9 Lec9 DSP Z Transform

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Lecture 9

DIGITAL SIGNAL

PROCESSING

(DSP)

1

Z-TRANSFORM


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Z-transform

Transform techniques are an important tool

in the analysis of signals and Linear time-

invariant (LTI) systems.

The z-transform plays the same role in the

analysis of discrete-time signals and LTI

systems as

The Laplace transform does in the analysis

of continuous-time signals and LTI systems.

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Z-transform3

The z-transform of a sequence x[n•] is

In both cases z is a continuous complex

variable.

n

n z n x z X ][][

Z

X[z]x[n]


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Example: Determine the z-transforms of the

following finite-duration signal.

x1[n]=(1, 2, 5, 7, 0, 1)

Z-transform4

5321

1

543210

1

5

01

1

7521][

107521][

][][

][][

z z z z z X

z z z z z z z X

z n x z X

z n x z X

n

n

n

n


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Z-transform

Example: Determine the z-transforms of the

following finite-duration signal

5

312

1

321012

1

3

21

1

1752][

107521][

][][

][][

z z z z z X

z z z z z z z X

z n x z X

z n x z X

n

n

n

n

x[n]= [1, 2, 5, 7, 0, 1]


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Z-transform & Fourier transform6

In general

We may obtain the Fourier transform from

the z-transform by making the substitution


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Z-transform & Fourier transform7

For r=1 this becomes the Fourier transform of

x[n•]

The Fourier transform therefore corresponds to

the z-transform evaluated on the unit circle:


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Region of Convergence (ROC)

The Fourier transform does not converge for

all sequences.

Similarly, the z-transform does not converge

for all sequences or for all values of z.

The set of values of z for which the z-transform

converges is called

the region of convergence (ROC).

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The z-transform therefore exists (or converges)

if

This leads to the condition for the existence of

the z-transform.

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If the ROC includes the unit circle

Then the Fourier transform will converge.

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Most useful z-transforms can be expressed in the form

where P(z) and Q(z) are polynomials in z.

The values of z for which P(z) = 0 are called the

zeros of X(z).

The values of z for which Q(z) = 0 are called thepoles of X(z).

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Right-sided exponential sequence12

Thissequence is

right-sided

because it

is nonzero

only for

n ≥ 0

0 < a < 1 x[n] = anu[n]


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ROC

ROC

0 < a < 1 x[n] = anu[n]


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ROC

0 < a < 1 x[n] = anu[n]


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ROCoutside

a circle of

radius (a)

ROC

0 < a < 1 x[n] = anu[n]


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The Fourier transform of x[n]•exists

ROC includes the unit circle

x[n] = anu[n]

ROC


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Right-sided exponential sequence

The Fourier transform of x[n]•doesn’t exist ROC doesn’t include the unit circle

x[n] is exponentially growing, and the sum

therefore does not converge.

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If

x[n] = anu[n]


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Left-sided exponential sequence18

This sequence is left-sided because it is

nonzero only for n ≤ -1

0 < a < 1

x[n] = -anu[-n-1]


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x[n] = -anu[-n-1]

ROC


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ROC

inside

a circle of

radius (a)

x[n] = -anu[-n-1]

ROC


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x[n] = -anu[-n-1]


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left-sided sequence & right-sided sequence

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Left-sided sequence

x[n] = -anu[-n-1]

Right-sided sequence

x[n] = anu[n]


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Sum of two Exponentials24

x[n] = (1/2)nu[n]+(-1/3)nu[n]


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x[n] = (1/2)nu[n]+(-1/3)nu[n]

ROC

ROC


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x[n] = (1/2)nu[n]+(-1/3)nu[n]


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× PolesO Zeros

ROC

│Z│> 1/2


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Region of Convergence (ROC)28

Left-sided sequence

ROC │Z││a│

x[n] = anu[n]


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Z-transform Pairs29


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Z-transform Pairs30


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Z-transform Pairs31

9 Lec9 DSP Z Transform

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