mit6 003s10 lec06 handout
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8/8/2019 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
0
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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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MIT OpenCourseWarehttp://ocw.mit.edu
6.003 Signals and Systems
Spring 2010
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