\mathsf{B}}{2},\frac{\mathsf{B}}{2}]$, then

$\displaystyle f(t)$
$\displaystyle=\sum_{k\in\mathbb{Z}}f(\tfrac{k}{\mathsf{B}})\operatorname*{sinc}(\mathsf{B}(t-\tfrac{k}{\mathsf{B}}))$
(4.23)
$\displaystyle=\sum_{k\in\mathbb{Z}}f(\mathsf{T}k)\operatorname*{sinc}\left(\frac{t-\mathsf{T}k}{\mathsf{T}}\right),$
(4.24)

where we set $\mathsf{T}=\frac{1}{\mathsf{B}}$. The value of $\mathsf{T}$ then
gives the time between two successive samples. The rate $\frac{1}{\mathsf{T}}$
is the sampling rate and in the previous case is exactly $\mathsf{B}$, which
is known as the Nyquist rate. This is the minimum rate at which we need to
sample in order to reconstruct $f$ completely (see, e.g., [55, Chap. 2]) by
means of a sampling function, in this case the $\operatorname*{sinc}$
function. In general, assuming a band-limited signal with band limit
$\frac{\mathsf{B}}{2}$, we distinguish 3 cases:

  1. (i)

$\mathsf{T}>\frac{1}{\mathsf{B}}$: undersampling, no guaranteed reconstruction

  2. (ii)

$\mathsf{T}=\frac{1}{\mathsf{B}}$: critical sampling, reconstruction is
possible with the WKNS sampling formula, but losing 1 sample leads to
undersampling

  3. (iii)

$\mathsf{T}<\frac{1}{\mathsf{B}}$: oversampling, reconstruction is possible,
but the samples are linearly dependent. Note that the reconstruction formula
$f(t)=\sum_{k\in\mathbb{Z}}f(\mathsf{T}k)\operatorname*{sinc}\left(\frac{t-\mathsf{T}k}{\mathsf{T}}\right)$
still holds, because we extended the frequency-band
$[-\frac{1}{2\mathsf{T}},\frac{1}{2\mathsf{T}}]\supset[-\frac{\mathsf{B}}{2},\frac{\mathsf{B}}{2}]$.
Due to the linear dependence of the samples it is still possible to
reconstruct if (some) samples are lost.

(a) Undersampling of $\operatorname*{sinc}(t)^{2}$. The sampling rate is only
half the Nyquist rate, hence we can only recover $\operatorname*{sinc}(t)$,
but not $\operatorname*{sinc}(t)^{2}$. Note that $\operatorname*{sinc}(t)$ and
$\operatorname*{sinc}(t)^{2}$ coincide at the sampling points
$k\in\mathbb{Z}$. (b) Reconstruction of $\operatorname*{sinc}(t)^{2}$ using
the WKNS sampling theorem.

(c) Reconstruction of $\operatorname*{sinc}(t)^{2}$ at the Nyquist rate with
the lost sample $f(0)$. (d) Reconstruction of $\operatorname*{sinc}(t)^{2}$ at
the Nyquist rate with the lost sample $f(\tfrac{1}{2})$.

Figure 11: The function $\operatorname*{sinc}(t)^{2}$ (gray, dashed) and some
un-/successful reconstructions (black, solid) using the WKNS sampling theorem.
Note that $\mathcal{F}(\operatorname*{sinc}(t)^{2})(\omega)$ is supported on
$[-1,1]$ and therefore the Nyquist rate is $\mathsf{B}=2$.

A further generalization of the WKNS sampling theorem is obtained by shifting
the spectrum of the band-limited function $f$ by $\omega_{0}$, i.e.,

$\operatorname*{supp}(T_{\omega_{0}}\widehat{f})\subset[\omega_{0}-\frac{\mathsf{B}}{2},\omega_{0}+\frac{\mathsf{B}}{2}].$
(4.25)

So, if
$\operatorname*{supp}(\widehat{f})\subset[-\frac{\mathsf{B}}{2},\frac{\mathsf{B}}{2}]$,
then $T_{\omega_{0}}\widehat{f}$ is supported on
$[\omega_{0}-\frac{\mathsf{B}}{2},\omega_{0}+\frac{\mathsf{B}}{2}]$. Since, by
(1.6) we have

$M_{\omega}=\mathcal{F}^{-1}T_{\omega}\mathcal{F},$ (4.26)

it follows that
$\mathcal{F}^{-1}T_{\omega_{0}}\widehat{f}=M_{\omega_{0}}\mathcal{F}^{-1}\widehat{f}=M_{\omega_{0}}f$.

Now, define the function $f_{1}(t)=M_{\omega_{0}}f(t)$, which has Fourier
transform $\widehat{f_{1}}(\omega)=T_{\omega_{0}}\widehat{f}(\omega)$. We may
apply the Poisson summation formula to $M_{-\omega_{0}}f_{1}(t)=f(t)$;

$f(t)=e^{-2\pi i\omega_{0}t}f_{1}(t)=\sum_{k\in\mathbb{Z}}e^{-2\pi
i\omega_{0}\mathsf{T}k}f_{1}(\mathsf{T}k)\operatorname*{sinc}\left(\tfrac{t-\mathsf{T}k}{\mathsf{T}}\right)$
(4.27)

Hence, the function $f_{1}$ which is band-limited within the interval
$[\omega_{0}-\frac{\mathsf{B}}{2},\omega_{0}+\frac{\mathsf{B}}{2}]$ can be
written as

$f_{1}(t)=\sum_{k\in\mathbb{Z}}f_{1}(\mathsf{T}k)\operatorname*{sinc}\left(\tfrac{t-\mathsf{T}k}{\mathsf{T}}\right)e^{2\pi
i\omega_{0}(t-\mathsf{T}k)}$ (4.28)

Now, assume that
$\operatorname*{supp}(\widehat{f})\subset\bigcup_{l=1}^{N}I_{l}$, where
$I_{l}=\omega_{l}+[-\frac{B_{l}}{2},\frac{B_{l}}{2}]$,
$\omega_{l}\in\mathbb{R}$ and

$|I_{m}\cap I_{n}|=0,$ (4.29)

for $l\neq m$. Then we can only apply the WKNS sampling theorem directly to a
single frequency band $\omega_{m}+I_{m}$. To apply it to the whole signal we
need to cut the spectrum into single frequency bands;

$\widehat{f}(\omega)=\sum_{l\in\mathbb{Z}}\widehat{f}(\omega)\raisebox{2.0pt}{$\chi$}_{I_{l}}(\omega)=\sum_{l\in\mathbb{Z}}\widehat{f_{l}}(\omega).$
(4.30)

By setting $f_{l}(t)=\int_{I_{l}}\widehat{f_{l}}(\omega)e^{2\pi
it\omega}\,d\omega$ we have a band-limited function with
$\operatorname*{supp}(\widehat{f_{l}})\subset I_{l}$ which can be recovered by
the sampling theorem.

$f_{l}(t)=\sum_{k\in\mathbb{Z}}f_{l}(\mathsf{T}_{l}k)\operatorname*{sinc}\left(\frac{t-\mathsf{T}_{l}k}{\mathsf{T}_{l}}\right)e^{2\pi
i\omega_{l}(t-\mathsf{T}_{l}k)}.$ (4.31)

To recover the original signal, we need to “glue” together the band-limited
functions $f_{l}$. If

$f(t)=\int_{\mathbb{R}}\widehat{f}(\omega)e^{2\pi it\omega}\,d\omega,$ (4.32)

then we also have
$f(t)=\sum_{l=1}^{N}\int_{I_{l}}\widehat{f_{l}}(\omega)e^{2\pi
it\omega}\,d\omega=\sum_{l=1}^{N}f_{l}(t)$ and we get

$f(t)=\sum_{l=1}^{N}\sum_{k\in\mathbb{Z}}f_{l}(\mathsf{T}_{l}k)\operatorname*{sinc}\left(\frac{t-\mathsf{T}_{l}k}{\mathsf{T}_{l}}\right)e^{2\pi
i\omega_{l}(t-\mathsf{T}_{l}k)}.$ (4.33)

Lastly, assume $f\in L^{1}(\mathbb{R})\cap C(\mathbb{R})$ is given as the
Fourier transform of an $L^{1}(\mathbb{R})(\cap L^{2}(\mathbb{R}))$ function
$\widehat{f}$ (so $f\in L^{2}(\mathbb{R})$ as well), so it need not be band-
limited. Then, we may slice the Fourier domain into frequency bands
$I_{l}=l+[-\frac{1}{2},\frac{1}{2}]$ of band-width 1, so
$\bigcup_{l\in\mathbb{Z}}I_{l}=\mathbb{R}$ (and $|I_{m}\cap I_{n}|=0$). As

$\widehat{f}(\omega)=\sum_{l\in\mathbb{Z}}\widehat{f}(\omega)\raisebox{2.0pt}{$\chi$}_{I_{l}}(\omega)=\sum_{l\in\mathbb{Z}}\widehat{f_{l}}(\omega),$
(4.34)

we may apply the WKNS sampling theorem to every frequency band $I_{l}$ and
obtain

$\displaystyle f(t)$
$\displaystyle=\sum_{l\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}f_{l}(k)\operatorname*{sinc}(t-k)e^{2\pi
il(t-k)}$ (4.35)
$\displaystyle=\sum_{l\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}f_{l}(k)\operatorname*{sinc}(t-k)e^{2\pi
ilt}\underbrace{e^{-2\pi ilk}}_{=1},$ (4.36)

where $f_{l}(\omega)=\int_{I_{l}}\widehat{f_{l}}(\omega)e^{2\pi
it\omega}\,d\omega$. So, we need to perform an inverse Fourier transform on
every frequency band to obtain a part of the signal $f$ which is then band-
limited on $I_{l}$. We note that a band-limited function actually has infinite
support in the time domain, as we will see in the next section. So, even if
the signal $f$ has a finite time duration, none of the pieces $f_{l}$ will
have a finite time duration as they are all band-limited.

In the $L^{2}(\mathbb{R})$-sense we can actually write

$f=\sum_{k,l\in\mathbb{Z}}f_{l}(k)M_{l}T_{k}\operatorname*{sinc}=\sum_{k,l\in\mathbb{Z}}\langle
f,M_{l}T_{k}\operatorname*{sinc}\rangle
M_{l}T_{k}\operatorname*{sinc}=\sum_{k,l\in\mathbb{Z}}V_{\operatorname*{sinc}}f(k,l)\,\pi(k,l)\operatorname*{sinc}.$
(4.37)

The equality of the coefficients $f_{l}(k)=\langle
f,M_{l}T_{k}\operatorname*{sinc}\rangle$ follows because the system
$\\{M_{l}T_{k}\operatorname*{sinc}\mid k,l\in\mathbb{Z}\\}$ is actually
orthonormal with dense span in $L^{2}(\mathbb{R})$. Thus, the coefficients
$f_{l}(k)$ in the WKNS expansion are therefore uniquely given by the inner
products of $f$ with the (integer) time-frequency shifted
$\operatorname*{sinc}$-function, or likewise, by sampling the STFT
$V_{\operatorname*{sinc}}f$ on (the integer lattice) $\mathbb{Z}^{2}$.
However, this can also be confirmed by a direct computation.

$\displaystyle f_{l}(t)$
$\displaystyle=\int_{\mathbb{R}}\widehat{f}(\omega)\raisebox{2.0pt}{$\chi$}_{l+[-\frac{1}{2},\frac{1}{2}]}(\omega)e^{2\pi
it\omega}\,d\omega=\mathcal{F}^{-1}(\widehat{f}\,T_{l}\raisebox{2.0pt}{$\chi$}_{[-\frac{1}{2},\frac{1}{2}]})(t)=f(t)*(M_{l}\operatorname*{sinc})(t)$
(4.38)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq_STFT_notation}}}{{=}}e^{2\pi
itl}V_{\operatorname*{sinc}}f(t,l),$ (4.39)

as $\overline{\operatorname*{sinc}}^{\vee}(t)=\operatorname*{sinc}(t)$. Hence,
sampling $f_{l}$ on $\mathbb{Z}$ yields

$f_{l}(k)=\underbrace{e^{2\pi ikl}}_{=1}V_{\operatorname*{sinc}}f(k,l),\quad
k\in\mathbb{Z}.$ (4.40)

######  Remark.

Beating the Nyquist rate and still obtaining perfect reconstruction is
possible, if further restrictions of the signal are known. One such
restriction might be that the (digitized) signal is “sparse”, i.e., having few
non-zero entries, in some domain (e.g., with respect to a certain basis). The
concept of sparsity is excessively used in the area of compressed sensing.
However, this would lead too far away from the intended topics of this course
and will hence not be discussed any further.

##  5 Uncertainty Principles

Classical uncertainty principles are statements about the pair
$(f,\widehat{f})$, including inequalities on their supports or a vanishing
theorem. We will introduce uncertainty principles in a non-rigorous and rather
descriptive way as stated in [20]. They can be summed up in the following
metatheorems.

######  Metatheorem A.

A function $f$ and its Fourier transform $\widehat{f}$ cannot both be
simultaneously small.

######  Metatheorem B.

A function $f$ occupies an area of at least 1 in the time-frequency plane (or
phase space).

######  Metatheorem C.

Every time-frequency representation comes with its own uncertainty principle.

The above metatheorems should be understood in a heuristic sense rather than
in a precise mathematical way. At this point, we remark that the Nyquist
sampling rate might be understood as a reverse uncertainty principle. We
called the case $\mathsf{T}>\frac{1}{\mathsf{B}}$ undersampling. Formulated
differently, we might put it as

$\mathsf{T}\mathsf{B}>1.$ (5.1)

Hence, on average we have less than 1 sample per unit square in the time-
frequency plane, which is too little information to recover a (band-limited)
signal.

In order to obtain rigorous mathematical statements, we need precise
definitions of the terms “small” and “to occupy an area”. Metatheorem C
stresses the fact that we cannot beat the “classical” uncertainty principles
by choosing a different “time-frequency representation”. To be more precise,
the idea is to take an uncertainty principle of type A or B for the pair
$(f,\widehat{f})$ and replace it by a statement about the STFT or the Wigner
distribution. The statement might be a (support) inequality or a vanishing
result.

The size of a function is usually measured by some $L^{p}$-norm. In the case
that $p=2$, we also speak of the energy. Also, decay conditions might be used.
In order to describe how much area a function occupies, we need the concept of
the essential support of a function and, already established, the notion of
the time-frequency plane.

We start with the very classical Heisenberg-Pauli-Weyl uncertainty principle.

######  Theorem 5.1 (Heisenberg-Pauli-Weyl Uncertainty Principle).

Let $f\in L^{2}(\mathbb{R})$ and $a,b\in\mathbb{R}$. Then

$\left(\int_{\mathbb{R}}(x-a)^{2}|f(x)|^{2}\,dx\right)^{1/2}\left(\int_{\mathbb{R}}(\omega-b)^{2}|\widehat{f}(\omega)|^{2}\,d\omega\right)^{1/2}\geq\frac{1}{4\pi}\lVert
f\rVert^{2}_{2}\,.$ (5.2)

###### Proof.

We note that by looking at the function

$f_{a,b}(x)=e^{-2\pi ib\cdot x}f(x+a)=M_{-b}T_{-a}f(x),$ (5.3)

we may assume that $a=b=0$. Further, we need some necessary technical
assumptions. For the following calculations we actually need to assume that
$xf(x)\in L^{2}(\mathbb{R})$ and $\omega\widehat{f}(\omega)\in
L^{2}(\mathbb{R})$, so $f^{\prime},\widehat{f}^{\prime}\in L^{2}(\mathbb{R})$
as well. However, on the other hand, if either of the assumptions is not met,
then the inequality is trivially true. An (unmotivated) integration by parts
shows that

$\int_{\mathbb{R}}\left(xf(x)\right)\overline{f^{\prime}(x)}\,dx=x|f(x)|^{2}\Bigg{|}_{x=-\infty}^{\infty}-\int_{\mathbb{R}}\left(|f(x)|^{2}+xf^{\prime}(x)\overline{f(x)}\right)\,dx.$
(5.4)

Assuming $\lim_{x\to\pm\infty}x|f(x)|^{2}=0$, which holds for any Schwarz
function, the above equation can be re-arranged in the following way

$\int_{\mathbb{R}}|f(x)|^{2}\,dx=-\int_{\mathbb{R}}xf(x)\overline{f^{\prime}(x)}\,dx-\int_{\mathbb{R}}x\overline{f(x)}f^{\prime}(x)\,dx=-2\Re\left(\int_{\mathbb{R}}xf(x)\overline{f^{\prime}(x)}\,dx\right),$
(5.5)

as $x\in\mathbb{R}$. We note the simple estimate

$\left|-2\Re\left(\int_{\mathbb{R}}xf(x)\overline{f^{\prime}(x)}\,dx\right)\right|\leq
2\left|\int_{\mathbb{R}}xf(x)\overline{f^{\prime}(x)}\,dx\right|.$ (5.6)

Now, we use the Cauchy-Schwarz inequality to obtain

$\left|\int_{\mathbb{R}}xf(x)\overline{f^{\prime}(x)}\,dx\right|=|\langle
xf,f^{\prime}\rangle|\leq\lVert xf\rVert_{2}\lVert
f^{\prime}\rVert_{2}=\left(\int_{\mathbb{R}}x^{2}|f(x)|^{2}\,dx\right)^{1/2}\left(\int_{\mathbb{R}}|f^{\prime}(x)|^{2}\,dx\right)^{1/2}.$
(5.7)

From the last three calculations it follows that

$\int_{\mathbb{R}}|f(x)|^{2}\,dx\leq
2\left(\int_{\mathbb{R}}x^{2}|f(x)|^{2}\,dx\right)^{1/2}\left(\int_{\mathbb{R}}|f^{\prime}(x)|^{2}\,dx\right)^{1/2}.$
(5.8)

By combining Plancherel’s theorem and the action of the Fourier transform on
derivatives (1.16) we get

$\left(\int_{\mathbb{R}}|f^{\prime}(x)|^{2}\,dx\right)^{1/2}=\left(\int_{\mathbb{R}}|(2\pi
i\omega)\widehat{f}(\omega)|^{2}\,d\omega\right)^{1/2}.$ (5.9)

Combining the last lines, we obtain

$\frac{\lVert f\rVert_{2}^{2}}{4\pi}\leq\lVert
xf\rVert_{2}\lVert\omega\widehat{f}\rVert_{2}$ (5.10)

∎

We note that in the statement of Theorem 5.1 we have a factor of $2\pi$ due to
our choice of the normalization of the Fourier transform. Often, the
uncertainty principle is stated with a factor $\frac{1}{2}$ due to a different
convention of normalizing the Fourier transform. In quantum mechanics, the
statement usually also involves Planck’s reduced constant $\hbar$. We will
make a little excursion into the field of quantum mechanics now.

Excursion. The following paragraphs are the author’s interpretation of quantum
mechanical results and it should be emphasized that the author does not
possess a solid background on the subject.

In quantum mechanics, the (energy – or density – of the) Fourier transform is
something which can be measured and, hence, depends on measure(able) units.
One way to write it is the following.

$\mathcal{F}_{\hbar}f(p)=\left(\frac{1}{2\pi\hbar}\right)^{d/2}\int_{\mathbb{R}^{d}}f(x)e^{-\frac{i}{\hbar}p\cdot
x}\,dx$ (5.11)

Now, assuming $\lVert f\rVert_{2}=1$, the expression $|f(x)|^{2}$ is the
probability density of the chance to find a particle at a certain position, or
rather in an interval. The measure unit is meter [m]. The expression
$\hbar=\frac{h}{2\pi}$ is Planck’s constant $h$ divided by $2\pi$. It is named
after physicist Max Planck555Max Planck was awarded the Nobel Prize in Physics
“in recognition of the services he rendered to the advancement of Physics by
his discovery of energy quanta” in 1919 for the year 1918. During the
selection process in 1918, the Nobel Committee for Physics decided that none
of the year’s nominations met the criteria as outlined in the will of Alfred
Nobel. According to the Nobel Foundation’s statutes, the Nobel Prize can in
such a case be reserved until the following year, and this statute was then
applied. Max Planck therefore received his Nobel Prize for 1918 one year
later, in 1919.
https://www.nobelprize.org/prizes/physics/1918/summary/ in recognition of his
discovery that the energy of harmonically oscillating systems can only change
in integer multiples of some smallest portion proportional to the oscillating
frequency. Planck’s constant describes this proportionality factor; $\Delta
E=\hbar\omega$ ($\Delta E$ being the change of energy, $\omega$ being the
circle frequency). Hence, $h$ measure energy times time, [J s], which can also
be expressed as [m2 kg/s]. Lastly, we note that $p$, hence, has measure unit
[m kg/s] which is velocity times mass, also known as momentum. Hence,
$|\widehat{f}|^{2}$ (with the normalization induced by $\mathcal{F}_{\hbar}$)
is the probability density that the particle has a certain momentum. For
$d=1$, Theorem 5.1 is usually stated as

$\Delta_{f}x\Delta_{f}p\geq\frac{\hbar}{2}.$ (5.12)

The interpretation is that a particle’s position and momentum cannot be
measured to arbitrary precision at the same time. It is highly important to
understand that this is not a restriction due to measurement errors, it is
imposed by a mathematical theorem! Admittedly, expressed as a mathematical
theorem the uncertainty principle is much more prosaic and loses some of its
mystery. It is a simple inequality for standard deviations.

A model where it finds applications is in a simple model of the hydrogen atom.
The hydrogen atom consists of a core with one proton and it has an electron
which oscillates around it. In the simplest model, the electron only moves
within one direction ($\mathbb{R}$) and satisfies the Schrödinger equation of
the harmonic oscillator. Then $a=\int_{\mathbb{R}}x|f(x)|^{2}\,dx$ is the
expected position of the electron and
$b=\int_{\mathbb{R}}p|\widehat{f}(p)|^{2}\,dp$ is its expected momentum.
Finally, $\Delta_{f}x$ and $\Delta_{f}p$ are the standard deviations of the
probability densities $|f(x)|^{2}$ and $|\widehat{f}(p)|^{2}$.

If $f\in L^{2}(\mathbb{R}^{d})$ for general $d\in\mathbb{N}$ (the physically
most relevant case being $d=3$ or $d=3N$ for $N$ particles), then the
uncertainty principle holds for conjugate coordinates, i.e.,

$\Delta_{f}x_{k}\Delta_{f}p_{k}\geq\frac{\hbar}{2},\qquad k=1,\ldots,d.$
(5.13)

We note that in time-frequency analysis the pair $(x,p)$ of position and
momentum is simply exchanged by the pair $(x,\omega)$. In quantum mechanics,
the point $(x,p)\in\mathbb{R}^{2d}$ is said to be a point in phase-space,
which is the same concept as the time-frequency plane. Finally, by setting
$\hbar=\frac{1}{2\pi}$ we obtain the Fourier transform as used in time-
frequency analysis. Mathematically, there is hence no big difference between
the fields of quantum mechanics and time-frequency analysis.

$\diamond$

The Heisenberg-Pauli-Weyl uncertainty principle can more generally be derived
from an inequality about (non-commuting) self-adjoint operators on a Hilbert
space. For this, let $A$ and $B$ be two linear operators on a Hilbert space
$\mathcal{H}$, i.e.,

$\displaystyle A:$ $\displaystyle\mathcal{H}\to\mathcal{H}$ $\displaystyle B:$
$\displaystyle\mathcal{H}\to\mathcal{H}$ (5.14) $\displaystyle f\mapsto Af$
$\displaystyle f\mapsto Bf$ (5.15)

We denote the commutator of these operators by

$[A,B]=AB-BA.$ (5.16)

######  Lemma 5.2.

Let $A$ and $B$ be (possibly unbounded) self-adjoint operators on
$\mathcal{H}$. Then

$\lVert(A-a)f\rVert_{\mathcal{H}}\lVert(B-b)f\rVert_{\mathcal{H}}\geq\tfrac{1}{2}|\langle[A,B]f,f\rangle_{\mathcal{H}}|,$
(5.17)

for all $a,b\in\mathbb{R}$ and for all $f$ in the domain of $AB$ and $BA$.
Equality holds if and only if $(A-a)f=ic(B-b)f$ for some $c\in\mathbb{R}$.

###### Proof.

We rewrite the commutator and use the fact that $A$ and $B$ are self-adjoint
to obtain

$\displaystyle\langle[A,B]f,f\rangle_{\mathcal{H}}$
$\displaystyle=\langle\left((A-a)(B-b)-(B-b)(A-a)\right)f,f\rangle_{\mathcal{H}}$
(5.18)
$\displaystyle=\langle(B-b)f,(A-a)f\rangle_{\mathcal{H}}-\langle(A-a)f,(B-b)f\rangle_{\mathcal{H}}$
(5.19)
$\displaystyle=\langle(B-b)f,(A-a)f\rangle_{\mathcal{H}}-\overline{\langle(B-b)f,(A-a)f\rangle_{\mathcal{H}}}$
(5.20)
$\displaystyle=2\Im\left(\langle(B-b)f,(A-a)f\rangle_{\mathcal{H}}\right).$
(5.21)

Now, we use the Cauchy-Schwarz inequality and get

$\displaystyle|\langle[A,B]f,f\rangle_{\mathcal{H}}|$ $\displaystyle\leq
2|\langle(B-b)f,(A-a)f\rangle_{\mathcal{H}}|$ (5.22) $\displaystyle\leq
2\lVert(A-a)f\rVert_{\mathcal{H}}\lVert(B-b)f\rVert_{\mathcal{H}}$ (5.23)

Equality in (5.22) holds if and only if
$\langle(B-b)f,(A-a)f\rangle_{\mathcal{H}}$ is purely imaginary, and equality
in (5.23) (Cauchy-Schwarz inequality) holds if and only if
$(A-a)f=\lambda(B-b)f$, $\lambda\in\mathbb{C}$. Together, this implies that
for equality to hold $\lambda=ic$ with $c\in\mathbb{R}$. ∎

To deduce Theorem 5.1 from Lemma 5.2 we need to introduce the position and
momentum operators666The factor $\frac{1}{2\pi i}$ for the momentum operator
$P$ clearly comes from the normalization of the Fourier transform. In quantum
mechanics, it is often stated as $P=-i\hbar\partial_{x}$. on