Images, sampling and frequency domain processing - Feature Extraction and Image Processing - page 42

Amplitude

Amplitude

Signal

Signal

Sampling
instants

Sampling
instants

Time

Time

∆ t
(a) Sampling at high frequency

∆ t
(b) Sampling at low frequency

Figure 2.7 Sampling at different frequencies

signal's spectrum exists only between these frequencies. This function is sampled every
∆ t s: this is a sampling function of spikes occurring every ∆ t s. The Fourier transform of the
sampling function is a series of spikes separated by f sample = 1/∆

t Hz. The Fourier pair of

this transform was illustrated earlier, Figures 2.6 (g) and (h).
The sampled signal is the result of multiplying the time-variant signal by the sequence
of spikes, this gives samples that occur every ∆ t s, and the sampled signal is shown in
Figure 2.9 (a). These are the outputs of the A/D converter at sampling instants. The frequency
domain analogue of this sampling process is to convolve the spectrum of the time-variant
signal with the spectrum of the sampling function. Convolving the signals, the convolution
process, implies that we take the spectrum of one, flip it along the horizontal axis and then
slide it across the other. Taking the spectrum of the time-variant signal and sliding it over
the spectrum of the spikes, results in a spectrum where the spectrum of the original signal
is repeated every 1/∆ t Hz, f sample in Figures 2.9 (b-d). If the spacing between samples is ∆ t ,
the repetitions of the time-variant signal's spectrum are spaced at intervals of 1/∆ t , as in
Figure 2.9 (b). If the sample spacing is small , then the time-variant signal's spectrum is
replicated close together and the spectra collide , or interfere, as in Figure 2.9 (d). The
spectra just touch when the sampling frequency is twice the maximum frequency in the
signal. If the frequency domain spacing, f sample , is more than twice the maximum frequency,
f max , the spectra do not collide or interfere, as in Figure 2.9 (c). If the sampling frequency
exceeds twice the maximum frequency then the spectra cannot collide. This is the Nyquist
sampling criterion:

In order to reconstruct a signal from its samples, the sampling frequency must
be at least twice the highest frequency of the sampled signal.

If we do not obey Nyquist's sampling theorem the spectra collide. When we inspect the
sampled signal, whose spectrum is within - f max to f max , wherein the spectra collided, the
corrupt spectrum implies that by virtue of sampling we have ruined some of the information.
If we were to attempt to reconstruct a signal by inverse Fourier transformation of the
sampled signal's spectrum, processing Figure 2.9 (d) would lead to the wrong signal whereas
inverse Fourier transformation of the frequencies between - f max and f max in Figures 2.9 (b)