Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 222

where the delay operation is given by
b n ¼ hðn MÞ for n ¼ 0 ; 1 ; / ; 2 M (7.12)

Similarly, we can obtain the design equations for other types of FIR filters, such as highpass, bandpass,
and bandstop, using their ideal frequency responses and Equation (7.8) . The derivations are omitted
here. Table 7.1 gives a summary of all the formulas for FIR filter coefficient calculations.
The following example illustrates the coefficient calculation for the lowpass FIR filter.

EXAMPLE 7.2
a. Calculate the filter coefficients for a 3-tap FIR lowpass filter with a cutoff frequency of 800 Hz and a sampling
rate of 8,000 Hz using the Fourier transform method.
b. Determine the transfer function and difference equation of the designed FIR system.
c. Compute and plot the magnitude frequency response for U ¼ 0; p=4; p=2; 3p=4; and p radians.
Solution:
a. Calculating the normalized cutoff frequency leads to
U c ¼ 2pf c T s ¼ 2p 800=8; 000 ¼ 0:2p radians

Since 2M þ 1 ¼ 3 in this case, using the equation in Table 7.1 results in
hð0Þ¼ U c
p

for n ¼ 0

hðnÞ¼ sinðU c n Þ
np

¼ sinð0:2p n Þ
np

for n s 1

The computed filter coefficients via the previous expression are listed as:
hð0Þ¼ 0:2p
p

¼ 0:2

hð1Þ¼ sin½0:2p 1
1 p

¼ 0:1871

Using the symmetry leads to

hð1Þ¼hð1Þ¼0:1871

Thus delaying hðnÞ by M ¼ 1 sample using Equation (7.12) gives

b 0 ¼ hð0 1Þ¼hð1Þ¼0:1871
b 1 ¼ hð1 1Þ¼hð0Þ¼0:2

b 2 ¼ hð2 1Þ¼hð1Þ¼0:1871

b. The transfer function is achieved as
HðzÞ¼0:1871 þ 0:2z 1 þ 0:1871z 2

Using the technique described in Chapter 6, we have
Y ð z Þ
X ðzÞ ¼ HðzÞ¼0:1871 þ 0:2z 1 þ 0:1871z 2