Example: Design of 9-coefficient LP FIR filter: Design a nine-coefficient (or 9-point or 9-tap) FIR digital filter to related an ideal low-pass filter with a cut-off frequency ω_c= 0.2p. The value response, ∠H_d(ω),is provided below. Take H_d(ω) = 0.

Solution The impulse response of the provided filter is 

Since h_d(n) ≠ 0 for n < 0 that is a noncausal filter. The rest of the design is aimed at taking up with a noncausal relation of the above impulse response.

For a rectangular window of size 9, the related impulse response is calculated by evaluating h_d(n) for -4 ≤ n ≤ 4 on a computer. In the MATLAB segment below, which creates the h_d(n) coefficients for -50 ≤ n ≤ 50, division by zero for n = 0 provides "NaN" (Not a Number), while all the other coefficients are correct. In preparing the frequency response |H(ω)| we paste and copy all the coefficients except for h_d(0) which is shown by hand.

Aside (MATLAB) The segment below creates and stem-plots the hd(n) coefficients.

%Calculate hdn = (sin (0.2*pi*n)) / (pi*n) and stem plot n = -50: 50, hdn = (sin(0.2*pi*n)) ./(pi*n), stem(n, hdn)

xlabel('n'), ylabel('hd(n)'); grid; title ('hd(n) = (sin (0.2*pi*n)) / (pi*n)')

n = -50 to 50

Warning: Divide by zero.

hdn = -0, -0.0038, -0.0063, -0.0064, -0.0041, 0, 0.0043    0.0070    0.0072,

0.0046   -0, -0.0048   -0.0080   -0.0082   -0.0052     0, 0.0055    0.0092,     0.0095

0.0060   -0, -0.0065   -0.0108   -0.0112   -0.0072     0, 0.0078,     0.0132    0.0138

0.0089   -0, -0.0098   -0.0168   -0.0178   -0.0117     0,    0.0134    0.0233    0.0252

0.0170   -0, -0.0208   -0.0378   -0.0432   -0.0312,      0, 0.0468    0.1009    0.1514

0.1871       NaN    0.1871    0.1514    0.1009, 0.0468    0.0000   -0.0312   -0.0432

-0.0378   -0.0208   -0.0000    0.0170    0.0252, 0.0233    0.0134    0.0000   -0.0117

-0.0178   -0.0168   -0.0098   -0.0000    0.0089, 0.0138    0.0132    0.0078    0.0000

-0.0072   -0.0112   -0.0108   -0.0065   -0.0000, 0.0060     0.0095    0.0092    0.0055

0.0000   -0.0052   -0.0082   -0.0080   -0.0048, -0.0000     0.0046    0.0072    0.0070

0.0043    0.0000   -0.0041   -0.0064   -0.0063, -0.0038   -0.0000

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