Sampling Theorem for Band-Limited Random Processes

Fundamentals

Definition: Band-Limited Signal
A deterministic real signal \( x(t) \) with Fourier transform \( X(f) \) is said to be band-limited if:

\[ X(f) = 0, \quad \text{for } |f| > W, \]

where \( W \) is the highest frequency contained in \( x(t) \).

Such a signal can be uniquely represented by discrete samples taken at a rate of:

\[ f_s \geq 2W \quad \text{samples per second}. \]

Definition: Nyquist Rate
The Nyquist rate, denoted as:

\[ f_N = 2W, \]

is the minimum sampling rate required to ensure that the signal can be perfectly reconstructed from its discrete samples.

Sampling of Complex-Valued Signals

For complex-valued signals, the bandwidth \( W \) is defined as half of the frequency support of the signal. That is, if \( W_1 \) and \( W_2 \) denote the lowest and highest frequency components, respectively, then:

\[ 2W = W_2 - W_1. \]

Perfect Signal Reconstruction

A band-limited signal can be perfectly reconstructed from its sampled values if and only if the sampling rate satisfies:

\[ f_s \geq 2W. \]

This fundamental result, known as the Nyquist-Shannon sampling theorem, provides the theoretical foundation for digital signal processing and data acquisition in communication systems.

Sampling and Reconstruction of Band-Limited Signals

Degrees of Freedom in Sampled Signals

In the case of complex signals, each sampled value is complex-valued, meaning that two real numbers are required to represent each sample. This leads to the following observations:

• A real signal can be fully described using \( 2W \) real numbers per second, meaning it has \( 2W \) degrees of freedom or real dimensions per second.

• A complex signal has \( 4W \) degrees of freedom per second, which is equivalent to \( 2W \) complex dimensions or \( 4W \) real dimensions per second.

Aliasing in Undersampling

If the sampling rate falls below the Nyquist rate (\( f_s < 2W \)), frequency aliasing occurs, where different frequency components become indistinguishable from one another. This results in irreversible distortion in the reconstructed signal.

Reconstruction from Samples

A band-limited signal that is sampled at the Nyquist rate can be perfectly reconstructed using the interpolation formula:

\[ x(t) = \sum_{n=-\infty}^{\infty} x\left(\frac{n}{2W}\right) \text{sinc} \left[ 2W \left( t - \frac{n}{2W} \right) \right], \]

where \( \{x(n/2W)\} \) are the discrete-time samples taken at \( t = n/2W \), with \( n = 0, \pm1, \pm2, \ldots \).

Reconstruction via Filtering

An equivalent reconstruction method is to pass the sampled signal through an ideal lowpass filter with an impulse response:

\[ h(t) = \text{sinc}(2Wt). \]

This filtering operation ensures that only the original frequency components of the signal are retained, enabling perfect signal reconstruction.

These principles form the foundation of digital signal processing, ensuring that continuous signals can be accurately represented and recovered from discrete samples.