Sunday, November 3, 2019

Sampling theorem, communication

The sampling theorem lays the very fundamentals of communication theory. It reveals the dimensionality of time & band-limited signal space, which in principle sets the limit on  the rate of communication (or the capacity)

Not only that, it also has its ultimate significance in the practical design of digital communication system, the concept of Inter-Symbol Interference (ISI)

Nyquist criteria

Suppose we'd like to transmit/modulate a sequence $\{a_0, a_1, ..., a_n\}$, the sampling theorem basically assures us that if we use the waveform of
\[ \sum a_n {\rm sinc}(t-n/2W) \]
and if the receiver knows exactly where to re-sample the waveform at the period of $1/2W$, the whole sequence could be recovered precisely (no noise presents).  This is mainly because the sinc function has the following property:
\[ {\rm sinc}(t-n/2W) = \left\{ \begin{array}{lr}
1 & \mbox{, when } t=n/2W \\
0 & \mbox{, when } t \neq n/2W \\ \end{array} \right. \]
which often is referred as no ISI condition, or met the Nyquist criteria.  The convolution kernel of ${\rm sinc}$ function here is usually referred as pulse shaping filter.

We know that sinc function is not the only function to satisfy Nyquist criteria, for example, ${\rm sinc}({2Wt \over M}) $ where $M$ is integer do as well.  However, in communication, they are not efficient enough (the data rate is reduced).  If we picture the no ISI condition in frequency domain, it may be more insightful.

The Fourier transform of ${\rm sinc}(2Wt)$ looks like:
and if we sample at $t=0, 1/2W,....n/2W$, the Fourier transform of the resulted sequence is:
which is the Fourier transform of impulse $\delta_n$.  It means if we sample the waveform at the right timing, the whole chain (modulation & sampling) is just:
\[ a_n * \delta_n \]

Now, if we allow the spectrum of $f(t)$ to be extended beyond W a bit, but put the constraint on the expansion that,  after aliasing,  the whole spectrum will  be purely flat, it should satisfy the Nyquist criteria as well.

One of the famous function has this property is raised cosine, which is the most commonly used pulse shaping filter in digital communication.  Notice that in communication system, the shaping filter is usually divided into to two equal parts, one in transmitter and one in receiver, namely, root raised cosine filters. The rationale behind this separation is the matched filter theory, which can maximize the signal to noise ration (SNR) at sampling point. It also simplifies the analysis of the digital system, as the noise will be filtered by the matched filter and its power will be proportional to the nominal bandwidth W, thus the system performance will be independent of the rate.

Nyquist filter bank

In signal processing, often we hear the name of Nyquist filter bank. It is the very same property (above) of the prototype filter gives the name. And it is this property make the synthesis extremely easy: just add all the samples from the filters together.