Bilinear Transform for Digital Filter Design

Core Idea

The bilinear transform s = (2/T)(z–1)/(z+1) maps analog filter designs to digital via a conformal transformation. It preserves stability and causality, mapping the imaginary axis to the unit circle. Frequency warping compresses high analog frequencies near fs/2, but pre-warping at a specific frequency can correct this effect.

Explainer

You already have a solid analog filter design in the s-domain — perhaps a Butterworth lowpass prototype or a Chebyshev bandpass filter. You also know the z-transform and how discrete-time systems work in the z-domain. The bilinear transform is the bridge between these two worlds: a substitution rule that converts your continuous-time transfer function H(s) into a discrete-time transfer function H(z) that you can implement as a digital filter.

Why not just discretize directly? The most naive approach — replacing derivatives with finite differences, i.e., s → (z−1)/T — leads to the forward Euler method, which maps the stable left half s-plane into a small circle near z = 1, not the unit circle. Poles that were barely stable in continuous time can end up outside the unit circle in discrete time: the method is conditionally stable and creates aliasing problems. The bilinear transform avoids this by using the substitution s = (2/T)(z−1)/(z+1), which maps the entire left half s-plane to the *interior* of the unit circle and the imaginary axis (jω) to the unit circle itself. Stability is preserved by construction: any causal, stable analog filter becomes a causal, stable digital filter after the bilinear transform.

Frequency warping: the unavoidable distortion. The bilinear transform is a conformal map, but it is not linear in frequency. The relationship between the analog frequency Ω and the digital frequency ω is: Ω = (2/T)·tan(ω/2). At low frequencies, ω and Ω track each other well. But as ω approaches π (the Nyquist frequency), the analog frequency Ω stretches toward infinity — the entire semi-infinite analog frequency axis is compressed into the finite range 0 to π. This frequency warping means that if your Butterworth filter had a −3 dB point at Ω₀ in the analog domain, the digital filter's −3 dB point will not be at the corresponding digital frequency ω₀ = Ω₀·T unless you correct for this compression.

Pre-warping: the fix. The standard procedure is to pre-warp the critical frequency before designing the analog prototype. You choose the digital frequency ω_c you want (in radians per sample), compute the pre-warped analog frequency Ω_c = (2/T)·tan(ω_c/2), design the analog prototype at Ω_c, and then apply the bilinear transform. After the transform, the warping maps Ω_c exactly back to ω_c. The filter's gain is exactly correct at that one frequency; all other frequencies are still warped, but for well-designed filters (especially equiripple designs), this is acceptable. The pre-warping step is essential whenever you have a specific cutoff, notch, or center frequency requirement — it is the step most commonly omitted by beginners, leading to digital filters whose cutoff frequency is significantly off from the target.

Putting it all together: the design recipe. (1) Specify the desired digital cutoff frequency ω_c. (2) Pre-warp: compute Ω_c = (2/T)·tan(ω_c/2). (3) Design an analog prototype H_a(s) with cutoff at Ω_c using Butterworth, Chebyshev, or Elliptic design tables. (4) Apply the bilinear substitution: replace every s in H_a(s) with (2/T)(z−1)/(z+1). (5) Expand and simplify to get H(z) in standard form. The result is a digital IIR filter with the desired frequency response characteristics — stable, causal, and directly implementable as a difference equation. This design path from analog prototype to digital filter is standard in DSP toolboxes; knowing the mathematics tells you exactly what the tool is doing and lets you debug when results are unexpected.