FIR filter design - Continuous and Discrete Time Signals and Systems - page 687

Fig. 15.13. Magnitude response
of the highpass FIR filter
designed in Example 15.5.

20 log 10 | H ( W )|

0

−20

−40

−60

W

0.25 p 0.5 p 0.75 p p

0

10 0 . 01 / 20 − 1 = 0 . 0012, while δ s should be less than 10

− 60 / 20 − 1 = 0 . 001. The
minimum attenuation A is therefore given by A = min(0 . 0012 , 0 . 001) = 0 . 001,
or 60 dB.
The shape parameter is evaluated from Eq. (15.20) as follows:

β = 0 . 1102( A − 8 . 7) = 5 . 6533 .

The transition band Ω c for the FIR filter is Ω p = 0 . 375 π . The nor-

malized transition band Ω n is therefore given by Ω c /π

−

Ω s

= 0 . 375. Using

Ω n = 0 . 375, the length N of the Kaiser window is given by

60 − 7 . 95
2 . 285 π 0 . 375

N ≥

= 19 . 3354 .

Rounding off to the higher closest odd number, we obtain N = 21.
The expression for the Kaiser window is given by

1 − [( k − 10) / 10] 2

I 0

5 . 6533

0 ≤ k ≤ 20

w kaiser [ k ] = (15.30)

I 0 [5 . 6533]

0 otherwise.

The impulse response of the highpass FIR filter is given by

h hp [ k ] = h ihp [ k ] w kaiser [ k ] ,

where h ihp [ k ] is specified in Eq. (15.29) with m = 10 and w kaiser [ k ]isgivenin
Eq. (15.30). The filter gain at Ω = π is given by

N − 1 N − 1

H hp ( π ) = h hp [ k ] − h hp [ k ] = 1 . 0002 .

k = 0 , 2 ,... k = 1 , 3 ,...

As H ( π ) ≈ 1, the coefficients of h [ k ] need not be normalized.
The magnitude response of the highpass FIR filter is plotted in Fig. 15.13,
which verifies that the initial specifications of the filter are satisfied.

In Example 15.5 we designed a highpass FIR filter directly from the given spec-
ifications. An alternative procedure to design a highpass FIR filter is to exploit