bilinear+prewarping

8/8/2019 bilinear+prewarping

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The bilinear transform (also known as Tustin's

method) is used in digital signal processing and

discrete-time control theory to transform continuous-

time system representations to discrete-time and viceversa.

The bilinear transform is a special case of a conformal

mapping (namely, the Mbius transformation), often

used to convert a transfer function of a linear, time-

invariant (LTI) filter in the continuous-time domain

(often called an analog filter) to a transfer function

of a linear, shift-invariant filter in the discrete-timedomain (often called a digital filteralthough there are

analog filters constructed with switched capacitors that

are discrete-time filters). It maps positions on the

axis, , in the s-plane to the unit circle, , in

the z-plane. Other bilinear transforms can be used to

warp the frequency response of any discrete-time linear

system (for example to approximate the non-linearfrequency resolution of the human auditory system) and

are implementable in the discrete domain by replacing a

system's unit delays with first orderall-pass filters.

The transform preserves stability and maps every point

of the frequency response of the continuous-time filter,

to a corresponding point in the frequencyresponse of the discrete-time filter, 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


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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 frequenciesbut is quite evident at frequencies close to theNyquist

frequency.


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Di c - i pp xi i

The bili eartransform is a first order approximation of

the nat rallogarithm function thatis an exact mapping

ofthe z-plane to the s-plane. When the Laplacetransformis performed on a discrete-time signal (witheach element ofthe discrete-time sequence attached to a

correspondingl delayed unitimpulse), the resultis

precisely the Z transform ofthe discrete-time sequence

with the substitution of

where is the sample time (the reciprocal ofthesampling frequency) ofthe discrete-time filter. The

above bilinear approximation can be solved for or a

similar approximation for can be

performed.

The inverse ofthis mapping (and its first-order bilinear

approximation) is


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The bilinear transform essentially uses this first order

approximation and substitutes into the continuous-time

transfer function,

That is


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Stability and minimum hase ro erty reserved

A continuous-time causal filter is stable if thepoles of

its transfer function fall in the left half of the complexs-

plane. A discrete-time causal filter is stable if the polesof 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 filtersthat are minimum-phase are converted to discrete-time

filters that preserve that property of being minimum-

phase.


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Ex pl

As an example take a simple low-passRC filter. This

continuous-time filter has a transfer function

If we wish to implementthis filter as a digital filter, wecan apply the bilineartransform by substituting forsthe

formula above; after some reworking, we getthe

following filter representation:

The coefficients ofthe denominator are the 'feed-backward' coefficients and the coefficients ofthe

numerator are the 'feed-forward' coefficients used to

implement a real-time digital filter.


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Frequenc warping

To determine the frequency response of a continuous-

time filter, the transfer function is evaluated at

which is on the axis. Likewise, to determine thefrequency response of a discrete-time filter, the transferfunction is evaluated at which is on the unit

circle, . When the actual frequency of is inputto

the discrete-time filter designed by use ofthe bilinear

transform, itis desired to know at what frequency, ,

forthe continuous-time filterthatthis is mapped to.

This shows that every point on the unit circle in thediscrete-time filter z-plane, is mapped to a point

on the axis on the continuous-time filter s-plane,


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. Thatis, the discrete-time to continuous-time

frequency mapping ofthe bilineartransform is

and the inverse mapping is

The discrete-time filter behaves at frequency the same

way thatthe continuous-time filter behaves at frequency

. Specifically, the gain and phase shiftthatthe discrete-time filter has at frequency is the same

gain and phase shiftthatthe continuous-time filter has

at frequency . This means that every

feature, every "bump" thatis visible in the frequencyresponse ofthe continuous-time filteris also visible in

the discrete-time filter, but at a different frequency. For

low frequencies (thatis, when or ),.

One can see thatthe entire continuous frequency range

is mapped onto the fundamental frequency interval

The continuous-time filter frequency correspondsto the discrete-time filter frequency and the


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continuous-time filter frequency correspond to

the discrete-time filter frequency

One can also see thatthere is a nonlinear relationship

between and This effect ofthe bilineartransform iscalledfrequency warping. The continuous-time filtercan be designed to compensate forthis frequency

warping by setting for every frequency

specification thatthe designer has control over (such as

corner frequency or center frequency). This is called

pre-warpingthe filter design.

The main advantage ofthe warping phenomenon is the

absence of aliasing distortion ofthe frequency response

characteristic, such as observed with Impulse

invariance. Itis necessary, however, to compensate for

the frequency warping by pre-warping the given

frequency specifications ofthe continuous-time system.

These pre-warped specifications may then be used inthe bilineartransform to obtain the desired discrete-time

system.

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bilinear+prewarping