Bilinear transform

Signal processing operation

The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used for converting a transfer function $H_{a}(s)$ of a linear, time-invariant (LTI) filter in the continuous-time domain (often named an analog filter) to a transfer function $H_{d}(z)$ of a linear, shift-invariant filter in the discrete-time domain (often named a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the $j\omega$ axis, $\mathrm {Re} [s]=0$ , in the s-plane to the unit circle, $|z|=1$ , in the z-plane. Other bilinear transforms can be used for warping the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays $\left(z^{-1}\right)$ with first order all-pass filters.

The transform preserves stability and maps every point of the frequency response of the continuous-time filter, $H_{a}(j\omega _{a})$ to a corresponding point in the frequency response of the discrete-time filter, $H_{d}(e^{j\omega _{d}T})$ although to a somewhat different frequency, as shown in the Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency.

Discrete-time approximation

The bilinear transform is a first-order Padé approximant of the natural logarithm function that is an exact mapping of the z-plane to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of

{\begin{aligned}z&=e^{sT}\\&={\frac {e^{sT/2}}{e^{-sT/2}}}\\&\approx {\frac {1+sT/2}{1-sT/2}}\end{aligned}}

where $T$ is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation;^[1] or, in other words, the sampling period. The above bilinear approximation can be solved for $s$ or a similar approximation for $s=(1/T)\ln(z)$ can be performed.

The inverse of this mapping (and its first-order bilinear approximation) is

{\begin{aligned}s&={\frac {1}{T}}\ln(z)\\&={\frac {2}{T}}\left[{\frac {z-1}{z+1}}+{\frac {1}{3}}\left({\frac {z-1}{z+1}}\right)^{3}+{\frac {1}{5}}\left({\frac {z-1}{z+1}}\right)^{5}+{\frac {1}{7}}\left({\frac {z-1}{z+1}}\right)^{7}+\cdots \right]\\&\approx {\frac {2}{T}}{\frac {z-1}{z+1}}\\&={\frac {2}{T}}{\frac {1-z^{-1}}{1+z^{-1}}}\end{aligned}}

The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, $H_{a}(s)$

s\leftarrow {\frac {2}{T}}{\frac {z-1}{z+1}}.

That is

H_{d}(z)=H_{a}(s){\bigg |}_{s={\frac {2}{T}}{\frac {z-1}{z+1}}}=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right).\

Stability and minimum-phase property preserved

A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane. A discrete-time causal filter is stable if the poles of its transfer function fall inside the unit circle in the complex z-plane. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.

Likewise, a continuous-time filter is minimum-phase if the zeros of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.

Transformation of a General LTI System

A general LTI system has the transfer function

H_{a}(s)={\frac {b_{0}+b_{1}s+b_{2}s^{2}+\cdots +b_{Q}s^{Q}}{a_{0}+a_{1}s+a_{2}s^{2}+\cdots +a_{P}s^{P}}}

The order of the transfer function N is the greater of P and Q (in practice this is most likely P as the transfer function must be proper for the system to be stable). Applying the bilinear transform

s=K{\frac {z-1}{z+1}}

where K is defined as either 2/T or otherwise if using frequency warping, gives

H_{d}(z)={\frac {b_{0}+b_{1}\left(K{\frac {z-1}{z+1}}\right)+b_{2}\left(K{\frac {z-1}{z+1}}\right)^{2}+\cdots +b_{Q}\left(K{\frac {z-1}{z+1}}\right)^{Q}}{a_{0}+a_{1}\left(K{\frac {z-1}{z+1}}\right)+a_{2}\left(K{\frac {z-1}{z+1}}\right)^{2}+\cdots +b_{P}\left(K{\frac {z-1}{z+1}}\right)^{P}}}

Multiplying the numerator and denominator by the largest power of (z + 1)⁻¹ present, (z + 1)^-N, gives

H_{d}(z)={\frac {b_{0}(z+1)^{N}+b_{1}K(z-1)(z+1)^{N-1}+b_{2}K^{2}(z-1)^{2}(z+1)^{N-2}+\cdots +b_{Q}K^{Q}(z-1)^{Q}(z+1)^{N-Q}}{a_{0}(z+1)^{N}+a_{1}K(z-1)(z+1)^{N-1}+a_{2}K^{2}(z-1)^{2}(z+1)^{N-2}+\cdots +a_{P}K^{P}(z-1)^{P}(z+1)^{N-P}}}

It can be seen here that after the transformation, the degree of the numerator and denominator are both N.

Consider then the pole-zero form of the continuous-time transfer function

H_{a}(s)={\frac {(s-\xi _{1})(s-\xi _{2})\cdots (s-\xi _{Q})}{(s-p_{1})(s-p_{2})\cdots (s-p_{P})}}

The roots of the numerator and denominator polynomials, ξ_i and p_i, are the zeros and poles of the system. The bilinear transform is a one-to-one mapping, hence these can be transformed to the z-domain using

z={\frac {K+s}{K-s}}

yielding some of the discretized transfer function's zeros and poles ξ'_i and p'_i

{\begin{aligned}\xi '_{i}&={\frac {K+\xi _{i}}{K-\xi _{i}}}\quad 1\leq i\leq Q\\p'_{i}&={\frac {K+p_{i}}{K-p_{i}}}\quad 1\leq i\leq P\end{aligned}}

As described above, the degree of the numerator and denominator are now both N, in other words there is now an equal number of zeros and poles. The multiplication by (z + 1)^-N means the additional zeros or poles are ^[2]

{\begin{aligned}\xi '_{i}&=-1\quad Q<i\leq N\\p'_{i}&=-1\quad P<i\leq N\end{aligned}}

Given the full set of zeros and poles, the z-domain transfer function is then

H_{d}(z)={\frac {(z-\xi '_{1})(z-\xi '_{2})\cdots (z-\xi '_{N})}{(z-p'_{1})(z-p'_{2})\cdots (z-p'_{N})}}

Example

As an example take a simple low-pass RC filter. This continuous-time filter has a transfer function

{\begin{aligned}H_{a}(s)&={\frac {1/sC}{R+1/sC}}\\&={\frac {1}{1+RCs}}.\end{aligned}}

If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for $s$ the formula above; after some reworking, we get the following filter representation:

$H_{d}(z)\$	$=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)\$
	$={\frac {1}{1+RC\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)}}\$
	$={\frac {1+z}{(1-2RC/T)+(1+2RC/T)z}}\$
	$={\frac {1+z^{-1}}{(1+2RC/T)+(1-2RC/T)z^{-1}}}.\$

The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used for implementing a real-time digital filter.

Transformation for a general first-order continuous-time filter

It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, first-order continuous-time filter with the given transfer function

H_{a}(s)={\frac {b_{0}s+b_{1}}{a_{0}s+a_{1}}}={\frac {b_{0}+b_{1}s^{-1}}{a_{0}+a_{1}s^{-1}}}

using the bilinear transform (without prewarping any frequency specification) requires the substitution of

s\leftarrow K{\frac {1-z^{-1}}{1+z^{-1}}}

where

K\triangleq {\frac {2}{T}}

However, if the frequency warping compensation as described below is used in the bilinear transform, so that both analog and digital filter gain and phase agree at frequency $\omega _{0}$ , then

K\triangleq {\frac {\omega _{0}}{\tan \left({\frac {\omega _{0}T}{2}}\right)}}

This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter:

H_{d}(z)={\frac {(b_{0}K+b_{1})+(-b_{0}K+b_{1})z^{-1}}{(a_{0}K+a_{1})+(-a_{0}K+a_{1})z^{-1}}}

Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation. This results in

H_{d}(z)={\frac {{\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}z^{-1}}{1+{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}z^{-1}}}.

The difference equation (using the Direct form I) is

y[n]={\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n]+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n-1]-{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}\cdot y[n-1]\ .

General second-order biquad transformation

A similar process can be used for a general second-order filter with the given transfer function

H_{a}(s)={\frac {b_{0}s^{2}+b_{1}s+b_{2}}{a_{0}s^{2}+a_{1}s+a_{2}}}={\frac {b_{0}+b_{1}s^{-1}+b_{2}s^{-2}}{a_{0}+a_{1}s^{-1}+a_{2}s^{-2}}}\ .

This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter:

H_{d}(z)={\frac {(b_{0}K^{2}+b_{1}K+b_{2})+(2b_{2}-2b_{0}K^{2})z^{-1}+(b_{0}K^{2}-b_{1}K+b_{2})z^{-2}}{(a_{0}K^{2}+a_{1}K+a_{2})+(2a_{2}-2a_{0}K^{2})z^{-1}+(a_{0}K^{2}-a_{1}K+a_{2})z^{-2}}}

Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding difference equation. This results in

H_{d}(z)={\frac {{\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-1}+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-2}}{1+{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-1}+{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-2}}}.

The difference equation (using the Direct form I) is

y[n]={\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n]+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-1]+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-2]-{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-1]-{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-2]\ .

Frequency warping

To determine the frequency response of a continuous-time filter, the transfer function $H_{a}(s)$ is evaluated at $s=j\omega _{a}$ which is on the $j\omega$ axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function $H_{d}(z)$ is evaluated at $z=e^{j\omega _{d}T}$ which is on the unit circle, $|z|=1$ . The bilinear transform maps the $j\omega$ axis of the s-plane (which is the domain of $H_{a}(s)$ ) to the unit circle of the z-plane, $|z|=1$ (which is the domain of $H_{d}(z)$ ), but it is not the same mapping $z=e^{sT}$ which also maps the $j\omega$ axis to the unit circle. When the actual frequency of $\omega _{d}$ is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, $\omega _{a}$ , for the continuous-time filter that this $\omega _{d}$ is mapped to.

H_{d}(z)=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)

$H_{d}(e^{j\omega _{d}T})$	$=H_{a}\left({\frac {2}{T}}{\frac {e^{j\omega _{d}T}-1}{e^{j\omega _{d}T}+1}}\right)$
	$=H_{a}\left({\frac {2}{T}}\cdot {\frac {e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)$
	$=H_{a}\left({\frac {2}{T}}\cdot {\frac {\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)$
	$=H_{a}\left(j{\frac {2}{T}}\cdot {\frac {\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)/(2j)}{\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)/2}}\right)$
	$=H_{a}\left(j{\frac {2}{T}}\cdot {\frac {\sin(\omega _{d}T/2)}{\cos(\omega _{d}T/2)}}\right)$
	$=H_{a}\left(j{\frac {2}{T}}\cdot \tan \left(\omega _{d}T/2\right)\right)$

This shows that every point on the unit circle in the discrete-time filter z-plane, $z=e^{j\omega _{d}T}$ is mapped to a point on the $j\omega$ axis on the continuous-time filter s-plane, $s=j\omega _{a}$ . That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is

\omega _{a}={\frac {2}{T}}\tan \left(\omega _{d}{\frac {T}{2}}\right)

and the inverse mapping is

\omega _{d}={\frac {2}{T}}\arctan \left(\omega _{a}{\frac {T}{2}}\right).

The discrete-time filter behaves at frequency $\omega _{d}$ the same way that the continuous-time filter behaves at frequency $(2/T)\tan(\omega _{d}T/2)$ . Specifically, the gain and phase shift that the discrete-time filter has at frequency $\omega _{d}$ is the same gain and phase shift that the continuous-time filter has at frequency $(2/T)\tan(\omega _{d}T/2)$ . This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency. For low frequencies (that is, when $\omega _{d}\ll 2/T$ or $\omega _{a}\ll 2/T$ ), then the features are mapped to a slightly different frequency; $\omega _{d}\approx \omega _{a}$ .

One can see that the entire continuous frequency range

-\infty <\omega _{a}<+\infty

is mapped onto the fundamental frequency interval

-{\frac {\pi }{T}}<\omega _{d}<+{\frac {\pi }{T}}.

The continuous-time filter frequency $\omega _{a}=0$ corresponds to the discrete-time filter frequency $\omega _{d}=0$ and the continuous-time filter frequency $\omega _{a}=\pm \infty$ correspond to the discrete-time filter frequency $\omega _{d}=\pm \pi /T.$

One can also see that there is a nonlinear relationship between $\omega _{a}$ and $\omega _{d}.$ This effect of the bilinear transform is called frequency warping. The continuous-time filter can be designed to compensate for this frequency warping by setting $\omega _{a}={\frac {2}{T}}\tan \left(\omega _{d}{\frac {T}{2}}\right)$ for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called pre-warping the filter design.

It is possible, however, to compensate for the frequency warping by pre-warping a frequency specification $\omega _{0}$ (usually a resonant frequency or the frequency of the most significant feature of the frequency response) of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system. When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at a specified frequency $\omega _{0}$ , as well as matching at DC, if the following transform is substituted into the continuous filter transfer function.^[3] This is a modified version of Tustin's transform shown above.

s\leftarrow {\frac {\omega _{0}}{\tan \left({\frac {\omega _{0}T}{2}}\right)}}{\frac {z-1}{z+1}}.

However, note that this transform becomes the original transform

s\leftarrow {\frac {2}{T}}{\frac {z-1}{z+1}}

as $\omega _{0}\to 0$ .

The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance.

References

^ Oppenheim, Alan (2010). Discrete Time Signal Processing Third Edition. Upper Saddle River, NJ: Pearson Higher Education, Inc. p. 504. ISBN 978-0-13-198842-2.
^ Bhandari, Ayush. "DSP and Digital Filters Lecture Notes" (PDF). Archived from the original (PDF) on 3 March 2022. Retrieved 16 August 2022.
^ Astrom, Karl J. (1990). Computer Controlled Systems, Theory and Design (Second ed.). Prentice-Hall. p. 212. ISBN 0-13-168600-3.