Undersampling

The individual frequency-shifted copies of the original transform are called aliases.

The frequency offset between adjacent aliases is the sampling-rate, denoted by fs.

The first and third graphs of Figure 1 depict a baseband spectrum before and after being sampled at a rate that completely separates the aliases.

The condition for a non-destructive sample rate is that the aliases of both bands do not overlap when shifted by all integer multiples of fs.

The fourth graph depicts the spectral result of sampling at the same rate as the baseband function.

For the case of a given sampling frequency, simpler formulae for the constraints on the signal's spectral band are given below.

and the reconstructive interpolation function, or lowpass filter impulse response, is

To accommodate undersampling, the bandpass condition is that X(f) = 0 outside the union of open positive and negative frequency bands The corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses: On the other hand, reconstruction is not usually the goal with sampled IF or RF signals.

Fig 1: The top 2 graphs depict Fourier transforms of 2 different functions that produce the same results when sampled at a particular rate. The baseband function is sampled faster than its Nyquist rate, and the bandpass function is undersampled, effectively converting it to baseband. The lower graphs indicate how identical spectral results are created by the aliases of the sampling process.
Plot of sample rates (y axis) versus the upper edge frequency (x axis) for a band of width 1; grays areas are combinations that are "allowed" in the sense that no two frequencies in the band alias to same frequency. The darker gray areas correspond to undersampling with the maximum value of n in the equations of this section.
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz ( n = 5) sampling. An anti-alias filter quite tight to the FM radio band is required, and there's not room for stations at nearby expansion channels such as 87.9 without aliasing.
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz ( n = 4) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (even n ).