Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 227

2

Combined two
sinusoidal input

0

-2

0 5 10 15 20 25 30 35 40 45 50

2

Linear phase filter output

0

M=8

Matching x(n)

-2

0 5 10 15 20 25 30 35 40 45 50

2

Output waveform shape is
different from the one of x(n)

0

90 degree phase shift
for x 1(n) an d x2(n)

-2

0 5 10 15 20 25 30 35 40 45 50

n

FIGURE 7.7

Comparison of linear and nonlinear phase responses.

The linear phase effect is shown in the middle plot of Figure 7.7 . We see that y 1 ðnÞ is the eight-sample
delayed version of xðnÞ . However, considering a unit gain filter with a phase delay of 90 degrees for all
the frequency components, we obtain the filtered output as

1
3 sin ð 0 : 15 pn p= 2 Þ

y 2 ðnÞ¼ sin ð 0 : 05 pn p= 2 Þ

where the first term has a phase shift of 10 samples (see sin ½ 0 : 05 pðn 10 Þ ), while the second term
has a phase shift of 10 = 3 samples see 1
3

sin 0 : 15 p

. Certainly, we do not have the

10
3

n

linear phase feature. The signal y 2 ðnÞ plotted in Figure 7.7 shows that the waveform shape is
different from that of the original signal xðnÞ , and hence has significant phase distortion. This phase
distortion is audible for audio applications and can be avoided by using an FIR filter, which has the
linear phase feature.
We now have finished discussing the coefficient calculation for the FIR lowpass filter, which has
a good linear phase property. To explain the calculation of filter coefficients for the other types of filters
and examine the Gibbs effect, we look at another simple example.