- Bilinear transform
The

**bilinear transform**(also known as**Tustin's method**) is used indigital signal processing and discrete-timecontrol theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear transform is aconformal map ping, often used to convert atransfer function $H\_a(s)$ of alinear ,time-invariant (LTI) filter in the continuous-time domain (often called ananalog filter ) to a transfer function $H\_d(z)$ of a linear, shift-invariant filter in the discrete-time domain (often called adigital filter although there are analog filters constructed withcharge-coupled device s that are discrete-time filters). It maps positions on the $j\; omega$ axis, $Re\; [s]\; =0$, in thes-plane to theunit circle , $|z|\; =\; 1$, in the z-plane. Other bilinear transforms can be used to warp thefrequency response of any discrete-time linear system (e.g., to approximate the human auditory's non-linear frequency resolution) and are implementable in the discrete domain by replacing a system's unit delays $left(\; z^\{-1\}\; ight)$ with first orderall-pass filter s.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 theNyquist frequency .The bilinear transform is a first-order approximation 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 theZ transform of the discrete-time sequence with the substitution of:$egin\{align\}z\; =\; e^\{sT\}\; \backslash \; =\; frac\{e^\{sT/2\{e^\{-sT/2\; \backslash \; approx\; frac\{1\; +\; s\; T\; /\; 2\}\{1\; -\; s\; T\; /\; 2\}end\{align\}$

where $T$ is the

sample time (the reciprocal of thesampling frequency ) of the discrete-time filter. 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

:$egin\{align\}s\; =\; frac\{1\}\{T\}\; ln(z)\; \backslash \; =\; frac\{2\}\{T\}\; left\; [frac\{z-1\}\{z+1\}\; +\; frac\{1\}\{3\}\; left(\; frac\{z-1\}\{z+1\}\; ight)^3\; +\; frac\{1\}\{5\}\; left(\; frac\{z-1\}\{z+1\}\; ight)^5\; +\; frac\{1\}\{7\}\; left(\; frac\{z-1\}\{z+1\}\; ight)^7\; +\; cdots\; ight]\; \backslash \; approx\; frac\{2\}\{T\}\; frac\{z\; -\; 1\}\{z\; +\; 1\}\; \backslash \; approx\; frac\{2\}\{T\}\; frac\{1\; -\; z^\{-1\{1\; +\; z^\{-1end\{align\}$

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

:$H\_d(z)\; =\; H\_a(s)\; igg|\_\{s\; =\; frac\{2\}\{T\}\; frac\{z\; -\; 1\}\{z\; +\; 1=\; H\_a\; left(\; frac\{2\}\{T\}\; frac\{z-1\}\{z+1\}\; ight).$

The bilinear transform is a special case of a

conformal map ping, namely, theMöbius transformation defined as:$z^\{prime\}\; =\; frac\{a\; z\; +\; b\}\{c\; z\; +\; d\}.$

**Stability and minimum-phase property preserved**A continuous-time filter is stable if the poles of its transfer function fall in the left half of the complex

s-plane . A discrete-time filter is stable if the poles of its transfer function fall inside theunit 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 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.**Example**As an example take a simple

low-pass RC filter . This continuous-time filter has a transfer function:$egin\{align\}H\_a(s)\; =\; frac\{1/sC\}\{R+1/sC\}\; \backslash =\; frac\{1\}\{1\; +\; RC\; s\}.end\{align\}$

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:

:

**Frequency warping**To determine the frequency response of a continuous-time filter, the

transfer function $H\_a(s)$ is evaluated at $s\; =\; j\; omega$ 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 $z\; =\; e^\{\; j\; omega\; T\}$ which is on the unit circle, $|z|\; =\; 1$. When the actual frequency of $omega$ is input to the discrete-time filter designed by use of the bilinear transform, it is desired to know at what frequency, $omega\_a$, for the continuous-time filter that this $omega$ is mapped to.:$H\_d(z)\; =\; H\_a\; left(\; frac\{2\}\{T\}\; frac\{z-1\}\{z+1\}\; ight)$

:

This shows that every point on the unit circle in the discrete-time filter z-plane, $z\; =\; e^\{\; j\; omega\; T\}$ 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

:$omega\_a\; =\; frac\{2\}\{T\}\; an\; left(\; omega\; frac\{T\}\{2\}\; ight)$

and the inverse mapping is

:$omega\; =\; frac\{2\}\{T\}\; arctan\; left(\; omega\_a\; frac\{T\}\{2\}\; ight).$

The discrete-time filter behaves at frequency $omega$ the same way that the continuous-time filter behaves at frequency $(2/T)\; an(omega\; T/2)$. Specifically, the gain and phase shift that the discrete-time filter has at frequency $omega$ is the same gain and phase shift that the continuous-time filter has at frequency $(2/T)\; an(omega\; T/2)$. 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 $omega\; ll\; 2/T$ or $omega\_a\; ll\; 2/T$), $omega\; approx\; omega\_a$.

One can see that the entire continuous frequency range

: $-infty\; <\; omega\_a\; <\; +infty$

is mapped onto the fundamental frequency interval

: $-frac\{pi\}\{T\}\; <\; omega\; <\; +frac\{pi\}\{T\}.$

The continuous-time filter frequency $omega\_a\; =\; 0$ corresponds to the discrete-time filter frequency $omega\; =\; 0$ and the continuous-time filter frequency $omega\_a\; =\; pm\; infty$ correspond to the discrete-time filter frequency $omega\; =\; pm\; pi\; /\; T.$

One can also see that there is a nonlinear relationship between $omega\_a$ and $omega.$ 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 $omega\_a\; =\; frac\{2\}\{T\}\; an\; left(\; omega\; frac\{T\}\{2\}\; ight)$ 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.The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with

Impulse invariance . It is necessary, however, to compensate for the frequency warping by pre-warping the given frequency specifications of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system.

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