Review on Z-Transforms

How to do Z-TransformsHow to do inverse Z-TransformsHow to infer properties of a signal from its Z-transform

Transfer FunctionsHow to obtain Transfer FunctionsHow to infer properties of a system from its Transfer Function

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raw readings from a noisy temperature sensor- Input Signal

smooth temperature values after filtering- Output Signal

FilterY[n]=1/2x[n]+1/2y[n-1]

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Z-Transform of a Signal not a Filter

X[n]Z-1

x(0)x(1)x(2)x(3)x(4)…

X(z)Z

x(0) · z0 +x(1) · z-1 +x(2) · z-2 +x(3) · z-3 +x(4) · z-4 …

∞

X(Z)=Σ x(k) . Z-k

k=0

2-137-5…

2· z0 -1 · z-1 3 · z-2 7 · z-3 -5· z-4 …

Z Transform of Unit Impulse Signal (z)

(k) (z)ZZ-1

(0) = 1(1) = 0(2) = 0(3) = 0(4) = 0…

1 · z0 +0 · z-1 +0 · z-2 +0 · z-3 +0 · z-4 …

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0.5

1

(z)=1

Z-Transform of Unit Step Signal

ustep(k) Ustep(z)ZZ-1

u(0) = 1u(1) = 1u(2) = 1u(3) = 1u(4) = 1…

1 · z0 +1 · z-1 +1 · z-2 +1 · z-3 +1 · z-4 …

...zzz1(z)U 321step

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Unit Step Signal - continued

,n 1,|a|

a1a1

a1)a...aaa)(1(1a...aa1

1n

n2n2

A little bit more math …

assuming

a11

a1a1

a1)a...aaa)(1(1...aa1

1n

n22

n

n

lim

lim

1-321

step z-11...zzz1(z)U

Z-Transform of Exponential Signal

uexp(k) Uexp(z)ZZ-1

u(0) = 1u(1) = au(2) = a2

u(3) = a3

u(4) = a4

…

1 · z0 +a · z-1 +a2 · z-2 +a3 · z-3 +a4 · z-4 …

1-

33221exp

az-11

...zazaaz1(z)U

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a=1.2

Remember this!

This result is from last slide where a->az-1

Important Theorem : Convolution in time doamin is the same as simple multiplication in z- domain

u(k) *(convolution)

v(k) = y(k)

U(z) V(z) Y(z)=·(multiplication)

Z Z-1 Z Z-1 Z Z-1

Time Domain

Z Domain

Z-Transform/Inverse Z-TransformConvolution in time is parallel to multiply in Z domain

yimpuse(k)=0.3k-1u (k)=0.7k y (k)?

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1 *(convolution)

10.7z11

Z

=

·(multiplication)

= )0.7z)(10.3z(1z

11

1

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Z-1

1

-1

0.3z1z

Z Transfer Function

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Delay the Unit Step Signal

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ustep (k) (k-1) u(k-1)*(convolution)

=

1z11

Z Transfer Function

·(multiplication)

Zz-1 =

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1

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1

-1

z1z

Z

(k-1)u (k) y (k)

convolution

Delayed Unit Step Signal – Cont’d

u(k-1) Udstep(z)ZZ-1

u(0) = 0u(1) = 1u(2) = 1u(3) = 1u(4) = 1…

0 · z0 +1 · z-1 +1 · z-2 +1 · z-3 +1 · z-4 …

1z1

z-1z

...zzz(z)U

1-

1-

321dstep

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Remember this!

Transfer Function

Transfer function provides a much more intuitive way to understand the input-output relationship, or system characteristics of an LTI systemStabilityAccuracySettling timeOvershoot…

properties of Z-Transform

Linearity means scaling and superposition

Time Domain Z-Transform

y(k)=au(k)

y(k)=u(k)+v(k)

Y(z)=aU(z)

Y(z)=U(z)+V(z)

Scaling

Superposition

sin? cos?

Use Exponential to get Trigonometric function’s z transform

isincose i

isincos)isin()cos(e

1-

33221exp

az-11

...zazaaz1(z)U

?

Euler Formula:

Z[cos(kθ)]?Z[sin(kθ)]?

2eecos

ii

2ieesin

ii

Z-Transform of sin/cos

ikeu(k) 1-i ze-11U(z)

2ee)cos(ku(k)

ikik

Time Domain Z-Transform

2iee)sin(ku(k)

ikik

-ikeu(k)1-i- ze-1

1U(z)

21

1

2121

1

1111

1i1-i

zz2cos1zcos1

)z(sin)zcos(1zcos1

)/2zisinzcos11

zisinzcos11(

)/2ze11

ze-11(U(z)

21-

-1

zz2cos-1zsinU(z)

Exponentially Modulated sin/cos

2)(ae)(ae)cos(ka(k)ukiki

kexpcos

2i)(ae)(ae)sin(ka(k)ukiki

kexpsin

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zazcos2a-1zsinaU(z)

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u(k)=c os(k*pi/6)*0.9k

221-

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zazcos2a-1zsinaU(z)

A damped oscillating signal – a typical output of a second order system