Handbook for Sound Engineers

DSP Technology 1169

Now the frequency response of the sampled contin-
uous-time signal becomes a collection of shifted copies
of the original frequency response of the analog signal
Xc( j:. Fig. 31-10 shows the frequency response of
Xc(j:, the impulse train, S( j:, and the resulting
frequency response of the sampled signal, Xs( j:.
This frequency response, XS(j:, can also be inter-
preted as the convolution in the frequency domain
between the frequency response of the continuous-time
signal and the frequency response of the impulse train,
S( j:.

(31-19)

From Fig. 31-10 it can be seen that as long as the
sampling frequency minus the highest frequency is
greater than the highest frequency, :S:N >:N, the
frequency copies do not overlap. This condition can be
rewritten as :S>2:N, which means that the sampling
frequency must be at least twice as high as the highest
frequency in the signal. If the sampling frequency is less
than the highest frequency in the signal, :S:N, then
the frequency copies overlap as shown in Fig. 31-11.
This overlap causes the frequencies of the adjacent
spectral copies to be added together, which results in the
loss of spectral information. It is impossible to remove
the effects of aliasing once aliasing has happened. The

overlap is caused because the sampling frequency, :S,

is not high enough relative to the highest frequency in

the continuous-time signal Xc( j:. As shown above, the

sampling frequency must be at least twice as high as the

highest frequency in the continuous-time signal in order

to prevent this overlap, or aliasing, of frequencies.

31.6.2 Reconstructing the Continuous-Time Signal

As seen from sampling a continuous-time signal, if the

signal is not sampled fast enough, then the resulting

frequency response of the sampled signal will have

overlapping copies of the frequency response of the

original signal. Assuming the signal is sampled fast

enough (at least twice the bandwidth of the signal), the

continuous-time signal can be reproduced by simply

removing all of the spectral copies except for the

desired one. This frequency separation can be per-

formed with an ideal low-pass filter with gain, T, and

cut-off frequency, :C, where the cut-off frequency is

higher than the highest frequency in the signal as well

as the frequency where the first frequency replica

starts,—i.e., :N<:C<:S:N. Fig. 31-12 shows the

repeated frequency spectrum and the ideal low-pass fil-

ter. Fig. 31-13 shows the result of applying the

low-pass filter to XS(j:

31.6.3 Sampling Theory

The requirements for sampling are summarized by the

Nyquist sampling theorem.^1 Let xc(t) be a band-limited

signal with Xc( j:)=0 for |:_ t :N. Then xc(t) is

uniquely determined by its samples, x[n]=xc(nT), if

:S=2Se 7 t 2 :N. The frequency :N is referred to as

the Nyquist frequency, and the frequency 2:N is referred

to as the Nyquist rate. This theory is significant because

it states that as long as a continuous-time signal is

band-limited and sampled at least twice as fast as the

highest frequency, then it can be exactly reproduced by

the sampled sequence.

Figure 31-10. The frequency response of the analog signal,
Xc(j:, the sampling function, S(j:, and the resulting
frequency response of the sampled signal, Xs(j:.

Xs j:^1

2 S

=------Xc j:^ S j:

(^7) S
1
7
c(jX^7 )
0
T
2 P
T
1
0
0
(jS 7 )
s(jX^7 )
(^7) N (^7) N
7S (^7) S (^7) S 7S
7S (^7) S (^7) N^7 N^7 S 7S 7
7
( (^7) N)
Figure 31-11. Sampling where the sampling frequency, :S,
is less than twice the highest frequency, :N.
 
(^7) N
7
T
1


(^7) S (^7) N
(^7) N
(^7) S (^7) S (^7) S (^7) N