Skip to content

Sampling

Aliasing

Given a continuous sinusoidal signal \(x(t) = \sin \left( 2 \pi f_0 t \right)\), the discretized signal can be written as:

\[ x(n) = \sin(2 \pi f_0 n t_s) = \sin \left( 2\pi f_0 n t_s + 2 \pi m \right) = \sin \left( 2 \pi \left( f_0 + \frac{m}{n t_s} \right)n t_s \right), \]

where \(t_s\) is the sampling interval. Let \(m\) be an integer multiple of \(n\), i.e., \(m = kn\). Then we have:

\[ x(n) = \sin(2 \pi \left( f_0 + k f_s \right) n t_s). \]

In order words, when sampling at a rate of \(f_s\) Hz, we cannot distinguish between the sampled values of a sinewave of \(f_0\) Hz and a sinewave of \(f_0 + k f_s\) Hz for \(k \in \mathbb{Z}\). The spectrum of any discrete series of samples values contains periodic replications of the original continuous spectrum. The period between these replicated spectra in the frequency domain will always be \(f_s\). Aliasing gets introduced when we do improper sampling. Every sampling operation inherently results in spectral replications.

Proper Sampling

The Nyquist sampling theorem 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.

Examples:

  1. A sampling rate of 2000 samples per second requires the analog signal to be composed of frequencies below 1000 cycles/second. If frequencies above this limit are present in the signal, they will be aliased to frequencies between 0 and 1000 cycles/second, combining with whatever information that was legitimately there.
  2. Sine wave has a frequency of 0.09 of the sampling rate. This might represent, for example, a 90 cycle/second sine wave being sampled at 1000 samples/second. There are 11.1 samples taken over each complete cycle of the sinusoid. This is proper sampling.
  3. Sine wave has a frequency to 0.31 of the sampling rate. This results in only 3.2 per sine wave cycle. This is proper sampling.
  4. 0.95 of the samping rate, with a mere 1.05 samples per sine wave cycle. This is improper sampling and introduces aliasing.

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.