Bilinear transform

Short description: Signal processing operation

File:Bilinear Transform Mapping with Shading.svg

How the bilinear transform maps the z-plane to the s-plane. The unstable regions for the poles of a linear control system are shaded.

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 $j ω$ axis, $R e [s] = 0$ , 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 $(z^{- 1})$ with first order all-pass filters.

The transform preserves stability and maps every point of the frequency response of the continuous-time filter, $H_{a} (j ω_{a})$ to a corresponding point in the frequency response of the discrete-time filter, $H_{d} (e^{j ω_{d} T})$ 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. The change in frequency 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^{s T} \\ = \frac{e^{s T / 2}}{e^{- s T / 2}} \\ \approx \frac{1 + s T / 2}{1 - s T / 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} [\frac{z - 1}{z + 1} + \frac{1}{3} {(\frac{z - 1}{z + 1})}^{3} + \frac{1}{5} {(\frac{z - 1}{z + 1})}^{5} + \frac{1}{7} {(\frac{z - 1}{z + 1})}^{7} + \dots] \\ \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) |_{s = \frac{2}{T} \frac{z - 1}{z + 1}} = H_{a} (\frac{2}{T} \frac{z - 1}{z + 1}) .

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 $H_{a} (s) = \frac{b_{0} + b_{1} s + b_{2} s^{2} + \dots + b_{Q} s^{Q}}{a_{0} + a_{1} s + a_{2} s^{2} + \dots + a_{P} s^{P}}$ 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 $H_{d} (z) = \frac{b_{0} + b_{1} (K \frac{z - 1}{z + 1}) + b_{2} {(K \frac{z - 1}{z + 1})}^{2} + \dots + b_{Q} {(K \frac{z - 1}{z + 1})}^{Q}}{a_{0} + a_{1} (K \frac{z - 1}{z + 1}) + a_{2} {(K \frac{z - 1}{z + 1})}^{2} + \dots + b_{P} {(K \frac{z - 1}{z + 1})}^{P}}$ Multiplying the numerator and denominator by the largest power of $(z + 1) -1$ present, $(z + 1) - N$ , gives $H_{d} (z) = \frac{b_{0} (z + 1)^{N} + b_{1} K (z - 1) (z + 1)^{N - 1} + b_{2} K^{2} (z - 1)^{2} (z + 1)^{N - 2} + \dots + b_{Q} K^{Q} (z - 1)^{Q} (z + 1)^{N - Q}}{a_{0} (z + 1)^{N} + a_{1} K (z - 1) (z + 1)^{N - 1} + a_{2} K^{2} (z - 1)^{2} (z + 1)^{N - 2} + \dots + a_{P} K^{P} (z - 1)^{P} (z + 1)^{N - P}}$ 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 $H_{a} (s) = \frac{(s - ξ_{1}) (s - ξ_{2}) \dots (s - ξ_{Q})}{(s - p_{1}) (s - p_{2}) \dots (s - p_{P})}$ 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 $p' i$ $\begin{aligned} ξ'_{i} & = \frac{K + ξ_{i}}{K - ξ_{i}} 1 \leq i \leq Q \\ p'_{i} & = \frac{K + p_{i}}{K - p_{i}} 1 \leq i \leq P \end{aligned}$ 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] $\begin{aligned} ξ'_{i} & = - 1 Q < i \leq N \\ p'_{i} & = - 1 P < i \leq N \end{aligned}$ Given the full set of zeros and poles, the z-domain transfer function is then $H_{d} (z) = \frac{(z - ξ'_{1}) (z - ξ'_{2}) \dots (z - ξ'_{N})}{(z - p'_{1}) (z - p'_{2}) \dots (z - p'_{N})}$

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 / s C}{R + 1 / s C} \\ = \frac{1}{1 + R C s} . \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:

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

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 ≜ \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 $ω_{0}$ , then

K ≜ \frac{ω_{0}}{\tan (\frac{ω_{0} T}{2})}

.

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}) + (2 b_{2} - 2 b_{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}) + (2 a_{2} - 2 a_{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{2 b_{2} - 2 b_{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{2 a_{2} - 2 a_{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{2 b_{2} - 2 b_{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{2 a_{2} - 2 a_{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 $s = j ω_{a}$ which is on the $j ω$ axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function $H_{d} (z)$ is evaluated at $z = e^{j ω_{d} T}$ which is on the unit circle, $| z | = 1$ . The bilinear transform maps the $j ω$ 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 $z = e^{s T}$ which also maps the $j ω$ axis to the unit circle. When the actual frequency of $ω_{d}$ is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, $ω_{a}$ , for the continuous-time filter that this $ω_{d}$ is mapped to.

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

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

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

ω_{a} = \frac{2}{T} \tan (ω_{d} \frac{T}{2})

and the inverse mapping is

ω_{d} = \frac{2}{T} \arctan (ω_{a} \frac{T}{2}) .

The discrete-time filter behaves at frequency $ω_{d}$ the same way that the continuous-time filter behaves at frequency $(2 / T) \tan (ω_{d} T / 2)$ . Specifically, the gain and phase shift that the discrete-time filter has at frequency $ω_{d}$ is the same gain and phase shift that the continuous-time filter has at frequency $(2 / T) \tan (ω_{d} T / 2)$ . 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 $ω_{d} ≪ 2 / T$ or $ω_{a} ≪ 2 / T$ ), then the features are mapped to a slightly different frequency; $ω_{d} \approx ω_{a}$ .

One can see that the entire continuous frequency range

- \infty < ω_{a} < + \infty

is mapped onto the fundamental frequency interval

- \frac{π}{T} < ω_{d} < + \frac{π}{T} .

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

One can also see that there is a nonlinear relationship between $ω_{a}$ and $ω_{d} .$ 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 $ω_{a} = \frac{2}{T} \tan (ω_{d} \frac{T}{2})$ 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 $ω_{0}$ (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 $ω_{0}$ , 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{ω_{0}}{\tan (\frac{ω_{0} T}{2})} \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 $ω_{0} \to 0$ .

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". http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/00800_TransIIR.pdf.
↑ Astrom, Karl J. (1990). Computer Controlled Systems, Theory and Design (Second ed.). Prentice-Hall. p. 212. ISBN 0-13-168600-3.

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Original source: https://en.wikipedia.org/wiki/Bilinear transform. Read more

[1] 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.

[2] Bhandari, Ayush. "DSP and Digital Filters Lecture Notes". http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/00800_TransIIR.pdf.

[3] Astrom, Karl J. (1990). Computer Controlled Systems, Theory and Design (Second ed.). Prentice-Hall. p. 212. ISBN 0-13-168600-3.

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v t e Digital signal processing
Theory	Detection theory Discrete signal Estimation theory Nyquist–Shannon sampling theorem
Sub-fields	Audio signal processing Digital image processing Speech processing Statistical signal processing
Techniques	Advanced Z-transform Bilinear transform Constant-Q transform Discrete cosine transform (DCT) Discrete Fourier transform (DFT) Discrete-time Fourier transform (DTFT) Impulse invariance Integral transform Laplace transform Matched Z-transform method Post's inversion formula Starred transform Z-transform Zak transform
Sampling	Aliasing Anti-aliasing filter Downsampling Nyquist rate / frequency Oversampling Quantization Sampling rate Undersampling Upsampling

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Discrete-time approximation

Stability and minimum-phase property preserved

Transformation of a General LTI System

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Transformation for a general first-order continuous-time filter

General second-order biquad transformation

Frequency warping

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Bilinear transform

Discrete-time approximation

Stability and minimum-phase property preserved

Transformation of a General LTI System

Example

Transformation for a general first-order continuous-time filter

General second-order biquad transformation

Frequency warping

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