How it works
For IIR design via bilinear transform, step one is pre-warping the digital cutoff: Ω_a = 2f_s·tan(πf_c/f_s). With f_c = 1 kHz and f_s = 8 kHz, that gives Ω_a = 2×8000×tan(π/8) ≈ 6628 rad/s. Next, find the required Butterworth filter order: N ≥ log(10^(A_s/10)−1)/(10^(A_p/10)−1)) / (2·log(Ω_s/Ω_p)). Round N up to the nearest integer. Determine the 3 dB cutoff Ω_c, form H_a(s), then substitute the bilinear transform. For FIR window design, simply calculate ideal h_d[n], multiply by window coefficients, and apply bit-reversal if needed for FFT-based implementation.
Key points to remember
Filter order N directly controls roll-off steepness: a 1st-order Butterworth falls at −20 dB/decade, each additional order adds another −20 dB/decade. Chebyshev Type I achieves the same stopband attenuation as Butterworth with a lower order, but introduces ripple up to the specified δ_p in the passband. Elliptic filters minimise order for given passband and stopband specs but have nonlinear phase. Direct form II transposed is the preferred structure for IIR implementation in fixed-point DSPs because it minimises delay memory. Linear phase — achievable only in FIR — is mandatory in applications like data modems and medical imaging where phase distortion corrupts the signal.
Exam tip
Every Anna University DSP exam has a 16-mark filter design question — practice the full design procedure for both Butterworth IIR using bilinear transform and FIR using Hamming window, because the examiner picks one or the other and gives partial credit for each correctly completed step.