FIR filter design - Continuous and Discrete Time Signals and Systems - page 672

h hamm [ k ]

h rect [ k ]

0.3183 0.3183

k

k

12 14

24 10 16 18 20 2 10 12 14 16 18 20

(a) (b)

Solution
Substituting Ω c

Fig. 15.4. Impulse response of
FIR filters obtained by truncating
the impulse response of the
ideal lowpass filter with (a) a
rectangular window and (b) a
Hamming window.

= 1 in Eq. (14.12), the impulse response of an ideal lowpass

filter is given by

h ilp [ k ] = sin( k − m )
( k − m ) π

= 1
π

k − m
π

sinc , (15.12)

where m = ( N − 1) / 2 = 10. The expressions for the rectangular and Hamming
windows with 21 taps are as follows:

10 ≤ k ≤ 20
0 otherwise;

rectangular window w rect [ k ] = (15.13)

2 π k
20

0 . 54 − 0 . 46 cos

0 ≤ k ≤ 20

Hamming window w hamm [ k ] =

0 otherwise.
(15.14)
The FIR filters are obtained by multiplying the impulse response of the ideal
lowpass filter by the expressions for the rectangular and Hamming windows.
The resulting impulse responses are as follows:

1
π

k − 10
π

sinc 0 ≤ k ≤ 20

rectangular window h rect [ k ] = (15.15)

0 otherwise;

Hamming window h hamm [ k ]

0 . 54 − 0 . 46 cos

1
π

( k − 10)
π

2 π k
20

sinc 0 ≤ k ≤ 20

= (15.16)

0 otherwise.

The impulse responses for FIR filters obtained by truncating the ideal lowpass
filter impulse response with the rectangular and Hamming windows are shown in
Fig. 15.4. Although the two impulse responses have the same value at k = 10,
the impulse response h hamm [ k ], shown in Fig. 15.4(b), decays more rapidly
as we move away from the central point ( k = 10) and is different from the
impulse response h rect [ k ], shown in Fig. 15.4(a). Typically, the pass-band gain
of the truncated FIR filters, obtained from the ideal lowpass filters using the
windowing method, is not unity, as desired. To prove this, we calculate the value