Finite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 270

Example 5.6

Design a bandpass filter that approximates the ideal magnitude response given in
Figure 5.5(c), in which ω c 2 = 0 . 6 π and ω c 1 = 0 . 2 π . Let us select a Hamming

window of length N 11 and plot the magnitude response of the filter.

The coefficients c BP (n) of the Fourier series for the magnitude response given
are computed from formula (5.35) given below:

=

⎨
⎩

(ω c 2 − ω c 1 )

; n = 0

π

c BP (n) =

sin (ω c 2 n)
πn

sin (ω c 1 n)
πn

| | ≥ 0

− ; n

But since the Hamming window function has a length of 11, we need to compute
the coefficients c BP (n) also, from n 5 only. So also we calculate

the 11 coefficients of the Hamming window, using the formula

=− 5to n =

= 0 . 54 + 0 . 46 cos 2 πn
N

w H (n) ;− 5 ≤ n ≤ 5

Their products h w (n) c BP (n)w H (n) are computed next. The 11 coefficients

c BP (n) , w H (n) and h w (n) for − 5 ≤

=

≤ 5 are listed below. Next the coefficients
h w (n) are delayed by five samples to get the coefficients of the FIR filter function
[i.e., h(n)

n

10 below. The
plot of the four sequences and the magnitude response of the FIR are shown in
Figures 5.10 and 5.11, respectively.

= h w (n − 5 ) ], and these are also listed for 0 ≤ n ≤

c BP (n) = 0 . 00 0 . 0289 − 0 . 1633 − 0 . 2449 0 . 1156 0 . 400 0 . 1156

− 0 . 2449 − 0 . 1633 0 . 0289 0 . 000

w H (n) = 0 . 08 0 . 1679 0 . 0379 0 . 6821 0 . 9121 1 . 00 . 9121

0 . 6821 0 . 0379 0 . 1679 0 . 0800

h w (n) = 0 . 00 0 . 0049 − 0 . 0650 − 0 . 1671 0 . 1055 0 . 4000 0 . 1055

− 0 . 1671 − 0 . 0650 0 . 0049 0 . 000

h(n) 0.0000 0.0049 − 0.0650 − 0.1671 0.1055 0.4000

0.1055 − 0.1671 − 0.0650 0.0049 0.0000

Example 5.7

Design a lowpass FIR filter of length 11, with a cutoff frequency ω c 0 . 3 π .

Using a Hamming window, find the value of the samples h( 3 ) and h( 9 ) of the
FIR filter given by H(z − 1 )

=

= 1 n = 0 h(n)z − n .
Since the length of the FIR filter is given as 11, its order is N = 10. The

coefficients h w (n) have to be known for − 5 ≤ n ≤ 5 and delayed by five samples.