Describe FIR design by window method

The window method designs an FIR filter by truncating the infinite impulse response of an ideal filter with a finite-length window.

Steps

1. Specify the ideal response H_d(e^jω), the pass-band edge ωp, the stop-band edge ωs and the stop-band attenuation αs.
2. Choose the cut-off ωc = (ωp + ωs)/2 for a low-pass filter.
3. Find the ideal impulse response by the inverse DTFT: h_d(n) = (1/2π) ∫ H_d(e^jω) e^{jωn} dω = sin(ωc(n − α)) / (π(n − α)), α = (M−1)/2 (for high-pass: δ(n−α) − sin(ωc(n−α))/(π(n−α)); for band-pass: [sin(ω₂(n−α)) − sin(ω₁(n−α))]/(π(n−α)).)
4. Choose a window from αs: rectangular (about 21 dB), Bartlett (25 dB), Hanning (44 dB), Hamming (53 dB), Blackman (74 dB), or Kaiser for any attenuation.
5. Choose the length M from the transition width Δω = ωs − ωp and the window's main-lobe width (for example, for Hamming and Hanning: M ≈ 8π/Δω). Take M odd for a Type-I filter, so that α is an integer.
6. Form the filter: h(n) = h_d(n) · w(n), 0 ≤ n ≤ M−1. The shift by α makes it causal.
7. Check the frequency response H(e^jω) against the specification; if it fails, increase M or use a better window.

Remarks

• Because h(n) = h(M−1−n), the phase is exactly linear: θ(ω) = −αω.
• Windowing multiplies in time and convolves in frequency, so the sharp edge of H_d is smoothed (a transition band appears) and ripples appear (Gibbs effect).
• Trade-off: a narrow main lobe gives a sharp transition; low side lobes give high attenuation. For a fixed length, a window cannot optimise both at once.

The method is simple and gives linear-phase, always-stable filters, but the pass-band and stop-band ripples cannot be controlled independently.