Module 5�The Z-Transform

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The z-Transform

• We derive the z-transform by applying a complex exponential input to an LTI system.

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We define transfer function

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and the inverse z-transform

ZT

----------------------------> (1)

----------------------------> (2)

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Convergence

• The z-transform exists for signals that do not have a DTFT.

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Fig 2: Illustration of a signal that has z-transform but do not have a DTFT

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The z-Plane

----------------------------> (1)

----------------------------> (2)

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DTFT corresponds to the z-transform evaluated on the unit circle.

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z-Transform of a causal exponential signal

Ex 1:

Determine the z-transform of the signal

Sol:

This is a geometric series of infinite length

ZT

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ROC is the shaded region of the z-plane.

ROC is exterior to the unit circle.

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z-Transform of an anticausal exponential signal

Ex 2:

Determine the z-transform of the signal

Sol:

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Hence

ZT

ROC is the shaded region of the z-plane.

ROC is interior to the unit circle.

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z-Transform of Two sided signal

Ex 3:

Determine the z-transform of the signal

Sol:

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The summations will converge if

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Properties of Region of Convergence (ROC)

1. ROC cannot contain any poles.

• This sum will converge provided each of its terms is finite.

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Fig 4: The relationship between the ROC and the time extent of the signal

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ROCs of two sided signals

Ex 4:

Sol:

The first series converges if

The second series converges if

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Sol:

The first series converges if

The second series converges if

The combined ROC is

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Fig 6: ROC is outside of the circle containing

the pole of largest magnitude

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Sol:

The first series converges if

The second series converges if

The combined ROC is

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Fig 6: ROC is inside the circle containing

the pole of smallest magnitude

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Most of the properties are analogous to those of DTFT

We assume that

and

The ROC is changed by certain operations. The effect of operation on ROC is described by a change in the radii of the ROC boundaries.

1a. Linearity property

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Pole-Zero Cancellation

Ex 5: Suppose

and

Sol:

According to linearity property,

ROC is the intersection of the individual ROCs

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ROC can be verified

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2. Time Reversal property

If

then

Proof:

----------------------------> (1)

----------------------------> (2)

Comparing (1) and (2), we get

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3. Time Shift property

If

then

Proof:

----------------------------> (1)

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4. Multiplication by an exponential sequence

If

then

Proof:

----------------------------> (1)

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5. Convolution

If

and

then

Proof:

----------------------------> (1)

----------------------------> (2)

Substitute (1) in (2), we get

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Rearranging the summations

Substitute

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If

then

Proof:

This operation does not change the ROC.

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Applying Multiple properties

Sol:

ZT

Recall the ZT pair

ZT

ZT

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----------------------------> (1)

ZT

ZT

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----------------------------> (2)

ZT

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Sol:

ZT

Recall the multiplication by an exponential property,

----------------------------> (1)

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Sol:

ZT

----------------------------> (1)

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Recall the multiplication by an exponential property,

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Sol:

Let

----------------------------> (1)

----------------------------> (2)

Using the time shift property,

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• A right sided signal has an ROC that lies outside the pole radius.

• A left sided signal has an ROC that lies inside the pole radius.

ZT

and

ZT

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Inversion by Partial Fraction Expansion

Sol:

Use the partial fraction expansion,

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ZT

(Right sided sequence)

ZT

(left sided sequence)

ZT

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Combining the individual terms,

Ex 11: Repeat the Ex 10 for the following ROCs:

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Inversion of an Improper Rational Function

Sol:

The poles are at

Using the long division method to reduce the order of numerator polynomial,

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----------------------------> (1)

Using partial fraction expansion, we get

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----------------------------> (2)

----------------------------> (3)

where

The ROC has a smaller radius than either pole,

Apply time shift property,

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• If the signal is known to be causal, then right sided inverse transforms need to be chosen.

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Sol:

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Power Series Expansion (Long Division Method)

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Sol:

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Expanding the standard equation, we get

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Sol:

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Sol:

Using power series,

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Causality and Stability

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• ROC must include the unit circle in the z-plane.

• LTI systems that are both stable and causal must have all their poles inside the unit circle.

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Fig: relationship between the location of pole and impulse response characteristics of stable system.

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