Q3Digital Signal Processing

The desired frequency response is an ideal lowpass filter with cutoff frequency ωc = 3π/4 and linear phase corresponding to a delay of 3 samples (from the e^(-j3ω) term):

Step 1 — find the ideal impulse response hd(n): taking the inverse Fourier transform of Hd(ejω):

This is the standard ideal-lowpass sinc impulse response, centered (peaking) at n=3 due to the linear-phase delay term, and is symmetric about n=3, i.e., hd(n) = hd(6-n), consistent with a Type I linear-phase design for M=7 taps (n=0 to 6).

Step 2 — evaluate hd(n) at n=0,...,6 using the formula above (with the value at n=3 found by L'Hôpital's rule / direct limit, since the sinc formula is 0/0 there):

Step 3 — apply the rectangular window: the rectangular window is defined as w(n) = 1 for 0 ≤ n ≤ M-1 = 6, and w(n) = 0 otherwise, which for M=7 simply truncates hd(n) to the range n=0 to 6 without any additional tapering (since w(n)=1 throughout this range, it does not modify the values, only limits the summation/support to these 7 samples). Therefore the designed FIR filter coefficients are h(n) = hd(n)·w(n) = hd(n) for n=0,...,6:

The resulting filter H(ejω) is the actual frequency response obtained by taking the DTFT of this truncated (windowed) impulse response, which approximates the ideal 'brick-wall' lowpass response Hd(ejω) but exhibits the well-known Gibbs phenomenon — ripples near the cutoff frequency (both in passband and stopband) caused by the abrupt truncation inherent to the rectangular window, since a rectangular window in the time domain corresponds to a Dirichlet-kernel convolution in the frequency domain, which does not decay quickly enough to avoid noticeable ringing artifacts near the sharp transition in Hd. This ripple could be reduced (at the cost of a wider transition band) by instead using a tapered window such as the Hamming window discussed elsewhere in this paper.

Interpretation of the resulting filter shape: the computed coefficients {0.0750, -0.1592, 0.2251, 0.75, 0.2251, -0.1592, 0.0750} exhibit the expected symmetric taper around the center tap h(3)=0.75 (the largest coefficient, corresponding to the peak of the sinc function at its center), with alternating-sign, decaying-magnitude side coefficients that are the hallmark of a truncated sinc/lowpass impulse response; the negative side-lobe values (h(1)=h(5)≈-0.1592) directly reflect the sidelobes of the underlying sinc function being sampled, and their relatively large magnitude (compared to what a tapered window would produce) is exactly why the rectangular-windowed design exhibits more pronounced Gibbs-phenomenon ripple in its frequency response than a Hamming- or Blackman-windowed design of the same length would.

General design procedure summary: this problem illustrates the complete classical window-design method for FIR filters: (1) specify the ideal (generally non-causal, infinite-duration) desired frequency response Hd(ejω), including any desired linear-phase delay; (2) compute its inverse DTFT to obtain the ideal impulse response hd(n), typically a sinc-based function; (3) select a window function w(n) of the desired finite length M matched to the required transition-bandwidth and ripple specifications; and (4) multiply hd(n) by w(n) to obtain the final, realizable FIR filter coefficients h(n) = hd(n)w(n). This same four-step procedure applies regardless of which specific window (rectangular, Hamming, Hanning, Blackman, Kaiser) is chosen, with the window choice only affecting step 3 and consequently the specific ripple/transition-bandwidth trade-off achieved in the final design.