On computing the number of independent values from a signal's bandwidth?

Question

Can anyone help me understand where the value of $2$ comes from in the following statement?

A time function t whose frequency response is white out to a bandwidth B has 
about 2 · B · T independent values when measured for the time interval T.

This is in the context of a serial stream of digital (e.g. binary) data. The function t represents this data stream.

Answer 1

It comes from the Nyquist rate. The Nyquist sample rate for a signal with bandwidth $B$ is $2B$. In other words, that is the lowest sample rate that will contain all of the information in the signal without distorting it (i.e. without losing information).

The question implicitly assumes that the sample rate is $2B$, and thus that in time $T$ you will get $2B*T$ independent samples. If the sample rate was greater than $2B$, then the signal would not be "white" anymore and the samples would have some correlation.

Answer 2

I don't think that whiteness is of any importance here, and in particular, no digital modulation scheme is going to create a signal whose power spectrum is anywhere close to that of band-limited white noise -- flat in bandwidth $B$ and $0$ outside the band. The result the OP is asking about is based on what is called the Landau-Pollak theorem.

The late U.S. Supreme Court Justice Potter Stewart once rendered a decision in which he memorably said

``I may not know how to define it legally, but I know it when I see it.''

He was, of course, speaking of pornography, but the notion of bandwidth of a signal is very similar. Every engineer understands the notion, perhaps only at an intuitive level, but a definition of bandwidth that is satisfactory for all purposes remains as elusive as ever.

A signal $s(t)$ is said to be strictly time-limited if $s(t)$ is 0 outside a time interval of finite length, e.g. $s(t) = 0$ if $t < 0$ or $t > T$. It is said to be strictly band-limited if its Fourier transform $S(f)$ is 0 outside a frequency interval of finite length, e.g. $S(f) = 0$ if $\vert f\vert > W$, or $S(f) = 0$ if $\vert f\vert > f_c+W/2$, or $\vert f \vert < f_c - W/2$ corresponding respectively to lowpass or bandpass signals of bandwidth $W$. Unfortunately, a signal cannot be both strictly time-limited and strictly band-limited. Now, let $s(t)$ be a unit-energy strictly time-limited function. In particular, suppose that $s(t) = 0$ if $t < 0$ or if $t > T$. Let $\eta$ denote a small positive number, that is, $0 < \eta \ll 1$. Let $W$ be a number such that $$ \int_{-W}^W \vert S(f)\vert^2 \, df > 1 - \eta, $$ that is, almost all of the energy in $s(t)$ is in the frequency band $[-W, W]$. We say that $s(t)$ is essentially band-limited to $W$ Hz or an essentially low-pass signal of bandwidth $W$ Hz. Now, suppose that $\{s(t)\}$ denotes a collection of unit-energy signals that are all strictly time-limited to $[0, T]$ and all essentially band-limited to $[-W, W]$. How many mutually orthogonal signals does the set $\{s(t)\}$ contain? The answer is given by a result called the Landau-Pollak Theorem. The number of orthogonal signals is less than $2WT/(1 - \eta)$. Since $\eta$ is very close to 0, the denominator is just slightly less than 1, and so we conclude that the number of orthogonal signals strictly time-limited to $[0, T]$ and essentially band-limited to $W$ Hz is approximately $M = 2WT$.

Duality says that if the signal is strictly band-limited to $W$ Hz, then it is essentially time-limited to a duration of $T$ seconds (that is, ($1-\eta$) of the energy is in that duration) and there are approximately $2WT$ orthogonal signals with this property.