# Quantum versus classical quenches and the broadening of wave packets

K. Schönhammer  Institut für Theoretische Physik, Universität Göttingen,
Friedrich-Hund-Platz 1, D-37077 Göttingen

###### Abstract

The time dependence of one-dimensional quantum mechanical probability
densities is presented when the potential in which a particle moves is
suddenly changed, called a quench. Quantum quenches are mainly addressed but a
comparison with results for the dynamics in the framework of classical
statistical mechanics is useful. Analytical results are presented when the
initial and final potentials are harmonic oscillators. When the final
potential vanishes the problem reduces to the broadening of wave packets. A
simple introduction to the concept of the Wigner function is presented which
allows a better understanding of the dynamics of general wave packets. It is
pointed out how special the broadening of Gaussian wave packets is, the only
example usually presented in quantum mechanics textbooks.

##  I Introduction

A quantum quench is a sudden change of the Hamilton operator of a system. The
initial operator $\hat{H}_{i}$ is changed to the final one $\hat{H}_{f}$. The
study of the dynamics after the quench is an active field of research in
quantum many-body systems CC  as this presents an interesting example of non-
equilibrium physics.

In this paper, we present one simple example of such a quench. Our discussion
for a single particle in one dimension complements standard courses as only a
few time-dependent problems are discussed in most quantum mechanics textbooks.

As the initial state we take the ground state $|\psi_{0}\rangle$ of the
initial Hamiltonian $\hat{H}_{i}=\hat{T}+\hat{V}_{i}$, where
$\hat{T}=\hat{p}^{2}/2m$ is the operator of the kinetic energy and $V_{i}(x)$
is the initial attractive potential. At $t=0$ the change to
$\hat{H}_{f}=\hat{T}+\hat{V}_{f}$, is performed. The time dependence can be
described in the Schrödinger or the Heisenberg picture. Which is more
appropriate depends on $H_{f}$ and the initial state, here taken as the
groundstate of $H_{i}$. For the special case of vanishing $V_{f}(x)$ the
dynamics is just the free wave packet evolution of the ground state
$|\psi_{0}\rangle$ of $\hat{H}_{i}$. The description of quenches in the
framework of classical statistical mechanics Huang  is also presented. A
comparison with the quantum mechanical results is presented in sections III
and IV.

A complete analytical description of the dynamics is possible when the initial
and final Hamiltonians are of the harmonic oscillator type. We also present
results for linear potentials $V_{f}(x)=-Fx$ in section III.

In section IV we assume that $V_{i}(x)$ is a square well potential and
$V_{f}(x)$ vanishes. We present a simple introduction to the concept of Wigner
functions Wigner ; Case . This allows the description of the broadening of
quantum mechanical wave packets similar to the classical case.

##  II Time evolution after the quench

In the following we mainly focus on the probability density

$\displaystyle\rho_{qm}(x,t)$ $\displaystyle=$
$\displaystyle\langle\psi(t)|\delta(x-\hat{x})|\psi(t)\rangle~{},$ (1)
$\displaystyle=$
$\displaystyle\int\langle\psi(t)|\delta(x-\hat{x})|x^{\prime}\rangle\langle
x^{\prime}|\psi(t)\rangle dx^{\prime}~{},$ $\displaystyle=$
$\displaystyle|\langle x|\psi(t)\rangle|^{2}~{}.$

to find the particle at position $x$ at time $t$, where we have introduced the
index $qm$ as we later compare quantum quenches with quenches in the framework
of classical statistical mechanics.

Instead of directly working with the delta function in Eq. (1) we use its
integral representation

$\rho_{qm}(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-ikx}\langle\psi(t)|e^{ik\hat{x}}|\psi(t)\rangle
dk~{}.$ (2)

This rarely used expression for $\rho_{qm}(x,t)$ turns out to be very useful
in the following.

The usual approach to obtain $\rho_{qm}(x,t)$ for times after the quench is to
calculate $|\psi(t)\rangle=e^{-i\hat{H}_{f}t/\hbar}|\psi\rangle$, using the
eigenstates of $\hat{H}_{f}$, where $|\psi\rangle$ is the initial state before
the quench. This is described for the free time evolution after the quench in
section IV.

However, it is sometimes useful to work in the Heisenberg picture in which
operators acquire the time dependence
$\hat{A}(t)=e^{i\hat{H}_{f}t/\hbar}\hat{A}e^{-i\hat{H}_{f}t/\hbar}$. Expanding
$e^{ik\hat{x}}$ in Eq. (2) and using $\hat{x}^{n}(t)=(\hat{x}(t))^{n}$ one
obtains

$\displaystyle\rho_{qm}(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-ikx}\langle\psi|e^{ik\hat{x}(t)}|\psi\rangle
dk~{},$ (3) $\displaystyle=$
$\displaystyle\langle\psi|\delta(x-\hat{x}(t))|\psi\rangle~{}.$

In classical statistical mechanics for a single particle the initial state at
time $t=0$ is described by the phase space probability density
$\rho_{0}(x_{0},p_{0})$ for the initial position $x_{0}$ and momentum $p_{0}$
of the particle Huang . The average classical particle density at finite times
is given by

$\displaystyle\rho_{cl}(x,t)$ $\displaystyle=$
$\displaystyle\int\delta(x-x_{x_{0},p_{0}}(t))\rho_{0}(x_{0},p_{0})dx_{0}dp_{0}$
(4) $\displaystyle\equiv$
$\displaystyle\langle\delta(x-x_{x_{0},p_{0}}(t))\rangle_{cl}~{},$

where $x_{x_{0},p_{0}}(t)$ is the particle trajectory. For the comparison with
the quantum mechanical results in section III we also make use of the integral
representation of the delta function

$\rho_{cl}(x,t)=\frac{1}{2\pi}\int e^{-ikx}\langle
e^{ikx_{x_{0},p_{0}}(t)}\rangle_{cl}dk~{}.$ (5)

If the particle is in thermal equilibrium with a bath at temperature $T$
before the quench the initial probability density $\rho_{0}(x_{0},p_{0})$ is
proportional to $e^{-\beta(p_{0}^{2}/2m+V_{i}(x_{0}))}$, with
$\beta=1/k_{B}T$. This is an example for a factorized initial distribution
$\rho_{0}(x_{0},p_{0})=\rho_{0}(x_{0})\tilde{\rho}_{0}(p_{0})$.

##  III Harmonic oscillators

###  III.1 The shifted harmonic potential

The Hamiltonian of a one-dimensional harmonic oscillator reads

$\hat{H}=\frac{1}{2m}\hat{p}^{2}+\frac{\lambda}{2}\hat{x}^{2},$ (6)

with $\lambda$ the spring constant. The angular frequency of the oscilator is
given by $\omega_{0}=\sqrt{\lambda/m}$.

The quantum mechanical description of the harmonic oscillator is especially
simple using the ladder operators. One defines the lowering operator $\hat{a}$
and its adjoint $\hat{a}^{\dagger}$

$\hat{a}=\sqrt{\frac{m\omega_{0}}{2\hbar}}\hat{x}+\frac{i}{\sqrt{2m\hbar\omega_{0}}}\hat{p}~{};~{}~{}\hat{a}^{\dagger}=\sqrt{\frac{m\omega_{0}}{2\hbar}}\hat{x}-\frac{i}{\sqrt{2m\hbar\omega_{0}}}\hat{p}~{},$
(7)

which obey the commutation relation $[\hat{a},\hat{a}^{\dagger}]=\hat{1}$. The
position operator $\hat{x}$ and the momentum operator $\hat{p}$ read in terms
of $\hat{a}$ and $\hat{a}^{\dagger}$

$\hat{x}=\sqrt{\frac{\hbar}{2m\omega_{0}}}\left(\hat{a}^{\dagger}+\hat{a}\right);~{}~{}\hat{p}=i\sqrt{\frac{m\hbar\omega_{0}}{2}}\left(\hat{a}^{\dagger}-\hat{a}\right).$
(8)

The ground state $|0\rangle$ is annihilated by $\hat{a}$, i.e.
$\hat{a}|0\rangle=0$ holds.

As $\hat{H}_{i}$ we use the Hamiltonian in Eq.(6) with
$\lambda\to\lambda_{i}$, and its ground state $|0\rangle_{i}$ as the initial
state. Averages of operators $\hat{A}$ in this ground state we denote by
$\langle\hat{A}\rangle$. Then $\langle\hat{x}\rangle=0$ holds as well as
$\langle\hat{p}\rangle=0$.

As the final potential we use

$V_{f}(x)=-Fx+\frac{\lambda_{f}}{2}x^{2}=\frac{\lambda_{f}}{2}\left(x-\frac{F}{\lambda_{f}}\right)^{2}-\frac{F^{2}}{2\lambda_{f}}~{},$
(9)

i.e., a harmonic oscillator with spring constant $\lambda_{f}$ shifted to the
position $a_{F}=F/\lambda_{f}$ and angular frequency
$\omega_{f}=\sqrt{\lambda_{f}/m}$.

The equations of motion and their solutions for the operators $\hat{x}(t)$ and
$\hat{p}(t)$ are identical in form to the classical ones for $x(t)$ and
$p(t)$. In the quantum mechanical case the classical initial conditions
$x_{0}$ and $p_{0}$ are replaced by $\hat{x}=\hat{x}(0)$ and
$\hat{p}=\hat{p}(0)$. With the definition
$\Delta\hat{A}\equiv\hat{A}-\langle\hat{A}\rangle$ one obtains

$\langle\hat{x}(t)\rangle=a_{F}(1-\cos{(\omega_{f}t)}),~{}~{}~{}\langle\hat{p}(t)\rangle=ma_{F}\omega_{f}\sin{\omega_{f}t}$
(10)

and

$\displaystyle\Delta\hat{x}(t)$ $\displaystyle=$
$\displaystyle\hat{x}\cos{\omega_{f}t}+\hat{p}\sin{(\omega_{f}t)}/m\omega_{f}$
(11) $\displaystyle\Delta\hat{p}(t)$ $\displaystyle=$
$\displaystyle-\hat{x}m\omega_{f}\sin{\omega_{f}t}+\hat{p}\cos{\omega_{f}t}~{}.$
(12)

In order to calculate $\rho_{qm}(x,t)$ it is useful to express
$\Delta\hat{x}(t)$ as a linear combination of the ladder operators
corresponding to the harmonic oscillator $\hat{H}_{i}$

$\Delta\hat{x}(t)=\alpha(t)\hat{a}^{\dagger}+\alpha(t)^{*}\hat{a}~{},$ (13)

where using Eq. (8), $\alpha(t)$ is given by

$\alpha(t)=\sqrt{\frac{\hbar}{2m\omega_{i}}}\left(\cos{\omega_{f}t}+i\frac{\omega_{i}}{\omega_{f}}\sin{\omega_{f}t}\right)~{}.$
(14)

Using $\hat{a}|0\rangle=0$ and $[\hat{a},\hat{a}^{\dagger}]=\hat{1}$ the
expectation value of $(\Delta\hat{x}(t))^{2}$ in the ground state of
$\hat{H}_{i}$ is given by $|\alpha(t)|^{2}$

$\displaystyle\langle(\Delta\hat{x}(t))^{2}\rangle$ $\displaystyle=$
$\displaystyle|\alpha(t)|^{2}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2m\omega_{i}}\left(\cos^{2}{\omega_{f}t}+\left(\frac{\omega_{i}}{\omega_{f}}\right)^{2}\sin^{2}{\omega_{f}t}\right)$

In order to calculate $\rho_{qm}(x,t)$ we use the Baker-Hausdorff-identity
Merzb . It states that if
$[\hat{A},[\hat{A},\hat{B}]]=0=[\hat{B},[\hat{A},\hat{B}]]$

$e^{\hat{A}+\hat{B}}=e^{\hat{A}}e^{\hat{B}}e^{-\frac{1}{2}[\hat{A},\hat{B}]}~{}.$
(16)

For operators $\hat{A}$ and $\hat{B}$ linear in the ladder operators the
requirements are fulfilled. Therefore the expectation value in Eq. (3) can
easily be calculated. In the integrand in Eq. (3) we write

$e^{-ikx}e^{ik\hat{x}(t)}=e^{-ik(x-\langle\hat{x}(t)\rangle)}e^{ik\Delta\hat{x}(t)}$
(17)

and evaluate the expectation value of $e^{ik\Delta\hat{x}(t)}$ using equations
(13) and (16)

$\displaystyle\langle e^{ik\hat{\Delta}x(t)}\rangle$ $\displaystyle=$
$\displaystyle\langle
e^{ik\alpha(t)\hat{a}^{\dagger}+ik\alpha(t)^{*}\hat{a}}\rangle$ (18)
$\displaystyle=$ $\displaystyle\langle
e^{ik\alpha(t)\hat{a}^{\dagger}}e^{ik\alpha(t)^{*}\hat{a}}\rangle
e^{-k^{2}|\alpha(t)|^{2}/2}$ $\displaystyle=$ $\displaystyle
e^{-k^{2}|\alpha(t)|^{2}/2}~{}.$

Because of $\hat{a}|0\rangle_{i}=0$ which implies ${}_{i}\langle
0|\hat{a}^{\dagger}=0$ the expectation value in the second equality equals
$1$. Putting this into the upper part of Eq. (3) the Gaussian integration can
be performed and one obtains

$\rho_{qm}(x,t)=\frac{1}{\sqrt{2\pi\langle(\Delta\hat{x}(t))^{2}\rangle}}\exp{\left(-\frac{(x-\langle\hat{x}(t)\rangle)^{2}}{2\langle(\Delta\hat{x}(t))^{2}\rangle}\right)}~{}.$
(19)

An analogous result is obtained for the momentum probability distribution

$\langle\delta(p-\hat{p}(t)\rangle=\frac{1}{\sqrt{2\pi\langle(\Delta\hat{p}(t))^{2}\rangle}}\exp{\left(-\frac{(p-\langle\hat{p}(t)\rangle)^{2}}{2\langle(\Delta\hat{p}(t))^{2}\rangle}\right)}~{}.$
(20)

with the width determined by

$\langle(\Delta\hat{p}(t))^{2}\rangle=\frac{m\hbar\omega_{i}}{2}\left(\cos^{2}{\omega_{f}t}+\left(\frac{\omega_{f}}{\omega_{i}}\right)^{2}\sin^{2}{\omega_{f}t}\right)~{}.$
(21)

Fig. 1 shows the time dependence of $\langle\hat{x}(t)\rangle$, $\Delta
x(t)=\sqrt{\langle(\Delta\hat{x}(t))^{2}\rangle}$, $\Delta
p(t)=\sqrt{\langle(\Delta\hat{p}(t))^{2}\rangle}$ and the uncertainty product.
The initial wave packet makes an oscillatory motion around the center of the
harmonic potential in $H_{f}$. The width $\Delta x(t)$ and $\Delta p(t)$
oscillate with double frequency and the uncertainty product $\Delta x(t)\Delta
p(t)$ equals $\hbar/2$ at positions $0,a_{F}$ and $2a_{F}$ and is larger at
the intermediate positions.

Figure 1: Time dependence of $\langle\hat{x}(t)\rangle/a_{F}$ (solid line),
$\Delta x(t)/\Delta x(0)$ (dashed line), $\Delta p(t)/\Delta p(0)$
($-\cdot-\cdot-$), and the uncertainty product $2\Delta x(t)\Delta p(t)/\hbar$
($--\cdot--\cdot$) as a function of $\omega_{f}t$ for
$\omega_{f}/\omega_{i}=1/2$.

The behaviour of $\rho_{qm}(x,t)$ in Eq. (19) is well known. It is usually
obtained by calculating $\langle x|0(t)\rangle_{i}$ introducing the concepts
of coherent and squeezed states Merzb ; Ball  as a typical quantum mechanical
behaviour. The solution presented here without introducing these states is
simpler.

Surprisingly Eq. (19) can also be obtained purely classically. To show this we
consider the classical quench dynamics switching from $H_{i}$ to $H_{f}$ when
$\rho_{cl}(x_{0},p_{0})=\rho_{0}(x_{0})\tilde{\rho}_{0}(p_{0})$ and
$\rho_{0}(x_{0})$ and $\tilde{\rho}_{0}(p_{0})$ are both Gaussians. Then the
integration in Eq. (5) can be performed analytically by first calculating

$\langle e^{ikx_{x_{0},p_{0}}(t)}\rangle_{cl}=e^{ik\langle x(t)\rangle}\langle
e^{ik(x_{0}\cos{\omega_{f}t}+p_{0}\sin{(\omega_{f}t)}/m\omega_{f})}\rangle_{cl}$
(22)

by Gaussian integrations. The remaining $k-$integration is also Gaussian and
one obtains

$\rho_{cl}(x,t)=\frac{1}{\sqrt{2\pi\langle(\Delta
x(t))^{2}\rangle}_{cl}}\exp{\left(-\frac{(x-\langle
x(t)\rangle)^{2}}{2\langle(\Delta x(t))^{2}\rangle_{cl}}\right)}$ (23)

with

$\langle(\Delta x(t))^{2}\rangle_{cl}=\langle
x_{0}^{2}\rangle\cos^{2}({\omega_{f}t})+\langle
p_{0}^{2}\rangle\sin^{2}({\omega_{f}t})/(m\omega_{f})^{2}~{}.$ (24)

For the case of the canonical ensemble discussed after Eq. (5) one has
$\langle p_{0}^{2}\rangle/\langle x_{0}^{2}\rangle=m^{2}\omega_{i}^{2}$ and
the time dependence of $\langle(\Delta x(t))^{2}\rangle_{cl}$, apart from the
prefactor is the same as in Eq.(III.1). For the special temperature choice
$k_{B}T=\hbar\omega_{i}/2$, also the prefactor is the same. This corresponds
to the initial condition

$\rho_{0}(x_{0},p_{0})=\psi_{0}(x_{0})^{2}\tilde{\psi}_{0}(p_{0})^{2}~{},$
(25)

where $\psi_{0}(x_{0})=\langle x_{0}|\psi_{0}\rangle$ is the real wave
function for the Gaussian initial state $|\psi_{0}\rangle=|0\rangle_{i}$
considered in this section and $\tilde{\psi}_{0}(p_{0})=\langle
p_{0}|\psi_{0}\rangle$ the corresponding real Gaussian momentum amplitude.

It is left as an exercise to repeat the caclulation of $\rho_{qm}(x,t)$ for
$|1\rangle_{i}=\hat{a}^{\dagger}|0\rangle_{i}$ and
$|2\rangle_{i}=(\hat{a}^{\dagger})^{2}|0\rangle_{i}/\sqrt{2}$ as the initial
state by expanding the exponential functions in the expectation value in the
second equality in Eq. (18). The results for the special case
$\hat{H}_{f}=\hat{T}$ can be found in reference 7. There the free time
evolution after the quench for the initial state $c\hat{p}^{2}|0\rangle_{i}$
is also discussed. It is a special linear combination of $|0\rangle_{i}$ and
$|2\rangle_{i}$. Results for the general linear combination of these two
states are presented in reference 8.

###  III.2 The linear potential

We next discuss the case $\lambda_{f}=0$ and a finite value of $F$, i.e. the
linear potential $V_{f}(x)=-Fx$. Performing the limit $\omega_{f}\to 0$ in
Eq.(10) leads to

$\langle(\Delta\hat{x}(t))^{2}\rangle=\langle\hat{x}^{2}\rangle(1+\omega_{i}^{2}t^{2})$
(26)

independent of the value of $F$ which only appears in the expression for
$\langle\hat{x}(t)\rangle$ in Eq. (10), given by

$\langle\hat{x}(t)\rangle_{F}=Ft^{2}/2m~{}.$ (27)

The result for $|\langle x|\psi(t)\rangle|^{2}$ for the linear potential is
given by Eq. (19) with the results of Eq. (26) and Eq. (27) inserted. The time
dependence of the broadening is identical to the free particle case.

This holds for linear potentials for arbitrary initial states
$|\psi(0)\rangle$. The solution of the Heisenberg equation of motion is given
by

$\hat{x}(t)=\hat{x}+\hat{p}t/m+Ft^{2}/2m~{}.$ (28)

Putting this into the second line of Eq. (3) yields

$|\langle x|\psi(t)\rangle_{F}|^{2}=|\langle
x-\langle\hat{x}(t)\rangle_{F}|\psi(t)\rangle_{F=0}|^{2}$ (29)

This is in accord with the (weak) equivalence principle Holstein ; Nauenberg ,
which states that all laws of a freely falling particle are the same as in an
unaccelarated reference frame. The presented proof for the broadening of wave
packets using the Heisenberg picture by directly adressing the measurable
probability density $|\langle x|\psi(t)\rangle|^{2}$ is much simpler than
using the Schrödinger picture and calculating $\langle x|\psi(t)\rangle$
first. Holstein ; Nauenberg .

In quantum mechanics textbooks the factor multiplying $t^{2}$ for the free
case $F=0$ in Eq. (26) is usually expressed differently as the broadening of
the Gaussian wave packet is treated before the harmonic oscillator. Instead of
$\omega_{i}^{2}$ the factor is written
$(\hbar/(2m\langle\hat{x}^{2}\rangle))^{2}$

For the case $\lambda_{f}=0$ and $F=0$ the free broadening of a Gaussian wave
packet using operator manipulations was presented in this journal recently Jim
.

##  IV Free time evolution after the quench

In the previous section we used the Heisenberg picture to calculate the time
dependence of the probability density $\rho_{qm}$ after the quench. Here we
use the Schrödinger picture to obtain $\rho_{qm}(x,t)$ for the quench in which
$V_{f}(x)\equiv 0$.

We begin with the usual approach to obtain $\rho_{qm}(x,t)$ by calculating
$\langle x|\psi(t)\rangle$ and taking its absolute square. In the second part
of this section we discuss the additional insight one can obtain by using Eq.
(2) instead.

In the first approach one uses the eigenstates of $\hat{H}_{f}$. For the case
$\hat{H}_{f}=\hat{T}$ those are given by the momentum states $|p\rangle$.
Inserting the unit operator expressed in terms of the momentum states and
using $\langle x|p\rangle=e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ yields

$\langle
x|\psi(t)\rangle=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{ipx/\hbar}\langle
p|\psi\rangle e^{-i\epsilon_{p}t/\hbar}dp~{},$ (30)

with $\epsilon_{p}=p^{2}/2m$ and the momentum representation

$\tilde{\psi}(p)\equiv\langle
p|\psi\rangle=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{-ipx/\hbar}\langle
x|\psi\rangle dx~{},$ (31)

of the arbitrary initial state $|\psi\rangle$.

If the wave function $\psi(x)=\langle x|\psi\rangle$ is Gaussian the same
holds for $\tilde{\psi}(p)$ and the integration in Eq. (30) can be performed
analytically. This is presented in almost all quantum mechanics textbooks. For
generic $\psi(x)$ the integration in Eq. (30) has to be performed numerically.

In this section we take as the initial state $|\psi_{0}\rangle$ the
groundstate of an attractive square well potential
$V_{i}(x)=V_{i}\Theta(a-|x|)$, with $\Theta$ the step function and $V_{i}<0$.
Its wave function is given by

$\psi_{0}(x)=\begin{cases}c_{0}\cos(k_{0}x)/\cos(k_{0}a)&|x|\leq a\\\
c_{0}e^{-\kappa(|x|-a)}&|x|\geq a\end{cases}$ (32)

with $k_{0}=\sqrt{2m(E_{0}-V_{i})}/\hbar$ and the groundstate energy $E_{0}$ .

In order to have continuous first derivatives at $x=\pm a$

$\kappa=k_{0}\tan(k_{0}a)$ (33)

has to hold with $\tan(k_{0}a)>0$. The normalization constant is given by
$c_{0}=[1/\kappa+a(1+\kappa^{2}/k_{0}^{2})+\kappa/k_{0}^{2}]^{-1/2}$. The
dependence of $E_{0}$ on $V_{i}$ is irrelevant in the following.

There are two interesting limiting cases:

i) The limit $a\to 0,V_{i}\to-\infty$ with arbitrary $\kappa>0$, i.e. an
attractive delta potential with the ground state wave function
$\psi_{0}(x)=\sqrt{\kappa}e^{-\kappa|x|}$.

ii) The infinitely deep potential: For $ak_{0}\to\pi/2$ the ratio
$\kappa/k_{0}$ tends to infinity and one obtains
$\psi_{0}(x)=\cos(k_{0}x)\Theta(a-|x|)/\sqrt{a}$.

The momentum representation of $|\psi_{0}\rangle$ can be calculated
analytically. Using Eqs. (31) and (32) one obtains with $p=\hbar k$

$\tilde{\psi}_{0}(\hbar k)=\frac{2c_{0}}{\sqrt{2\pi\hbar}}(\kappa\cos ka-k\sin
ka)\left(\frac{1}{k^{2}+\kappa^{2}}-\frac{1}{k^{2}-k_{0}^{2}}\right)~{}.$ (34)

Inserting this result into the integral in Eq.(30) it can be calculated
numerically to obtain $\langle x|\psi(t)\rangle$. Its absolute value squared
is shown in Fig. 2 for $k_{0}a=\pi/2$, i.e. the ground state of the infinitely
deep well, for four different times. It shows that the probability density to
find the particle at the origin is larger for $t=0.14t_{0}$ than at the
initial time $t=0$, where $t_{0}=ma^{2}/\hbar$. For $t=0.07t_{0}$ it has a
minimum at the origin.

Figure 2: Probability density $\rho_{qm}$ times $a$ as a function of $x/a$ for
four different times: $t=0$ (solid line), $t=0.07t_{0}$ ($--\cdot--\cdot$)
$t=0.14t_{0}$ (dashed line) and $t=0.28t_{0}$ ($-\cdot-\cdot-$). Note that
$a\rho_{qm}(0,0.14)$ is larger than $a\rho_{qm}(0,0)$.

To elucidate this surprising effect in more detail $a\rho_{qm}$ is shown for
$x=0$ as function of $t/t_{0}$ for three different values of $k_{0}a$ in Fig.
3.

Figure 3: Probability density $\rho_{qm}$ times $a$ for $x=0$ as a function of
$t/t_{0}$ for three different values of $k_{0}a$: $k_{0}a=\pi/2$, which
corresponds to the infinitely deep well (solid line), $k_{0}a=\pi/2.5$ (dashed
line) and $k_{0}a=\pi/3$ ($-\cdot-\cdot-$) .

The short time oscillatory behaviour is more pronounced when the well is
deeper.

The fact that the probability to find the particle at the origin for a $t>0$
is larger than in the initial state is a purely quantum mechanical effect. It
is easy to see that this cannot happen in the classical case when the
probablity to find the particle at $t=0$ has its maximum at $x_{0}=0$. For a
free particle the trajectory is given by $x_{x_{0}p_{0}}(t)=x_{0}+p_{0}t/m$
and using Eq. (4) one obtains

$\rho_{cl}(x,t)=\int\rho_{0}(x-\frac{p_{0}t}{m},p_{0})dp_{0}~{}.$ (35)

In order to be as close as possible to the quantum mechanical case for a
general initial state $|\psi\rangle$ the phase space density
$\rho_{0}(x_{0},p_{0})$ should yield the quantum mechanical space probability
$|\langle x_{0}|\psi\rangle|^{2}$ by integration over $p_{0}$ and the momentum
probability $|\langle p_{0}|\psi\rangle|^{2}$ by integration over $x_{0}$. By
chosing

$\rho_{0}(x_{0},p_{0})=|\langle x_{0}|\psi\rangle|^{2}|\langle
p_{0}|\psi\rangle|^{2}~{}.$ (36)

this is obviously fulfilled. We call this the “classical approximation” in
quotation marks as the dynamics using Eq. (35) is classical, but this initial
condition involves quantum mechanical probability densities.

For $|\psi\rangle$ the ground state of the square well potential $|\langle
x_{0}|\psi\rangle|^{2}$ has its maximum at $x_{0}=0$. For $t>0$ one has
$\rho_{0}(-\frac{p_{0}t}{m},p_{0})<\rho_{0}(0,p_{0})$ which using Eq. (35)
implies $\rho_{cl}(0,t)<\rho_{cl}(0,0)$.

It turns out that the “classical approximation” works rather well for larger
times. This is shown in Fig. 4 for $k_{0}a=\pi/2$.

For a better understanding of this we show that the “classical approximation”
gives the exact result for the time dependence of the quantum mechanical width
of the wave packet. With the initial condition in Eq.(36) one obtains
$\langle(x(0))^{n}\rangle_{cl}=\langle(\hat{x}(0))^{n}\rangle_{qm}$ and
$\langle(p(0))^{n}\rangle_{cl}=\langle(\hat{p}(0))^{n}\rangle_{qm}$. As we
consider initial states $|\psi\rangle$ with even wave functions $\langle
x(0)\rangle_{cl}=0$ and $\langle(x(0)p(0)\rangle_{cl}=0$ holds. This leads to

$\langle(\Delta
x(t))^{2}\rangle_{cl}=\langle\hat{x}^{2}\rangle_{qm}+t^{2}\frac{\langle\hat{p}^{2}\rangle_{qm}}{m^{2}}~{}.$
(37)

To calculate the expectation value
$\langle\hat{x}\hat{p}\rangle+\langle\hat{p}\hat{x}\rangle$ for a square
integrable wave function $\psi(x)$ one can use that the integral over the
derivative of $x|\psi(x)|^{2}$ vanishes as $x|\psi(x)|^{2}\to 0$ for
$x\to\pm\infty$. For $\psi(x)=f(x)e^{i\phi(x)}$ with $f$ and $\phi$ real
functions one obtains (exercise)

$\langle\hat{x}\hat{p}\rangle+\langle\hat{p}\hat{x}\rangle=2\hbar\int_{-\infty}^{\infty}x\phi^{\prime}(x)f^{2}(x)dx~{}.$
(38)

Therefore the sum vanishes for real $\psi(x)$. This leads to

$\langle(\Delta\hat{x}(t))^{2}\rangle_{qm}=\langle(\Delta
x(t))^{2}\rangle_{cl}~{}.$ (39)

In order to discuss the long time behaviour of $\rho_{cl}(x,t)$ one can
alternatively perform the $p_{0}$ integration in Eq. (4) first or change the
integration variable in Eq. (35), both leading to

$\rho_{cl}(x,t)=\frac{m}{t}\int\rho_{0}(x_{0},\frac{mx}{t}-\frac{mx_{0}}{t})dx_{0}~{}.$
(40)

If $|\psi(x_{0})|^{2}$ in Eq. (36) decays fast enough like for the square well
potential Eq. (32) the term $mx_{0}/t$ in the integrand can be neglected in
the long time limit leading to

$t\rho_{cl}(x,t)\to m|\tilde{\psi}(\frac{mx}{t})|^{2}$ (41)

for finite $x/t$.

Figure 4: Probability density $\rho_{qm}$ times $a$ for $k_{0}a=\pi/2$ as a
function of $t/t_{0}$ for three different values of $x/a$ (solid lines),
compared to the “classical approximation” described in the text (dashed
lines). Figure 5: Long time behaviour of $(t/t_{0})a\rho_{qm}$ for the
infinitely deep square well as a function of $(x/a)/(t/t_{0})$ for different
values of $t/t_{0}$. It quickly approaches
$\hbar|\tilde{\psi}((x/a)/(t/t_{0}))|^{2}/a$ (long dashed curve).

In Fig. 5 we show numerical results for the quantum mechanical long time
behaviour using Eqs. (30) and (34). The function $(t/t_{0})a\rho_{qm}$ is
shown for various values of $t/t_{0}$ as a function of $(x/a)/(t/t_{0})$. It
quickly approaches $\hbar|\tilde{\psi}((x/a)/(t/t_{0}))|^{2}/a$, in agreement
with Eq. (41).

Before we present an explanation why the quantum mechanical and the
“classical” long time behaviour agree we switch to Eq. (2) to calculate
$\rho_{qm}(x,t)$ without computing $\langle x|\psi(t)\rangle$ first.

For the later treatment of the free particle time dependence it turns out to
be useful to factorize $e^{ik\hat{x}}=e^{ik\hat{x}/2}e^{ik\hat{x}/2}$ and
insert the unit operator expressed in terms of the momentum states in between.
Using $e^{ik\hat{x}/2}|p\rangle=|p+\hbar k/2\rangle$ as can be proven by
multiplying both sides by $\langle x|$ this leads to

$e^{ik\hat{x}}=\int|p+\hbar k/2\rangle\langle p-\hbar k/2|dp~{}.$ (42)

Putting this into Eq. (2) and changing the integration variable
$k=\tilde{p}/\hbar$ one obtains

$\rho_{qm}(x,t)=\int W(x,p,t)dp~{},$ (43)

with

$W(x,p,t)=\frac{1}{2\pi\hbar}\int
e^{-i\tilde{p}x/\hbar}\langle\psi(t)|p+\tilde{p}/2\rangle\langle
p-\tilde{p}/2|\psi(t)\rangle d\tilde{p}~{}.$ (44)

This is a real function as seen by changing the integration variable
$\tilde{p}\to-\tilde{p}$. Integration over $x$ gives a factor
$2\pi\delta(\tilde{p}/\hbar)$ leading to

$\int W(x,p,t)dx=|\langle p|\psi(t)\rangle|^{2}~{}.$ (45)

It is left as an exercise to show that after inserting unit operators in terms
of position states in the integral in Eq. (44) the function $W(x,p,t)$ can
also be written as

$W(x,p,t)=\frac{1}{2\pi\hbar}\int
e^{-ipy/\hbar}\langle\psi(t)|x-y/2\rangle\langle x+y/2|\psi(t)\rangle dy~{}.$
(46)

In 1932 Wigner Wigner  introduced the quantum phase-space distribution
$W(x,p,t)$ for arbitrary time dependent states $|\psi(t)\rangle$, which has
similarities to the classical phase-space distribution Huang , but allows to
obain exact quantum mechanical results, as shown above. The higher dimensional
generalization of the Wigner function is widely used as a tool in various
areas, e.g. quantum optics Schleich ; WFST . However, it is introduced only in
a few quantum mechanics textbooks, e.g. Merzb ; Ball . A “pedestrian”
introduction has been published in this journal Case .

For $W(x,p,0)$ to be a probability density, i.e. it is positive everywhere,
$\psi(x)$ has to be the exponential of a quadratic polynomial Hudson . A well
known example is a real-valued Gaussian. The integration in Eq. (46) can then
be easily performed and one obtains $W(x,p,0)=\psi(x)^{2}\tilde{\psi}(p)^{2}$.

The form of the Wigner function in Eq. (44) is well suited to determine its
time dependence for free particles discussed in this section. The momentum
states are the eigenstates for $\hat{H}_{f}=\hat{T}$. With
$\epsilon_{p+\tilde{p}/2}-\epsilon_{p-\tilde{p}/2}=\tilde{p}p/m$ one obtains

$W(x,p,t)=W(x-\frac{pt}{m},p,0)~{}.$ (47)

If this is inserted into Eq. (43) it leads to

$\rho_{qm}(x,t)=\int W(x-\frac{pt}{m},p,0)dp~{}.$ (48)

This result for $\rho_{qm}(x,t)$ has the form as in the classical case in Eq.
(35), with $W(x,p,0)$ instead of $\rho_{0}(x,p)$. Therefore the arguments
leading to Eq. (41) can be generalized to obtain

$t\rho_{qm}(x,t)\to m|\tilde{\psi}(\frac{mx}{t})|^{2}~{}.$ (49)

if $W(x,p,0)$ has a well-localized $x$-dependence, which is the case for the
groundstate of a deep square well potential.

The long time behaviour of the broadening of wave packets has been discussed
previously in two papers in this journal Andrews ; Mita . In reference 7 the
result Eq. (49) is presented using the free particle propagator, shortly
discussed in the appendix. The statement in reference 17, that all wave
packets become “approximately Gaussian” after a long enough time, is
critically discussed.

The Wigner function for the ground state of the infinitely deep square well
can be obtained by simple integration using Eq. (46)Thomas

$\displaystyle W(x,\hbar k,0)$ $\displaystyle=$
$\displaystyle\frac{\Theta(a-|x|)}{2\pi\hbar}\Biggl{[}\cos(\pi
x/a)\frac{\sin(2k(a-|x|))}{ak}$ (50) $\displaystyle+$
$\displaystyle\frac{\sin((2ak+\pi)(1-|x|/a))}{2ak+\pi}$ $\displaystyle+$
$\displaystyle\frac{\sin((2ak-\pi)(1-|x|/a))}{2ak-\pi}\Biggr{]}~{}.$

It is shown in Fig. 6 as function of momentum for two different values of $x$.
One can see that $W$ is negative in some intervals. This is different for
$|\psi(0)|^{2}|\tilde{\psi}(p)|^{2}$ shown as the dashed curve.

Figure 6:  Wigner function $\hbar W(x,\hbar k,0)$ as a funcion of $ak$ for
$x=0$ and $x=a/2$. The dashed curve is the positive function
$\hbar|\psi(0)|^{2}|\tilde{\psi}(\hbar k)|^{2}$.

The oscillatory behaviour of $\rho_{qm}(0,t)$ shown in Fig. 3 can be
understood without performing a numerical integration by plotting the
integrand in Eq. (48) $W(-\hbar kt/m,\hbar k,0)=W(ka^{2}t/t_{0},\hbar k,0)$
(symmetric in $k$). In Fig. 7 this is shown for the times of the deepest
minimum ($t/t_{0}\approx 0.071$) and highest maximum ($t/t_{0}\approx 0.128$)
of the full line in Fig 3. One clearly sees that the integral over the full
curve yields a result larger than that of the dotted curve.

Figure 7:  Integrand $W(-\hbar kt/m,\hbar k,0)$ for the calculation of
$\rho_{qm}(0,t)$ using Eq. (48) for $t=0.14t_{0}$ (full curve) and
$t=0.07t_{0}$ (dashed curve).

What we called the “classical approximation” can be given a different
interpretation. Comparing Eqs. (35) and (48) it can be viewed as the following
approximation for the Wigner function

$W(x,p,0)\approx|\langle x|\psi\rangle|^{2}|\langle p|\psi\rangle|^{2},$ (51)

i.e. a factorized probability distribution. As discussed above this holds
exactly only for a Gaussian wave packet.

It should be mentioned that various other wave packets have been studied which
are not groundstate wave functions of an initial potential, e.g. a rectangular
initial wave function Andrews ; Schleich2  which at short times leads to a
strongly oscillatory behaviour. In reference 17 this is discussed also using
the Wigner function.

##  V Conclusions

The broadening of a free Gaussian wave packet is one of the few time-dependent
problems treated in courses on quantum mechanics. The wave function $\langle
x|\psi(t)\rangle$ can be calculated analytically to obtain the probability
density $|\langle x|\psi(t)\rangle|^{2}$. It is little known that the time
dependence of other wave packets can differ considerably from the smooth
Gaussian broadening.

It is important to include the treatment of time-dependent Hamiltonians when
teaching quantum mechanics. A simple case is the sudden change from
$\hat{H}_{i}$ to $\hat{H}_{f}$ treated in this paper.

It turned out that directly adressing $|\langle x|\psi(t)\rangle|^{2}$ using
Eq. (3) allowed a simple way to obtain results for harmonic oscillator
systems. It is little known that the oscillatory behaviour in Fig. 1 can also
happen in the framework of classical statistical mechanics. For the free time
evolution after the quench directly addressing $|\langle
x|\psi(t)\rangle|^{2}$ provides additional insight, as the introduction of the
concept of the Wigner function is straightforward. This function allows an
understanding of the oscillatory behaviour of the broadening of the initial
ground state of an infinitely deep well without performing a numerical
integration. This is an argument for presenting the concept of the Wigner
function in quantum mechanics courses.

##  VI Acknowledgements

The author would like to thank T. Dittrich, G. Hegerfeld, F. Heidrich-Meisner,
H. Leschke, S. Manmana and V. Meden for a critical reading of the manuscript
and useful suggestions. Special thanks go to W. Schleich for bringing Refs.
$8$ and $17$ to my attention and for a stimulating discussion about the use of
Wigner functions.

##  Appendix A  The free propagator

In this appendix we shortly present the alternative derivation Andrews  of the
result for the quantum mechanical long time limit Eq.(49) for the broadening
of wave packets.

The time dependent wave function $\langle x|\psi(t)\rangle$ is calculated by
inserting position states

$\displaystyle\langle x|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\langle
x|e^{-i\hat{T}t/\hbar}|x^{\prime}\rangle\langle x^{\prime}|\psi\rangle
dx^{\prime}$ (52) $\displaystyle\equiv$
$\displaystyle\int_{-\infty}^{\infty}K(x.x^{\prime},t)\psi(x^{\prime})dx^{\prime}~{}.$

The propagator $K(x.x^{\prime},t)$ can be calculated inserting momentum
eigenstates Merzb

$\displaystyle K(x.x^{\prime},t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\langle x|p\rangle
e^{-i\epsilon_{p}t/\hbar}\langle p|x^{\prime}\rangle dx^{\prime}$ (53)
$\displaystyle=$ $\displaystyle\sqrt{\frac{m}{2\pi i\hbar
t}}e^{im(x-x^{\prime})^{2}/2\hbar t}~{}.$

If one writes

$\frac{m(x-x^{\prime})^{2}}{2t\hbar}=\frac{mx^{2}}{2t\hbar}-\frac{mxx^{\prime}}{t\hbar}+\frac{mx^{\prime
2}}{2t\hbar}~{},$ (54)

the last term can be neglected in the long time limit for a sufficiently
localized wave function $\psi(x^{\prime})$ and one obtains

$|\psi(x,t)|^{2}\to\frac{m}{t}|\tilde{\psi}(\frac{mx}{t})|^{2}~{}.$ (55)

## References

  * (1) P. Calabrese and J. Cardy “Quantum quenches in extended sysytems”, J. Stat. Mech. 2007, P6008, 1-33
  * (2) K. Huang, Introduction to Statistical Physics (Taylor and Francis, London, 2001)
  * (3) E. Wigner, “On the quantum correction for thermodynamic equlibrium”, Phys. Rev. 40, 749-759 (1932)
  * (4) W.B. Case, “Wigner functions and Weyl transforms for pedestrians”, Am. J. Phys. 76, 937-946 (2008)
  * (5) E. Merzbacher, Quantum Mechanics, 3rd edition (John Wiley and Sons, New York, 2000)
  * (6) L.E. Ballentine, Quantum Mechanics: A Modern Development, 2nd edition (World Scientific Publishing)
  * (7) M. Andrews, “The evolution of free wave packets” ’ Am. J. Phys. 76, 1102-1107 (2008)
  * (8) K. Vogel, F. Gleisberg, N.L. Harshman, P. Kazemi, R. Mack, L. Plimak, W.P. Schleich “Optimally focusing wave packets”, Chemical Physics, 375, 133-143 (2010)
  * (9) B.R. Holstein, “Topics In Advanced Quantum Mechanics”, (Adison-Wesley, Redwood City, 1992), p. 108-110
  * (10) M. Nauenberg, “Einstein’s equivalence principle in quantum mechanic revisited”, Am. J. Phys. 84, 879-882 (2016)
  * (11) A.M. Orjuela and J.K. Freericks, “Free expansion of a Gaussian wavepacket using operator manipulations”, Am. J. Phys. 91, 463 (2023)
  * (12) W. P. Schleich, Quantum optics in Phase Space, (John Wiley and Sons, New York, 2001)
  * (13) D. K. Ferry and M. Nedjalkov, The Wigner function in Science and Technology (IOP Publishing, 2018)
  * (14) R.L. Hudson, “When is the Wigner quasi-probability density non-negative?”, Reports on Mathematical Physics, 6, 249-252 (1974)
  * (15) K. Mita “Dispersion of non-Gaussian free particle wave packets” Am. J. Phys. 76, 950-953 (2007)
  * (16) T. Dittrich, INFORMATION DYNAMICS In Classical and Quantum Systems (Springer Nature Switzerland, 2022) , p. 276
  * (17) M.R. Goncalves, W.B. Case, A. Arie, W.P. Schleich, “Single-slit focusing and its representations”, Appl. Phys. B, 123:121 (2017)


# Deep Neural Network Augmented Wireless Channel Estimation for  Preamble-
based OFDM PHY on Zynq System on Chip

Syed Asrar ul haq, Abdul Karim Gizzini, Shakti Shrey, Sumit J. Darak, Sneh
Saurabh and Marwa Chafii This work is supported by the funding received from
core research grant (CRG) awarded to Dr. Sumit J. Darak from DST-SERB, GoI.
Syed Asrar ul haq, Shakit Shrey, Sumit J. Darak and Sneh Saurabh are with
Electronics and Communications Department, IIIT-Delhi, India-110020 (e-mail:
<EMAIL_ADDRESS>Abdul Karim Gizzini is with ETIS,
UMR8051, CY Cergy Paris Université, ENSEA, CNRS, France (e-mail:
abdulkarim.gizzini@ensea.fr). Marwa Chafii is with the Engineering Division,
New York University (NYU) Abu Dhabi, 129188, UAE, and NYU WIRELESS, NYU Tandon
School of Engineering, Brooklyn, 11201, NY (e-mail: marwa.chafii@nyu.edu).

###### Abstract

Reliable and fast channel estimation is crucial for next-generation wireless
networks supporting a wide range of vehicular and low-latency services.
Recently, deep learning (DL) based channel estimation has been explored as an
efficient alternative to conventional least-square (LS) and linear minimum
mean square error (LMMSE) approaches. Most of these DL approaches have not
been realized on system-on-chip (SoC), and preliminary study shows that their
complexity exceeds the complexity of the entire physical layer (PHY). The high
latency of DL is another concern. This paper considers the design and
implementation of deep neural network (DNN) augmented LS-based channel
estimation (LSDNN) for  preamble-based orthogonal frequency-division
multiplexing (OFDM) physical layer (PHY) on SoC. We demonstrate the gain in
performance compared to the conventional LS and LMMSE approaches. Via
software-hardware co-design, word-length optimization, and reconfigurable
architectures, we demonstrate the superiority of the LSDNN over the LS and
LMMSE for a wide range of signal-to-noise ratio (SNR), number of pilots,
preamble types, and wireless channels. Further, we evaluate the performance,
power, and area (PPA) of the LS and LSDNN application-specific integrated
circuit (ASIC) implementations in 45 nm technology. We demonstrate that word-
length optimization can substantially improve PPA for the proposed
architecture in ASIC implementations.

###### Index Terms:

Channel estimation, Deep learning, Least-square, Linear minimum mean square
error, System on chip, FPGA, ASIC, hardware software co-design

##  I Introduction

Accurate channel state information estimation is crucial for reliable wireless
networks [1, 2]. Conventional least square (LS) based channel estimation is
widely used in long-term evolution (LTE) and WiFi networks due to its low
complexity and latency. However, it performs poorly at a low signal-to-noise
ratio (SNR) [3]. Other techniques like linear minimum mean square error
(LMMSE) offer better performance than LS but need prior knowledge of noise,
and second-order channel statistics [4]. Furthermore, it is computationally
intensive [4, 5]. The limited sub-6 GHz spectrum has led to the co-existence
of heterogeneous devices with limited guard bands. This has resulted in poor
SNR due to high noise floor and adjacent-channel interference. This has led to
significant interest in improving the performance of the conventional LS and
LMMSE channel estimation approaches. With the integration of vehicular and
wireless networks, the radio environment has become increasingly dynamic,
posing serious challenges in establishing reliable communication links.
Furthermore, high-speed ultra-reliable services demand ultra-low latency of
the order of milliseconds [6].

Recent advances in artificial intelligence, machine and deep learning (AI-MDL)
have been explored to improve the performance of wireless physical layer such
as channel estimation, symbol recovery, link adaptation, localization, etc.
[7, 8, 9, 10, 11, 12, 13, 14, 15]. Various studies have shown that data-driven
AI-MDL approaches have potential to significantly improve the performance and
offer increased robustness towards imperfections, non-linearity, and dynamic
nature of the wireless environment and radio front-end [16, 17, 18, 6, 19]. In
the June 2021 3GPP workshop, various industry leaders presented the framework
for the evolution of intelligent and reconfigurable wireless physical layer
(PHY), emphasizing the commercial potential of the AI-MDL based academic
research in the wireless domain. Though intelligent and reconfigurable PHY may
take a few years to get accepted commercially, some of these initiatives are
expected to be included in the upcoming 3GPP Rel-18.

Few works have replaced the LS, and LMMSE channel estimation with
computationally complex deep learning (DL) architectures [11, 20, 21].
However, most existing works have not been realized on system-on-chip (SoC).
In this context, this work focuses on improving the channel estimation
performance in preamble-based orthogonal frequency division multiplexing
(OFDM) wireless PHY used in IEEE 802.11p and similar standards. Compared to
existing works, we augment the LS with a Fully Connected Feedforward DNN
instead of replacing it with computationally intensive DL networks. We
demonstrate the performance gain over the conventional LS and LMMSE
approaches. We extensively study the effects of training SNR on the
performance of the LSDNN channel estimation.

We have mapped the proposed and existing algorithms on two Zynq multi-
processor system-on-chip (ZMPSoC) boards from AMD-Xilinx: ZC706 and ZCU111.
The ZC706 comprises a dual-core ARM Cortex-A9 processor and Xilinx 7 series
Kintex FPGA. The ZCU111 comprises a quad-core ARM Cortex A53 processor and an
ultra-scale FPGA. Through software-hardware co-design, word-length
optimization, and  adaptable architectures, we demonstrate the superiority of
the proposed LSDNN architecture over LS and LMMSE architectures for a wide
range of SNRs, number of pilots, preamble types, and wireless channels.
Experimental results show that the proposed LSDNN architecture offers lower
complexity and a huge improvement in latency over the LMMSE. Next, we evaluate
the performance, power, and area (PPA) of the LS and LSDNN on application-
specific integrated circuit (ASIC) implementations in 45 nm technology. We
demonstrate that the word-length optimization can substantially improve PPA
for the proposed architecture in ASIC implementations. The AXI-compatible
hardware IPs and PYNQ-based graphical user interface (GUI) demonstrating the
real-time performance comparison on the ZMPSoC are physical deliverables of
this work.  Please refer to [22] for source codes, datasets, and hardware
files used in this work. The work presented in this paper is a significant
extension of the conference paper [11]. Here, we improve the LSDNN algorithm
as per the memory constraints of the SoC, two LSDNN models, and detailed
performance analysis, including the effect of training SNR. The hardware
architectures on ZMPSoC (Section V onwards) have not been discussed
previously.

The rest of the paper is organized as follows. Section II reviews the work
related to applications of AI-MDL approaches for wireless PHY. In Section III,
we present the system model and introduce LS, and LMMSE-based channel
estimation approaches. The proposed LSDNN algorithm and simulation results
comparing the performance of LS, LMMSE, and LSDNN are given in Section IV
followed by their architectures in Section V. The functional performance and
complexity results of fixed-point architectures are analyzed in Section VI.
The ASIC implementation and results are presented in Section VII. Section VIII
concludes the paper along with directions for future works.

Notations: Throughout the paper, vectors are defined with lowercase bold
symbols $\bm{x}$ whose $k$-th element is $\bm{x}[k]$. Matrices are written as
uppercase bold symbols $\bm{X}$.

##  II Literature Review: DL in Wireless PHY

Various studies and experiments have demonstrated that conventional frameworks
such as Shannon theory [23], detection theory [24], and queuing theory [25]
based wireless PHY suffer from performance degradation due to randomness and
diversity of wireless environments. Instead of exploring approximate
frameworks, recent advances in AI-MDL approaches and their ability to address
hard-to-model problems offer a good alternative for making the wireless PHY
robust [26, 27]. Numerous works have shown that traditional ML techniques have
been successful in combating complex wireless environments and hardware non-
linearities [28, 29]. However, they need manual feature selection. The DL
approaches offer the capability to extract features from the data itself,
thereby eliminating the need for manual feature extraction. This has lead to
numerous AI-MDL based approaches for various problems such as modulation
classification [30, 31], spectrum sensing [32, 15], multiple-input multiple-
output (MIMO) high-resolution channel feedback [33],mobility and beam
management [34], link adaptation [35].With the evolution of heterogeneous
large-scale networks, existing frameworks suffer from scaling issues due to
the high cost of exhaustive searching and iterative heuristic algorithms
[36].Such scaling issues can be handled more efficiently using AI-MDL
approaches [37]. To bring DL-aided intelligent and reconfigurable wireless PHY
into reality, issues such as potential use cases, achievable gain and
complexity trade-off, evaluation methodologies, dataset availability, and
standard compatibility impact need to be addressed [16, 18, 6, 19].

The DL-based PHY can be categorized in two approaches: 1) Complete transmitter
and receiver are replaced with respective DL architectures [38, 39], 2)
Independent DL blocks for each sub-block or combination of sub-blocks in PHY
[40, 31, 20, 41, 21]. The drawback of the first approach is that it may not
provide intermediate outputs such as channel status indicator, precoding, and
channel rank feedback, making them incompatible with existing standards.
Furthermore, existing works can not support high throughput requirements. In
this paper, we focus on the second approach addressing the channel estimation
task in wireless PHY.

ChannelNet [20] treats the LS estimated values at pilot sub-carriers in an
OFDM frame as a low-resolution 2D image and develops a CNN-based image
enhancement technique to increase the resolution to obtain accurate channel
estimates at data sub-carriers. Its performance is sensitive to the SNR
conditions, which demand SNR-specific models resulting in high reconfigurable
time and large on-chip memory. This model is further enhanced [21] by
replacing the two CNN models with a single deep residual neural network
(ReEsNet) to reduce its complexity and improve performance. Interpolation-
ResNet in [41] replaces the transposed convolution layer of ReEsNet with a
bilinear interpolation block to reduce the complexity further and make the
network flexible for any pilot pattern. The computational complexity and
memory requirements of these models are high, even higher than the complexity
of PHY. In this work, we focus on augmenting the conventional channel
estimation with DL instead of DL-based channel estimation [20, 21, 41].

The validation on synthetic datasets is another concern in DL-based
approaches. This has been addressed in some of the recent works, such as [42]
where DL-based channel estimation is performed using real radio signals,
demonstrating the feasibility in practical deployment. This works provides a
detailed process and addresses the concerns regarding dataset creation from
the RF environment and training and testing the models using real radio
signals. The authors in [39] used the concept of transfer learning in [43] to
devise a two-phase training strategy to train a DL-based PHY using real-world
signals and demonstrated over-the-air communication. It offered similar
performance as conventional PHY validating the feasibility of DL-based
approaches in a real-radio environment. However, limited efforts have been
made on hardware realization of the DL-based wireless PHY [44, 45, 46, 47,
48]. Hence, there is limited knowledge of their performance on fixed-point
hardware, latency, and computation complexity. The impact of the hardware-
specific constraints such as high cost and area of on-chip memory and a
limited number of memory ports on the training approaches have not been
highlighted in the literature yet. The work presented in this paper addresses
these issues and offers innovative solutions at algorithm and architecture
levels for channel estimation in wireless PHY.

##  III System Model

This paper considers orthogonal frequency-division multiplexing (OFDM) based
transceivers with frame structure based on IEEE 802.11p standard shown in Fig.
1.  The transmitted frame consists of a preamble header including ten short
training sequences and two long training symbols (LTS), followed by the signal
and data fields. The LTS consists of predefined fixed symbols known to both
transmitter and receiver and used for channel estimation at the beginning of
the frame. There are total $K=64$ sub-carriers with the sub-carrier spacing of
$156.25$ KHz, and the corresponding transmission bandwidth is $10$ MHz. Each
LTS consists of $64$ sub-carriers in a single OFDM symbol, of which $12$ are
NULL sub-carriers, and $52$ are sub-carriers carrying BPSK modulated preamble
sequence. Each data symbol carries $48$ data and $4$ pilot sub-carriers. In
this work, we assume that the channel is static over the course of frame
transmission. In cellular PHY, the preamble is absent, and comb-pattern-based
pilots are embedded in OFDM data symbols [40, 31, 20, 41, 21, 47]. Due to
different system models and channel estimation approaches, we limit our
discussion to preamble-based channel estimation.

Figure 1: IEEE 802.11p frame structure and sub-carrier mapping.

The block diagram of the IEEE 802.11p PHY transceiver is shown in Fig. 2. The
data to be transmitted is modulated using an appropriate modulation scheme
such as QPSK/QAM, followed by sub-carrier mapping. The set of 64 sub-carriers
is passed through OFDM waveform modulation comprising IFFT and cyclic prefix
addition. The length of cyclic prefix is $K_{cp}=16$. The scheduler inserts
the preamble OFDM symbols at the beginning of each frame, comprising a certain
number of data symbols. Next, the complete OFDM frame is transmitted through
the wireless channel. At the receiver, OFDM symbols containing the LTS
preamble are detected and extracted, followed by OFDM demodulation. Then,
channel estimation and equalization operations are performed.

Figure 2: Building blocks of the IEEE 802.11p based OFDM transmitter and
receiver PHY.

The frequency-domain input-output relation between the transmitted and the
received OFDM frame at the receiver can be expressed as follows

$\bm{Y}[k,i]={\bm{H}}[k,i]\bm{X}[k,i]+\bm{V}[k,i].$ (1)

Here, $\bm{X}[k,i]$, ${\bm{H}}[k,i]$, $\bm{Y}[k,i]$, and $\bm{V}[k,i]$ denote
the transmitted OFDM frame, the frequency domain channel gain, the received
OFDM frame, and the addictive white Gaussian noise (AWGN) with zero mean and
variance, $\mathcal{N}_{0}$. $k$ and $i$ denote the sub-carrier and OFDM
symbol indices within the received frame, respectively.

Recall that the channel estimation is performed once per frame using the
received preamble symbols, therefore (1) can be rewritten as

$\bm{Y}[k,p]={\bm{H}}[k,p]\bm{D}[k,p]+\bm{V}[k,p],$ (2)

where $\bm{D}[k,p]=\bm{d}_{p}\left[k\right]$ denote the $p$-th predefined
transmitted preamble symbol, where $1\leq p\leq K_{p}$ and $K_{p}$ is the
total number of the transmitted preamble symbols within the frame.

In this context, the LS channel estimation [4] can be expressed as

$\hat{\tilde{\bm{H}}}_{\text{LS}}[k,p]=\frac{\sum_{q=1}^{K_{p}}\bm{Y}[k,q]}{K_{p}\bm{D}[k,p]}$
(3)

The LS estimation is easy to implement in hardware and does not require prior
channel and noise information. On the other hand, the LMMSE channel estimation
[4] needs prior knowledge of the second order channel and noise statistics.
Mathematically, the estimated channel gain at the $k$-th sub-carrier can be
expressed as

$\hat{\tilde{\bm{H}}}_{\text{LMMSE}}[k,p]={\bm{R}_{h}}_{p}\left({\bm{R}_{h}}_{p}+\left(\frac{K\mathcal{N}_{0}}{E_{p}}\right)\bm{I}\right)^{-1}\hat{\tilde{\bm{H}}}_{\text{LS}}[k,p],$
(4)

where ${R_{h}}_{p}$ is the channel auto-correlation matrix, and ${E}_{p}$
denotes the power per preamble symbol. It can be observed that the LMMSE
improves the performance of the LS, and the improvement depends on the prior
knowledge of the channel and noise statistics.

##  IV Deep Neural Networks Based Channel Estimation

In this section, we present LSDNN design details and simulation results using
floating-point arithmetic comparing various channel estimation approaches.

###  IV-A LSDNN based Channel Estimation

Figure 3: Fully connected neural network. Figure 4: (a) NMSE and (b) BER
comparisons of various channel estimation approaches using floating point
arithmetic.

Deep Neural Networks (DNN) is a subset of DL that tries to mimic how
information is passed among biological neurons. A fully-connected neural
network is shown in Fig. 3. It has a layered structure, with each layer
feeding to the next layer in a forward direction. The first layer is called an
input layer representing the input to the DNN, and the last layer is called an
output layer from which the outputs are taken. The layers between the input
layer and output layer are called hidden layers. The output and hidden layers
contain parallel processing elements called neurons, which take the inputs
from the previous layer and send output to the next layer. A neuron performs
the summation of weighted inputs followed by a non-linear activation function,
as shown in Fig.3. These functions include Sigmoid, tanh, ReLU, leaky ReLU
[49]. For DNN with $L$ layers, including input and output layers, the output
of $j$-th neuron in layer $l$, $1<l<L$ is given as

$\bm{y}_{j}^{l}=f^{(l)}\left(\sum_{i=0}^{N_{l}}\left(\bm{w}_{i,j}^{l}\bm{y}_{i,j}^{l-1}\right)+\bm{b}_{j}^{l}\right),$
(5)

where $N_{l}$ is the total number of neurons in $l$-th layer, $b_{j}^{l}$ is
the bias of $j$-th neuron of $l$-th layer, $w_{i,j}$ is the weight from $i$-th
neuron of $(l-1)$-th layer to $j$-th neuron of $l$-th layer, and
$f^{(l)}\left(.\right)$ is the activation function of $l$-th layer.

The proposed DNN augmented LS estimation approach, an extension of [11],
processes the output of the LS estimator using the DNN, thereby reducing the
effect of noise on LS estimates. As shown in Fig. 2, the proposed LSDNN
channel estimator proceeds as follows

  * •

LS channel estimation as illustrated in (3), where
$\hat{\tilde{\bm{H}}}_{\text{LS}}\in\mathbb{R}^{K_{\text{on}\times 1}}$ and
$K_{\text{on}}$ refers to the active subcarriers within the transmitted
preamble symbols.

  * •

$\hat{\tilde{\bm{H}}}_{\text{LS}}\in\mathbb{R}^{K_{\text{on}\times 1}}$ is
converted from complex to real-valued domain by stacking the real and
imaginary values in one vector such that
$\hat{\tilde{\bm{H}}}^{(R)}_{\text{LS}}\in\mathbb{R}^{2K_{\text{on}\times
1}}$.

  * •

$\hat{\tilde{\bm{H}}}^{(R)}_{\text{LS}}\in\mathbb{R}^{2K_{\text{on}\times 1}}$
is fed as an input to the employed DNN.

  * •

Finally, the processed LS complex-valued channel estimates
$\hat{\tilde{\bm{H}}}^{(DNN)}_{\text{LS}}\in\mathbb{R}^{K_{\text{on}\times
1}}$ is obtained by converting the real-valued output vector of DNN back to
the complex-valued vector.

A training dataset of 30,000 samples is generated using MATLAB-based
IEEE802.11p PHY. The transmitter transmits the LTS over the channel for a wide
range of SNR. The LS estimate of received LTS and the corresponding channel
frequency response are used as the training data. The dataset is then divided
into a training set of 24,000 samples and a validation set of 6,000 samples in
an 80:20 ratio. During inference, the OFDM frame consisting of LTS and data
symbols is transmitted over the channel. The LS estimate of LTS is passed as
input to the trained DNN, and the estimates thus obtained are used to equalize
the data symbols for subsequent BER analysis. Please refer to [22] for source
codes, datasets, and hardware files used in this work.

The DNN is trained to reduce the loss function
$J_{\Omega,b}(\tilde{H},y_{\hat{\tilde{\bm{H}}}_{\text{LS}}}^{L})$ over one
long training symbol, where $\bm{H}$ is the original channel frequency
response and $\hat{\tilde{\bm{H}}}_{\text{LS}}$ is the channel estimated using
the LS approach. Training is an iterative process where the model parameters
are updated using optimization functions to minimize the loss over time.
Various optimization functions such as stochastic gradient descent, root mean
square prop, and adaptive moment estimation (ADAM) can be used [50].

Hyper-parameters for the DNN architecture is chosen manually. Multiple
iterations were performed with different hyper-parameters until the one with
the best accuracy and low latency was found. We restricted our search to only
a few hidden layers to reduce the complexity of the overall design. For
illustrations, we consider two DNN architectures: 1) LSDNN1: DNN with a single
hidden layer of size $K_{on}$, and 2) LSDNN2: DNN with two hidden layers of
size $2\times K_{on}$. Each model is trained for 500 epochs with Mean Square
Error (MSE) as a loss function and ADAM as an optimizer. Table I summarises
the specifications of neural network. After training, the next step is
inference, in which a trained DNN model is tested on new data to evaluate its
performance. The hardware realizations of DNN focus on accelerating the
inference phase.

TABLE I: DNN Parameters

Parameters | Values
---|---
LSDNN1 (hidden layers ; neurons per layer) | (1 ; $K_{on}$)
LSDNN2 (hidden layers ; neurons per layer) | (2; $2\times K_{on}$ )
Activation Function | ReLU
Epochs | 500
Optimizer | ADAM
Loss Function | MSE

Figure 5: Magnitude plot for channel estimation corresponding to subcarriers
in an OFDM symbol for (a) 0dB SNR, (b) 20dB SNR, (c) 45dB SNR.

###  IV-B Simulation Results on Floating Point Arithmetic

For the system model discussed in Section III, we have used six channel models
considering vehicle-to-vehicle (V2V) and roadside-to-vehicle (RTV)
environments as discussed in [51]. We have considered two performance metrics:
1) Normalized mean square error (NMSE) and 2) Bit-error-rate (BER) to analyze
the performance of channel estimation and wireless PHY, respectively.

In Fig. 4(a), we compare the average NMSE of LS, LMMSE, and LSDNN
architectures (LSDNN1 and LSDNN2) for a wide range of SNR, and the
corresponding BER results on end-to-end wireless transceiver are shown in Fig.
4(b). We assume Rayleigh fading channel with AWGN. We consider an LSDNN
architecture trained on a single SNR referred to as training SNR. Here, the
training SNR is selected as 10 dB, and the testing SNR range is from -50 dB to
50 dB. Please refer to Section VI-C for more details on the selection of
training SNR.

As shown in Fig. 4, the performance of the conventional LS and LMMSE
architectures matches with their analytical results, and the error for the
LMMSE estimator is less than that of the LS estimator. It can be observed that
the LMMSE improves the performance of the LS, and the improvement depends on
the prior knowledge of the channel and noise statistics. When prior channel
statistics are not known accurately, the LMMSE performance degrades, and the
NMSE becomes worse than LS at high SNR, as shown in Fig. 4. The BER of the
LSDNN1 and LSDNN2 are close to that of the ideal channel estimation approach
and significantly outperforms the LS and LMMSE at all SNRs. The NMSE
performance of the LSDNN improves with the increase in SNR until the training
SNR of 10 dB. Even though LSDNN1 has less complexity, it offers superior
performance than LSDNN2. The NMSE is also a function of the number of training
samples; hence, the dataset should be sufficiently large. Similar results are
also observed for different wireless channels and preamble types.
Corresponding plots are not included to avoid the repetition of results.

The improved performance of the LSDNN can be attributed to the capability of
the DNN to extract channel features and learn a meaningful representation of
the given dataset. Specifically, it accurately learns to distinguish between
the AWGN noise and the channel gain. The LS estimation does not consider
noise, resulting in noisy estimates at low SNRs. As SNR increases, the noise
content in the LS estimation reduces, resulting in improved performance. This
is evident from Fig. 5 where we have analyzed the plots of estimated channel
gain magnitudes at different OFDM sub-carriers for three different SNRs. It
can be observed that LSDNN and LMMSE performance is close to ideal channel
estimation even at low SNR, with the former better than the latter at all sub-
carriers. The erroneous knowledge of channel parameters leads to degradation
in LMMSE performance, as shown in Fig. 5. Such prior knowledge is not needed
in the LSDNN.

##  V Channel Estimation on SoC

In this section, we discuss the algorithm to architecture mapping of three
channel estimation approaches, LS, LSDNN, and LMMSE, on FPGA (hardware) part
of the ZMPSoC. The software implementation of these algorithms on an ARM
processor is straightforward, and corresponding results are given in Section
VI. We begin with the LSDNN architecture, which includes LS estimation as
well.

Figure 6: Various building blocks of the LSDNN architecture.

###  V-A LSDNN Architecture

At the receiver, the LTS samples are extracted from the data obtained after
the OFDM demodulation, as shown in Fig. 2. LSDNN uses these symbols for
channel estimation, and it involves five tasks: 1) Data extraction, 2) LS-
based channel estimation, 2) Pre-processing, 3) DNN, and 4) Post-processing,
as shown in Fig. 6. In this section, we present the architectures to
accomplish these tasks.

The serial data with complex samples received after OFDM demodulation are
extracted with independent real and imaginary parts. For the chosen system
model with an OFDM symbol comprising of 52 sub-carriers, the input to LS
estimation is a vector with 104 real samples. Another input to LS estimation
is a reference LTS sequence of the same size. The LS estimation involves the
complex division operation of the received ($y_{p}$) and reference ($x_{p}$)
versions of the LTS. The complex division can be mathematically expressed as
shown below, and the corresponding architecture is shown in Fig. 6

$\displaystyle\hat{\tilde{\bm{H}}}_{\text{LS}}$
$\displaystyle=\frac{y_{p}}{x_{p}}$ (6) $\displaystyle=\frac{x\\_r\times
y\\_r+x\\_i\times y\\_i}{x\\_r^{2}+x\\_i^{2}}+\frac{x\\_r\times
y\\_i+x\\_i\times y\\_r}{x\\_r^{2}+x\\_i^{2}}i$

where x_r, x_i, y_r, and y_i denote the real and imaginary parts of received
and reference versions of the LTS, respectively.

The LS estimation of each sub-carrier requires six real multiplications, one
real division, three additions, and one subtraction operation. We have
explored various architectures of simultaneous calculation of the LS
estimation of multiple sub-carriers by appropriate memory partitioning to have
parallel access to data and pipelining to improve the utilization efficiency
of hardware resources. Please refer to Section VI for more details. Further
optimizations can be carried out depending on the modulation type of the LTS
sequence. For instance, the LS estimation of QPSK modulated LTS sequence needs
only one complex multiplication operation. In the case of IEEE 802.11p, the
LTS and pilot symbols are BPSK modulated. The LS estimation is reduced to
choosing either the received complex value or its two’s compliment based on
whether the LTS is +1 or -1, respectively.

The output of the LS channel estimation is further processed using DNN
architecture. Before DNN, pre-processing is needed to normalize the DNN input
to have zero mean and unit standard deviation. The estimation of mean and
standard deviation during inference is not feasible due to limited on-chip
memory and latency constraints. Hence, we use the pre-calculated values of the
mean and standard deviation using the training dataset. Similarly, the DNN
output is de-normalized so that transmitted data can be demodulated for BER-
based performance analysis of the end-to-end transceiver. At the architecture
level, pre and post-processing incur additional costs in terms of multipliers
and adders, as shown in Fig. 6.

Figure 7: Scheduling of operations inside the single DNN layer.  Figure 8:
Architecture of the LMMSE based channel estimation.

As shown in Eq. 5 and Fig. 6, each neuron, i.e. processing element (PE), in
DNN consists of a multiplier and adder unit to perform element-wise
multiplications between outputs of all PEs of the previous layer and layer-
specific weights stored in internal memory. In the end, the output is added
with the layer-specific bias term. This means each PE needs to store $N_{l-1}$
weights and one bias value in its memory, where $N_{l-1}$ is the number of PEs
in the previous layer. The output of each layer is stored in memory which is
completely partitioned to allow parallel access from the PEs in the subsequent
layer. The PE functionality is realized sequentially using the counter and
multiplexers. Though the parallel realization of all multiplication and
addition operations inside the PE is possible due to memory partitioning at PE
output, this significantly increases resource utilization and power
consumption. ReLU block is added at the end of each hidden layer to introduce
non-linearity. It is a hardware-friendly activation function that replaces the
negative PE output with zero. As shown in Fig. 7, the output of the first PE
of the previous layer is processed by all PEs in the layer simultaneously, and
the output is stored in their respective buffers. Next, the output of the
second PE of the previous layer is processed, followed by accumulation with
the previous intermediate output in the buffer. This process is repeated until
the outputs of all PEs in the previous layer have been processed. After bias
addition and ReLU operation, the subsequent layer is activated.

In a completely parallel DNN architecture, all the PEs in a layer are realized
using the dedicated hardware resource, and hence the output of all PEs is
computed simultaneously. On the other hand, in a completely serial
architecture, all PEs in a layer share the hardware resources; hence, the
outputs of PEs are computed serially. Depending on resource and latency
constraints, various architectures such as fully serial, fully parallel, and
serial-parallel architectures for various layers have been explored.

Figure 9: Proposed architecture on ZMPSoC and screenshot of Pynq based GUI
depicting the real-time demonstration of channel estimation.

###  V-B LMMSE Architecture

The LMMSE channel estimation requires prior knowledge of the channel
correlation matrix ($R_{h}$) and SNR in addition to LTS symbols. The LMMSE
architecture is shown in Fig. 8 and is based on Eq. 4. In the beginning,
$R_{h}$ is separated into real and imaginary matrices, and the term (1/SNR) is
added with each diagonal element of the real matrix of $R_{h}$. Then, the
inverse of the $R_{h}$ matrix is performed. This is followed by various matrix
multiplication and addition operations. We have modified Xilinx’s existing
matrix multiplication and matrix inversion reference examples to support the
complex number arithmetic since the baseband wireless signal is represented
using complex samples. The well-known lower-upper (LU) decomposition method is
selected for matrix inversion. We parallelize individual operations like
element-wise division and Matrix Multiplication on the FPGA. Every element in
the matrix is parallelly processed to compute division, and every row column
dot product in matrix multiplication is performed in parallel to speed up the
computation. In the end, multiple instances of these IPs are integrated to get
the desired LMMSE functionality, as shown in Fig. 8.

###  V-C Hardware Software Co-design and Demonstration

In Fig. 9, various building blocks of the channel estimation tasks such as LS,
LMMSE, LSDNN, scheduler, DMA, interrupt, and GUI controller is shown. For
illustration, we have shown all channel estimation approaches on the FPGA. To
enable this, we have developed AXI-stream compatible hardware IPs and
interconnected them with PS via direct-memory access (DMA) in the scatter-
gather mode for efficient data transfers. Later in Section VI, we considered
various architectures via hardware-software co-design by moving the blocks
between ARM Processor and FPGA. We have deployed PYNQ based operating system
on the ARM processor of the  ZCU111 platform, which takes care of various
scheduling and controlling operations. It also enables graphical user
interface (GUI) development for real-time demonstration. As shown in Fig. 9,
GUI allows the user to choose various parameters such as channel estimation
schemes, SNRs, and word length.

Figure 10: (a) NMSE and (b) BER comparisons of various WL architectures of the
LSDNN on the ZSoC.

##  VI Performance and Complexity Analysis

This section presents the performance analysis of different channel estimation
architectures implemented on the  Xilinx ZC706 board for a wide range of SNRs,
word length, and wireless parameters such as channels, preamble type, etc.
The FPGA fabric used in the ZC706 board consists of 350 logic cells with
218,600 lookup tables (LUT) and 437,200 flip-flops. It also includes 19.1 Mb
of BRAM memory and 900 DSP48 slices. The resource utilization, execution time,
and power consumption of various architectures obtained via word length
optimization, hardware-software co-design, and  adaptability are compared. The
need for  adaptable architectures and challenges in training DNN for wireless
applications are highlighted at the end.

###  VI-A Word Length Impact on Channel Estimation Performance

TABLE II: Resource utilization and latency comparison of LS, LMMSE, and DNN-
Augmented LS channel estimation for different word length implementations.

Sr. No | Architectures | Word Length | Execution time (ms) | BRAMs | DSPs | LUTs | FFs | PL Power (W) | SoC Power (W)
---|---|---|---|---|---|---|---|---|---
1 | LS | SPFL | 0.0133 | 14 | 96 | 14721 | 15094 | 0.438 | 1.969
2 | HPFL | 0.0146 | 5 | 72 | 8371 | 10369 | 0.296 | 1.827
3 | FP (18,9) | 0.0155 | 7 | 24 | 14665 | 16649 | 0.373 | 1.904
4 | MMSE | DPFL | 248.41 | 191.5 | 522 | 44444 | 43485 | 1.158 | 2.689
5 | SPFL | 214.7 | 99.5 | 194 | 24249 | 24778 | 0.621 | 2.152
6 | DP(INV)-SP | 219.75 | 139.5 | 338 | 36225 | 34898 | 0.918 | 2.449
7 |  LSDNN (104 Parallel PE) | SPFL | 0.0266 | 88 | 322 | 62895 | 58064 | 1.078 | 2.647
8 | HPFL | 0.0298 | 32 | 260 | 30436 | 32286 | 0.569 | 2.138
9 | FP (32,8) | 0.0186 | 88 | 520 | 157130 | 151544 | 2.46 | 4.029
10 | FP (24,8) | 0.0179 | 11 | 260 | 85155 | 82721 | 1.318 | 2.849
11 |  | LSDNN
---
(52 Parellel PE)
FP (24,8) | 0.125 | 20 | 32 | 23534 | 23455 | 0.465 | 1.996
12 |  | LSDNN
---
(1 PE)
FP (24,8) | 0.236 | 22.5 | 17 | 15450 | 16025 | 0.373 | 1.904

TABLE III: Execution times for different Hw/Sw codesign approaches for DNN
Augmented LS Estimation

Sr. No | Blocks in PS | Blocks in PL | Execution time (us) | Acceleration factor | BRAM | DSP | LUT | FF | PL Power(W) | SoC Power (W)
---|---|---|---|---|---|---|---|---|---|---
1 | B1,B2,B3 | NA | 558 | 1 | NA
2 | B1,B2 | B3 | 49 | 11 | 80 | 130 | 26622 | 25807 | 0.453 | 2.021
3 | B1 | B2,B3 | 30 | 18 | 82 | 130 | 26456 | 28136 | 0.453 | 2.022+
4 | NA | B1,B2,B3 | 26 | 21 | 88 | 322 | 62895 | 58064 | 1.078 | 2.647

In Section IV-B, we have analyzed the performance of various channel
estimation approaches on floating-point arithmetic in Matlab.

Conventionally, floating-point architectures using double-precision floating
point (DPFP) and single-precision floating point (SPFP) WLs offer excellent
functional accuracy. However, they require high resources, power, and
execution time. Since the extremely-large dynamic range offered by floating-
point arithmetic may not be needed due to the presence of fixed-width analog-
to-digital converters (ADCs), fixed-point WL architectures are preferred. The
FPGA offers complete flexibility in the WL selection; detailed analysis is
needed to choose the appropriate WL. Here, we use $W$ bits to represent each
variable in fixed point WL. Out of $W$ bits, we use $I$ and ($W-I$) bits to
represent the integer and fractional parts, respectively. To identify
appropriate values of $W$ and $I$ for the given architecture, we first use
sufficient large $W$ and analyze the functional accuracy for different values
of $I$ for all datasets and choose the smallest $I$, that offers the same
performance as that of SPFL architecture. For the selected $I$, we find the
lowest $W$ using the same process.

In Fig. 10(a) and (b), we compare the NMSE and BER performance of the various
LSDNN architectures, respectively, for six different WLs along with the SPFP
architecture of the LS and DPFP architecture of the LMMSE. The architectures
with the half-precision floating point (HPFL) and fixed-point architectures
with WL below 24 and integer length of 8 (FP(24,8)) lead to significant
degradation in the NMSE. However, degradation in NMSE may not correspond to
degradation in the BER due to data modulation in the wireless transceiver.
This is evident from Fig. 10(b), where the architecture with HPFL offers the
BER performance same as that of the SPFL architecture. On the other hand, the
BER of the fixed-point architecture with WL below  FP(24,8) leads to
significant degradation in the performance.

Figure 11: NMSE Comparison of various WL architectures of (a) LS and (b) LMMSE
on the ZSoC. Figure 13: (a) NMSE and (b) BER comparisons of various LSDNN
architectures trained on different training SNRs.

The WL selection is architecture-dependent; hence, the WL for the LS may not
be the same as that of LSDNN. Based on the detailed experimental analysis, the
LS architecture with fixed-point WL of 18 bits  (FP(18,9)) or higher offers
the same performance as its SPFL architecture, as shown in Fig 11(a). In the
case of LMMSE, fixed-point architecture is not feasible. Even SPFL
architecture fails to offer the desired performance, especially at high SNRs;
hence, DPFL is needed. Further analysis revealed that only the matrix inverse
sub-block needs DPFL WL while the rest of the sub-blocks can be realized in
SPFL WL as shown in Fig. 11(b). Such analysis highlights the importance of
experiments on the SoC for different WL since such results can not be obtained
using simulations.

TABLE IV: Resource utilization and latency comparison of LSDNN for wireless
PHY with different FFT size

Sr. No | FFT Size | Execution time (us) | Acceleration factor | BRAM | DSP | LUT | FF | PL Power(W) | SoC Power (W)
---|---|---|---|---|---|---|---|---|---
1 | 64 | 26 | 21 | 88 | 322 | 62895 | 58064 | 1.078 | 2.647
2 | 128 | 48 | 28 | 195 | 512 | 83193 | 80679 | 1.474 | 3.005
3 | 256 | 85 | 31 | 339 | 832 | 119043 | 118606 | 2.206 | 3.737

###  VI-B Resource, Power and Execution Time Comparison

In this section, we compare the resource utilization, execution time, and
power consumption of various architectures on the ZSoC platform. In Table II,
we consider three different WL architectures of the LS and LMMSE and six
different WL architectures of the LSDNN. It is assumed that the complete
architecture is realized in the FPGA (PL) part of the SoC. The execution time
of the LS architecture is the lowest, while the execution time of the LMMSE
architecture is the highest. The execution time of the LSDNN is significantly
lower than that of LMMSE and is around 1.5-2 times that of the LS
architecture. Similarly, resource utilization and power consumption of the
LSDNN are higher than that of the LS. This is the penalty paid to gain
significant improvement in channel estimation performance. However, the LSDNN
offers a significantly lower execution time than the LMMSE, and resource
utilization is also lower, especially in terms of BRAM and DSP units which are
limited in numbers on FPGA. The use of fixed-point architectures leads to
significant improvement in all three parameters. In case of LSDNN, we also
considered two additional architectures: 1) Serial-parallel architecture where
2 PEs share the same hardware resources, and 2) Fully serial architecture
where all PEs share the same hardware resources. As we move from a completely
parallel architecture to these two architectures, we can gain significant
savings in the number of resources, especially DSPs. However, the execution
time increases, which is still significantly lower than the LMMSE. Further
improvement is possible if the input data type of the architecture is changed
to fixed-point so that fixed-to-floating point conversion overhead at the
input and output can be removed.

Next, we consider various LSDNN architectures obtained via hardware-software
co-design. For this purpose, LSDNN is divided into three blocks – LS (B1),
normalization and denormalization (B2), and DNN (B3). In Table III, we
consider four architectures: 1) B1, B2, B3 in PS, 2) B1, B2 in PS, and B3 in
PL, 3) B1 in PS, and B2, B3 in PL, and 4) B1, B2, B3 in PL (Same as Table II).
Moving DNN to PL gives $11\times$ improvement in execution time with respect
to PS implementation as FPGA can exploit inherent parallelism in DNNs, as
explained in section V-A, to speed up the processing. Normalization and
denormalization can also be performed in parallel as there is no data
dependency for these operations and thus can be accelerated by moving to PL.
Execution time can further be reduced by moving LS to PL and processing
multiple subcarriers in parallel, thus achieving an overall acceleration
factor of $21\times$ when all the data processing blocks are implemented in PL
compared to complete PS implementation. Reducing execution time by moving
blocks to PL and introducing parallelism increases the FPGA fabric’s resource
utilization and power consumption. Thus, when choosing a particular
architecture, there is a trade-off between latency and resource
utilization/power consumption.

With the evolution of wireless networks, the transmission bandwidth and hence,
the FFT size of the OFDM PHY is also increasing. The increase in FFT size
leads to an increase in the complexity of the channel estimation architecture,
which in turn demands efficient acceleration via FPGA on the SoC. In Table IV,
we compare the effect on execution time and corresponding acceleration factor
for different FFT sizes. It can be observed that the gain in acceleration
factor increases with the increase in the FFT size mainly due to more
opportunities for parallel arithmetic operations. Thus, LSDNN architecture has
the potential to offer a significant improvement in performance and execution
time for next-generation wireless PHY.

###  VI-C Effect of Training SNR

For the results presented till now, the LSDNN model is obtained using the
training SNR of 10 dB. Training a DNN on a single SNR allows it to converge
faster and model the channel better as the DNN does not have to encounter
multiple noise distributions during the training phase, thus reducing training
complexity. However, faster training should not compromise the functional
performance of the LSDNN. For in-depth performance analysis, we analyze the
effect of training SNR on the performance of the LSDNN model.

Figure 12: BER performance of the LSDNN architecture trained with dataset
containing samples from complete SNR range of -50dB to 50dB. Figure 15: (a)
NMSE and (b) BER comparisons of various channel estimation approaches on
different models. The LSDNN model is trained on model M1.

We initially consider various architectures of the LSDNN model trained using
the dataset comprising all SNR samples ranging from -50 dB to 50 dB. As shown
in Fig.12, the NMSE and BER performance is poor due to the reasons mentioned
before. However, the selection of training SNR is not trivial since optimal
training SNR may vary depending on the testing SNR, i.e., the deployment
environment. For instance, as shown in Fig. 13 (a) and (b), single training
SNR affects the performance of LSDNN considerably, especially at high testing
SNRs. To address this issue, we consider the selection of training SNR based
on the testing SNR range, and corresponding results are shown in Fig. 14. It
shows the NMSE for different testing SNRs plotted against the selected
training SNR. For each testing SNR, the NMSE first decreases as the training
dataset SNR is increased till it reaches its minimum value and then starts
increasing. The best training SNR is where the NMSE is minimum, e.g., for the
testing SNR range of 0 dB - 20dB, the lowest NMSE corresponds to a training
SNR of 10dB. Similarly, for testing SNR of -10 dB to 10 dB, the optimal
training SNR is 0 dB. Note that training SNR is irrelevant at low SNRs due to
the significant impact of noise on the received signal, and DNN is unable to
learn the channel properties, resulting in high NMSE and BER. Such dependence
on the testing SNR demands reconfigurable architecture that can switch between
various LSDNN models depending on the given testing SNR range. Such
architecture is presented in the next section.

Figure 14: Comparison of effect of training SNR on the NMSE for various ranges
of the testing SNR.

###  VI-D Adaptable Architecture

Though the LSDNN approach offers better performance than LMMSE without the
need of prior channel knowledge, LSDNN performance depends on the efficacy of
the training dataset with respect to the validation environment. For instance,
the performance of LSDNN may degrade if the parameters such as testing SNR,
preamble type, pilot type, etc., of the training and validation environment do
not match. For instance, the performance of the LSDNN trained on model $M1$
degrades significantly on two different models, $M2$ and $M3$, as shown in
Fig. 15. This is expected because for a DNN to perform satisfactorily, the
training and testing conditions should remain the same. To address this
challenge, we need an  adaptable architecture for LSDNN to enable an on-the-
fly switch between various LSDNN models.

We propose an  adaptable architecture for LSDNN where the parameters of
various LSDNN-trained models are pre-computed and stored in memory. Here, it
is assumed that the DNN architecture is fixed for various channel conditions.
Thus, the same hardware architecture can be used across multiple testing
environments.  In Table V,we consider two architectures based on the type of
memory in the SoC. In the first architecture, external DDR memory stores the
model parameters, and the parameters of the selected LSDNN model are
transferred to the PL via DMA. Such architecture suffers from high execution
time since the parameters are read from external DDR memory. However, on-field
adaptability is possible since the parameters corresponding to new channel
conditions can be added in DDR using PS without needing hardware
reconfiguration. The low cost of the DDR allows the storage of a large number
of DNN parameters. In the second approach, model parameters are stored in the
BRAM of the PL, resulting in lower execution time. Since the BRAM is dual-port
memory, we can perform some of the operations in parallel, thereby
significantly reducing the execution time at the cost of a slight increase in
resource utilization. Due to limited BRAM, the second approach limits the
number of supported DNN models and hence, different channel conditions. The
support for new channel conditions demands updation of BRAM content which
requires FPGA configuration either completely or partially. Complete
reconfiguration is needed when DNN architecture is different and partial
reconfiguration of BRAM is needed when DNN architecture is the same, but
parameters are different. Note that FPGA reconfiguration can still be done
remotely without requiring product recall. As part of future works, the second
architecture can be enhanced to optimize BRAM utilization via dynamic partial
reconfiguration, thereby making the BRAM utilization independent of the number
of models. Such reconfigurable architecture also needs intelligence to detect
the change in the environment and reconfigure the hardware. The design of
intelligent and reconfigurable channel estimation architecture is an important
research direction.

TABLE V: Comparison of various  adaptable LSDNN architectures | External DDR | On-chip BRAM
---|---|---
Memory | 4 models | 8 models
Execution time (ms) | 0.257 | 0.0264 | 0.029
LUT | 65926 | 68424 | 70634
FF | 70300 | 63802 | 63814
BRAM | 92 | 221.5 | 253.5
DSP | 300 | 324 | 324
PL power (W) | 0.854 | 1.099 | 1.121
SoC power (W) | 2.423 | 2.668 | 2.69

##  VII ASIC Implementation

In addition to FPGA, ASIC is another popular hardware platform for wireless
PHY. In this section, we assess the PPA delivered by various architectures in
the ASIC implementation using 45 nm CMOS technology. First, we synthesize the
Verilog code used for FPGA implementation, using suitable constraints, to
obtain the gate level netlist. We verify the functional correctness of the
design using a combinational equivalence checker and ensure its temporal
safety using static timing analysis. The validated gate-level netlist is
converted into the final Graphic Data System (GDS) using the physical design
steps such as floor-planning, placement, clock-tree synthesis, and routing.
Finally, we carry out various sign-off checks on the ASIC implementation using
physical verification and timing analysis tools. We report the PPA of the
implementations with various channel estimation architectures in Fig. 16. To
obtain the maximum operable frequency for a given architecture, we have
carried out an ASIC implementation flow with multiple timing constraints. We
gradually increased the target frequency until no more improvement was
possible.

We found that for 32-bit implementation, LS achieves the operating frequency
of 350 MHz, while the LSDNN achieves the operating frequency of 200 MHz, and
the LSDNN implementation consumes 4.8$\times$ more area and 2.5$\times$ higher
power compared to the LS implementation at their corresponding frequencies of
operation. These results are expected because LSDNN architecture performs more
computation than the corresponding LS implementation.

Figure 16: Comparative analysis of PPA in ASIC implementations of channel
estimation techniques (Vdd = 1V, technology = 45 nm)

Further, we study the effect of word-length optimizations on the ASIC
implementations of different channel estimation techniques. We found that upon
word-length optimization of LS from 32 bits to 18 bits, the 18-bits LS
achieves 1.5$\times$ higher operating frequency compared to 32-bits LS (from
350 MHz. to 500 MHz.). Moreover, the 18-bits LS takes 3.3$\times$ less area
and consumes 1.8$\times$ less power than the 32-bits LS at their peak
operating frequencies. Likewise, upon word-length optimization of LSDNN from
32 bits to 24 bits, the 24-bits LSDNN achieves 1.4$\times$ higher operating
frequency compared to 32-bits LSDNN (from 200 MHz to 285.7 MHz). Moreover, the
24-bits LSDNN takes 1.8$\times$ less area and 1.2$\times$ less power compared
to the 32-bits LSDNN at their peak operating frequencies. Thus, the word-
length optimization delivers PPA improvements in ASIC implementations also,
both for the LS and LSDNN architectures.

##  VIII Conclusions and Future Directions

In this work, we proposed a deep neural network (DNN) augmented least-square
(LSDNN) based channel estimation for wireless physical layer (PHY). We have
compared the performance, resource utilization, power consumption, and
execution time of LSDNN with conventional LS and linear minimum mean square
estimation (LMMSE) approach on Zynq multiprocessor system-on-chip (ZMPSoC) and
application-specific integrated circuits (ASIC) platforms. Numerous
experiments validate the superiority of the LSDNN over LS and LMMSE. The AXI-
compatible hardware IPs and PYNQ-based graphical user interface (GUI)
demonstrating the real-time performance comparison on the ZMPSoC are a
physical deliverable of this work.

Future works include an extension of the proposed approach for cellular OFDM
PHY with comb-based pilots instead of the preamble in IEEE 802.11. The
proposed approach can offer a significant reduction in complexity for
millimeter-wave massive MIMO networks. The demonstration of the proposed
approach in a real radio environment is critical for commercial deployment.

## Acknowledgement

We thank Mr. Mohammed Sajjad Jafri (Research Intern, IIIT Delhi) for his
assistance with developing GUI to demonstrate the proposed work on ZMPSoC.

## References

  * [1] S. Zeadally, M. A. Javed, and E. B. Hamida, “Vehicular communications for its: Standardization and challenges,” IEEE Communications Standards Magazine, vol. 4, no. 1, pp. 11–17, 2020.
  * [2] S. Mumtaz, V. G. Menon, and M. I. Ashraf, “Guest editorial: Ultra-low-latency and reliable communications for 6g networks,” IEEE Communications Standards Magazine, vol. 5, no. 2, pp. 10–11, 2021.
  * [3] Y. Liu, Z. Tan, H. Hu, L. J. Cimini, and G. Y. Li, “Channel estimation for ofdm,” IEEE Communications Surveys & Tutorials, vol. 16, no. 4, pp. 1891–1908, 2014.
  * [4] K. K. Nagalapur, F. Brännström, E. G. Ström, F. Undi, and K. Mahler, “An 802.11p cross-layered pilot scheme for time- and frequency-varying channels and its hardware implementation,” IEEE Transactions on Vehicular Technology, vol. 65, no. 6, pp. 3917–3928, 2016.
  * [5] Z. Yuanjin, “A novel channel estimation and tracking method for wireless ofdm systems based on pilots and kalman filtering,” in 2003 IEEE International Symposium on Circuits and Systems (ISCAS), vol. 2, pp. II–II, 2003\.
  * [6] F. Restuccia and T. Melodia, “Deep learning at the physical layer: System challenges and applications to 5g and beyond,” IEEE Communications Magazine, vol. 58, no. 10, pp. 58–64, 2020.
  * [7] J. Liu, J. Chen, S. Luo, S. Li, and S. Fu, “Deep learning driven non-orthogonal precoding for millimeter wave communications,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 10, no. 2, pp. 164–176, 2020.
  * [8] Y. Yang, F. Gao, X. Ma, and S. Zhang, “Deep learning-based channel estimation for doubly selective fading channels,” IEEE Access, vol. 7, pp. 36579–36589, 2019.
  * [9] X. Wei, C. Hu, and L. Dai, “Deep learning for beamspace channel estimation in millimeter-wave massive mimo systems,” IEEE Transactions on Communications, vol. 69, no. 1, pp. 182–193, 2020.
  * [10] X. Ma and Z. Gao, “Data-driven deep learning to design pilot and channel estimator for massive mimo,” IEEE Transactions on Vehicular Technology, vol. 69, no. 5, pp. 5677–5682, 2020.
  * [11] A. K. Gizzini, M. Chafii, A. Nimr, and G. Fettweis, “Enhancing least square channel estimation using deep learning,” in 2020 IEEE 91st Vehicular Technology Conference (VTC2020-Spring), pp. 1–5, IEEE, 2020.
  * [12] A. K. Gizzini, M. Chafii, A. Nimr, and G. Fettweis, “Deep learning based channel estimation schemes for ieee 802.11 p standard,” IEEE Access, vol. 8, pp. 113751–113765, 2020.
  * [13] J. Pan, H. Shan, R. Li, Y. Wu, W. Wu, and T. Q. Quek, “Channel estimation based on deep learning in vehicle-to-everything environments,” IEEE Communications Letters, vol. 25, no. 6, pp. 1891–1895, 2021.
  * [14] C.-F. Teng and A.-Y. Wu, “A 7.8–13.6 pj/b ultra-low latency and reconfigurable neural network-assisted polar decoder with multi-code length support,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 68, no. 5, pp. 1956–1965, 2021.
  * [15] R. Rajesh, S. J. Darak, A. Jain, S. Chandhok, and A. Sharma, “Hardware–software co-design of statistical and deep-learning frameworks for wideband sensing on zynq system on chip,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 31, no. 1, pp. 79–89, 2023.
  * [16] D. Gündüz, P. de Kerret, N. D. Sidiropoulos, D. Gesbert, C. R. Murthy, and M. van der Schaar, “Machine learning in the air,” IEEE Journal on Selected Areas in Communications, vol. 37, no. 10, pp. 2184–2199, 2019.
  * [17] T. Wang, C.-K. Wen, H. Wang, F. Gao, T. Jiang, and S. Jin, “Deep learning for wireless physical layer: Opportunities and challenges,” China Communications, vol. 14, no. 11, pp. 92–111, 2017.
  * [18] S. Zhang, J. Liu, T. K. Rodrigues, and N. Kato, “Deep learning techniques for advancing 6g communications in the physical layer,” IEEE Wireless Communications, vol. 28, no. 5, pp. 141–147, 2021.
  * [19] K. Mei, J. Liu, X. Zhang, N. Rajatheva, and J. Wei, “Performance analysis on machine learning-based channel estimation,” IEEE Transactions on Communications, vol. 69, no. 8, pp. 5183–5193, 2021.
  * [20] M. Soltani, V. Pourahmadi, A. Mirzaei, and H. Sheikhzadeh, “Deep learning-based channel estimation,” IEEE Communications Letters, vol. 23, no. 4, pp. 652–655, 2019.
  * [21] L. Li, H. Chen, H.-H. Chang, and L. Liu, “Deep residual learning meets ofdm channel estimation,” IEEE Wireless Communications Letters, vol. 9, no. 5, pp. 615–618, 2020.
  * [22] S. A. U. Haq, “Demonstration video. Source codes and datasets will be shared after paper acceptance..” Link, 2022\.  [Online; accessed 5-January-2023].
  * [23] C. E. Shannon, “A mathematical theory of communication,” The Bell system technical journal, vol. 27, no. 3, pp. 379–423, 1948.
  * [24] S. M. Kay, “Fundamentals of statistical signal processing. detection theory, volume ii,” Printice Hall PTR, vol. 1545, 1998.
  * [25] M. K. Karray and M. Jovanovic, “A queueing theoretic approach to the dimensioning of wireless cellular networks serving variable-bit-rate calls,” IEEE Transactions on Vehicular Technology, vol. 62, no. 6, pp. 2713–2723, 2013.
  * [26] N. Singh, S. S. Santosh, and S. J. Darak, “Toward intelligent reconfigurable wireless physical layer (phy),” IEEE Open Journal of Circuits and Systems, vol. 2, pp. 226–240, 2021.
  * [27] S. J. Darak and M. K. Hanawal, “Multi-player multi-armed bandits for stable allocation in heterogeneous ad-hoc networks,” IEEE Journal on Selected Areas in Communications, vol. 37, no. 10, pp. 2350–2363, 2019.
  * [28] X. Liu, J. Sha, H. Xie, F. Gao, S. Jin, Z. Zhang, X. You, and C. Zhang, “Efficient channel estimator with angle-division multiple access,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 66, no. 2, pp. 708–718, 2019.
  * [29] O. Castañeda, T. Goldstein, and C. Studer, “Vlsi designs for joint channel estimation and data detection in large simo wireless systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 3, pp. 1120–1132, 2018.
  * [30] R. Zhou, F. Liu, and C. W. Gravelle, “Deep learning for modulation recognition: A survey with a demonstration,” IEEE Access, vol. 8, pp. 67366–67376, 2020.
  * [31] S. Zheng, S. Chen, and X. Yang, “Deepreceiver: A deep learning-based intelligent receiver for wireless communications in the physical layer,” IEEE Transactions on Cognitive Communications and Networking, vol. 7, no. 1, pp. 5–20, 2021.
  * [32] S. Chandhok, H. Joshi, A. V. Subramanyam, and S. J. Darak, “Novel deep learning framework for wideband spectrum characterization at sub-nyquist rate,” Wireless Networks (Springer), vol. 27, pp. 4727–4746, 2021.
  * [33] T. Wang, C.-K. Wen, S. Jin, and G. Y. Li, “Deep learning-based csi feedback approach for time-varying massive mimo channels,” IEEE Wireless Communications Letters, vol. 8, no. 2, pp. 416–419, 2018.
  * [34] P. Zhou, X. Fang, X. Wang, Y. Long, R. He, and X. Han, “Deep learning-based beam management and interference coordination in dense mmwave networks,” IEEE Transactions on Vehicular Technology, vol. 68, no. 1, pp. 592–603, 2019.
  * [35] R. C. Daniels, C. M. Caramanis, and R. W. Heath, “Adaptation in convolutionally coded mimo-ofdm wireless systems through supervised learning and snr ordering,” IEEE Transactions on Vehicular Technology, vol. 59, no. 1, pp. 114–126, 2010.
  * [36] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weighted mmse approach to distributed sum-utility maximization for a mimo interfering broadcast channel,” IEEE Transactions on Signal Processing, vol. 59, no. 9, pp. 4331–4340, 2011.
  * [37] X. Zhang, H. Zhao, J. Xiong, X. Liu, L. Zhou, and J. Wei, “Scalable power control/beamforming in heterogeneous wireless networks with graph neural networks,” in 2021 IEEE Global Communications Conference (GLOBECOM), pp. 01–06, IEEE, 2021.
  * [38] T. O’Shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Transactions on Cognitive Communications and Networking, vol. 3, no. 4, pp. 563–575, 2017.
  * [39] S. Dörner, S. Cammerer, J. Hoydis, and S. t. Brink, “Deep learning based communication over the air,” IEEE Journal of Selected Topics in Signal Processing, vol. 12, no. 1, pp. 132–143, 2018.
  * [40] H. Ye, G. Y. Li, and B.-H. Juang, “Power of deep learning for channel estimation and signal detection in ofdm systems,” IEEE Wireless Communications Letters, vol. 7, no. 1, pp. 114–117, 2018.
  * [41] D. Luan and J. Thompson, “Low complexity channel estimation with neural network solutions,” in WSA 2021; 25th International ITG Workshop on Smart Antennas, pp. 1–6, 2021.
  * [42] Y. Zhang, A. Doshi, R. Liston, W.-T. Tan, X. Zhu, J. G. Andrews, and R. W. Heath, “Deepwiphy: Deep learning-based receiver design and dataset for ieee 802.11ax systems,” IEEE Transactions on Wireless Communications, vol. 20, no. 3, pp. 1596–1611, 2021.
  * [43] S. J. Pan and Q. Yang, “A survey on transfer learning,” IEEE Transactions on Knowledge and Data Engineering, vol. 22, no. 10, pp. 1345–1359, 2010.
  * [44] P. K. Chundi, X. Wang, and M. Seok, “Channel estimation using deep learning on an fpga for 5g millimeter-wave communication systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 69, no. 2, pp. 908–918, 2022.
  * [45] F. Restuccia and T. Melodia, “Deep learning at the physical layer: System challenges and applications to 5g and beyond,” IEEE Communications Magazine, vol. 58, no. 10, pp. 58–64, 2020.
  * [46] P. Sahoo, R. Rajoria, S. Chandhok, S. J. Darak, D. P. Pau, and H. Dabral, “Resource constrained neural networks for direction-of-arrival estimation in micro-controllers,” in The First International Conference on AI-ML-Systems, AIMLSystems 2021, (New York, NY, USA), Association for Computing Machinery, 2021.
  * [47] S. Brennsteiner, T. Arslan, J. Thompson, and A. McCormick, “A real-time deep learning ofdm receiver,” ACM Trans. Reconfigurable Technol. Syst., vol. 15, dec 2022.
  * [48] S. V. S. Santosh and S. J. Darak, “Multiarmed bandit algorithms on zynq system-on-chip: Go frequentist or bayesian?,” IEEE Transactions on Neural Networks and Learning Systems, pp. 1–14, 2022.
  * [49] V. Sze, Y.-H. Chen, T.-J. Yang, and J. S. Emer, “Efficient processing of deep neural networks: A tutorial and survey,” Proceedings of the IEEE, vol. 105, no. 12, pp. 2295–2329, 2017.
  * [50] S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv preprint arXiv:1609.04747, 2016.
  * [51] G. Acosta-Marum and M. A. Ingram, “Six time- and frequency- selective empirical channel models for vehicular wireless lans,” IEEE Vehicular Technology Magazine, vol. 2, no. 4, pp. 4–11, 2007.