Xử lý tín hiệu số Z - transform

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Z - transform

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• Z- transform • Properties of Z-transform • Inversion of Z- transform • Analysis of LTI systems in Z domain

4.1. Z - transform

• Given a discrete-time signal x(n), its z-transform is defined as

the following series:

where z is a complex variable. • Writing explicitly a few of the terms:

values of z for this series converges.

• Z-transform is an infinite power series, it exists only for those

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• The region of convergence (ROC) of X(z) is the set of all

values of z for which X(z) attains a finite value.

4.1. Z - transform

• Example: Determines the z-transform of the following finite

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duration signals

4.1. Z - transform

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• Solution

4.1. Z - transform

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• Example

4.1. Z - transform

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Recall that

4.1. Z - transform

• Example

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• Solution

4.1. Z - transform

• We have (l = -n),

• Using the formula (when A<1)

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• Example

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4.1. Z - transform

• We have identical closed-form expressions for the z

transform

• A closed-form expressions for the z transform does not

uniquely specify the signal in time domain.

• The ambiguity can be resolved if the ROC is specified.

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• Z – transform = closed-form expressions + ROC

4.1. Z - transform

• Solution

• Example

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– The first power series converges if |z| > |a| – The second power series converges if |z| < |b|

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• Case 1

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• Case 2

• Characteristics families of signals with their corresponding

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ROC

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4.1. Z - transform

• The z-transform of the impulse response h(n) is called the

transfer function of a digital filter:

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• Determine the transfer function H(z) of the two causal filters

4.2. Properties of Z-transform

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• Example

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• Example

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Exercise

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4.2. Properties of Z-transform

if k>0 and z=∞ if k<0.

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• The ROC of z-k X(z) is the same as that of X(z) except for z=0

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• Solution

4.2. Properties of Z-transform

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• Example

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4.2. Properties of Z-transform

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• Example

4.2. Properties of Z-transform

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4.2. Properties of Z-transform

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• Example

4.2. Properties of Z-transform

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4.2. Properties of Z-transform

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• Example

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4.2. Properties of Z-transform

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• Convolution in Z domain

4.3. RATIONAL Z-TRANSFORM

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4.3.1. Poles and Zeros

• An important family of z-transforms are those for which X(z)

is a rational function.

• Poles and Zeros

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– Zeros: value of z for which X(z) = 0; – Poles: value of z for which X(z) = ∞

4.3.1. Poles and Zeros

• Solution

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• Example

4.3.2. Causality and Stability

• A causal signal of the form

the common ROC of all the terms will be

will have z-transform

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•

4.3.2. Causality and Stability

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if the signal is completely anticausal the ROC is in this case

4.3.2. Causality and Stability

• Causal signals are characterized by ROCs that are outside

the maximum pole circle.

• Anticausal signals have ROCs that are inside the minimum

pole circle.

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• Mixed signals have ROCs that are the annular region between two circles—with the poles that lie inside the inner circle contributing causally and the poles that lie outside the outer circle contributing anticausally.

4.3.2. Causality and Stability

• Stability can also be characterized in the z-domain in terms of

the choice of the ROC.

• A necessary and sufficient condition for the stability of a signal x(n) is that the ROC of the corresponding z-transform contain the unit circle.

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• A signal or system to be simultaneously stable and causal, it is necessary that all its poles lie strictly inside the unit circle in the z-plane.

4.3.2. Causality and Stability

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4.3.3. System function of LTI

• System function

• From a linear constant coefficient equation

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• We have,

4.3.3. System function of LTI

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• Or equivalently

4.3.3. System function of LTI

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• Example

4.3.3. System function of LTI

• Solution

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• The unit sample response

4.4. Inverse Z- transform

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• By contour integration. • By power series expansion. • By partial fraction expansion.

• The partial fraction expansion method can be applied to z-

transforms that are ratios of two polynomials

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• The partial fraction expansion of X(z) is given by

• The two coefficients are obtained as follows:

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• Example

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If the degree of the numerator polynomial N(z) is exactly equal to the degree M of the denominator D(z), then the PF expansion must be modified

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• Example • Compute all possible inverse z-transforms of

• Solution

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• Where

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|z| > 0.5: |z|<0.5:

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• Example • Determine all inverse z-transforms of

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• Solution

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there are only two ROCs I and II:

MORE ABOUT INVERSE Z-TRANSFORM

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• Distinct poles

• Solution

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• Multiple order poles

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In such a case, the partial fraction expansion is:

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Exercise

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• a) • b) • c)

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