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.