# Statistical characterization of the chordal product determinant of
Grassmannian codes

Javier Álvarez-Vizoso, Carlos Beltrán, Diego Cuevas, Ignacio Santamaría,
Vít Tuček and Gunnar Peters

###### Abstract

We consider the chordal product determinant, a measure of the distance between
two subspaces of the same dimension. In information theory, collections of
elements in the complex Grassmannian are searched with the property that their
pairwise chordal products are as large as possible. We characterize this
function from an statistical perspective, which allows us to obtain bounds for
the minimal chordal product and related energy of such collections.

##  1 Introduction and statement of the main results

Let $T\geq 2M$ be two positive integers and consider the complex Grassmannian
$\mathbb{G}r(M,{\mathbb{C}}^{T})$, i.e. the space of $M$–dimensional complex
vector subspaces of $\mathbb{C}^{T}$. Finite collections of points (also
called codes or packings) in $\mathbb{G}r(M,{\mathbb{C}}^{T})$ with different
desired separation properties have been investigated by several authors (in
Section 1.3 we describe some relevant references). The most frequent criterium
for “well–separated” codes is the maximization of the minimal mutual squared
chordal distance, which is the sum the squared sines of the principal angles
of two subspaces. However, following [HM00, MBV02, CAVB+22] (see also Section
1.1 below), a more relevant measure for its application to information theory
is given by the chordal product energy, related to the product of the squared
sines of the principal angles, which justifies its name. Given a code $[{\bf
X}_{1}],\ldots,[{\bf X}_{k}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$, its chordal
product energy with parameter $N$ is

$\mathcal{E}({\bf X}_{1},\ldots,{\bf X}_{K})=\sum_{i\neq
j}\operatorname{det}({\bf I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf X}_{j}^{H}{\bf
X}_{i})^{-N},$ (1)

where ${\bf I}_{M}$ is the identity matrix and we have chosen representatives
${\bf X}_{i}$ of each point $[{\bf X}_{i}]$ satisfying ${\bf X}_{i}^{H}{\bf
X}_{i}={\bf I}_{M}$. Note that the energy is well defined in the sense that it
does not change if other representatives with that property are chosen. Recall
that the SVD of ${\bf X}_{i}^{H}{\bf X}_{j}$, e.g. ${\bf U}{\bf D}{\bf
V}^{H}$, is given in terms of the cosines of the principal angles between the
subspaces $[{\bf X}_{i}]$ and $[{\bf X}_{j}]$,
$\cos\theta_{1},\ldots,\cos\theta_{M}$, cf. [HR06], so

$\operatorname{det}\left({\bf I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf
X}_{j}^{H}{\bf X}_{i}\right)=\operatorname{det}\left({\bf I}_{M}-{\bf
D}^{2}\right)=\prod_{i=1}^{M}\sin^{2}\theta_{i},$ (2)

while the squared chordal distance between the two subspaces $[{\bf X}_{i}]$
and $[{\bf X}_{j}]$ is given by $\sum_{i=1}^{M}\sin^{2}\theta_{i}$. The sum in
(1) is a pairwise interaction energy in the spirit of the well–studied Riesz
or logarithmic energies of importance in Potential Theory (see [BHS19] for a
complete monograph dedicated to energy minimization in the sphere and other
spaces). We refer to the function (again, choosing representatives ${\bf A}$
and ${\bf B}$ such that ${\bf A}^{H}{\bf A}={\bf B}^{H}{\bf B}={\bf I}_{M}$)

$[{\bf A}],[{\bf
B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})\mapsto\operatorname{det}({\bf
I}_{M}-{\bf A}^{H}{\bf B}{\bf B}^{H}{\bf A})=\operatorname{det}({\bf
I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B}),$ (3)

as the chordal product determinant or, simply, the chordal product, and note
that is not a metric in $\mathbb{G}r(M,{\mathbb{C}}^{T})$, for it may happen
that $[{\bf A}]\neq[{\bf B}]$ and yet $\operatorname{det}({\bf I}_{M}-{\bf
A}^{H}{\bf B}{\bf B}^{H}{\bf A})=0$, if the intersection of $[{\bf A}]$ and
$[{\bf B}]$ is nontrivial.

In this paper we perform the first theoretical study of the chordal product
energy, for numerical results, see [CAVB+22] and references therein. We start
describing the context where the problem arises, following [HM00].

###  1.1 The importance of Grassmannian codes in information theory

Consider a transmitter, i.e. some device that is able to send a signal, which
is indeed a collection of numbers ordered in a complex $T\times M$ matrix
${\bf X}$. Physically, this corresponds to the setting where the transmitter
has $M$ antennas and there is a total amount of $T$ time slots where the
communication channel is assumed to be constant (i.e. the contour conditions
of the communication are considered constant during the time that these $TM$
numbers are sent). The receiver is another device, that we consider equipped
with $N$ antennas, and the signal it receives is

$\mathbf{Y}=\mathbf{X}\mathbf{H}+\sqrt{\frac{M}{T\rho}}\mathbf{W},$

where $\mathbf{H}$ is an unknown $M\times N$ matrix (termed the channel),
$\mathbf{W}$ describes the noise and $\rho$, called the signal-to-noise-ratio
(SNR), measures the magnitude of the signal against the noise.

####  1.1.1 The zero–noise case

Since $\mathbf{H}$ is unknown (it is common to assume that it has random
complex Gaussian entries), even in the event that $\mathbf{W}=0$ the receiver
cannot recover the whole matrix ${\bf X}$:

  * •

If two matrices ${\bf X}_{1}$ and ${\bf X}_{2}$ have the same column span,
then one can easily find an full–rank matrix $\mathbf{H}$ such that
$\mathbf{X}_{1}\mathbf{H}=\mathbf{X}_{2}\mathbf{H}$, hence the receiver just
cannot distinguish which of these two matrices was the original signal.

  * •

On the other hand, if two matrices ${\bf X}_{1}$ and ${\bf X}_{2}$ have the
property that the intersection of the column span of ${\bf X}_{1}$ and ${\bf
X}_{2}$ is trivial, the the receiver can easily distinguish if a given matrix
${\bf Y}$ has been constructed by ${\bf X}_{1}\mathbf{H}$ or by ${\bf
X}_{2}\mathbf{H}$: if the column span of ${\bf Y}$ intersected with the column
span of ${\bf X}_{1}$ (resp. ${\bf X}_{2}$) is nontrivial, then ${\bf X}_{1}$
(resp. ${\bf X}_{2}$) was sent.

Summarizing, if a previously agreed code of possible signals $[{\bf
X}_{1}],\ldots,[{\bf X}_{K}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed with
the property that the column spans of ${\bf X}_{i}$ and ${\bf X}_{j}$ have
trivial intersection for $i\neq j$, the receiver will be able to recover, at
least in the zero–noise scenario, the element of the Grassmannian represented
by the sent signal, but not the concrete representative of that element.
Hence, collections of points in $\mathbb{G}r(M,{\mathbb{C}}^{T})$ are searched
with that property.

####  1.1.2 The general case

In the more realistic context of the presence of non–zero noise, the analysis
is quite more involved since there is always a non–zero probability of error
in the detection procedure. The pioneer work [HM00] showed that, in order to
recover the element ${\bf X}_{i}$ of $\mathbb{G}r(M,{\mathbb{C}}^{T})$ just by
knowing ${\bf Y}$, the optimal method is to use the so called
maximum–likelihood decoder:

$i=\operatorname{argmax}_{j=1,\ldots,K}\operatorname{tr}{({\bf Y}^{H}{\bf
X}_{j}{\bf X}_{j}^{H}{\bf
Y})}=\operatorname{argmax}_{j=1,\ldots,K}\operatorname{tr}{({\bf
X}_{j}^{H}{\bf Y}{\bf Y}^{H}{\bf X}_{j})},$

where $\cdot^{H}$ holds for Hermitian conjugate. Then, [BV01] showed that if
only $2$ codewords are permitted, i.e. if $K=2$, and assuming that the entries
of $\mathbf{H}$ and $\mathbf{W}$ are complex Gaussian $\mathcal{N}(0,1)$
numbers, then the probability $P_{e}({\bf X}_{1},{\bf X}_{2},\rho)$ of
erroneously decoding ${\bf X}_{1}$ if ${\bf X}_{2}$ was sent can be given by a
(quite complicated) formula involving the residues of a certain rational
function. Luckily, the asymptotic expansion of this Pairwise Error Probability
(PEP) in the case $\rho\to\infty$, called the high-SNR asymptotic analysis,
admits a much more concise expression, see [MBV02, CAVB+22]:

$P_{e}({\bf X}_{1},{\bf X}_{2},\rho)\approx C\rho^{-NM}\operatorname{det}({\bf
I}_{M}-{\bf X}_{1}^{H}{\bf X}_{2}{\bf X}_{2}^{H}{\bf
X}_{1})^{-N},\quad\rho\to\infty,$ (4)

where $C=\frac{1}{2}\left(\frac{4M}{T}\right)^{NM}\frac{(2NM-1)!!}{(2NM)!!)}$,
it is assumed that any two distinct points have trivial intersection as linear
subspaces, and the representatives ${\bf X}_{i}$ of each $[{\bf X}_{i}]$ are
such that ${\bf X}_{i}^{H}{\bf X}_{i}={\bf I}_{M}$. If we have $K$ elements
$[{\bf X}_{1}],\ldots,[{\bf X}_{K}]$ in the code of possible signals and we
assume that we send one of them at random, all with equal probability $1/K$,
then the total probability of erroneously decoding a signal is bounded above
by

$\frac{1}{K}\sum_{i\neq j}P_{e}({\bf X}_{i},{\bf
X}_{j},\rho)\approx\frac{C}{K}\rho^{-NM}\sum_{i\neq j}\operatorname{det}({\bf
I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf X}_{j}^{H}{\bf X}_{i})^{-N}.$ (5)

The determinant in (4) is the chordal product (3) and the sum in the
right–hand side in (5) is the energy (1).

####  1.1.3 Criteria for the design of Grassmannian codes

It follows from the previous discussion that reasonable criteria for the
design of a code $[{\bf X}_{1}],\ldots,[{\bf X}_{K}]$ would be to maximize the
pairwise chordal product (3), or to minimize the chordal product energy (1).
In [CAVB+22] these approaches are considered, numerically showing that the
obtained codes are very well suited for their use in non–coherent
communications, with a slight advantage in the use of the chordal product
energy. Yet, little or no theory exists about the behavior of the optimal
pairwise chordal product or energy. The main purpose of this paper is to put
the basis for the study of this question.

###  1.2 Main results of the paper

We will start our study by computing the moments of the chordal product when
$[{\bf B}]$ is fixed and $[{\bf A}]$ is chosen at random uniformly in
$\mathbb{G}r(M,{\mathbb{C}}^{T})$, w.r.t. the unique, standard
rotation–invariant probability measure. This yields a complete statistical
characterization of the chordal product as a product of beta–distributed
random variables:

######  Theorem 1.

Assume that $T\geq 2M$. Let $p\in(2M-T-1,\infty)$ (notice that $p$ may be
negative and/or noninteger). Let $[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$
be any fixed element and let $[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ be
uniformly distributed on the Grassmannian. Then, the $p$–th moment of
$\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$ is:

${\rm E}_{{\bf A}}[\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf
A}^{H}{\bf
B})^{p}]=\prod_{m=1}^{M}\frac{\Gamma(T-m+1)\Gamma(T+p-m-M+1)}{\Gamma(T-m-M+1)\Gamma(T+p-m+1)},$
(6)

where $\Gamma(\cdot)$ is Euler’s Gamma function. Moreover,
$\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$ is
distributed as the product of $M$ independent beta random variables, $z_{m}$,
with parameters $\alpha_{m}=T-M+1-m$ and $\beta_{m}=M$, $m=1,\ldots,M$, i.e.

$\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf
B})\sim\prod_{m=1}^{M}z_{m},\quad z_{m}\sim{\rm Beta}(T-M+1-m,M).$ (7)

An immediate consequence is that, at least for moderate values of $N$, we can
upper bound the energy (1) and hence the probability of error (5) of random
codes $[{\bf X}_{1}],\ldots,[{\bf X}_{K}]$ when they are all independently and
uniformly distributed:

######  Corollary 1.

Assume that $N\leq T-2M$. For i.i.d. chosen $[{\bf X}_{1}],\ldots,[{\bf
X}_{K}]$, the expected value of the chordal product energy (1) is

$K(K-1)\prod_{m=1}^{M}\frac{(T-m)!(T-N-m-M)!}{(T-m-M)!(T-N-m)!},$

In particular, there exists a code such that the union bound (5) is at most:

$C(K-1)\rho^{-NM}\prod_{m=1}^{M}\frac{(T-m)!(T-N-m-M)!}{(T-m-M)!(T-N-m)!},$

where $C=\frac{1}{2}\left(\frac{4M}{T}\right)^{NM}\frac{(2NM-1)!!}{(2NM)!!)}$.

In Section 3 we use Theorem 1 to compute exactly the probability density
function and the cumulative density function of the random variable
$\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$ in the
same hypotheses of the theorem. The expressions we get are exact and can be
obtained in closed form for any fixed value of $M$. A reduced version of that
result for $M=2$ is now shown:

######  Corollary 2.

Fix any $[{\bf B}]\in\mathbb{G}r(2,\mathbb{C}^{T})$ with $T\geq 4$. The
probability that a randomly chosen $[{\bf A}]\in\mathbb{G}r(2,\mathbb{C}^{T})$
satisfies $\operatorname{det}({\bf I}_{2}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf
B})\leq\delta\in(0,1]$ is exactly:

$F_{2}(\delta,T)=\frac{1}{2}(T-1)(T-2)^{2}(T-3)\delta^{T-3}\left(\frac{1}{T-3}-\frac{2\delta}{(T-2)^{2}}-\frac{\delta^{2}}{T-1}+\frac{2\delta\log\delta}{T-2}\right),$

and similar formulas can be computed for the probability density functions
$F_{M}(\delta,T)$ for higher values of $M$ (the case $M=1$ which yields
$F_{1}(\delta,T)=\delta^{T-1}$ is quite trivial but it also follows from our
approach).

See the complete result in Corollary 4. As an illustrative example, Figs. 1
and 2 depict, respectively, the computed pdf and cdf of the chordal product
for different values of $T$ and $M$.

Figure 1: Probability density functions of $\operatorname{det}({\bf
I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$, when $[{\bf
B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed and $[{\bf
A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is uniformly distributed on the
Grassmannian. Figure 2: Cumulative distribution function of
$\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$, when
$[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed and $[{\bf
A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is uniformly distributed on the
Grassmannian.

Using the statistical characterization above, we have derived a lower bound on
the number of elements in any code in the Grassmannian with a given minimum
value of chordal product $\delta$. Following [BN02], we call this result a
Gilbert-Varshamov bound since its proof mimics the argument of that classical
result.

######  Corollary 3 (Gilbert–Varshamov lower bound).

Assume that $T\geq 2M$. For any fixed $K\geq 2$, there exists a code $[{\bf
X}_{1}],\ldots,[{\bf X}_{K}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ such that
$\operatorname{det}({\bf I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf X}_{j}^{H}{\bf
X}_{i})\geq\delta$ where $\delta$ is the unique solution of the equation:

$F_{M}(\delta;T)=\frac{1}{K};\text{ that is
}\delta=F_{M}^{-1}\left(\frac{1}{K};T\right)$

Equivalently, given $\delta\in(0,1)$, there exists a code consisting of
$K\geq\frac{1}{F_{M}(\delta;T)}$ elements and satisfying
$\operatorname{det}({\bf I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf X}_{j}^{H}{\bf
X}_{i})\geq\delta$ for $i\neq j$.

######  Example 1.

In the case $M=1$ we have that

$\operatorname{det}({\bf I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf X}_{j}^{H}{\bf
X}_{i})=\sin^{2}\theta,$

where $\theta\in[0,\pi/2]$ is the principal angle between the one-dimensional
subspaces $[{\bf X}_{i}]$ and $[{\bf X}_{j}]$ in
$\mathbb{G}(1,\mathbb{C}^{T})$. That is to say, the chordal product coincides
with the squared chordal distance. For uniformly distributed subspaces, the
squared sine of the pairwise principal angle has cdf
$F_{1}(\delta,T)=\delta^{T-1}$. The Gilbert-Varshamov bound shows that, for
$\delta\in(0,1)$, there exist codes with cardinality $K$ and minimum chordal
product $\delta=\sin^{2}\theta$ such that

$K>\delta^{-(T-1)}=\left(\sin\theta\right)^{-2(T-1)}.$ (8)

######  Example 2.

Let us now take $T=10,M=2$. Assume that we want to allocate $K=2^{9}$ points
in $\mathbb{G}r(M,{\mathbb{C}}^{T})$. Then, Corollary 3 says that there exists
a code $[{\bf X}_{1}],\ldots,[{\bf X}_{K}]$ such that for all these points the
chordal product is at least $\delta$, the unique solution of:

$\frac{1}{2}(T-1)(T-2)^{2}(T-3)\delta^{T-3}\left(\frac{1}{T-3}-\frac{2\delta}{(T-2)^{2}}-\frac{\delta^{2}}{T-1}+\frac{2\delta\log\delta}{T-2}\right)=\frac{1}{K},$

that is

$\log_{2}\left(2016\delta^{7}\left(\frac{1}{7}-\frac{2\delta}{64}-\frac{\delta^{2}}{9}+\frac{\delta\log\delta}{4}\right)\right)=-9,$

which yields $\delta\approx 0.2129$. The numerical algorithm in [CAVB+22]
produces in this case $[{\bf X}_{1}],\ldots,[{\bf X}_{2^{9}}]$ with minimum
determinantal value $0.3958>0.2129$.

###  1.3 Historical discussion

There exist several results on packings on Grassmannian spaces but they are
rather centered in finding codes such that the mutual chordal distance between
different elements $[{\bf X}_{i}],[{\bf X}_{j}]$ is close to maximal. For
example, in [Ran55] we find bounds for the mutual distance of any code with a
fixed number of elements (this is known as the Rankin bound).
Gilbert–Varshamov bounds have also been obtained for that chordal distance by
resorting to calculations of the volume of a metric ball of radius $\delta$ in
$\mathbb{G}r(M,{\mathbb{C}}^{T})$, see [BN02] and [DLR08]. The case that
$\delta$ is sufficiently small was analyzed in [DLR08], [Hen05], and the real
case has also been studied, see [CHS96] and references therein. But to our
knowledge our results are the first theoretical bounds on codes focusing on
the explicit use of the chordal product, which is the key figure of merit in
non–coherent communications.

In the case $M=1$ the Grassmannian becomes the projective space, the chordal
product equals the squared chordal distance, and the literature is much more
prolific, going back to [Sha59] (although Shannon studied the case of the
geodesic, not chordal, distance), [BN02] for the chordal and the geodesic
distance and more recently [JKM19] where a more complete set of references can
be found. The optimal value of (1) in that case has been studied in [BE18] and
[ADG+22] as a case of Riesz energy, showing that the minimum value is equal to
the average computed in Corollary 1, minus a term of the form

$O\left(K^{1+\frac{N}{T-1}}\right)=o(K^{2}),\quad\text{ for }N\leq T-2.$

##  2 Proof of Theorem 1

First assume that $p$ is an integer in the range of the hypotheses. Let
${\mathrm{E}}(p,T)$ be the expected value in the theorem (we omit the
dependence on $M$ in the notation). By unitary invariance, we can assume that
${\bf B}=\binom{{\bf I}_{M}}{\bf 0}$. If we write the expected value using
Proposition 1 and we pass to polar coordinates we get

$\displaystyle{\rm E}(p,T)=$ $\displaystyle C(T)\int_{\tilde{{\bf
A}}\in\mathbb{C}^{(T-M)\times M}}\frac{\operatorname{det}\left({\bf
I}_{M}-({\bf I}_{M}+\tilde{\bf A}^{H}\tilde{\bf
A})^{-1}\right)^{p}}{\operatorname{det}({\bf I}_{M}+\tilde{\bf
A}^{H}\tilde{\bf A})^{T}}\,d\tilde{{\bf A}}$
$\displaystyle=C(T)\int_{\tilde{{\bf A}}\in\mathbb{C}^{(T-M)\times
M}}\frac{\operatorname{det}\left({\bf A}^{H}\tilde{\bf
A}\right)^{p}}{\operatorname{det}({\bf I}_{M}+\tilde{\bf A}^{H}\tilde{\bf
A})^{T+p}}\,d\tilde{{\bf A}}$ $\displaystyle=$ $\displaystyle
C(T)\int_{0}^{\infty}\rho^{2M(T-M)+2pM-1}\int_{\underset{\|\tilde{{\bf
A}}\|_{F}=1}{\tilde{\bf A}\in\mathbb{C}^{(T-M)\times
M}}}\frac{\operatorname{det}(\tilde{\bf A}^{H}\tilde{\bf
A})^{p}}{\operatorname{det}({\bf I}_{M}+\rho^{2}\tilde{\bf A}^{H}\tilde{\bf
A})^{T+p}}\,d\tilde{{\bf A}}\,d\rho,$

where we omit the dependence on $M$ in the constant:

$C(T)=\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))}\stackrel{{\scriptstyle
Lemma\leavevmode\nobreak\
\ref{lem:volumenG}}}{{=}}\frac{(T-M)!\cdots(T-1)!}{\pi^{M(T-M)}1!\cdots(M-1)!}.$

Since the integrand of the inner integral depends only on the singular values
of $\tilde{\bf A}$, we can take it to the set $\mathbb{S}_{M}^{+}$ consisting
of ordered tuples of positive numbers $\sigma_{1}>\ldots>\sigma_{M}$ with the
property that $\sigma_{1}^{2}+\cdots+\sigma_{M}^{2}=1$, see for example
[Bel11, Th. 3.3], that yields

${\rm E}(p,T)=D(T)\times\\\
\int_{0}^{\infty}\rho^{2M(T-M)+2pM-1}\int_{\mathbb{S}_{M}^{+}}\frac{(\sigma_{1}\cdots\sigma_{M})^{2p+2T-4M+1}\prod_{j\neq
k}(\sigma_{k}^{2}-\sigma_{j}^{2})^{2}}{\prod_{m=1}^{M}(1+\rho^{2}\sigma_{m}^{2})^{T+p}}\,d\sigma_{1}\cdots
d\sigma_{M}\,d\rho,$

where

$\displaystyle D(T)=$
$\displaystyle\frac{C(T)Vol({\mathcal{U}}_{T-M})Vol({\mathcal{U}}_{M})}{Vol({\mathcal{U}}_{T-2M})2^{M(T-M)}\pi^{M}}.$

It follows immediately that ${\rm E}(p,T)/D(T)={\rm
E}(0,T+p)/D(T+p)=1/D(T+p)$, that is,

$\displaystyle{\rm E}(p,T)=$ $\displaystyle\frac{D(T)}{D(T+p)}$
$\displaystyle=$
$\displaystyle\frac{C(T)Vol({\mathcal{U}}_{T-M})Vol({\mathcal{U}}_{M})}{2^{M(T-M)}Vol({\mathcal{U}}_{T-2M})}\frac{2^{M(T+p-M)}Vol({\mathcal{U}}_{T+p-2M})}{C(T+p)Vol({\mathcal{U}}_{T+p-M})Vol({\mathcal{U}}_{M})}$
$\displaystyle=$
$\displaystyle\frac{(2\pi)^{Mp}(T-M)!\cdots(T-1)!Vol({\mathcal{U}}_{T-M})Vol({\mathcal{U}}_{T+p-2M})}{Vol({\mathcal{U}}_{T-2M})(T+p-M)!\cdots(T+p-1)!Vol({\mathcal{U}}_{T+p-M})}.$

The volume of the unitary group is known (see [Bel11, p. 28]):

$Vol({\mathcal{U}}_{k})=\frac{(2\pi)^{k(k+1)/2}}{1!\cdots(k-1)!}.$

The theorem (for integer $p$ in the range) follows by substituting the known
values in the constants above.

On the other hand, it is known that the $p$th moment of a beta distributed
random variable with parameters $\alpha>0$ and $\beta>0$ denoted as $x\sim{\rm
Beta}(\alpha,\beta)$ is [SK79]

${\rm
E}[x^{p}]=\frac{\Gamma(\alpha+\beta)\Gamma(\alpha+p)}{\Gamma(\alpha+\beta+p)\Gamma(\alpha)},$
(9)

so the $m$th product term in (6) corresponds to the $p$th moment of a beta
distributed random variable with parameters $\alpha_{m}=T-M+1-m$ and
$\beta_{m}=M$, thus proving that the distribution of $\operatorname{det}({\bf
I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$ is equivalent to the
distribution of the product of $M$ independent beta random variables (this is
an instance of the Hausdorff moments problem, hence the distribution is
uniquely determined by its moments). Notice that (9) is valid for
$p+\alpha>0$, and since $m$ can get up to $M$ this entails to $p>2M-T-1$. Now
that we have characterized $\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf
A}{\bf A}^{H}{\bf B})$ as a product of beta distributed random variables, we
can write down the formula for its moments for noninteger $p>2M-T-1$,
finishing the proof of the theorem.

##  3 Probability density function of the chordal product

Can we effectively recover the pdf of the random variable
$x=\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$ from
its moments? If the density function is $f(x)$ and the moments are
$\mathcal{M}_{n}$ then we have the classical formula:

$f(x)=\int_{-\infty}^{\infty}e^{2i\pi xs}\sum_{n=0}^{\infty}\frac{(-2i\pi
s)^{n}}{n!}\mathcal{M}_{n}\,ds.$ (10)

Following [ST70, Th. 7] a closed-form expression for the pdf can actually be
written down in terms of certain special functions called Meijer
$G$–functions. However, this expression is quite involved and requires extra
work in practice for the derivation of bounds. In the following, we show that
we can obtain simpler closed-form formulas for small values of $M=1,2,3$. They
represent the most practical use cases in noncoherent communications.
Moreover, we also provide a general recursive procedure to obtain the pdfs for
higher values:

######  Corollary 4.

Let $T\geq 2M$. The probability density function (pdf) of
$\operatorname{det}({\bf I}_{M}-{\bf B}^{H}{\bf A}{\bf A}^{H}{\bf B})$, when
$[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed and $[{\bf
A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is uniformly distributed on the
Grassmannian, for $M=1,2,3$ is:

$\displaystyle M=1\,\to\,f_{1}(x;T)=$ $\displaystyle(T-1)x^{T-2},$
$\displaystyle M=2\,\to\,f_{2}(x;T)=$
$\displaystyle\frac{1}{2}(T-1)(T-2)^{2}(T-3)x^{T-4}\left(1-x^{2}+2x\log
x\right),$ $\displaystyle M=3\,\to\,f_{3}(x;T)=$
$\displaystyle\frac{1}{288}(T-1)(T-2)^{2}(T-3)^{3}(T-4)^{2}(T-5)x^{T-6}\times$
$\displaystyle\left(1+80x-162x^{2}+80x^{3}+x^{4}+24x\log x-24x^{3}\log
x-36x^{2}\log^{2}x\right)$

The cumulative distribution function (cdf)
$F_{M}(x,T)=\int_{0}^{x}f_{M}(s,T)\,ds$ for these three cases is respectively:

$\displaystyle M=1\to$ $\displaystyle F_{1}(x;T)=x^{T-1}$ $\displaystyle
M=2\to$ $\displaystyle
F_{2}(x;T)=\frac{1}{2}(T-1)(T-2)^{2}(T-3)x^{T-3}\left(\frac{1}{T-3}-\frac{2x}{(T-2)^{2}}-\frac{x^{2}}{T-1}+\frac{2x\log
x}{T-2}\right)$ $\displaystyle M=3\to$ $\displaystyle
F_{3}(x;T)=\frac{1}{288}(T-1)(T-2)^{2}(T-3)^{3}(T-4)^{2}(T-5)x^{T-5}\times Q,$

with

$Q=\frac{1}{T-5}+\frac{80x}{T-4}-\frac{24x}{(T-4)^{2}}-\frac{162x^{2}}{T-3}-\frac{72x^{2}}{(T-3)^{3}}+\frac{80x^{3}}{T-2}\\\
+\frac{24x^{3}}{(T-2)^{2}}+\frac{x^{4}}{T-1}+\frac{24x\log
x}{T-4}+\frac{72x^{2}\log x}{(T-3)^{2}}-\frac{24x^{3}\log
x}{T-2}-\frac{36x^{2}\log^{2}x}{T-3}.$

For arbitrary higher values of $M$ the pdf has the form:

$\displaystyle
f_{M}(x;T)=(T-M)^{M}\prod_{m=1}^{M-1}(T-m)^{m}(T-M-m)^{M-m}\cdot$
$\displaystyle\left[\sum_{m=1}^{M}\sum_{l=1}^{m}\frac{A_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-1}\log^{l-1}x+\sum_{m=1}^{M-1}\sum_{l=1}^{M-m}\frac{B_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-M-1}\log^{l-1}x\right]$

where the $M^{2}$ coefficients $A_{ml},B_{ml}$ can be obtained (e.g. with the
aid of symbolic computation software) by solving the linear system of $M^{2}$
equations resulting from equating coefficients on both sides for the
polynomial identity:

$\displaystyle\sum_{m=1}^{M}\sum_{l=1}^{m}A_{ml}(x-m)^{m-l}\prod_{i\neq
m}^{M}(x-i)^{i}\prod_{i=1}^{M-1}(x-M-i)^{M-i}+$
$\displaystyle+\sum_{m=1}^{M-1}\sum_{l=1}^{M-m}B_{ml}(x-M-m)^{M-m-l}\prod_{i=1}^{M}(x-i)^{i}\prod_{i\neq
m}^{M-1}(x-M-i)^{M-i}=1.$

Proof.

For $M=1$ and integer $p\geq 0$ note that

$\int_{0}^{1}x^{p}\underbrace{(T-1)x^{T-2}}_{f_{1}(x;T)}\,dx=\frac{T-1}{T+p-1},$

and hence the claimed pdf satisfies Theorem 1 and must be the searched
distribution. With the help of some integral formulas for the $\log$ function
it is easy to check that

$\int_{0}^{1}x^{p}f_{2}(x;T)\,dx=\frac{(T-1)(T-2)^{2}(T-3)}{(T+p-1)(T+p-2)^{2}(T+p-3)},$

which again satisfies Theorem 1 and we are done. A more lengthy but trivial
computation gives the case $M=3$.

These formulas and the general case for higher values of $M$ can be derived
from the following procedure. The moments of the chordal product determinant
from Theorem 1 are

$\mathcal{M}_{p}(T,M)=\prod_{m=1}^{M}\frac{(T-m)!(T+p-m-M)!}{(T-m-M)!(T+p-m)!}.$

By expanding the factorials and collecting terms in the product, this can be
rewritten as

$\mathcal{M}_{p}(T,M)=\prod_{m=1}^{M}\left(\frac{T-m}{T+p-m}\right)^{m}\;\cdot\;\prod_{m=1}^{M-1}\left(\frac{T-M-m}{T+p-M-m}\right)^{M-m}$

so the denominator $D$ is the value at $x=T+p$ of the polynomial

$D(x)=\prod_{m=1}^{M}(x-m)^{m}\prod_{m=1}^{M-1}(x-M-m)^{M-m}.$

Notice that this is a product of all-different real root factors $(x-\alpha)$
with varying multiplicities, so its inverse has a partial fraction
decomposition

$\frac{1}{D}=\sum_{m=1}^{M}\sum_{l=1}^{m}\frac{A_{ml}}{(x-m)^{l}}+\sum_{m=1}^{M-1}\sum_{l=1}^{M-m}\frac{B_{ml}}{(x-M-m)^{l}},$

for some coefficients $A_{ml},\,B_{ml}\in\mathbb{R}$. Following one of the
usual procedures to solve for these coefficients, for general $x$, multiplying
by $D(x)$ on both sides yields the polynomial equation of order $M^{2}-1$:

$\displaystyle\sum_{m=1}^{M}\sum_{l=1}^{m}A_{ml}(x-m)^{m-l}\prod_{i\neq
m}^{M}(x-i)^{i}\prod_{i=1}^{M-1}(x-M-i)^{M-i}+$
$\displaystyle+\sum_{m=1}^{M-1}\sum_{l=1}^{M-m}B_{ml}(x-M-m)^{M-m-l}\prod_{i=1}^{M}(x-i)^{i}\prod_{i\neq
m}^{M-1}(x-M-i)^{M-i}=1.$

Expanding and gathering terms by powers of $x$, one can equate the coefficient
of $x^{0}$ to $1$ and the coefficients of $x^{n}$, for $n=1,\dots,M^{2}-1$, to
$0$ to obtain a linear system of $M^{2}$ equations in the $M^{2}$ coefficients
$A_{ml},\,B_{ml}$, and solve for them.

Now, notice that by the Laplace transform properties for $a,b$ nonnegative
integers

$\int_{0}^{1}x^{a}\log^{b}x\,dx=(-1)^{b}\mathcal{L}[t^{b}](a+1)=(-1)^{b}\frac{b!}{(a+1)^{b+1}},$

and thus

$\frac{1}{(T+p-m)^{l}}=\frac{(-1)^{l-1}}{(l-1)!}\int_{0}^{1}x^{p}x^{T-m-1}\log^{l-1}x\,dx.$

Hence, by expressing every term of the partial fraction decomposition in this
integral form, the function $f_{M}(x;T)$ can be identified inside the moment
function written as an integral:

$\displaystyle\mathcal{M}_{p}(T,M)=$
$\displaystyle(T-M)^{M}\prod_{m=1}^{M-1}(T-m)^{m}(T-M-m)^{M-m}\cdot$
$\displaystyle\cdot\int_{0}^{1}x^{p}\left[\sum_{m=1}^{M}\sum_{l=1}^{m}\frac{A_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-1}\log^{l-1}x+\right.$
$\displaystyle\left.\sum_{m=1}^{M-1}\sum_{l=1}^{M-m}\frac{B_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-M-1}\log^{l-1}x\right]dx,$

where the factors in front of the integral all come from the numerator over
$D$ in $\mathcal{M}_{h}(T,M)$. This finishes the proof of the corollary.

##  4 Proof of Corollary 3

Let $\delta$ be the unique solution of the equation $F_{M}(\delta;T)=K^{-1}$.
Let $G$ be the maximum number of points in $\mathbb{G}r(M,{\mathbb{C}}^{T})$
that can be allocated with the claimed property. We must prove that $G\geq K$.
Indeed, assume that $G<K$ an let ${\bf X}_{1},\ldots,{\bf X}_{G}$ be a code
with $\operatorname{det}({\bf I}_{M}-{\bf X}_{i}^{H}{\bf X}_{j}{\bf
X}_{j}^{H}{\bf X}_{i})\geq\delta$ for all $1\leq i,j\leq G$. We note that

$\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))}Vol\left(\cup_{i=1}^{G}\\{[{\bf
A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T}):\operatorname{det}({\bf I}_{M}-{\bf
X}_{i}^{H}{\bf A}{\bf A}^{H}{\bf X}_{i})\leq\delta\\}\right)\leq\\\
\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))}\sum_{i=1}^{G}Vol\left(\\{[{\bf
A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T}):\operatorname{det}({\bf I}_{M}-{\bf
X}_{i}^{H}{\bf A}{\bf A}^{H}{\bf X}_{i})\leq\delta\\}\right)=\\\
GF_{M}(\delta;T)=\frac{G}{K}<1,$

and we thus deduce that there exists $[{\bf
A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ such that

$\operatorname{det}({\bf I}_{M}-{\bf X}_{i}^{H}{\bf A}{\bf A}^{H}{\bf
X}_{i})>\delta\quad\forall\;1\leq i\leq G.$

But then the code ${\bf X}_{1},\ldots,{\bf X}_{G},{\bf X}_{G+1}$ with ${\bf
X}_{G+1}={\bf A}$ also satisfies the claimed property and has $G+1$ points,
which contradicts the definition of $G$.

##  Appendix A Alternative parameterization of the Grassmannian and the
density function of $\tilde{\bf A}$ in $\binom{{\bf I}_{M}}{\tilde{\bf
A}}\in\mathbb{G}r(M,{\mathbb{C}}^{T})$

We recall the volume of the Grassmannian for completeness.

######  Lemma 1.

The volume of the Grassmannian $\mathbb{G}r(M,{\mathbb{C}}^{T})$ is:

$Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))=\frac{\pi^{M(T-M)}1!\cdot
2!\cdots(M-1)!}{(T-M)!\cdot(T-M+1)!\cdots(T-1)!}$

Proof.  This is a classical fact: since the Grassmannian is formally defined
as a quotient of the Stiefel manifold $\mathbb{S}t(M,{\mathbb{C}}^{T})$ (i.e.
the set of $T\times M$ complex matrices ${\bf X}$ such that ${\bf X}^{H}{\bf
X}={\bf I}_{M}$) by the unitary group $\mathcal{U}_{M}$, the volume of
$\mathbb{G}r(M,{\mathbb{C}}^{T})$ is the quotient of the volumes of the
Stiefel and unitary matrices which is well–known, see for example [Hua63]
(note that there exist several normalizations for the Riemannian structure of
the classical groups, leading to different volume formulas. We use the
standard that considers $\mathbb{S}t(M,{\mathbb{C}}^{T})$ and
$\mathcal{U}_{M}$ as submanifolds of their ambient affine spaces, with the
inherited structure).

Recall that a $p\times n$ complex matrix ${\bf X}$ is distributed as a complex
matrix-variate $t$ distribution with $\nu$ degrees of freedom when its density
is given by

$p({\bf X})=C^{-1}\operatorname{det}({\bf I}_{p}+{\bf X}^{H}{\bf
X})^{-(\nu+p+n-1)}.$ (11)

######  Proposition 1.

Let $T\geq 2M$. If $[{\bf A}]$ is uniformly distributed in
$\mathbb{G}r(M,{\mathbb{C}}^{T})$ and we write $[{\bf A}]=\begin{bmatrix}{{\bf
I}_{M}}\\\ {\tilde{\bf A}}\end{bmatrix}$ (note that there exists a unique
representative of that form), then $\tilde{\bf A}$ has density

$\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))\operatorname{det}({\bf
I}_{M}+\tilde{\bf A}^{H}\tilde{\bf A})^{T}}.$ (12)

Hence, $\tilde{\bf A}\in\mathbb{C}^{(T-M)\times M}$ follows a matrix–variate
$t$ distribution with $\nu=1$ degrees of freedom. In other words, for any
measurable non–negative or integrable function
$f:\mathbb{G}r(M,{\mathbb{C}}^{T})\to\mathbb{C}$,

$\displaystyle\int_{[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})}f([{\bf
A}])\,d[{\bf A}]=$ $\displaystyle\int_{\tilde{{\bf
A}}\in\mathbb{C}^{(T-M)\times M}}\frac{f\binom{{\bf I}_{M}}{\tilde{\bf
A}}}{\operatorname{det}({\bf I}_{M}+\tilde{\bf A}^{H}\tilde{\bf
A})^{T}}\,d\tilde{{\bf A}}$ $\displaystyle=$ $\displaystyle\int_{\tilde{{\bf
A}}\in\mathbb{C}^{(T-M)\times M}}\frac{f\binom{({\bf I}_{M}+\tilde{\bf
A}^{H}\tilde{\bf A})^{-1/2}}{\tilde{\bf A}({\bf I}_{M}+\tilde{\bf
A}^{H}\tilde{\bf A})^{-1/2}}}{\operatorname{det}({\bf I}_{M}+\tilde{\bf
A}^{H}\tilde{\bf A})^{T}}\,d\tilde{{\bf A}}.$

Proof.  This result has been proved in [AVCB+, Prop. 1 and Cor. 1], by showing
that both sides of the equality are equal to

$\int_{{\bf W}\in\mathbb{C}^{(T-M)\times M},\|{\bf
W}\|_{op}<1}f\left(\begin{bmatrix}\sqrt{{\bf I}_{M}-{\bf W}^{H}{\bf W}}\\\
{\bf W}\end{bmatrix}\right)\,d{\bf W},$

with $\|\cdot\|_{op}$ the operator norm. Note that the two integrals on ${\bf
C}^{(T-M)\times M}$ are equal since $f$ is a function defined in the
Grassmannian and hence its value is independent of the choice of
representatives. Moreover, the advantage of the last expression in the
proposition is that

${\bf X}=\binom{({\bf I}_{M}+\tilde{\bf A}^{H}\tilde{\bf
A})^{-1/2}}{\tilde{\bf A}({\bf I}_{M}+\tilde{\bf A}^{H}\tilde{\bf A})^{-1/2}}$

is a Stiefel matrix, i. e. it satisfies ${\bf X}^{H}{\bf X}={\bf I}_{M}$.

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Department of Communications Engineering, Universidad de Cantabria, Santander,
Spain

E-mail addresses<EMAIL_ADDRESS>diego.cuevas<EMAIL_ADDRESS>
Department of Mathematics, Statistics and Computing, Universidad de Cantabria,
Santander, Spain

E-mail address<EMAIL_ADDRESS>
Department of Wireless Algorithms, Huawei Technologies, Kista, Sweden

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>