b 0:110280 À 8:7 7:875:
The required window length L is computed as
80 À 7:5910
14:362:5 À 2
1 % 101:85:
Thus we choose the filter order L 103.
The procedures of designing FIR filters using windows are summarized as follows:
1. Determine the window type that will satisfy the stopband attenuation requirements.
2. Determine the window size L based on the given transition width.
3. Calculate the window coefficients w(n), n 0, 1, . . . , L À 1.
4. Generate the ideal impulse response h(n) using (5.3.3) for the desired filter.
5. Truncate the ideal impulse response of infinite length using (5.3.5) to obtain
n, À M n M.
6. Make the filter causal by shifting the result M units to the right using (5.3.7) to
, l 0, 1, . . . , L À 1.
7. Multiply the window coefficients obtained in step 3 and the impulse response
coefficients obtained in step 6 sample-by-sample. That is,
Á wl, l 0, 1, . . . , L À 1:
Applying a window to an FIR filter's impulse response has the effect of smoothing the
resulting filter's magnitude response. A symmetric window will preserve a symmetric
FIR filter's linear-phase response.
The advantage of the Fourier series method with windowing is its simplicity. It does
not require sophisticated mathematics and can be carried out in a straightforward
manner. However, there is no simple rule for choosing M so that the resulting filter
will meet exactly the specified cut-off frequencies. This is due to the lack of an exact
relation between M and its effect on leakage.
5.3.5 Frequency Sampling Method
The frequency-sampling method is based on sampling a desired amplitude spectrum and
obtaining filter coefficients. In this approach, the desired frequency response H! is
first uniformly sampled at L equally spaced points !
L , k 0, 1, . . . , L À 1. The
frequency-sampling technique is particularly useful when only a small percentage of the
DESIGN AND IMPLEMENTATION OF FIR FILTERS