Finite Impulse Response Digital Filter Design - Practical Analog and Digital Filter Design - page 173

A = − 20 ⋅ log[min( δ , δ )] (7.30)

p s

In addition, Kaiser developed empirical estimates of the filter length required
to satisfy a given set of filter specifications, as indicated in (7.31). It should be
emphasized here that FIR filter design is not as precise as IIR design. The
truncation/modification of coefficients results in responses that may or may not
meet the requirements. Therefore, the N value of (7.31) is just an estimate and the
filter responses must be checked carefully to determine if all requirements are met.
If they are not, the value on N should be adjusted (usually up, but sometimes
decreasing N can result in a better filter).

A − 7 95

⎧

, for A > 21

⎪
⎨

2 285 ⋅ ∆Ω

N =

5 794

⎩

, for A < 21

∆Ω

(7.31)

In (7.31), ∆Ω represents the normalized radian transition band for lowpass
and highpass filters and the smaller of the two normalized transition bands in the
case of bandpass and bandstop filters.

∆Ω = ω − ω / f

(7.32)

stop pass s

Once the desired window function has been selected and the adjustments
made to the ideal coefficients, the causal coefficients can be determined as
indicated in the previous section.

Example 7.2 Determining Hamming Coefficients for an FIR Filter

Problem: Determine the coefficients for a lowpass filter using a Hamming
window of length 21 to satisfy the specifications shown below:
ω pass = 2π⋅3,000 rad/sec, ω stop = 2π⋅4,000 rad/sec, and f s = 20 kHz
Solution: The ideal coefficients have been determined in Example 7.1. We
can use (7.24) to determine the noncausal Hamming window coefficients as
shown. After multiplication and shifting the coefficients by 10 sampling periods,
the causal windowed coefficients result.
Figure 7.11 shows the frequency response for both Examples 7.1 and 7.2. The
rectangular window produces a filter that emphasizes transition band roll-off over
ripple in the stopband. On the other hand, the filter produced by using Hamming
coefficients has no noticeable ripple, but does not have a rapid roll-off in the
transition band.