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4ConvolutionIn Lecture 3 we introduced and defined a variety of system properties towhich we will make frequent reference throughout the course. Of particularimportance are the properties of linearity and time invariance, both becausesystems with these properties represent a very broad and useful class and be-cause with just these two properties it is possible to develop some extremelypowerful tools for system analysis and design.A linear system has the property that the response to a linear combina-tion of inputs is the same linear combination of the individual responses. Theproperty of time invariance states that, in effect, the system is not sensitive tothe time origin. More specifically, if the input is shifted in time by someamount, then the output is simply shifted by the same amount.

The importance of linearity derives from the basic notion that for a linearsystem if the system inputs can be decomposed as a linear combination ofsome basic inputs and the system response is known for each of the basic in-puts, then the response can be constructed as the same linear combination ofthe responses to each of the basic inputs. Signals (or functions) can be decom-posed as a linear combination of basic signals in a wide variety of ways. Forexample, we might consider a Taylor series expansion that expresses a func-tion in polynomial form. However, in the context of our treatment of signalsand systems, it is particularly important to choose the basic signals in the ex-pansion so that in some sense the response is easy to compute. For systemsthat are both linear and time-invariant, there are two particularly usefulchoices for these basic signals: delayed impulses and complex exponentials.In this lecture we develop in detail the representation of both continuous-time and discrete-time signals as a linear combination of delayed impulsesand the consequences for representing linear, time-invariant systems. The re-sulting representation is referred to as convolution.Later in this series of lec-tures we develop in detail the decomposition of signals as linear combina-tions of complex exponentials (referred to as Fourieranalysis) and theconsequence of that representation for linear, time-invariant systems.

In developing convolution in this lecture we begin with the representa-tion of discrete-time signals and linear combinations of delayed impulses. Aswe discuss, since arbitrary sequences can be expressed as linear combina-tions of delayed impulses, the output for linear, time-invariant systems can be

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Signals and Systems4-2

expressed as the same linear combination of the system response to a delayedimpulse. Specifically, because of time invariance, once the response to oneimpulse at any time position is known, then the response to an impulse at anyother arbitrary time position is also known.

In developing convolution for continuous time, the procedure is muchthe same as in discrete time although in the continuous-time case the signal isrepresented first as a linear combination of narrow rectangles (basically astaircase approximation to the time function). As the width of these rectan-gles becomes infinitesimally small, they behave like impulses. The superposi-tion of these rectangles to form the original time function in its limiting formbecomes an integral, and the representation of the output of a linear, time-in-variant system as a linear combination of delayed impulse responses also be-comes an integral. The resulting integral is referred to as the convolutionin-tegral and is similar in its properties to the convolution sum for discrete-timesignals and systems. A number of the important properties of convolution thathave interpretations and consequences for linear, time-invariant systems aredeveloped in Lecture 5. In the current lecture, we focus on some examples ofthe evaluation of the convolution sum and the convolution integral.Suggested ReadingSection 3.0, Introduction, pages 69-70Section 3.1, The Representation of Signals in Terms of Impulses, pages 70-75Section 3.2, Discrete-Time LTI Systems: The Convolution Sum, pages 75-84Section 3.3, Continuous-Time LTI Systems: The Convolution Integral, pages88 to mid-90

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Tkiv%

\ r - L ,tY%3- InAvr

- causal

'4s~w

STM rEcY:% lecorpose pt 5 pwLinvo GLineer comet'ncL4Zo1.

o C 0'sMaSic Vina

-tha. respolase eqs to

LT I SWs ens-e g Convo +Co,

x[-I] x[0] 1 x[2]-l OJr2x[0] x[O]8a[n]

-.- e--.-0- n-1 0 I2X[1] x[1]8[n-1]

-1 0 I 2x[-I]8[n+1]-0 -- *- n

-1 0 I 2X[-] x [-2]8[n +2]0--0-0 n-1 0 1 2

x[o]8[n]+x(I] 8[n -1]+ x [-I]8[n+ ]+.--+X kr

=2 x[k]8[n-k]k= -c

TRANSPARENCY4.1A general discrete-time signal expressedas a superposition ofweighted, delayed unitimpulses.

Convolutio

MARKERBOARD4.1

/9 vA

6.~i

I

rkv"O' "TVNVQI;Q,,c.I,

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Signals and Systems

TRANSPARENCY4.2Th e convolution sumfor linear, time-invariant discrete-timesystems expressingthe system output as aweighted sum ofdelayed unit impulseresponses.

jj x[ x[2]-1 0 I 2X10{ x[0]8[n]

n-I 0 1 2X x[1]8[n+1]

x[ 2 ] x[-2]8[n+2]1T n-1 0 i 2

-0-0-0-0-- m- x 0] h n]00

0

0

x [ 1] h [n-I]

x [-I] h [n+1.

x [-2] h [n.2]

TRANSPARENCY4.3One interpretation ofthe convolution sumfor an LTI system.Each individualsequence value can beviewed as triggering aresponse; all theresponses are addedto form the totaloutput.

x[n] =Ek = -o

x[k] S[n-k]

Linear System: +0ny [n] =E

k = - 010x[k] hk [n]

If Time-invariant:hk [n] = h [n-k]

LTI: y[n] + 0=E x[k] h[n - k]IConvolution Sum

5 [n - k] - h [n]

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Convoluti

x(-2A) x(-2A)&a(t+2A)A

-2A -A t

-A 0 t

I X 0)X(0) BA tA

0 A t

- xA )x(A) 86 (t-A)

A 2A t

x(o) 6A(t) A +x(A) 6A(t+ x(- A) 6A(t + A)A+...

x(k A) 6,(t - k A) A

= lim 1 x(k A) S(t - k A) AA+O k=-oc

TRANSPARENCY4.4Approximation of acontinuous-time signalas a linearcombination ofweighted, delayed,rectangular pulses.[The amplitude of thefourth graph has beencorrected to readx(O).]

TRANSPARENCY4.5As the rectangularpulses in Trans-parency 4.4 becomeincreasingly narrow,the representationapproaches anintegral, often referredto as the siftingintegral.

x(t)

X(t)

x(t)+W

f x(,r) 6(t -,r) d-r--00

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Signals and Systems4-6

TRANSPARENCY4.6Derivation of theconvolution integralrepresentation forcontinuous-time LTIsystems.

x(t) = Eim x(k A)'L+0 k=-o

Linear System: +oy(t) = 0 x(kA)+O k=- o

+00=f xT) hT(t) dr

If Time-Invariant:hkj t) = ho(t - kA)h,(t) = he (t - r)

LTI:

56(t - kA) A

hk(t)

+01v(t) f x(r) h(t-7) dr-

1 -0

Convolution Integral

TRANSPARENCY4.7Interpretation of theconvolution integral asa superposition of theresponses from eachof the rectangularpulses in therepresentation of theinput.

x(t)

0 ti

x 0)

x(A)

x(kA)AA

0 tx(t)

0 t

x (0) h Mt

kA ty(t)

oA

y(t)

oA

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Convolut

Convolution Sum:+0o

x[n] =E x[k]k= -0ok

y [n] = x [k]k= -o00

S[n-k]

h[n-k] =x[n] * h[n]

Convolution Integral:x(t)

y(t)

+00=f x(-) 6(t-r) dr

+fd= X(r) h(t-,r) dr-= x(t) -*h(t)

y [n] Z x[k]h [n-k]x [n]= u [n]h[n]=an u[n]

0 n

t k

n k

x [n]

h [n]

x [k]

h [n-k]

TRANSPARENCY4.8Comparison of theconvolution sum fordiscrete-time LTIsystems and theconvolution integralfor continuous-timeLTI systems.

TRANSPARENCY4.9Evaluation of theconvolution sum foran input that is a unitstep and a systemimpulse response thatis a decayingexponential for n > 0.

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Signals and Systems

TRANSPARENCY4.10Evaluation of theconvolution integralfor an input that is aunit step and a systemimpulse response thatis a decayingexponential for t > 0.

y(t)f x(r)h(t-r)dr

x(t) u (t)h (t )=e~43 u t)

O t

0 r

t T

MARKERBOARD4.2

x (t)

h (t)

x (r)

h(t-r)

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Convoluti

MARKERBOARD4.3

v4egva.z: t).Ct k Lt-T)aT t (0

ov0 C-t

Te Lk (t -CTo-

eoverkp ~3et ee

h*O t,

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