The Engineer's Guide to Standards Conversion - Snell

Sampling period
Sampling
frequency
a)
DFT
b)
0
Frequency
Time
d)
FT
c)
e)
Fig 3.6.1 a) The windowed impulse response of a filter.
b) The Discrete Fourier Transform of the impulse contains as many
frequencies as the window has points.
c) Each discrete frequency in the DFT represents a sinx/x spectrum in
a continuous transform.
d) The sinx/x pulse is the transform of the rectangular window.
e) The continuous spectrum is obtained by adding the sinx/x curves of
each of the discrete spectral lines. The origin of stop-band ripple
should be clear.
The values of the samples in the window can describe an impulse response as
shown in Fig 3.6.1a). Fourier analysis tells us that the spectrum of discrete signals
must also be discrete, and the number of different frequencies in the spectrum is
equal to the number of samples in the window. The spectrum of a) is shown in b).
As a consequence, the frequency response of the filter can be specified in a finite
number of evenly spaced places. In a two dimensional filter these places will form a
rectangular grid. In order to return to the continuous time domain from discrete
samples, each sample is replaced by a sinx/x impulse. The same principle holds in
the discrete frequency domain.
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