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Applications of Z-Transform in Signal Processing and Discrete Systems

Z -transform of the autocorrelation function r x (m)

If: Then:

ZT [ x ( n )] X ( z )


xx

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R ( z ) ZT [ r ( m )] X ( z ) X ( z 1 )

Applications of Z-Transform in Signal Processing and Discrete Systems


(2.52)

2.4. Application of Z- transform in signal processing and discrete systems

2.4.1. Transfer function of discrete system

We know that in the n- domain , a linear invariant system is characterized by its impulse response h ( n ) or by a constant coefficient linear difference equation, but system analysis often encounters inconveniences such as convolution, solving the difference equation, and considering stability.

To solve the difficulties in the n- domain, we will transfer the system representation to the Z- domain , specifically we introduce the concept of system transfer function.

a. Definition: The transfer function of a discrete system is the Z- transform of the impulse response and is denoted by H ( z ):

H z ZT [ h ( n )]

H zY z

X z

h(n)

In the discrete time domain n , the input-output relationship of the system is expressed by the convolution:

x(n) y(n)


The output response y ( n ) of the discrete system is given by the convolution y ( n ) x(n) * h(n)

We can find:

X ( z ) ZT [ x ( n )] and

H ( z ) ZT [ h ( n )]

According to the convolution property of the Z transform, we have:

Y ( z ) ZT [ y ( n ) x ( n ) * h ( n )] X ( z ) H ( z )


From that we can deduce:

H ( z ) Y ( z )

X ( z )


H(z)

X(z) Y(z)


In the Z domain the convolution has been transformed into a regular algebraic multiplication, which is one of the advantages of the Z transform .

Take the inverse Z- transform of the transfer function H ( z ) of the invariant linear system

causally, obtain the impulse characteristic h ( n ) of the system:

h ( n ) IZT H ( z )

In the Z- domain the input-output relationship of the system is realized by the usual algebraic multiplication instead of convolution, which leads to high computational efficiency. H ( z ): The transfer function of a discrete-time system is the Z- transform of the impulse response or it is also defined as the ratio of the Z- transform of the output signal to the Z -transform of the input signal. H ( z ) is the transfer function of the system that completely characterizes the system in the Z- domain and plays a similar role to the impulse response h ( n ) in the discrete-time domain.

b. Determine the transfer function H ( z ) of a discrete system from the difference equation

Consider a discrete linear system that is invariant and causal and is described by a general difference equation of order N :

NM

a ky ( n k ) b rx ( n r )

k 0

N

r 0


M

Take ZT on both sides:

ZT a ky ( n k ) ZT b rx ( n r )

k 0r 0

According to the linearity and hysteresis properties of the Z- transform, we get:


az Y ( z ) bz X ( z )

NM

kr

kr


Hence:

k 0

r 0


MM

bz rbz ( M r )

Y ( z )

rr

( N )

H ( z ) r 0r 0zM

(2.53)

X ( z ) N

N

k (

k )

azaz N


If a 0 = 1 will be:

kk

k 0 k 0


bz

M

r

Y ( z ) r

H ( z ) r 0

(2.54)

N

X ( z )

1 a kz

k

k 1

c. Represent the transfer function H ( z ) by poles and zeros.

Just like a discrete signal, the transfer function of a discrete system can be represented by its poles and zeros as follows:

(1 zz )

M

1

0

N

H ( z ) ck 1

pk

(1 zz 1 )

(2.55)

k 1


For example:


M

(1 z 0 )

N

cz N M k 1

(1 z pk )

k 1


(2.56)

Given a causal invariant linear system with the difference equation:

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

Determine the transfer function H ( z ) and the impulse characteristic h ( n ) of the system.

Prize:

Take the Z- transform of both sides of the above difference equation:

2 Y ( z ) 4 z 1 Y ( z ) 2 z 2 Y ( z ) X ( z ) 3 z 1 X ( z )

Or:

Y ( z )(2 4 z 1 2 z 2 ) X ( z )(1 3 z 1 )

Y ( z ) (1 3 z 1 )

z 1 ( z 3)

z ( z 3)

H ( z )

X ( z ) (2 4 z 1 2 z 2 ) 2 z 2 ( z 2 2 z 1 ) 2( z 2 2 z 1 )


So the transfer function is:

H ( z ) Y ( z )

z ( z 3)

z ( z 3)

X ( z ) 2( z 2 2 z 1 ) 2( z 1) 2

Take the inverse Z -transform of the transfer function H ( z ), and find the impulse characteristic h ( n ):

h ( n ) IZT H z IZT z ( z 3) with RC H z : z 1

2( z 1) 2


To find h(n) , analyze the function:

H ( z ) ( z 3 ) C 1 C 2

z 2( z 1) 2

( z 1) ( z 1) 2

1 3

( z 3 ) ( z 1 ) 2

2( z 1 ) 2

In which: C 2 z 1 C 2

d ( z 3 ) ( z 1 ) 2

1

2

1

C 1 dz

2( z 1 ) 2

z 1

C 1 2


So:

H ( z ) ( z 3 ) 1 1

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

H ( z ) 1 z z

2 ( z 1) ( z 1) 2

With:

RC H :| z | 1,

h ( n ) 0.5 u ( n ) nu ( n ) . Or:

h ( n ) (0.5 n ) u ( n )

2.4.2. System analysis in Z domain

a. System implementation elements

In chapter 1 we presented the representation of the implementation elements in the domain n . From this representation, taking the input and output Z- transforms of the implementation elements, we will have the following representation in the domain Z :

- Additive element

Let x i ( n ) be the inputs and y ( n ) be the outputs, we have the following relationship:

M

y ( n ) x i n

i 1

Taking the Z transform we have:


Z T y n ZT Mx n

i

i 1


M

YzX iz

i 1

The additive element in the Z domain is used to add two or more image functions X i ( z ) and is denoted as in Figure 2.11.

X 1 ( z )

2

X 1 ( z ) Y ( z ) X ( z )


Y ( z )


X 2 ( z )

X i ( z )


X M ( z )

M

a. Y (z) = X 1 ( z ) + X 2 ( z ) b. Y ( z ) X i ( z )

i 1

Figure 2.11. Additive element symbol in domain Z.

- Unit delay element

According to the delay property of the Z- transform : Y ( z ) ZT [ x ( n 1)] z 1 X ( z )

Therefore, the unit delay element in the Z domain has a transfer function as shown in Figure 2.12.

z 1

X ( z ) Y ( z )

H ( z ) z 1 and it is



Figure 2.12. Unit delay element symbol in Z domain

- Element multiplied by constant

Let x ( n ) be the input and y ( n ) be the outputs, we have the following relationship:

y ( n ) ax n

Taking the Z transform we have:


Z T y n ZT ax n

YzaXz

The multiplier by constant is used to multiply the function X ( z ) by the constant a , it is denoted as in Figure 2.13.

a

X ( z ) Y ( z )


Figure 2.13. Element symbol multiplied by a constant in the Z domain

Example 1:

Let's build a structural diagram in the Z domain of the digital system with input-output relationship:

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

Prize:

Take the Z -transform on both sides of the above equation to get:

Y ( z ) 2 X ( z ) 3 z 1 X ( z ) 0.5 z 1 Y ( z )

From there, we can construct a structural diagram in the Z domain of the system in Figure 2.14.


2

X ( z )

Y ( z )


z 1

z 1

3 - 0.5


Figure 2.14. Structural diagram of the system of example 1.

b. Discrete system analysis

A complex digital system can be described by a structure diagram or block diagram consisting of many interconnected blocks, where each block is characterized by a transfer function H i (z). Once the structure diagram or block diagram and the component transfer functions H i ( z ) are known , the transfer function H ( z ) of the entire system can be determined .

Discrete system analysis is based on the following general principles:

- Analyze the overall system into smaller systems (or smaller blocks).

- Find the connections between these smaller blocks.

- Find the transfer function Hi ( z ) of each of these small blocks.

- Combine the transfer functions Hi ( z ) of the found small blocks according to the above analysis rule .

How to connect the system diagram in the Z domain :

- Transfer function H ( z ) of series connected blocks

Consider the digital system consisting of m series-connected blocks in Figure 2.15. Then the output response of the system will be determined according to the expression:


From that we can deduce:

Y ( z ) X ( z ) H 1 ( z ) H 2 ( z )...... H m ( z ) X ( z ) H ( z )


m

H ( z ) H i( z )

i 1


(2.57)

The transfer function H ( z ) of the series-connected blocks is equal to the product of the component transfer functions Hi ( z )


H 1 (z)

H 2 (z)

H m (z)

X(z) Y(z)


Figure 2.15. Schematic diagram of series-connected H i ( z ) blocks .

- Transfer function H ( z ) of parallel connected blocks

Consider the number system consisting of m parallel connected blocks in Figure 2.16.

H 1 (z)

H 2 (z)

H m (z)

X(z) Y(z)



Figure 2.16. Diagram of parallel-linked H i ( z ) blocks .

Then the output response of the system will be determined according to the expression:

Y ( z ) X ( z ) H 1 ( z ) X ( z ) H 2 ( z ) ... X ( z ) H m ( z )

Or:

Y ( z ) X ( z ) H 1 ( z ) H 2( z ) ... H m( z ) X ( z ) H ( z )

From that we can deduce:


m

H ( z ) H i ( z )

i 1


(2.58)

The transfer function H ( z ) of the parallel interconnected blocks is equal to the sum of the component transfer functions Hi ( z ) .


- Transfer function H ( z ) of the feedback loop

Consider the digital system with feedback loop in figure 2.17, according to the block diagram:

X 2 ( z ) Y ( z ) H 2 ( z )

and :

X 1 ( z ) X ( z ) X 2 ( z ) X ( z ) Y ( z )] H 2 ( z )

Y ( z ) X 1 ( z ) H 1 ( z ) X ( z ) Y ( z ) H 2( z ) H 1 ( z )


From that we can deduce

Y ( z ) X ( z ) H 1 ( z ) Y ( z ) H 1 ( z ) H 2 ( z )

Y ( z ) 1 H 1 ( z ) H 2( z ) X ( z ) H 1 ( z )

H ( z )

Y ( z ) H 1 ( z )


(2.59)


X(z)

X ( z ) 1 H 1 ( z ) H 2 ( z )


X 2 (z)

H 2 (z)

H 1 (z)

X 1 (z)


Y(z)


Figure 2.17. Block diagram of the feedback loop.

Example 2:

Find the transfer function H ( z ) of the causal invariant linear system with the structural diagram shown in Figure 2.18.


X z

z 1

2

z 1

5

z 1

Y z

z 1

z 1

-3

z 1

Figure 2.18. Structural diagram of the number system of example 2.

Prize:

To find the transfer function H ( z ) of a given digital system, we can do it in the following order: First, find the transfer function of the series and parallel connected blocks and the transfer function of the feedback loops that only include one block, from there gradually reduce the block diagram of the system to one block, the transfer function of that block is the transfer function H ( z ) that needs to be found.

However, it is possible to find the transfer function H ( z ) of a given system more quickly by moving all the feedback loops to a common adder as shown in Figure 2.19, and then performing the reduction steps.

After determining the transfer function of the series and parallel connected blocks in the block diagram of Figure 2.19, reduce the diagram to the form of Figure 2.19, with H 2 ( z ) being the transfer function of the parallel connected feedback blocks:

H 2 ( z ) z 5 z 1 3 z 1 3 z 1 z 5 z 1 9 z 2


Y z

2

3z 1​

z

z  1

z  1

5z  1

3z  1

3z 1​

X z


Figure 2.19. After bringing the feedback loops to an adder.

3z 1​

H 2 (z)

z 2

X(z) Y(z)


Figure 2.20. After calculating the transfer function of series and parallel blocks.

Continue to find the transfer function of the feedback loop on the diagram in Figure 2.20:

z 2 z 2 z 2

2

H 3 ( z ) 1 z 2 H

( z ) 1 z 2 ( z 5 z 1 9 z 2 ) z 4 z 3 5 z 9

The block diagram can be reduced to the form shown in Figure 2.21. Calculate the transfer function of two series-connected blocks to obtain the transfer function H ( z ) of the given digital system:

3

H ( z ) 3 z 1 H

( z )

3z​

H 3 (z)

3z 1​

H(z)

z 4 z 3 5 z 9


X ( z )

Y ( z )

X ( z ) Y ( z )



Example 3:

Figure 2.21. After calculating the transfer function of the feedback link and two series-connected blocks, H(z) is obtained .

For a discrete system with a detailed diagram as shown below:

z  1

X(z)

X 1 (z)


Y(z)


Figure 2.22. System diagram in example 3

Prize:

Add an additional variable X 1 ( z ), then find the relationship between Y( z ) and X( z )

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