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DSP Lectures

Prof. A.H.M. Asadul Huq, Ph.D.

http://asadul.drivehq.com/students.htm

[email protected]

1/22/2014 6:13 PM A.H. 1

ULAB ETE 315


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The Z-Transform [Ifeachor P. 174]

INTRODUCTION

The Z-Transform provides a method for analysis of discrete-

time (D-T) signal and systems in the frequency domain.

The frequency domain analysis is more efficient than timedomain analysis in some cases.

The frequency response of a D-T system can be evaluated

using the transform.

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Definition of z-Transform [P.174]

Z{x(n)}x(n) X(z)

Direct Z-Transform Equation

In the above equation x(n) is the D-T sequence and X(z) is the transform.

Here, z is a continuouscomplex variable, i.e., z = Re(z) + jIm(z). In the polar

form, z = rej.

Actually zis a point in the complex z-plane as shown the fig in the

following slide.

Since the z-transform is a power series, it (the transform) exists only for

those values of z for which the X(z) series is notinfinite.

1/22/2014 6:13 PM A.H. 3

n

n eqnznxzX 6.4.)()(


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The Geometrical interpretation of the Z-

Transform

z is a complex variable that takes on continuous values.

z=Re(z) + j Im(z) = rej; where, r=|z|, and is the angle of z.

The contour |z|=1 is a circle in the z-plane of unity radius. The circleis called the Unit Circle (UC).

Fo , r=1 (|z|=1), the z-transform X(z) of x(n) reduces to its Fourier

Transform F(ejw).

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The Geometrical interpretation of

the Z-Transform-2

At z=1+j0, X(z) = X(ej0), i.e., the value of X(ej) (i.e., the

Fourier Transform) at =0;

At z=0+j1, X(z) = X(ej/2), i.e., the value of X(ej) at =/2;

If we evaluate X(z) on the UC counter clock wise at all values

of beginning at z=1+j0 and terminating back at the samepoint, effectively we have computed X(ej) in the frequency

range 0


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Region of Convergence (ROC) of

the Z-Transform

The z-transform is a power series with an infinite number of

terms and so many not converge for ALL values of z.

The region (in the z-plane) constituted by the set R of values

of z for which its ZT converges (finite) is called the ROC. Convergence of X(z) depends only on |z|, since the condition

is-

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n

nznx )(


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the Z-Transform-2

If some value of z, say, z=z1, is in the ROC, then all values of z

on the circle defined by |z|=|z1| will also be in the ROC.

For above reason, the ROC will consist of a RING in the z-plane

centered about the origin. The ring can extend outward to infinity or its inner boundary

can extend toward the center.

The area excluding the ROC corresponds to a region where

the transform does not exist (diverging or infinite).

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the Z-Transform-2

In general, the ROC of a ZT is an annular region of the z-plane:R1


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Computing z-Transforms

1. z-Transform of a finite duration causal (starts from zero) signal

1/22/2014 6:13 PM A.H. 9

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ROC: ROC is everywhere except at z=0, because X(z) is

infinite for z=0.


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Computing z-transforms

2. z-transform of a finite duration signal

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Here, X(z) is infinite if z=0 or if z=infinity. Thus ROC is

everywhere except at z=0 and z= infinity


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Computing z-transform

3. z-transform of an infinite series

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1/22/2014 6:13 PM 11A.H.


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The Comparison of Transforms

X() X(z) X(k)

FT of D-T Signal Z-Transform DFT

Transformer

x(n) Time domain

X( ) Frequency domain

1/22/2014 6:13 PM A.H. 12

X(w) and X(z) both apply to -

Infinite length sequences

Functions of continuous variablesThese two transforms are troublesome for numerical

computations because of infiniteness and continuous nature.

DFT is on the other hand numerically computable.


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POLESANDZEROSOFTHEZ-TRANSFORM

The rational z-transform will have the following structure:

The two polynomials, P(z) and Q(z), allow us to find the poles

and zeros of the Z-Transform.

Definition of zeros

The value(s) for z where P(z)=0. [The complex frequencies that

make the overall gain of the filter transfer function zero.]

Definition of poles

The value(s) for z where Q(z)=0. [The complex frequencies

that make the overall gain of the filter transfer function

infinite].

1/22/2014 6:13 PM 13A.H.


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Example of Zeros and Poles

A simple transfer function with the poles andzeros shown below

The zeros are: {-1}

The poles are:

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The Z-Plane

Once the poles and zeros have been found for agiven Z-Transform, they can be plotted onto theZ-Plane.

The Z-plane is a complex plane with an imaginary

and a real axis. The position on the complex plane is given by and

the angle from the positive, real axis around theplane is denoted by rej.

On the z-plane, poles are denoted by an "x" andzeros by an "o".

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Pole Zero Plot

The figure shows the Z-Plane, and examples ofplotting zeros and poles.

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Example: SimplePole/Zero Plot

The zeros are: {0}. The poles are :

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DSP Lecturez-Transform

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THE END

THANK YOUThis ppt may be downloaded from my web site:

http://asadul.drivehq.com/students.htm

Password email address): [email protected] password does not live long

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