Bài giảng Xử lý tín hiệu số - Chương 6: Transfer functions and digital filter realization - Hà Hoàng Kha

Chapter 6
Transfer functions 
and Digital Filter Realization
Click to edit Master subtitle styleHa Hoang Kha, Ph.D.
Ho Chi Minh City University of Technology
Email: hhkha@hcmut.edu.vn
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™With the aid of z-transforms, we can describe the FIR and IIR filters 
in se eral mathematicall eq i alent a v y u v w y
Ha H. Kha 2 Transfer functions 
and Digital Filter Realizations
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Content
1 T f f ti. rans er unc ons
‰ Impulse response
‰ Difference equation 
‰ Impulse response
‰ Frequency response
2 Digital filter realization
‰ Block diagram of realization
. 
‰ Direct form
‰ Canonical form 
‰ Cascade form
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1. Transfer functions
™ Given a transfer functions H(z) one can obtain:
( ) th i l h( )a e mpu se response n 
(b) the difference equation satisfied the impulse response
( ) h / diff i l i h ( ) h ic t e I O erence equat on re at ng t e output y n to t e nput 
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)
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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
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and Digital Filter Realizations
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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):
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and Digital Filter Realizations
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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 
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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
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and Digital Filter Realizations
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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:
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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
b i i f h i dwe o ta n an express on or t e magn tu e response 
‰ Drawing peaks when 
passing near poles
‰ Drawing dips when 
passing near zeros
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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
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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.
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Direct form realization
™ Use the I/O difference equation
‰ The b-multipliers are feeding forward
‰ The a-multipliers are feeding backward
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Example
™ Consider IIR filter with h(n)=0.5nu(n)
) D th di t f li ti f thi di it l filt ?a raw e rec orm rea za on o s g a er 
b) Given x=[2, 8, 4], find the first 6 samples of the output by using the 
sample processing algorithm ? 
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Canonical form realization
™ Note that )(
)(
1)()()()( zX
zD
zNzXzHzY ==
‰ The maximum number of
common delays: K=max(L,M)
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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.
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Cascade form
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Homework
™ Problems: 6.1, 6.2, 6.5, 6.16, 6.18, 6.19
™ Problems: 7.1, 7.3, 7.5, 7.10
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