NationMaster - Encyclopedia: Bilinear transform

In digital signal processing, the bilinear transform is a conformal mapping, often used to convert a transfer function $H_a(s)$ of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function $H_d(z)$ of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with charge-coupled devices that are discrete-time filters). It maps positions on the $j omega$ axis, $Re[s]=0$ , in the s-plane to the unit circle, $|z| = 1$ , in the z-plane. Other bilinear transforms can be used to warp the frequency 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 $left( z^{-1} right)$ with first order all-pass filters. Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... In mathematics, a conformal map is a function which preserves angles. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... The word linear comes from the Latin word linearis, which means created by lines. ... A time-invariant system is one whose output does not depend explicitly on time. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... An analog filter handles analog stimuli (e. ... A discrete signal is a signal that has been sampled from a continuous signal. ... An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital math operations on an intermediate form of a signal. ... A specially developed CCD used for ultraviolet imaging in a wire bonded package. ... The S plane is a mathematical domain, where instead of viewing processes in the time domain, modelled with time based functions they are viewed, as equations, in the frequency domain. ... Illustration of a unit circle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... An all-pass filter is an electronic filter that passes all frequencies equally, but changes the phase relationship between various frequencies. ...

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. The word stability has a number of technical meanings, all related to the common meaning of the word. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... The Nyquist frequency, named after the Nyquist-Shannon sampling theorem, is half the sampling frequency for a signal. ...

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 the Z transform of the discrete-time sequence with the substitution of In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

$z$	$= e^{sT}$
	$= frac{e^{sT/2}}{e^{-sT/2}}$
	$approx frac{1 + s T / 2}{1 - s T / 2}$

where $T$ is the sample time (the reciprocal of the sampling frequency) of the discrete-time filter. The above bilinear approximation can be solved for $s$ or a similar approximation for $s = (1/T) log(z)$ can be performed. The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ...

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

$s$	$= frac{1}{T} log(z)$
	$= frac{2}{T} left[frac{z-1}{z+1} + frac{1}{3} left( frac{z-1}{z+1} right)^3 + frac{1}{5} left( frac{z-1}{z+1} right)^5 + frac{1}{7} left( frac{z-1}{z+1} right)^7 + ldots right]$
	$approx frac{2}{T} frac{z - 1}{z + 1}$
	$approx frac{2}{T} frac{1 - z^{-1}}{1 + z^{-1}}$

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

$H_d(z) = H_a(s) bigg|_{s = frac{2}{T} frac{z - 1}{z + 1}}= H_a left( frac{2}{T} frac{z-1}{z+1} right).$

The bilinear transform is a special case of a conformal mapping, namely, the Möbius transformation defined as In mathematics, a conformal map is a function which preserves angles. ... In mathematics, a MÃ¶bius transformation is a bijective conformal mapping of the extended complex plane (i. ...

$z^{prime} = frac{a z + b}{c z + d}.$

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 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 the discrete-time domain that preserve that stability. In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... The S plane is a mathematical domain, where instead of viewing processes in the time domain, modelled with time based functions they are viewed, as equations, in the frequency domain. ... Illustration of a unit circle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

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. In control theory and signal processing, a linear, time-invariant system is minimum-phase if the system and its inverse are causal and stable. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...

Example

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

$H_a(s) = frac{1}{1 + RC s}.$

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:

$H_d(z)$	$=H_a left( frac{2}{T} frac{z-1}{z+1}right)$
	$= frac{1}{1 + RC left( frac{2}{T} frac{z-1}{z+1}right)}$
	$= frac{1 + z}{(1 - 2 RC / T) + (1 + 2RC / T) z}.$
	$= frac{1 + z^{-1}}{1 + (2RC / T) + (1 - 2RC / T) z^{-1}}.$

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.

$H_d(z) = H_a left( frac{2}{T} frac{z-1}{z+1}right)$

$H_d(e^{ j omega T})$	$= H_a left( frac{2}{T} frac{e^{ j omega T} - 1}{e^{ j omega T} + 1}right)$
	$= H_a left( frac{2}{T} cdot frac{e^{j omega T/2} left(e^{j omega T/2} - e^{-j omega T/2}right)}{e^{j omega T/2} left(e^{j omega T/2} + e^{-j omega T/2 }right)}right)$
	$= H_a left( frac{2}{T} cdot frac{left(e^{j omega T/2} - e^{-j omega T/2}right)}{left(e^{j omega T/2} + e^{-j omega T/2 }right)}right)$
	$= H_a left(j frac{2}{T} cdot frac{ left(e^{j omega T/2} - e^{-j omega T/2}right) /(2j)}{left(e^{j omega T/2} + e^{-j omega T/2 }right) / 2}right)$
	$= H_a left(j frac{2}{T} cdot frac{ sin(omega T/2) }{ cos(omega T/2) }right)$
	$= H_a left(j frac{2}{T} cdot tan left( omega frac{T}{2} right) right)$
	$= H_a left(j omega_a right).$

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

$omega_a = frac{2}{T} tan left( omega frac{T}{2} right)$

and the inverse mapping is

$omega = frac{2}{T} arctan left( omega_a frac{T}{2} right).$

The discrete-time filter behaves at frequency $omega$ the same way that the continuous-time filter behaves at frequency $(2/T) tan(omega T/2)$ . Specifically, the gain and phase shift that the discrete-time filter has at frequency $omega$ the same gain and phase shift that the continuous-time filter has at frequency $(2/T) tan(omega T/2)$ . 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 slightly different frequency. For low frequencies $omega approx omega_a$ .

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 $omega_a = frac{2}{T} tan left( omega frac{T}{2} right)$ 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.

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

Categories: Digital signal processing | Transforms

Results from FactBites:

THE BILINEAR TRANSFORM (450 words)
	By inspection of Figures 20 and 21, it is found that the bilinear approximation (42) or (43) also maps the exterior of the unit circle into the lower half-plane.
	Thus, although the bilinear approximation is an approximation, it turns out to exactly preserve the minimum-phase property.
	Clearly, the folding theorem is too generous for applications involving the bilinear transform.

The Bilinear Transform (175 words)
	The formula for a general first-order (bilinear) conformal mapping of functions of a
	Bilinear transformations map circles and lines into circles and lines (lines being viewed as circles passing through the point at infinity).
	``The Bark and ERB Bilinear Transforms'', by Julius O. Smith III and Jonathan S. Abel, preprint of version accepted for publication in the IEEE Transactions on Speech and Audio Processing, December, 1999.

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Example

Frequency warping

Stability and minimum-phase property preserved