Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 68

Due to the periodicity of X s (f), there is redundant frequency information in the
spectrum or ''discrete-time Fourier transform'' (DTFT) of x s (t). All the non-
redundant information in X s (f) is fully contained in the interval f 2f f s = 2 ; f s = 2 g
Hz, or equivalently f [ {0,f s }. In the literature X s (f) is normally plotted as a
function of the normalized frequency m = f/f s , or the normalized radian frequency,
X ¼ 2pm ¼ 2pf = f s ¼ xT s [see Fig. ( 2.3 )]. Note that m and X have no units, and the
period of X s (f)is1.
For real signals the frequency content (information) in the interval [-f s /2,0) is a
reflected copy of the frequency content in [0, +f s /2), and therefore many authors
display only the frequency interval [0, +f s /2) or the normalized frequency interval
[0, 1/2] of discrete-time signal spectra.
Note also that X s (f) is still continuous in the frequency variable. From a prac-
tical perspective one cannot compute X s (f) for a continuous range of frequency
values on a digital computer—one can only compute it for a finite number of
frequency positions. For this reason, X s (f) is usually only evaluated in practice at a
finite set of discrete frequencies. This discretization of the frequency domain of the
DTFT gives rise to the so-called discrete Fourier transform (DFT), which will be
studied later in Sect. 2.4.1 .
2.2.2 Ideal Reconstruction
Consider the discrete-time Fourier transform (DTFT) of the sampled signal
x s (t) shown graphically in Fig. ( 2.3 ). It is not hard to see that one can reconstruct
the original spectrum X(f) from X s (f) [and hence, the original signal x(t) from x s (t)]
by filtering the sampled signal with an ideal low-pass filter whose cutoff frequency
is B Hz.
Often in practice reconstruction occurs in a two stage process—the first
involves using a digital to analog converter (DAC) which applies a sample and
hold function, and the second stage involves applying a low-pass reconstruction
filter. Both stages are considered in more detail below.
2.2.2.1 Stage 1
When a digital signal is ready to be converted back to the analog domain, the
sequence of digitally stored values is usually fed synchronously into a DAC.
Almost all DACs apply a zero-order hold (sample-and-hold) function to the
sequence of input values [see Fig. ( 2.4 )]. The sample and hold function is effec-
tively a filter with a rectangular impulse response h ð t Þ¼ P T s ð t T s = 2 Þ: This
circuit simply holds the sample x(nT s ) for T s seconds. The corresponding filter
transfer function is a sinc function, as shown in Fig. ( 2.4 ).