Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 242

the filter length using the Hamming window can be determined by

3 : 3
Df

N ¼ (7.23)

Note that the passband ripple is defined as

d p dB ¼ 20 $ log 10 ð 1 þ d p Þ (7.24)

while the stopband attenuation is defined as

d s dB ¼ 20 log 10 ðd s Þ

(7.25)

The cutoff frequency used for the design will be chosen at the middle of the transition band, as
illustrated for the lowpass filter case shown in Figure 7.14 .
As a rule of thumb, the cutoff frequency used for design is determined by

f c ¼ðf pass þ f stop Þ= 2 (7.26)

Note that Equation (7.23) and formulas for other window lengths in Table 7.7 are empirically derived
based on the normalized spectral transition width of each window function. The spectrum of each
window function appears to be shaped like the lowpass filter magnitude frequency response with
ripples in the passband and side lobes in the stopband. The passband frequency edge of the spectrum
is the frequency where the magnitude just begins to drop below the passband ripple and where the
stop frequency edge is at the peak of the first side lobe in the spectrum. With the passband ripple and
stopband attenuation specified for a particular window, the normalized transition width of the
window is in inverse proportion to the window length N multiplied by a constant. For example, the
normalized spectral transition Df for the Hamming window is 3 : 3 =N . Hence, matching the FIR filter
transition width with the transition width of the window spectrum gives the filter length estimation
listed in Table 7.7 .
The following examples illustrate the determination of each filter length and cutoff frequency/
frequencies for the design of lowpass, highpass, bandpass, and bandstop filters. Application of each
designed filter to the processing of speech data is included, along with an illustration of filtering effects
in both the time domain and frequency domain.

EXAMPLE 7.8
A lowpass FIR filter has the following specifications:
Passband 0e1,850 Hz
Stopband 2,150e4,000 Hz
Stopband attenuation 20 dB
Passband ripple 1 dB
Sampling rate 8,000 Hz

Determine the FIR filter length and the cutoff frequency to be used in the design equation.