Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 49

H ( f ) h ( t )

2BA

A

f , Hz t , sec

− B 0 B 0

−1/(2B) 1/(2B)

Fig. 1.30 Transfer function and impulse response of an ideal low-pass filter

h ( t − T )

2BA

0

t , sec

t = T

Fig. 1.31 Quasi-ideal low-pass filter impulse response—it is a time shifted version of the ideal
low-pass filter impulse response

1.5.1 The Ideal Low-Pass Filter

The transfer function of an ideal LPF with cutoff frequency B Hz and gain A is a
rectangular box function in the frequency domain of width 2B. Using Tables, its
impulse response is found to be a sinc function:

H ð f Þ¼ 2BP 2B ð f Þ F h ð t Þ¼ 2AB sinc ð 2Bt Þ

This impulse response is shown in Fig. 1.30 .
Since h(t) includes negative values, the ideal LPF is non-causal, and hence
unrealizable. If one allows a time-delay T in the system, the delayed impulse
response is shifted right by T, and one gets h(t - T), as shown in Fig. 1.31 .This
introduces a linear phase into the transfer function. Since linear phase is a result of
only delaying the signal, it corresponds to a harmless phase changes in the transfer
function. Note that phase distortion only occurs when there are different delays
for different frequencies, while amplitude distortion occurs when there are dif-
ferent gains for different frequencies. Since h(t) has infinite length, there is still a
negative-time portion of h(t - T) despite the delay, but it is of small magnitude,
and approximation is possible by truncation.
There are many approximations to the ideal LPF that can be implemented in
practice. Two of these, the Butterworth and Chebychev-I filters will be studied in
the following subsections.