Page 36 - Analog and Digital Filter Design

Introduction 33

SOURCE SIGNALS

;

FREQUENCY
Figure 1.13 (a and b)

Multiplying Signals in the Frequency Domain

The relationship between time and frequency domains for multiplied signals is
important for digital filter designers. When analog signals are sampled, they are
effectively multiplied by an impulsive sampling signal. An periodic sampling
pulse that is very short has spectral energy at multiples (harmonics) of the sam-
pling frequency. The energy of every harmonic is equal to that of the lowest
(fundamental) frequency, Fs. This means that the analog signal “A” is multiplied
by the fundamental and every harmonic of the sampling signal. Thus spectral
spreading occurs with energy appearing at Fs k A, 2Fs k A, 3Fs k A, 4Fs k A,
and so on. When converting the sampled signal back into analog form, a further
sampling operation reverses the frequency spreading process and results in all
the spectral energy being concentrated at frequency “A.”

The analog signal must be frequency limited prior to sampling, to less than
half the sampling frequency. Otherwise the resultant spectral energy from
mixing products will overlap in the frequency domain (which is known as alias-
ing and illustrated in Figure 1.14). If this happens, when signals are converted
back into analog form, they have the wrong frequency. Filters are therefore
placed before the sampling device to prevent aliasing, and these are known as
anti-alias filters.