the z-transform

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1 The z-Transform ECON 397 Macroeconometrics Cunningham

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The z-Transform. ECON 397 Macroeconometrics Cunningham. z-Transform. The z-transform is the most general concept for the transformation of discrete-time series. The Laplace transform is the more general concept for the transformation of continuous time processes. - PowerPoint PPT Presentation

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Page 1: The z-Transform

1

The z-Transform

ECON 397

Macroeconometrics

Cunningham

Page 2: The z-Transform

2

z-Transform

The z-transform is the most general concept for the transformation of discrete-time series.

The Laplace transform is the more general concept for the transformation of continuous time processes.

For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.

The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.

Page 3: The z-Transform

3

The Transforms

The Laplace transform of a function f(t):

0

)()( dtetfsF st

The one-sided z-transform of a function x(n):

0

)()(n

nznxzX

The two-sided z-transform of a function x(n):

n

nznxzX )()(

Page 4: The z-Transform

4

Relationship to Fourier Transform

Note that expressing the complex variable z in polar form reveals the relationship to the Fourier transform:

n

nii

n

nini

n

n

ii

enxXeX

rifandernxreX

orrenxreX

)()()(

,1,)()(

,))(()(

which is the Fourier transform of x(n).

Page 5: The z-Transform

5

Region of ConvergenceThe z-transform of x(n) can be viewed as the Fourier transform of x(n) multiplied by an exponential sequence r-n, and the z-transform may converge even when the Fourier transform does not.

By redefining convergence, it is possible that the Fourier transform may converge when the z-transform does not.

For the Fourier transform to converge, the sequence must have finite energy, or:

n

nrnx )(

Page 6: The z-Transform

6

Convergence, continued

n

nznxzX )()(

The power series for the z-transform is called a Laurent series:

The Laurent series, and therefore the z-transform, represents an analytic function at every point inside the region of convergence, and therefore the z-transform and all its derivatives must be continuous functions of z inside the region of convergence.

In general, the Laurent series will converge in an annular region of the z-plane.

Page 7: The z-Transform

7

Some Special Functions

First we introduce the Dirac delta function (or unit sample function):

0,1

0,0)(

n

nn

This allows an arbitrary sequence x(n) or continuous-time function f(t) to be expressed as:

dttxxftf

knkxnxk

)()()(

)()()(

or

0,1

0,0)(

t

tt

Page 8: The z-Transform

8

Convolution, Unit Step

These are referred to as discrete-time or continuous-time convolution, and are denoted by:

)(*)()(

)(*)()(

ttftf

nnxnx

We also introduce the unit step function:

0,0

0,1)(or

0,0

0,1)(

t

ttu

n

nnu

Note also:

k

knu )()(

Page 9: The z-Transform

9

Poles and ZerosWhen X(z) is a rational function, i.e., a ration of

polynomials in z, then:

1. The roots of the numerator polynomial are referred to as the zeros of X(z), and

2. The roots of the denominator polynomial are referred to as the poles of X(z).

Note that no poles of X(z) can occur within the region of convergence since the z-transform does not converge at a pole.

Furthermore, the region of convergence is bounded by poles.

Page 10: The z-Transform

10

Example

)()( nuanx n

The z-transform is given by:

0

1)()()(n

n

n

nn azznuazX

Which converges to:

azforaz

z

azzX

11

1)(

Clearly, X(z) has a zero at z = 0 and a pole at z = a.

a

Region of convergence

Page 11: The z-Transform

11

Convergence of Finite Sequences

Suppose that only a finite number of sequence values are nonzero, so that:

2

1

)()(n

nn

nznxzX

Where n1 and n2 are finite integers. Convergence requires

.)( 21 nnnfornx

So that finite-length sequences have a region of convergence that is at least 0 < |z| < , and may include either z = 0 or z = .

Page 12: The z-Transform

12

Inverse z-Transform The inverse z-transform can be derived by using Cauchy’s integral theorem. Start with the z-transform

n

nznxzX )()(

Multiply both sides by zk-1 and integrate with a contour integral for which the contour of integration encloses the origin and lies entirely within the region of convergence of X(z):

transform.-z inverse the is)()(21

21

)(

)(21

)(21

1

1

11

nxdzzzXi

dzzi

nx

dzznxi

dzzzXi

C

k

n C

kn

C n

kn

C

k

Page 13: The z-Transform

13

Properties

z-transforms are linear:

The transform of a shifted sequence:

Multiplication:

But multiplication will affect the region of convergence and all the pole-zero locations will be scaled by a factor of a.

)()()()( zbYzaXnbynax Z

)()( 00 zXznnx nZ

)()( 1zaZnxan Z

Page 14: The z-Transform

14

Convolution of Sequences

both. of econvergenc of regions the inside of values for)()()(

)()()(

let

)()(

)()()(

Then

)()()(

zzYzXzW

zzmykxzW

knm

zknykx

zknykxzW

knykxnw

k

k m

m

n

k n

n

n k

k

Page 15: The z-Transform

15

More DefinitionsDefinition. Periodic. A sequence x(n) is periodic with period if and only if x(n) = x(n + ) for all n.

Definition. Shift invariant or time-invariant. Consider a sequence y(n) as the result of a transformation T of x(n). Another interpretation is that T is a system that responds to an input or stimulus x(n): y(n) = T[x(n)].

The transformation T is said to be shift-invariant or time-invariant if:

y(n) = T [x(n)] implies that y(n - k) = T [x(n – k)]

For all k. “Shift invariant” is the same thing as “time invariant” when n is time (t).

Page 16: The z-Transform

16

k

k

k

kk

k

nhnxknhkxny

T

nhkxny

knTkxknkxTny

kn

knnh

).(*)()()()(

then , transform the of invariance time have weIf

).()()(

)()()()()(

:Then . at occurring

shock or spike"" a ),( to system the of response the be )( Let

This implies that the system can be completely characterized by its impulse response h(n). This obviously hinges on the stationarity of the series.

Page 17: The z-Transform

17

Definition. Stable System. A system is stable if

k

kh )(

Which means that a bounded input will not yield an unbounded output.

Definition. Causal System. A causal system is one in which changes in output do not precede changes in input. In other words,

. for)()( then

for)()( If

021

021

nnnxTnxT

nnnxnx

Linear, shift-invariant systems are causal iff h(n) = 0 for n < 0.

Page 18: The z-Transform

18

.)()(

that so )()(Let

)()()(

Then. for )( let

is, That .sinusoidal be)( let)()()( Given

)(

nii

k

kii

k

kini

k

kni

ni

k

k

eeHny

ekheH

ekheekhny

nenx

nxnhkxny

Here H(ei) is called the frequency response of the system whose impulse response is h(n). Note that H(ei) is the Fourier transform of h(n).

Page 19: The z-Transform

19

We can generalize this state that:

. of function continuous a touniformly converges

and convergent absolutely is transform the then,)(

)(21

)(

)()(

n

nii

n

nii

nxIf

deeXnx

enxeX

This implies that the frequency response of a stable system always converges, and the Fourier transform exists.

These are the Fourier transform pair.

Page 20: The z-Transform

20

If x(n) is constructed from some continuous function xC(t) by sampling at regular periods T (called “the sampling period”), then x(n) = xC(nT) and 1/T is called the sampling frequency or sampling rate.

If 0 is the highest radial frequency of sinusoids comprising x(nT), then

21

or 2 0

0w

TT

Is the sampling rate required to guarantee that xC(nT) can be used to fully recover xC(t), This sampling rate 0 is called the Nyquist rate (or frequency). Sampling at less than this rate will involve losing information from the time series.

Assume that the sampling rate is at least the Nyquist rate.

Page 21: The z-Transform

21

deekTxT

tx

ekTxeX

deeTXtx

deiXtx

TTiX

TeX

tiT

T k

Tkicc

k

Tkic

Ti

tiTiT

T

c

tiT

T

cc

cTi

)(2

)(

have we,)()( Since

.)(21

)(

:Combining

.)(21

)(

:transform Fourier time continuous the From

),(1

)(

Page 22: The z-Transform

22

.)sin

)()(

:integral the Evaluating

2)()(

n,integratio and summation of order the Changing

)(

k

cc

k

T

T

kTticc

kTtT

kTtTktxtx

deT

kTxtx

NOTE: This equation allows for recovering the continuous time series from its samples. This is valid only for bandlimited functions.