Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 82

θ , rad/s

| H ( e j 2 π v ) |
2

π

ν = f / f s ν = f / f s

− 1 − 0.5 0.5 1

− π

− 1 − 0.5 0.5 1

Fig. 2.13 Magnitude and phase response of a leaky integrator

the phase response is \H ð e jxT s Þ¼ 0 : 458 rad. Figure ( 2.13 ) shows the magnitude
and phase response of this system as a function of the normalized frequency.
If one needs to recover the actual response versus frequency from the normalized
frequency response one simply scales the frequency axis by f s .
From Fig. ( 2.13 ) one can see that the shape of the transfer function is that of a
LPF; the circuit is in fact what is sometimes referred to as a leaky integrator. If
b = 1, it would be an integrator, as will be seen later.

2.4 A Discrete-Time and Discrete-Frequency Representation

The DTFT of a discrete-time signal x(n) is still continuous in the frequency var-
iable, and hence it is not possible to implement with practical digital technologies
such as computers. It is necessary, then, to find a discrete-time and discrete-
frequency representation for practical analysis. This doubly discrete representation
will be referred to as the Discrete Fourier Transform (DFT).

2.4.1 The Discrete Fourier Transform

In practice one has to restrict oneself to computing the Fourier transform at a
limited number of representative sample frequencies. It was seen previously that
with appropriate precautions no information is lost provided that certain require-
ments on the sampling rate are met. In particular, it is necessary that the sample
rate in the time domain is at least twice as high as the highest frequency com-
ponent present in the analog signal. It will be seen here that there is an analogous
result for frequency domain sampling—one does not lose any information by
sampling in the frequency domain, provided that certain conditions are met on the
frequency sampling rate.