Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 226

This clearly indicates that within the passband, all frequency components passing through the FIR
filter will have the same constant delay at the output, which equals M samples. Hence, phase distortion
is avoided.
Figure 7.6 verifies the linear phase property using an FIR filter with 17 taps. Two sinusoids of the
normalized digital frequencies 0 : 05 p and 0 : 15 p radians, respectively, are used as inputs. These two
input signals are within the passband, so their magnitudes are not changed. As shown in Figure 7.6 ,
beginning at the ninth sample the output matches the input, which is delayed by eight samples for
each case.
What would happen if the filter phase were nonlinear? This can be illustrated using the following
combined sinusoids as the filter input:

1
3 sin ð 0 : 15 pnÞuðnÞ
The original xðnÞ is the top plot shown in Figure 7.7 . If the linear phase response of a filter is
considered, such as f ¼MU 0 , where M ¼ 8 in our illustration, we have the filtered output as

xðnÞ¼x 1 ðnÞþx 2 ðnÞ¼ sin ð 0 : 05 pnÞuðnÞ

1
3 sin ½ 0 : 15 pðn 8 Þ

y 1 ðnÞ¼ sin ½ 0 : 05 pðn 8 Þ

2

0

-2

0 5 10 15 20 25 30 35 40 45 50

2

0

Match i ng x1(n)

M=8

-2

0 5 10 15 20 25 30 35 40 45 50

2

0

-2

0 5 10 15 20 25 30 35 40 45 50

2

0

M=8

Matching x2(n)

-2

0 5 10 15 20 25 30 35 40 45 50

n

FIGURE 7.6

Illustration of FIR filter linear phase property (constant delay of eight samples).