signal and system - hansungkwangho/lectures/signal_system/2015... · signal and system ( chapter 9:...

1 Dept. Information and Communication Eng. 1

Signal and System( Chapter 9: The z-Transform )

Prof. Kwang-Chun [email protected]: 02-760-4253 Fax:02-760-4435


What is the sampling of signals?Analog signals are continuous time signals

Can exist as voltage or current waveforms, or as mathematical functions

Modern systems must convert continuous time signals into discrete digital signals to be compatible with digital storage, digital transmission, and digital signal processing.

Sampling is the process in which continuous time signals are converted into discrete digital samples.Sampling is usually performed at a constant frequency

or sampling rate.

Sampling of Signals

( ) cos 2v t ft


Sampling theorem of continuous signals:First of all, consider an ideal sampler, formed from a

periodic impulse waveform, as follows:

A discrete-time signal g(t) can be obtained by sampling f(t) with the ideal sampler

This equation is equivalent to the following since the delta function has the effect of making g(t) nonzero only at times .

Sampling of Signals

( ) ( )T sn

t t nT

0( ) ( ) ( )s

ng t f t t nT

st nT

0( ) ( ) ( )s s

ng t f nT t nT


Sampling of Signals

n1 2 3 4


Taking Laplace transform of the sampled signal, it yields

A much simpler expression results if the following substitutions are made:

Then, the definition of the z-transform is

If the sampling time is fixed, the z-transform can also be written

Sampling of Signals

0( ) ( ) ssnT

sn

G s f nT e

, ( ) ( )ssTz e G s G z

0( ) ( ) n

sn

G z f nT z

sT

( ) ( ) n

nG z f n z

Is the same as Discrete-Time FT for discrete time signal ( )f n

( ) ( ) sj T ns

n

G f nT e

(DTFT)


A digital signal is superior to an analog signal because it is more robust to noise and can easily be recovered, corrected and amplified. For this reason, the tendency today is to change an

analog signal to digital data.Pulse code modulation (PCM) is essentially analog-

to-digital conversion of a special type, where the information contained in the instantaneous

samples of an analog signal is represented by digital words in a serial bit stream.

Digital Transmission


PCM consists of three steps to digitize an analog signal:Sampling, Quantization, Binary encoding


(analog-to-digital converter)


Sampling: Analog signal is sampled every sec.

Quantization: Quantizing operation approximates the analog values by

using a finite number of levels.

Binary encoding:Maps the quantized values to digital words that are

bits long.


sT

n


PCM DecoderTo recover an analog signal from a discrete digital

signal



Example 9.1:Determine the z-transform of unit step signal

Solution:

A continuous step function shown above is plotted in blueand the sampled step in red.

The z-Transform

0 , 0( )

1 , 0t

x tt

Determine the Discrete-Time FT for discrete time signal ( )x n


When a step function is sampled, each sample has a constant value of 1.

The z-transform can be written as a sum of terms as indicated below

The z-Transform

10

1( ) ( )1

n n

n nX z x n z z

z

1 1z If , this infinite series will converge !!

The set of values of z for which z-transform convergesis called the region of convergence (ROC).

X(z) exists for z-values in this ROC.

11 ( )ssT

s je

Spectrum for discrete time signal ( )x n


Example 9.2:Determine the z-transform of a step function delayed

by 1 sampling interval.

Solution:The z-transform is given by

The z-Transform

1 1 2 11 1

1

1( ) ( 1) 11

n n

n nX z x n z z z z z z

z


The z-transform of can be written as where is the z-transform of .

The z-Transform

( 1)x n 1 ( )z X z

( )X z ( )x n

[ Block diagram of delay theorem ]

[Discrete-time system expressed in the z-domain ]

AdderAdder

Multiplier

Unit delay


The general form of difference equation in LTI (causal and time-invariant) system is

If the are all zero, the systems are called 1st-order finite impulse response (FIR) systemsFIR: Each output is the sum of a finite number of

weighted samples of the input sequence

If at least one of the is nonzero, the systems are called -th order infinite impulse response (IIR) systems

The Difference Equation

mb

mbN

0 0

( ) ( ) ( )K N

m mm m

y n a x n m b y n m


IIR: It is a class of feedback system, and the output is computed in terms of previously computed values of the output as well as values of the input signal

Example 9.3:Find the first five output values for the difference

equation subjected to a discrete unit step input

Solution:


( ) ( ) ( 2) 0.5 ( 1)y n x n x n y n if0 0

( )1 0

nx n

n

Use iteration table:


Since the input is zero prior to n=0, the output y(n)=0 for n<0. This is causal system

Thus, the forced response is expressed as y(n)=0 if n<0, y(0)=1 and if

Example 9.4:Determine the complete response for of a

discrete-time system

for a step input x(n)=8u(n) and with initial conditions

( ) 3 0.5 ny n 1n

0n

( ) ( 1) 6 ( 2) ( )y n y n y n x n

( 1) 1, ( 2) 1y y

It is not easy to determine the general expression. We will see it later in z-transform.



Solution:First, determine the natural response by setting x(n)=0

and (eigenfunction), which yields

Thus, the roots of characteristic equation are

and the complementary solution is of the form

For the particular solution, we assume Substituting this one into the system equation, we get

which for yields


( ) ny n

1 2 2 2

26 6

3 2 0

n n n n

n

1 23, 2

1 2( ) 3 2n nhy n c c

( ) , 0 py n n

6 8 ( )u n 0n 2


Then, the nonhomogeneous solution becomes

We obtain the complete response of the system by solving for the unknown constants so that the initial conditions and are satisfied

Thus, the complete response is given by


1 2( ) ( ) ( ) 3 2 2, 0n nnh h py n y n y n c c n

2 21 2

1 11 2

( 2) 3 2 2 1,

( 1) 3 2 2 1

y c c

y c c

1 21.8, 4.8c c

( 1) 1y ( 2) 1y

( ) 1.8 3 4.8 2 2, 0n ny n n


Example 9.5:Determine the complete response of a discrete-time

system with initial conditionsSolution:

The characteristic equation is given by

so that is a repeated root. The complementary solution is

Applying the initial conditions, the natural response is


(0) 1, (1) 3y y ( ) 2 ( 1) ( 2) 0y n y n y n

22 2 1 1 0,

1 2 1

1 2( ) 1 1 .n ny n c c n

( ) 1 4 1 .n ny n n


It is the discrete-time counterpart of the Laplace transform for continuous-time signal.

The z-transform is defined as Example 9.6:Determine the z-transform of a system

Solution:It is the sum of two systems

First of all, consider the z-transform of

( ) ( ) n

n

G z f n z

The z-Transform

( ) ( ) ( 1) n nx n u n u n

1 2( ) ( ), ( ) ( 1) n nx n u n x n u n

1( )x n

1 10

1( )1

n n

n

X z zz

( ROC is , that is )1 1 z z


Next, for it is

Thus, the region of convergence for X(z)is the intersection of two regions and X(z) is

2 ( )x n1

12 1

1 0

1( )1

n n n n n n

n n n

X z z z z zz

( ROC is )1 1 z

z

1 2

1 1

( ) ( ) ( )1 1 ,

1 1

X z X z X z

z z

The z-Transform

z (ROC)


A important property of z-transform:Time Shifting:

Let x(n) be a causal sequence and let X(z) denote its transform. Then, for any integer

Similarly,

0 0n

0

0

0

00

0

( )

0

1

0

0 0

1

0 0

( ) ( ) ( )

(

( )

) ( )

( )

m nn

m n

nn m m

m

m

m

nmn

n

Z x n n x n n z x m z

z x m z x m

z

z

X mz xz

0

0

1

0( ) ( ) ( )

n m

m n

Z x n n z X z x m z

Can be called initial conditions !

The z-Transform


z-transform Properties:

The z-Transform


Example 9.7:Find the complete response of system with zero initial

conditions

Solution:Taking z-transform of both sides gives

1( ) ( ) ( 2) ( 1)2

y n x n x n y n if0 0

( )1 0

nx n

n

2 2

1( ) ( ) ( ) ( 2) ( 1)

0.5 ( ) ( 1)

Y z X z z X z x z x zz Y z y z

The z-Transform


Setting all the initial conditions to zero and substituting , we have

Using the inverse z-transform and the time-shift theorem, it becomes

1 3 / 22( 1) 3( ) 1 12 1 2 1 1/ 2

zzY z zz z z

3 1 ( )2 2

n

u n

13 1( ) ( ) ( 1)2 2

n

y n n u n

The z-Transform

( ) /( 1) X z z z


Example 9.8:Find the complete response of the system

for y(n), , if x(n)=u(n), y(1)=1 and y(0)=1Solution:

Applying the property of time-shifting, we have

Substituting and using the given initial conditions, we get

2( 2) ( 1) ( ) ( )9

y n y n y n x n

0n

2 1 2( ) (0) (1) ( ) (0) ( ) ( )9

z Y z y y z z Y z y Y z X z

( ) / 1 X z z z

The z-Transform


Expanding the fractional term in partial fractions yields

Consequently, the complete response is given by

The z-Transform

2

2 2

2

(0) (1)( )( )2 29 9

1 (1/ 3) (2 / 3) (1/ 3) (2 / 3)

z z y zyX zY zz z z z

z zz z z z z

Forced response Natural response

(9 / 2) (7 / 2) 7( )1 (1/ 3) (2 / 3)

z z zY zz z z

9 7 1 2( ) 7 ( )2 2 3 3

n n

y n u n


Example 9.9:Find the transfer function of the system shown in

Figure.Solution:

The z-Transform


From the block diagram, we have

Substituting and into yields

Thus, the transfer function is given by

The z-Transform

1( 1)w n 2 ( 2)w n ( )y n

0 1 1 2 2( ) ( ) ( 1) ( 1) ( 2) ( 2)y n b x n b x n a y n b x n a y n

1 2 20 1 2 0 1 2

1 2 21 2 1 2

( )1b b z b z b z b z bH z

a z a z z a z a

0 1

1 1 1 2

2 2 2

( ) ( ) ( 1),( ) ( ) ( ) ( 1),( ) ( ) ( ).

y n b x n w nw n b x n a y n w nw n b x n a y n

1 1 1 2

2 2 2

( 1) ( 1) ( 1) ( 2),( 2) ( 2) ( 2).

w n b x n a y n w nw n b x n a y n


A better approach is to check the poles of natural response or transfer function.

If , the natural response will grow. Such a system is unstable

If , the natural response neither grows or decays. Such a system is conditionally stable

If , the natural response is a transient response. Such a system is stable (as Example 9.8)

Stability in the z-Domain

s s s

s

sT T j T

Ts p s

z e e e

e T z T

1pz

1pz

1pz


Problem 9.1:Find the transfer function represented by the

following difference equations(a)(b)

Problem 9.2:Determine the difference equation for each of the

following systems

(a) (b)

Homework Assignments

( ) ( ) ( 1) 0.75 ( 1) 0.5 ( 3)y n x n x n y n y n

( ) 0.25 ( ) 0.5 ( 1) ( 2) 0.5 ( 3) 0.25 ( 4)y n x n x n x n x n y n

4 3 2

4

4 2 1( ) z z z zH zz

4

4

1( )0.5

zH zz


Problem 9.3:Determine the natural response of a system whose

difference equation is

Problem 9.4:Using Matlab, plot the output given that

is passed through a filter with a transfer function of


3( ) ( ) 2 ( 2) ( 1) ( 2)4

y n x n x n y n y n

( )y n( ) sin(0.05 )x n n

2 1( )

0.5zH z

z z


Problem 9.5:Find the difference-equation descriptions for the

systems depictedDetermine the complete solution. Assume that all the

initial conditions are zero, and .


( ) ( )x n u n

signal and system - hansungkwangho/lectures/signal_system/2015... · signal and system ( chapter 9:...

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