A universal identifier for communication channels

Abstract

Pseudo-differential operators, viewed as superpositions of time-frequency shifts, are natural models for communication channels. Channel identification is thus to find a proper input signal that induces an injective map on certain spaces of pseudo-differential operators. It is known that this is possible for channels with finite energy and spreading support area less than 1 (resp. 1/2) if the location and shape of the support area is known (resp. unknown) and the identifier depends on the spreading support. We will construct a universal input signal, which is independent of the spreading support, that identifies all such spaces of channels. The novelty of this result lies in the universality of this identifier.

Appendix: Rectification and discretization

This part contain exposition material of the method used in [9] to discretize the identification map on irregular compact underspread Jordan measurable sets. The decomposition formula in Proposition 1 below is only vaguely implied there but not explicitly stated. We adopted a different approach which uses the adjoint relation between the identification map and the short time Fourier transform, as a result, the derivation here should be simpler and clearer.

The Zak transform $Z_r$ of a function $f\in L^2(\mathbb R)$ is defined as

$$\begin{aligned} (Z_rf)(x,w)=\sum _{k\in \mathbb Z}f(x+kr)e^{-2\pi ikr\cdot w}, \end{aligned}$$

(6)

where the right hand side is defined a.e. and $r>0$ is a parameter.

The short time Fourier transform (STFT) $V_{\phi }$ with respect to a window $\phi$ on $\mathbb R^n$ can be written in several ways:

$$\begin{aligned} (V_{\phi }f)(t,v)=\langle f, M_{v}T_t\phi \rangle =e^{-2\pi it\cdot v}\int _{\mathbb R}f(x+t)\overline{\phi (x)}e^{-2\pi ix\cdot v}\;dx. \end{aligned}$$

(7)

The integral form is well defined for $f,\phi$ being $L^2(\mathbb R)$ functions, while the bracket form can be applied to any dual pairing.

Comparing the definition in (6) and (7), one can see that $g_r$ links STFT to Zak transforms (see also [6, Section 3.3]):

$$\begin{aligned} V_{g_r}f=e^{-2\pi it\cdot v}Z_rf. \end{aligned}$$

Moreover, for nice functions such as Schwartz class funcitons $f,g,\eta$, we have

$$\begin{aligned}&\iiint _{\mathbb R^3}\eta (t,v)\overline{f(x)\overline{(M_vT_tg_r)(x)}}\;dx\;dv\;dt\\&\quad =\iiint _{\mathbb R^3}\eta (t,v)(M_vT_tg_r)(x)\overline{f(x)}\;dv\;dt\;dx, \end{aligned}$$

which can be further written as

$$\begin{aligned} \langle \eta , V_{g}f\rangle =\langle \Phi _{g}\eta , f\rangle , \end{aligned}$$

(8)

under proper dual pairing. In particular [6, Theorem 4.1] (which is based on kernel theorems in [7, Chapter 14.4] and [36, Lemma 4.1]) shows that (8) holds for $g\in W^{A',\ell ^{\infty }}(\mathbb R)$, $f\in W^{A,\ell ^1}(\mathbb R)$ and $\eta \in L^2(U)$ where $U\subset \mathbb R^2$ is compact. See also [11, 37].

Now for $\eta \in L^2(U)$, we can define $\Phi _{g_{\vec c}}$ to be the element in $L^2(\mathbb R)$ that satisfies (8), then by the density in (2) we get that

$$\begin{aligned} \langle \Phi _{g_{\vec c}}\eta ,f\rangle =\langle \eta , V_{g_{\vec c}}f\rangle , \end{aligned}$$

(9)

holds for any $f\in L^2(\mathbb R)$ and $\eta \in L^2(U)$ with $U\subset \mathbb R^2$ compact.

The original case that Kailath considered becomes somewhat trivial under this perspective. Indeed, if the spreading function is supported in a rectangle with width r and height 1/r, then (9) already shows $\Phi _{\sqrt{r}g_r}$ is unitary from $L^2(U_r)$ to $L^2(\mathbb R)$, since its adjoint, $V_{\sqrt{r}g_r}$ restricted to $U_r$, is essentially the corresponding Zak transform, which is unitary onto such a rectangle.

Hence we now consider a bit more complicated case where the channel is still underspread but the spreading support can not be included in a rectangle of area 1. Let U be a compact underspread Jordan measurable set, we include it in a $\sqrt{n}\times \sqrt{n}$ square where $n\in \mathbb N$ is large enough, and view the time-frequency plane as a torus with this $\sqrt{n}\times \sqrt{n}$ square being its fundamental domain. Under this setting without loss of generality we may assume that U is in the first sector of the time-frequency plane and take the square to be $[0,\sqrt{n}]\times [0,\sqrt{n}]$. The discretization procedure, proposed in [9] and presented with a different and simpler proof here, consists of three steps: rectification, vectorization and assembling the matrix.

Rectification:

We split the this square into a grid consists of cells with size $1/\sqrt{n}\times 1/\sqrt{n}$, so that the grid has $n\times n$ cells in total. We index each cell by a member in the additive group $\mathbb Z_n\times \mathbb Z_n$. The intersection cell of the j-th column from left to right and kth-row from bottom to top will be indexed as $E_{(j-1,k-1)}^{(n)}$, the superscript (n) indicates grid and cell sizes.

If $\Lambda \subseteq \mathbb Z_n\times \mathbb Z_n$, we use the notation $E^{(n)}_{\Lambda }$ to denote the union of all cells indexed by $\Lambda$, i.e.,

$$\begin{aligned} E^{(n)}_{\Lambda }=\{E_{(j,k)}^{(n)}: (j,k)\in \Lambda \subseteq \mathbb Z_n\times \mathbb Z_n\}. \end{aligned}$$

(10)

The full grid can thus be written as $E^{(n)}_{\mathbb Z_n\times \mathbb Z_n}$. In particular, the area of $E^{(n)}_{\Lambda }$ is $|\Lambda |/n$. We consider all cells that intersects U and set

$$\begin{aligned} \Gamma =\left\{ (j,k):\; E_{(j,k)}^{(n)}\bigcap U\ne \emptyset \right\} . \end{aligned}$$

(11)

Vectorization:

Given $\Lambda \subset \mathbb Z_n\times \mathbb Z_n$, define the vectorization operator

$$\begin{aligned} S_{\Lambda }: L^2(E_{\Lambda }^{(n)}) \rightarrow L^2(E_{(0,0)}^{(n)})^{|\Lambda |}, \end{aligned}$$

so that if $(j,k)\in \Lambda$, then the (j, k)-th entry of $S_{\Lambda }\eta$ is $\eta$ restricted to the cell $E_{(j,k)}^{(n)}$, i.e.,

$$\begin{aligned} (S_{\Lambda }\eta )_{(j,k)}(t,v)=\eta |_{E_{(j,k)}^{(n)}}=\eta \left( t+\frac{j}{r}, v+\frac{k}{r}\right) , \end{aligned}$$

(12)

where $(t,v)\in E_{(0,0)}^{(n)}$, and $r=\sqrt{n}$. The action of $S_{\Lambda }$ is best described by the figure below (Fig. 3):

Fig. 3

Rectification and Vectorization

Assembling the matrix:

Take $\vec c\in \mathbb C^n$, for any $\eta \in L^2(U)$, we instead view it as an element in $L^2(E_{\Gamma }^{(n)})$, then since

$$\begin{aligned} (V_{\vec g_c}f)(t,v)=\sum _{k=0}^{n-1}c_k\left\langle {f, M_vT_{t+\frac{k}{r}}g_r}\right\rangle =\sum _{k=0}^{n-1}c_k(V_{g_r}f)\left( t+\frac{k}{r},v\right) , \end{aligned}$$

we obtain

$$\begin{aligned} (V_{\vec g_c}f)|_{E^{(n)}_{(0,0)}}=\sum _{k=0}^{n-1}c_kh_k=\langle \vec h, {\bar{c}}\rangle , \end{aligned}$$

where ${\bar{c}}$ denotes the complex conjugate of $\vec c$ , and

$$\begin{aligned} \vec h=S_{\mathbb Z_n\times \{0\}}V_{g_r}f. \end{aligned}$$

(13)

Now if we move horizontally by one cell, with quasi-periodicity and $r^2=n$ we get

$$\begin{aligned} (V_{\vec g_c}f)|_{E^{(n)}_{(1,0)}}&=(V_{\vec g_c}f)|_{E^{(n)}_{(0,0)}}\left( t+\frac{1}{r}, v\right) =\sum _{k=0}^{n-1}c_k(V_{g_r}f)|_{E^{(n)}_{(0,0)}}\left( t+\frac{k}{r}+\frac{1}{r},v\right) \nonumber \\&=\langle \vec h, T{\bar{c}}\rangle , \end{aligned}$$

(14)

alternatively if we move vertically by one cell, then with similar reasoning we obtain

$$\begin{aligned} (V_{\vec g_c}f)|_{E^{(n)}_{(0,1)}}=\sum _{k=0}^{n-1}c_k(V_{g_r}f)|_{E^{(n)}_{(0,0)}}\left( t+\frac{k}{r},v+\frac{1}{r}\right) =e^{-2\pi i\frac{t}{r}}\langle \vec h, M{\bar{c}}\rangle , \end{aligned}$$

(15)

combing the above altogether we get the following conclusion:

Proposition 1

Let $\vec c\in \mathbb C^n$, denote its complex conjugate as $\bar{c}$. Let U be a compact Jordan measurable set, let $r=\sqrt{n}$, and $\Gamma$ be as defined in (11), then for any $\eta \in L^2(U)$, we have

$$\begin{aligned} \Phi _{g_{\vec c}}\eta =\Phi _{g_r}S_{\mathbb Z_n\times \{0\}}^{-1}G_{\Gamma }({\bar{c}})DS_{\Gamma }\eta , \end{aligned}$$

where D is a unitary diagonal scaling, and the ordering of columns in $G_{\Gamma }({\bar{c}})$ is the same as the ordering of entries in $S_{\Gamma }$. Moreover, we also have

$$\begin{aligned} \frac{1}{\root 4 \of {n}}\sigma _{\min }\left( G_{\Gamma }({\bar{c}})\right) \Vert \eta \Vert _{L^2(U)}\le \Vert \Phi _{g_{\vec c}}\eta \Vert _{L^2(\mathbb R)}\le \frac{1}{\root 4 \of {n}}\sigma _{\max }\left( G_{\Gamma }({\bar{c}})\right) \Vert \eta \Vert _{L^2(U)}, \end{aligned}$$

where $\sigma _{\max }\left( G_{\Gamma }({\bar{c}})\right)$ and $\sigma _{\min }\left( G_{\Gamma }({\bar{c}})\right)$ are respectively the largest and the smallest singular values of of the Gabor matrix $G_{\Gamma }({\bar{c}})$.

Proof

With above derivations (14) and (15) we may decompose and vectorize $V_{g_{\vec c}}f$ on $E^{(n)}_{\Gamma }$ into

$$\begin{aligned} S_{\Gamma }V_{g_{\vec c}}f=D^*G_{\Gamma }^*({\bar{c}})S_{\mathbb Z_n\times \{0\}}V_{g_r}f, \end{aligned}$$

where $*$ denotes the adjoint, and $D^*$ is the unitary diagonal scaling that collects the exponential factor $e^{-2\pi i\frac{t}{r}}$ emerged in (15). The adjoint relation in (9) then gives the decomposition formula (It is easy to see that $S_{\Gamma }$ and $S_{\mathbb Z\times \{0\}}$ are unitary, thus their adjoints and their inverses coincide).

The norm estimate follows by noticing that $S_{\mathbb Z_n\times \{0\}}^{-1}, D, S_{\Gamma }$ are unitary, while the range of $S_{\mathbb Z_n\times \{0\}}^{-1}$ is $L^2([0,r]\times [0,1/r])$, on which $\Phi _{g_r}$ ($r=\sqrt{n}$) is, up to a scaling factor $\sqrt{r}$, also unitary. $\square$

Consequently, injectivity of $\Phi _{g_{\vec c}}$, and in particular the upper and lower bound of $\Phi _{g_{\vec c}}$ depends solely on the Gabor matrix $G_{\Gamma }({\bar{c}})$. Notice that it requires $|\Gamma |=n$ for $G_{\Gamma }({\bar{c}})$ to be a square matrix, which means $E_{\Gamma }^{(n)}$ has area 1. On the other hand, for any compact underspread Jordan measurable set U we can always choose n large enough to cover it by such a set $E_{\Gamma }^{(n)}$.

This decomposition also complies with the definition of discretized channels in application. A discrete channel on $\mathbb C^{n\times n}$ is a weighted superposition of discrete translations and modulations, it takes the form of a linear combination $\sum _{(j,k)\in \Gamma }a_{jk}M^jT^k$ with $\Gamma \subset \mathbb Z_n\times \mathbb Z_n$. Therefore its response on an input $\vec c$ is simply $\sum _{(j,k)\in \Gamma }a_{jk}M^jT^k\vec c$, which can be viewed as the Gabor matrix $[M^jT^k\vec c]_{(j,k)\in \Gamma }$ multiplying the vector $\vec a=(a_{jk})_{(j,k)\in \Gamma }^T$.