I stumbled upon the notion of “time varying channels” in signal processing and after reading a bit I realized that these are really interesting objects and appear in many different parts of mathematics. In this post I collect of few of their realizations and relations.
In signal processing a “time varying channel” is a mapping which maps a signal to an output via
Before we explain the name “time varying channel” we fix the notation for the Fourier transform I am going to use in this post:
Definition 1 For we define the Fourier transform by
and denote the inverse Fourier transform by or . For a function we denote with and the Fourier transform with respect to the first and second -component, respectively.
Remark 1 In all what follows we do formal calculations with integral not caring about integrability. All calculation are justified in the case of Schwartz-functions and often hold in a much broader context of tempered distributions (this for example happens if the integrals represent Fourier transforms of functions).
The name “time varying channel” can be explained as follows: Consider a pure frequency as input: . A usual linear channel gives as output a damped signal and the damping depends on the frequency : . If we send the pure frequency in our time varying channel we get
Hence, the time varying channel also damps the pure frequencies but with a time dependent factor.
Let’s start quite far away from signal processing:
1. Pseudo-differential operators
A general linear differential operator of order on functions on is defined with multiindex notation as
with coefficient functions . Using Fourier inversion we get
For a general function (usually obeying some restrictions) we call the corresponding a pseudo-differential operator with symbol .
2. As integral operators
By integral operator I mean something like
We plug the definition of the Fourier transform into and obtain
3. As “time varying convolution”
The convolution of two functions and is defined as
and we write “the convolution with ” as an operator .
we deduce from (1)
4. As superposition of time-frequency shifts
Using that iterated Fourier transforms with respect to components give the Fourier transform, i.e. , we obtain
From (1) we get
Before plugging this into we define time shifts and frequency shifts (or modulations) as
With this we get
Hence, is also a weighted superposition of (time-frequency shifts) with weight .
5. Back to time varying channels
Simple substitution brings us back the situation of a time varying channel
6. As superposition of product-convolution operators
Finally, I’d like to illustrate that this kind of operators can be seen as superposition of product convolution operators.
Introducing product operators we define a product-convolution operator as a convolution with a function followed by a multiplication with another function : .
To express with product-convolution operators we choose an orthonormal basis which consists of tensor-products of functions and develop into this basis as
and we obtain
Remark 2 The integral operators of the form are indeed general objects as can be seen from the Schwartz kernel theorem. Every reasonable operator mapping Schwartz functions linearly onto the tempered distributions is itself a generalized integral operator.
I tried to capture the various relation between the first five representations time varying channels in a diagram (where I went out of motivation before filling in all fields…):