6Systems Represented byDifferential andDifference Equations

An important class of linear, time-invariant systems consists of systems rep-resented by linear constant-coefficient differential equations in continuoustime and linear constant-coefficient difference equations in discrete time.Continuous-time linear, time-invariant systems that satisfy differential equa-tions are very common; they include electrical circuits composed of resistors,inductors, and capacitors and mechanical systems composed of masses,springs, and dashpots. In discrete time a wide variety of data filtering, time se-ries analysis, and digital filtering systems and algorithms are described by dif-ference equations.

In this lecture, we review the time-domain solution for linear constant-coefficient differential equations and show how the same basic strategy ap-plies to difference equations. While this review is presented somewhat quick-ly, it is assumed that you have had some prior exposure to differentialequations and their time-domain solution, perhaps in the context of circuitsor mechanical systems. In any case, in Lecture 9 after we have developed theFourier transform, we will see some more efficient (at least mathematically)ways of obtaining the solution.

In considering the time-domain solution to linear constant-coefficientdifferential and difference equations, we should recognize a number of impor-tant features. Foremost is the fact that the differential or difference equationby itself specifies a family of responses only for a given input x(t). In particu-lar we can always add to any solution another solution that satisfies thehomogeneous equation corresponding to x(t) or x(n) being zero. Thus, forunique specification of a system, in addition to the differential or differenceequation some auxiliary conditions (for example, a set of initial conditions)are needed that will specify the arbitrary constants present in the homo-geneous solution.

In Lecture 5 we showed that a linear, time-invariant system has the prop-erty that if the input is zero for all time, then the output will also be zero for alltime. Consequently, a linear, time-invariant system specified by a linear con-stant-coefficient differential or difference equation must have its auxiliary

Signals and Systems6-2

conditions consistent with that property. If, in fact, the system is causal in ad-dition to being linear and time-invariant, then the auxiliary conditions willcorrespond to the requirement of initial rest; that is, if the input is zero prior tosome time, then the output must be zero at least until the same time. In thecontext of RLC circuits, for example, this would correspond to an assumptionof no initial capacitor voltages or inductor currents prior to the time at whichthe input becomes nonzero.

An important distinction between linear constant-coefficient differentialequations associated with continuous-time systems and linear constant-coef-ficient difference equations associated with discrete-time systems is that forcausal systems the difference equation can be reformulated as an explicit re-lationship that states how successive values of the output can be computedfrom previously computed output values and the input. This recursive proce-dure for calculating the response of a difference equation is extremely usefulin implementing causal systems. However, it is important to recognize that ei-ther in algebraic terms or in terms of block diagrams, a difference equationcan often be rearranged into other forms leading to implementations that mayhave particular advantages. For example, as illustrated in the lecture, themost direct representation of a difference equation in terms of a block dia-gram or algorithm is often not the most efficient. Since the order in which lin-ear, time-invariant systems are cascaded is not important to the overall input-output response, the most direct representation can be rearranged so that itsimplementation requires significantly less memory or, equivalently, delay reg-isters. In addition, there are many other rearrangements, each having particu-lar advantages and disadvantages. Similar kinds of rearrangements of theblock diagrams also apply to the block diagram realizations of linear con-stant-coefficient differential equations for continuous-time systems.

Suggested ReadingSection 3.5, Systems Described by Differential and Difference Equations,

pages 101-111

Section 3.6, Block-Diagram Representations of LTI Systems Described by Dif-ferential and Difference Equations, pages 111-119

Differential and Difference Equations

LINEAR CONSTANT-COEFFICIENT DIFFERENTIAL EQUATION

Nth Order:N

E: a kk=0

dk y(t)

dtk

M

k=0

dkx(t)

dt k

LINEAR CONSTANT-COEFFICIENT DIFFERENCE EQUATION

Order: ak y[n-k]N

Ek=0

M

=Ek=0

bk x[n-k]

TRANSPARENCY6.1Nth-order linearconstant-coefficientdifferential anddifference equations.

N

Ek=0

N

k=0

dky(t) Ma = -E

dtk k=O

dkyh(t)k k 0

dt

dkx(t)

k dtk (1)

(Homogeneous Equation) (2)

Given x(t), if y (t) satisfies (1) then so does

y,(t) + yh(t) where yh (t) satisfies (2)

yp(t) A Particular solution

yh (t) = Homogeneous solution

TRANSPARENCY6.2Family of solutions fora linear constant-coefficient differentialequation.

Nth

Signals and Systems

6-4

TRANSPARENCY6.3Form of thehomogeneoussolution.

HOMOGENEOUS SOLUTION

N dkYh(t) 0ak dtk

k0O

"guess" solution of the form

Yh (t) = Aest

N

E ak A s k est = 0k=0

N

ak s k = 0k=0

N roots s i i = 1, ... , N

Yh(t) = A1 esit + A2 es2t + ... + ANeSNt

TRANSPARENCY6.4Requirements on theauxiliary conditionsfor a linear constant-coefficient differentialequation to corre-spond to a linearsystem or to a causalLTI system.

Need N auxiliary conditions, e.g.

dy(t)y(t), dt

N -1 V(t)

, "" ' ..N - 1

e Linear system <=> auxiliary conditions = 0

eCausal, LTI <=> initial rest:

if x(t)

then y(t)

at t = t

= 0 t <to

= 0 t<t 0

Differential and Difference Equations

e + a o

dt~

A e

C% :SOk"4

t'Ov% ts

ct I ' ikt LL

CLo

+ LV

ukt) S CO L

Lt)

C L o

Ct CO~&

'K Ot 1 t~~43L&L*

TRANSPARENCY6.5Family of solutions fora Nth-order linearconstant-coefficientdifference equation.

MARKERBOARD6.1

Signals and Systems6-6

TRANSPARENCY6.6Form of thehomogeneoussolution.

TRANSPARENCY6.7Requirements on theauxiliary conditionsfor a linear constant-coefficient differenceequation to corre-spond to a linearsystem or to a causalLTI system.

Differential and Difference Equations6-7

ENCE EQUATIONTRANSPARENCY6.8Recursive causalsolution to a linearconstant-coefficientdifference equation.

MARKERBOARD6.2

Signals and Systems

6-8

x [n] - y [n]

D D

TRANSPARENCY6.9 x [n1-|A y [n -1)Development of theblock diagramrepresentation of therecursive solution. D DDelays associated withthe input and output.

x [n-2] y [n-2]

D D

x [n-N] y[n-N]

x[n] b- I y[n]

D D

TRANSPARENCY b6.10 x[n-] 11 y[n-1)Incorporation of thecoefficientmultiplication.

D D

x [n-2] b-a 2 y [n -2)

x [n -N] y [n -N]

Differential and Difference Equations

TRANSPARENCY6.11Forming the sums ofthe weighted, delayedsequences.

TRANSPARENCY6.12Complete blockdiagram. This form isoften referred to asthe direct form Iimplementation.

x [n y [n]

x [n [n-N]

Signals and Systems

6-10

TRANSPARENCY6.13Interchanging theorder in which thetwo segments of theblock diagram ofTransparency 6.12are cascaded.

TRANSPARENCY6.14Result of combiningthe two chains ofdelays in Trans-parency 6.13. Thisform is often referredto as the direct form IIimplementation.

y n

Differential and Difference Equations6-11

MARKERBOARD6.3 (a)

TRANSPARENCY6.15Direct form I blockdiagram represen-tation of a linearconstant-coefficientdifferential equation.

Signals and Systems

6-12

TRANSPARENCY6.16Resulting blockdiagram when theorder of the twosubsystems inTransparency 6.15 isreversed.

TRANSPARENCY6.17The result ofcombining the twochains of integratorsin Transparency 6.16.This form is oftenreferred to as thedirect form II.

/ N bN

y( t)x (t

Differential and Difference Equations

MARKERBOARD6.3 (b)

CA~<Vt

ci vt~ cii~L*~<5 7~ tsKt4Y&~4t)

S I

K C

vat.L~i I Lii5

-t

((K A

ej$Adit*w 6, icl sti t 0'

- Needt wkber

kinit A C-&Aflmde%5

7.TI %46

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