mit6 003s10 lec06 handout

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6.003: SignalsandSystemsZ TransformZ transform isdiscrete-timeanalogofLaplace transform.Z transformmapsa functionofdiscrete timen toa functionof z.

X(z) = x[n]znn

Thereare

two

important

variants:

Unilateral

X(z) = x[n]zn

n=0Bilateral

X(z) = x[n]zn

n=Differencesareanalogous to those for theLaplace transform.

Check YourselfFind theZ transformofadelayedunit-sample signal.

n

x[n]

Shape of ROCRegionsofconverge forZ transformaredelimitedbycircles.Example: x[n] = nu[n]

X(z) = nu[n]zn = nzn

n= n=0=

1 z1||zx[n] = nu[n]

zz

n4321 0 1 2 3 4

z-planeROC

Lecture6 February23,201Check Yourself

Find theZ transformof theunit-sample signal.

n

[n]

Z TransformsExample: Find theZ transformof the following signal.

x[n] = 8

7nu[n]

n4321 0 1 2 3 4

n n 7 7 1 zX(z) =

8 znu[n] = 8 zn = 17 1 =z7n= n=0 8z 8

provided 8

7z18

7.x[n] =

8

7 nu[n]z

z78

n4321 0 1 2 3 4

78

z-planeROC

Shape of ROCRegionsofconvergeforLaplacetransformdelimitedbyverticallinesExample: x(t) = etu(t)

X(s) = etu(t)estdt= etestdt 0

1= ; Re(s) >Re()s

x(t) = etu(t)1

s

s-plane

ROCt

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6.003: SignalsandSystemsDistinguishing Features of TransformsMost-important featureofLaplace transforms is thederivative rule:

x(t) X(s)x(t) sX(s)

allowsustouseLaplacetransformstosolvedifferentialequations.Similarly,most-important featureofZ transforms is thedelay rule:

x[n] X(z)x[n1] z1X(z)

allowsus touseZ transforms to solvedifferenceequations.

Check YourselfWhatDT signalhas the followingZ transform?

78

z-planeROCz

z78

; |z|


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index shift

Delay R

6.003: SignalsandSystems Lecture6 February23,201Properties of Z Transforms Check YourselfTheuseofZTransforms to solvedifferentialequationsdependsonseveral importantproperties.Property x[n] X(z) ROCLinearity ax1[n] + bx2[n] aX1(z) + bX2(z) (R1R2)Delay x[n 1] z1X(z) R

dX(z)MultiplybynConvolve inn

nx[n]

x1[m]x2[nm]z

dz RX1(z)X2(z) (R1R2)

m=

Find the inverse transformofY(z) =

zz 1

2; |z| >1.

Concept Map: Discrete-Time SystemsRelationsamong representations.

Block Diagram System Functional

Difference Equation System Function

Unit-Sample Response

+Delay

+Delay

X Y YX =

11RR2

y[n] = x[n] + y[n1]+y[n2] H(z) = Y(z)X(z) =

z21 z z2

h[n] : 1,1,2,3,5,8,13,21,34,55, . . .

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6.003 Signals and Systems

Spring 2010

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mit6 003s10 lec06 handout

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