23Mapping Continuous-Time Filters toDiscrete-Time Filters

In Lecture 22 we introduced the z-transform. In this lecture we discuss someof the properties of the z-transform and show how, as a result of these proper-ties, the z-transform can be used to analyze systems described by linear con-stant-coefficient difference equations. Toward this end, the three most signifi-cant properties are the linearity property, the time-shifting property, and theconvolution property. As a consequence of the convolution property, the z-transform of the output of an LTI system is the product of the z-transform ofthe input and the z-transform of the system impulse response, referred to asthe system function. The system function, for any specific value of z, say zo,corresponds also to the change in (complex) gain of the eigenfunction zo as itpasses through the system. That is, H(z) represents the spectrum of eigenval-ues for discrete-time LTI systems, just as H(s) represents the spectrum ofeigenvalues for continuous time LTI systems.

Based on the linearity, time-shifting, and convolution properties, apply-ing the z-transform to a linear constant-coefficient difference equation con-verts it to an algebraic equation that can be solved for the system function.Again, closely paralleling the discussion for continuous time, this specifies thealgebraic expression for the system function but does not explicitly specifythe associated ROC. However, if in addition the system is specified to be caus-al, then the ROC must lie outside the circle bounded by the outermost pole.Alternatively, if the system is known to be stable, the ROC of the system func-tion must include the unit circle. It then extends inward and outward, in bothcases until it reaches a pole (or the origin and/or infinity). Typically (but notalways) in discussing systems described by linear constant-coefficient differ-ence equations, we assume causality of the system. Just as with continuous-time systems, first- and second-order discrete-time difference equations playa particularly important role as building blocks for higher-order differenceequations.

In designing a discrete-time system, a variety of design procedures isavailable for obtaining a linear constant-coefficient difference equation to

23-1

Signals and Systems23-2

meet or approximate a given set of system specifications. One particularly im-portant class of such procedures corresponds to mapping continuous-timedesigns to discrete-time designs. This approach is motivated in part by thefact that continuous-time filter design has a long and rich history; to the ex-tent that well-developed design procedures for continuous-time systems canbe exploited in the design of discrete-time systems, they should be. Further-more, in many applications discrete-time systems are used to process con-tinuous-time signals by exploiting the concepts of sampling. In such cases, thediscrete-time system to be designed and implemented is closely associatedwith a corresponding continuous-time system.

An often used but not highly desirable approach to mapping continuous-time systems to discrete-time systems is to replace derivatives in the differen-tial equation describing the continuous-time system by simple forward orbackward differences to obtain a discrete-time difference equation. The limi-tations of this approach are perhaps best understood by examining the corre-sponding mapping from the s-plane to the z-plane, from which it is evidentthat the frequency response can be severely distorted. In addition, with theuse of forward differences, unstable discrete-time filters can result, evenwhen the continuous-time filter from which it is derived is stable.

A second approach discussed is the impulse-invariant design procedure,whereby the discrete-time system function is determined in such a way thatthe impulse response of the discrete-time system corresponds to samples ofthe impulse response of the continuous-time system. This procedure canequivalently be interpreted as a mapping of the poles of the system function.In terms of the corresponding frequency responses, the discrete-time fre-quency response is identical in shape to the continuous-time frequency re-sponse except for possible distortion due to aliasing. Consequently, it is use-ful only for mapping continuous-time systems for which the frequencyresponse is bandlimited.

Suggested ReadingSection 10.5, Properties of the z-Transform, pages 649-654

Section 10.7, Analysis and Characterization of LTI Systems Using z-Trans-forms, pages 655-658

Section 10.4, Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot, pages 646-648

Section 10.8, Transformations Between Continuous-Time and Discrete-TimeSystems, pages 658-665

Mapping Continuous-Time Filters to Discrete-Time Filters23-3

z-TRANSFORM PROPERTIES

SIGNAL

ax, [n] + bx 2 [n]

x[n-no]

x, [n] * x2 [n]

TRANSFORM

aX, (z) + bX 2 (z)

z-o X(z)

ROC

at leastRi n R2

R

at leastR1 n R2

TRANSPARENCY23.1Some properties of thez-transform.

MARKERBOARD23.1 (a)

3 -Trans io'r m

4.

Ce~ve.-es~ *r SOW&4 4es '=i R OCj."

oiluers -- OC

I

Signals and Systems

23-4

TRANSPARENCY23.2Equivalence of theconstraints on asystem impulseresponse for stabilityand for the existenceof the Fouriertransform.

TRANSPARENCY23.3Relationship betweenthe properties ofstability and causalityand correspondingconstraints on theROC of the systemfunction.

x[n]

X(z)

y[n]

h[n] y[n]

H (z) Y (z)

= h [n] * x [n]

Y(z) = H (z) X(z)

+00

stable <=>n=- 00

+00

n=- 00

* stable <=>

lh[n] <o

jh[n] 1<oo

ROC of H(z) includesunit circle in z-plane

* causal => h [n] right-sided=> ROC of H(z) outside

outermost pole

e causal and stable <=> All poles insideunit circle

Th[n]

Mapping Continuous-Time Filters to Discrete-Time Filters23-5

TRANSPARENCY23.4Transparencies 23.4-23.6 show a specifiedsystem function and

rfit circle the relationshipbetween the threechoices for the ROCand the properties of

z-plane system stability andcausality. Here, thesystem is causal andunstable.

Re

/ "

TRANSPARENCY23.5The system is unstableand not causal.

Signals and Systems

23-6

TRANSPARENCY23.6The system is stableand not causal.

TRANSPARENCY z-TRAN23.7Some properties of thez-transform.[Transparency 23.1 SIGNALrepeated]

Mapping Continuous-Time Filters to Discrete-Time Filters23-7

S irst -or der

diferevWA eavVpO,

H ()

+ l

ECln eo a C)

secaow -erder

dsfewrevC ep tliw

Y()= .,xos 3' r%3

+2.rcos eg r j]

H(3)

Cos e <1 -> Cofles eles

|OIes t V e

TRANSPARENCY23.8Frequency responsefor an underdampedsecond-order system.

MARKERBOARD23.1 (b)

Ire

Va V-os e asi net

Unit circle z-plane

I H(ei) I

-01*7(3 - xt"35,

Signals and Systems

23-8

M-inos Tiwe V 14e

.

D iser..teTime -V lters

c-T StigoIS

* Exploi ch+66 she 4det procevdres SerC--T ;11er-s

*

~4CiN*

C-T D-T

1-Lt ) -- +b 9

Ft~L A~ 1c~

T

Fay- ~ ~ I~-.1 1

jq zit T

tL

Os I-y (S

We (3)-, Cvcic

TRANSPARENCY23.9Mapping from the s-plane to the z-planethat results when adifferential equationis mapped to adifference equation byreplacing derivativeswith differences.

Unit circle

s-plane

MARKERBOARD23.2

z-plane

Mapping Continuous-Time Filters to Discrete-Time Filters23-9

Aki

Op'lee0t pee at

14d ('

k-0

N I

14ccs) S.,

He (i o)

WC C

- c wc T

1/THd(ejS2) = T 00He

k=-oo L

/J\T

2rk)T/

-27r ir

- (wc T)

TRANSPARENCY23.10Illustration of spectraassociated withimpulse invariance.

MARKERBOARD23.3

p re.,erveA

Signals and Systems23-10

TRANSPARENCY23.11Continuous-timesecond-order transferfunction mapped to adiscrete-time systemfunction using impulseinvariance.

(s+a+jor) (s+a-jwr)

(s+a+jcor) +

TRANSPARENCY23.12Discrete-time systemfunction and pole-zeroplot resulting fromimpulse invarianceapplied to thesystem function inTransparency 23.9.

Hd(z)1 _e-aT e-jwrT Z- 1

liII

/ -z-plane

Re

Im

s-plane

-wr

2cirHe(s)

-j(s+a - jor

i i

1 -e-aOT ejorT z-1

X_ -

X--

Mapping Continuous-Time Filters to Discrete-Time Filters23-11

IHe(jW)I

TRANSPARENCY23.13Transparencies 23.13-23.15 show a

-r .5 x 104 comparison of thefrequency response

1 when a second-order3 - r continuous-time

system is mapped to asecond order discrete-time system usingimpulse invarianceand backwarddifferences. Shownhere is a continuous-time frequencyresponse.

0 .5 x 104 104

IH 0(j) I

I Hd (jS2)1

Impulse invariant design TRANSPARENCY23.14Discrete-timefrequency responseresulting from impulseinvariance.

n .5 x 104 104

0I-

iTir/2

Signals and Systems

23-12

TRANSPARENCY23.15Discrete-timefrequency responseresulting from theuse of backwarddifferences.

I HOW)I

I I I I I CoI

|Hd (jE2)IBackward differences

.5 x 104

7r/2

104

127T

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