Sampling Theorem

Proper Sampling

To describe a full revolution around a unit circle for the fastest sinusoid with a positive frequency (\(w = \pi\)), it requires two samples. If the system clock is \(t_s\) (\(f_s = 1 / t_s\)), to sample the real-world fastest sinusoid, the overall period required is \(2 t_s\) or \(f_s / 2\) frequency. Hence, the highest frequency spectral content that can be captured is \(f_s / 2\) which corresponds to the analysis frequency \(k = N / 2\) where \(N\) is the number of samples. This gives the basis for the Nyquist sampling theorem.

The Nyquist sampling criterion indicates that a continuous signal can be properly sampled, only if it does not contain frequency components above one-half of the sampling rate (the Nyquist frequency, the folding frequency, or the cutoff frequency). In order words, given a band-limited signal with band \(\pm B\), \(f_s\) must be greater than \(2B\) to separate spectral replications at the folding frequencies of \(\pm f_s / 2\).

In most systems, the frequency band between about 0.4 and 0.5 of the sampling frequency is an ususable wasteland of filter roll-off and aliased signals. This is a direct result of the limitations of analog filters.

Sample Rate Conversion

Changing the sample rate of a digital signal can be done via decimation (reduction in sample rate) and interpolation (increase in sample rate). A common interpolation scheme is via zero-insertion followed by a lowpass filter.

Aliasing and Spectral Replica

Aliasing gets introduced when we do improper sampling. Consider a continuous sinusoidal signal \(x(t) = \sin \left( 2 \pi f t \right)\). Sampling at \(f_s = 1 / t_s\) yields to a discrete samples:

\[ \begin{align} x(n) &= \sin(2 \pi f n t_s) \\ &= \sin(2 \pi f n t_s + 2 \pi m) \\ &= \sin \left( 2\pi \left(f + \frac{m}{nt_s} \right) nt_s \right). \end{align} \]

Let \(m\) be an integer multiple of \(n\), i.e., \(m = kn\). Then we have:

\[ x(n) = \sin\left( 2 \pi \left( f + k f_s \right) n t_s \right). \]

This equation means that when the sampling rate is \(f_s\), we cannot distinguish between the sampled values of a sinewave of frequency \(f\), and a sinewave of \(f + k f_s\) for \(k \in \mathbb{Z}\). The spectrum of any discrete series of sampled values contains periodic replications of the original continuous spectrum. The period between these replicated spectra in the frequency domain will always be \(f_s\). Every sampling operation inherently results in spectral replications.

Bandpass Sampling

Bandpass sampling (IF sampling or undersampling) is a technique where we can sample a bandpass-filtered signal at a sample rate below its Nyquist frequency without introducing aliasing. In bandpass sampling, we are more concerned with the signal's bandwidth than its highest frequency component.

Consider a continuous bandpass signal with bandiwth \(B\) and carrier frequency \(f_c\) (center frequency). Given an arbitrary number of replications, \(m\), to avoid aliasing, we can choose \(f_s\) as:

\[ \frac{2f_c - B}{m} \geq f_s \geq \frac{2f_c + B}{m + 1}, \quad f_s \geq 2B. \]

The upper limit is because if we increase the sampling rate \(f_s\), the original spectra do not shift, but all the replications will shift, causing them to overlap and introduce aliasing.

Function Approximation

Under appropriate "slowness" conditions for a continuous signal \(x(t)\), we have:

\[ x(t) = \sum^{\infty}_{n = -\infty} x(n t_s) \text{sinc} \left( \frac{t - nt_s}{t_s} \right), \]

where \(t_s\) is the sampling interval.