4
D. Rocchesso: Sound Processing
band-limited
reconstruction filter
Nyquist frequency
holder
sampler
sample and hold
/2
F
s
F
s
f
-F
0
X(f)
F
b
b
s
-F
Figure 1: Frequency spectrum of a sampled signal
Theorem 1.1 A continuous-time signal x(t), whose spectral content is limited
to frequencies smaller than F
b
(i.e., it is band-limited to F
b
) can be recovered
from its sampled version ^
x(n) = x(nT ) if the sampling rate F
s
= 1/T is such
that
F
s
> 2F
b
.
(8)
It is also clear how such recovering might be obtained. Namely, by a linear
reconstruction filter capable to eliminate the periodic images of the base band
introduced by the sampling operation. Ideally, such filter doesn't apply any
modification to the frequency components lower than the Nyquist frequency,
defined as F
N
= F
s
/2, and eliminates the remaining frequency components
completely.
The reconstruction filter can be defined in the continuous-time domain by
its impulse response, which is given by the function
h(t) = sinc(t) =
sin (t/T )
t/T
,
(9)
which is depicted in fig. 2.
-5
0
5
-1
0
1
time in sampling intervals
Impulse response of the Reconstruction Filter
sinc
Figure 2: sinc function, impulse response of the ideal reconstruction filter
Ideally, the reconstruction of the continuous-time signal from the sampled
signal should be performed in two steps:
· Conversion from discrete to continuous time by holding the signal con-
stant in time intervals between two adjacent sampling instants. This is
achieved by a device called a holder. The cascade of a sampler and a
holder constitutes a sample and hold device.
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