Chapter 6

Transfer function

and Digital Filter Realization

Nguyen Thanh Tuan, Click

M.Eng.

to edit Master subtitle style

Department of Telecommunications (113B3)

Ho Chi Minh City University of Technology

Email: nttbk97@yahoo.com

With the aid of z-transforms, we can describe the FIR and IIR filters

in several mathematically equivalent way

Digital Signal Processing

2

Transfer function

and Digital Filter Realization

Content

1. Transfer functions

Impulse response

Difference equation

Impulse response

Frequency response

Block diagram of realization

2. Digital filter realization

Direct form

Canonical form

Cascade form

Digital Signal Processing

3

Transfer function

and Digital Filter Realization

1. Transfer functions

Given a transfer functions H(z) one can obtain:

(a) the impulse response h(n)

(b) the difference equation satisfied the impulse response

(c) the I/O difference equation relating the output y(n) to the input

x(n).

(d) the block diagram realization of the filter

(e) the sample-by-sample processing algorithm

(f) the pole/zero pattern

(g) the frequency response H(w)

Digital Signal Processing

4

Transfer function

and Digital Filter Realization

Impulse response

Taking the inverse z-transform of H(z) yields the impulse response

h(n)

Example: consider the transfer function

To obtain the impulse response, we use partial fraction expansion to

write

Assuming the filter is causal, we find

Digital Signal Processing

5

Transfer function

and Digital Filter Realization

Difference equation for impulse response

The standard approach is to eliminate the denominator polynomial

of H(z) and then transfer back to the time domain.

Example: consider the transfer function

Multiplying both sides by denominator, we find

Taking inverse z-transform of both sides and using the linearity and

delay properties, we obtain the difference equation for h(n):

Digital Signal Processing

6

Transfer function

and Digital Filter Realization

I/O difference equation

Write

then eliminate the denominators and go back

to the time domain.

Example: consider the transfer function

We have

which can write

Taking the inverse z-transforms of both sides, we have

Thus, the I/O difference equation is

Digital Signal Processing

7

Transfer function

and Digital Filter Realization

Block diagram

One the I/O difference equation is determined, one can mechanize it

by block diagram

Example: consider the transfer function

We have the I/O difference equation

The direct form realization is given by

Digital Signal Processing

8

Transfer function

and Digital Filter Realization

Sample processing algorithm

From the block diagram, we assign internal state variables to all the

delays:

We define v1(n) to be the content of the x-delay at time n:

Similarly, w1(n) is the content of the y-delay at time n:

Digital Signal Processing

9

Transfer function

and Digital Filter Realization

Frequency response and pole/zero pattern

Given H(z) whose ROC contains unit circle, the frequency response

H(w) can be obtained by replacing z=ejw.

Example:

Using the identity

we obtain an expression for the magnitude response

Drawing peaks when

passing near poles

Drawing dips when

passing near zeros

Digital Signal Processing

10

Transfer function

and Digital Filter Realization

Example

Consider the system which has the I/O equation:

a) Determine the transfer function

b) Determine the casual impulse response

c) Determine the frequency response and plot the magnitude response

of the filter.

d) Plot the block diagram of the system and write the sample

processing algorithm

Digital Signal Processing

11

Transfer function

and Digital Filter Realization

2. Digital filter realizations

Construction of block diagram of the filter is called a realization of

the filter.

Realization of a filter at a block diagram level is essentially a flow

graph of the signals in the filter.

It includes operations: delays, additions and multiplications of signals

by a constant coefficients.

The block diagram realization of a transfer function is not unique.

Note that for implementation of filter we must concerns the

accuracy of signal values, accuracy of coefficients and accuracy of

arithmetic operations. We must analyze the effect of such

imperfections on the performance of the filter.

Digital Signal Processing

12

Transfer function

and Digital Filter Realization

Direct form realization

Use the I/O difference equation

The b-multipliers are feeding forward

The a-multipliers are feeding backward

Digital Signal Processing

13

Transfer function

and Digital Filter Realization

Example

Consider IIR filter with h(n)=0.5nu(n)

a) Draw the direct form realization of this digital filter ?

b) Given x=[2, 8, 4], find the first 6 samples of the output by using the

sample processing algorithm ?

Digital Signal Processing

14

Transfer function

and Digital Filter Realization

Canonical form realization

Note that

Y ( z) H ( z) X ( z) N ( z)

1

1

X ( z)

N ( z) X ( z)

D( z )

D( z )

The maximum number of

common delays: K=max(L,M)

Digital Signal Processing

15

Transfer function

and Digital Filter Realization

Cascade form

The cascade realization form of a general functions assumes that the

transfer functions is the product of such second-order sections

(SOS):

Each of SOS may be realized in direct or canonical form.

Digital Signal Processing

16

Transfer function

and Digital Filter Realization

Cascade form

Digital Signal Processing

17

Transfer function

and Digital Filter Realization

Review

Digital Signal Processing

18

Transfer function

and Digital Filter Realization

Homework 1

Digital Signal Processing

19

Transfer function

and Digital Filter Realization

Homework 2

Digital Signal Processing

20

Transfer function

and Digital Filter Realization

Homework 3

Digital Signal Processing

21

Transfer function

and Digital Filter Realization

Homework 4

Digital Signal Processing

22

Transfer function

and Digital Filter Realization

Homework 5

Digital Signal Processing

23

Transfer function

and Digital Filter Realization

Homework 6

Digital Signal Processing

24

Transfer function

and Digital Filter Realization

Homework 7

Digital Signal Processing

25

Transfer function

and Digital Filter Realization

Transfer function

and Digital Filter Realization

Nguyen Thanh Tuan, Click

M.Eng.

to edit Master subtitle style

Department of Telecommunications (113B3)

Ho Chi Minh City University of Technology

Email: nttbk97@yahoo.com

With the aid of z-transforms, we can describe the FIR and IIR filters

in several mathematically equivalent way

Digital Signal Processing

2

Transfer function

and Digital Filter Realization

Content

1. Transfer functions

Impulse response

Difference equation

Impulse response

Frequency response

Block diagram of realization

2. Digital filter realization

Direct form

Canonical form

Cascade form

Digital Signal Processing

3

Transfer function

and Digital Filter Realization

1. Transfer functions

Given a transfer functions H(z) one can obtain:

(a) the impulse response h(n)

(b) the difference equation satisfied the impulse response

(c) the I/O difference equation relating the output y(n) to the input

x(n).

(d) the block diagram realization of the filter

(e) the sample-by-sample processing algorithm

(f) the pole/zero pattern

(g) the frequency response H(w)

Digital Signal Processing

4

Transfer function

and Digital Filter Realization

Impulse response

Taking the inverse z-transform of H(z) yields the impulse response

h(n)

Example: consider the transfer function

To obtain the impulse response, we use partial fraction expansion to

write

Assuming the filter is causal, we find

Digital Signal Processing

5

Transfer function

and Digital Filter Realization

Difference equation for impulse response

The standard approach is to eliminate the denominator polynomial

of H(z) and then transfer back to the time domain.

Example: consider the transfer function

Multiplying both sides by denominator, we find

Taking inverse z-transform of both sides and using the linearity and

delay properties, we obtain the difference equation for h(n):

Digital Signal Processing

6

Transfer function

and Digital Filter Realization

I/O difference equation

Write

then eliminate the denominators and go back

to the time domain.

Example: consider the transfer function

We have

which can write

Taking the inverse z-transforms of both sides, we have

Thus, the I/O difference equation is

Digital Signal Processing

7

Transfer function

and Digital Filter Realization

Block diagram

One the I/O difference equation is determined, one can mechanize it

by block diagram

Example: consider the transfer function

We have the I/O difference equation

The direct form realization is given by

Digital Signal Processing

8

Transfer function

and Digital Filter Realization

Sample processing algorithm

From the block diagram, we assign internal state variables to all the

delays:

We define v1(n) to be the content of the x-delay at time n:

Similarly, w1(n) is the content of the y-delay at time n:

Digital Signal Processing

9

Transfer function

and Digital Filter Realization

Frequency response and pole/zero pattern

Given H(z) whose ROC contains unit circle, the frequency response

H(w) can be obtained by replacing z=ejw.

Example:

Using the identity

we obtain an expression for the magnitude response

Drawing peaks when

passing near poles

Drawing dips when

passing near zeros

Digital Signal Processing

10

Transfer function

and Digital Filter Realization

Example

Consider the system which has the I/O equation:

a) Determine the transfer function

b) Determine the casual impulse response

c) Determine the frequency response and plot the magnitude response

of the filter.

d) Plot the block diagram of the system and write the sample

processing algorithm

Digital Signal Processing

11

Transfer function

and Digital Filter Realization

2. Digital filter realizations

Construction of block diagram of the filter is called a realization of

the filter.

Realization of a filter at a block diagram level is essentially a flow

graph of the signals in the filter.

It includes operations: delays, additions and multiplications of signals

by a constant coefficients.

The block diagram realization of a transfer function is not unique.

Note that for implementation of filter we must concerns the

accuracy of signal values, accuracy of coefficients and accuracy of

arithmetic operations. We must analyze the effect of such

imperfections on the performance of the filter.

Digital Signal Processing

12

Transfer function

and Digital Filter Realization

Direct form realization

Use the I/O difference equation

The b-multipliers are feeding forward

The a-multipliers are feeding backward

Digital Signal Processing

13

Transfer function

and Digital Filter Realization

Example

Consider IIR filter with h(n)=0.5nu(n)

a) Draw the direct form realization of this digital filter ?

b) Given x=[2, 8, 4], find the first 6 samples of the output by using the

sample processing algorithm ?

Digital Signal Processing

14

Transfer function

and Digital Filter Realization

Canonical form realization

Note that

Y ( z) H ( z) X ( z) N ( z)

1

1

X ( z)

N ( z) X ( z)

D( z )

D( z )

The maximum number of

common delays: K=max(L,M)

Digital Signal Processing

15

Transfer function

and Digital Filter Realization

Cascade form

The cascade realization form of a general functions assumes that the

transfer functions is the product of such second-order sections

(SOS):

Each of SOS may be realized in direct or canonical form.

Digital Signal Processing

16

Transfer function

and Digital Filter Realization

Cascade form

Digital Signal Processing

17

Transfer function

and Digital Filter Realization

Review

Digital Signal Processing

18

Transfer function

and Digital Filter Realization

Homework 1

Digital Signal Processing

19

Transfer function

and Digital Filter Realization

Homework 2

Digital Signal Processing

20

Transfer function

and Digital Filter Realization

Homework 3

Digital Signal Processing

21

Transfer function

and Digital Filter Realization

Homework 4

Digital Signal Processing

22

Transfer function

and Digital Filter Realization

Homework 5

Digital Signal Processing

23

Transfer function

and Digital Filter Realization

Homework 6

Digital Signal Processing

24

Transfer function

and Digital Filter Realization

Homework 7

Digital Signal Processing

25

Transfer function

and Digital Filter Realization

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