Bilinear transformation

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The bilinear transformation is the substitution

The bilinear transformation

used to convert the representation of a continuous (or "analog") system into the representation of a discrete (or "digital") system.

For example, the second order low pass Butterworth filter with cutoff frequency ωc is usually given by the Laplace transform transfer function

Laplace transform transfer function of the second order low pass Butterworth filter

With the bilinear transformation, the Z transform transfer function of the same filter is

Z transform transfer function of the second order low pass Butterworth filter

Scaling to obtain 1 in the denominator, we have

Scaled Z transform transfer function of the second order low pass Butterworth filter

and the impulse response of the filter (its actual implementation in practice) is

Impulse response of the second order low pass Butterworth filter

where x(n) are the samples of the input signal and y(n) are the samples of the output signal.

Derivation of the bilinear transformation

The bilinear transformation follows from the Taylor series expansion of the function esT/2.

Deriving the bilinear transformation

Frequency warping of the bilinear transformation

Note that the properties of the resulting filter are evaluated at s = -jω or z = e-jω. However,

Warping of the frequency domain with the bilinear transformation

Thus, the discrete filter H(z), obtained after the bilinear transformation of H(s), will behave, at the discrete frequency ωd, the same way the continuous filter H(s) behaves at the frequency ωa = 2 tan(ωd/2). (Alternatively, ωd = 2 arctan(ωa/2)).

Take, for example, the Butterworth filter above. Its magnitude response, computed from H(s), is

Magnitude response of the second order low pass Butterworth filter with the Laplace transform

The magnitude response of the filter after bilinear transformation can be computed from

Computing the magnitude response of the second order low pass Butterworth filter with the Z transform after the bilinear transformation

and is

Magnitude response of the second order low pass Butterworth filter with the Z transform after the bilinear transformation

This warping of the frequency domain is small for small ω and increases as ω increases (ω is between 0 and π). At ωa = 1, for example, ωd ≈ 0.927.

When designing digital filters, this frequency warping can be remedied in one of two ways. First, a discrete filter with the cutoff frequency ωd can be designed with the bilinear transformation on the continuous filter with the cutoff frequency ωa = 2 tan(ωd/2). Alternatively, instead of the bilinear transformation, one can use the biquad transformation

Biquad transform

For example, the second order low-boost shelving filter with cutoff frequency ωc and gain G

Transfer function of the second order low-boost shelving filter with the Laplace transform

can be written with the bilinear transformation as

Transfer function of the second order low-boost shelving filter with the Z transform after the bilinear transformation

or with the biquad transform as

Transfer function of the second order low-boost shelving filter with the Z transform after the biquad transformation

The corresponding magnitude responses for the example cutoff frequency ωc = 2 in both cases, K = tan(ωc / 2), sampling rate of 2000 Hz, and gain of G = 2 (≈ 6 dB) are below.

Magnitude response of the second order low-boost shelving filter with the bilinear and biquad transformations

Filter stability and the bilinear transformation

Note that, for s = σ – j ω, if Re(s) = σ < 0, then

The bilinear transformation preserves stability

Thus, if we have the Laplace transform transfer function of a stable filter with roots of the denominator in the left part of the s- complex plane, the transfer function that we will obtain with the bilinear transformation would have roots that are inside the unit circle and the filter will still be stable. The bilinear transformation preserves stability.



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