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Bài giảng Xử lý tín hiệu số: Chapter 6 - Hà Hoàng Kha

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Chapter
p 6

Transfer functions
g
Filter Realization
and Digital
Ha Hoang Kha, Ph.D.Click to edit Master subtitle style
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
several
eral mathematically
mathematicall equivalent
eq i alent way
a

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Content

1 Transfer
1.
T
f functions
f ti
‰ Impulse response
‰ Difference equation
‰ Impulse response
‰ Frequency
q
y response
p
‰ Block diagram of realization

2 Digital filter realization
2.
‰ Direct form
‰ Canonical form
‰ Cascade form

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1. Transfer functions
™ Given a transfer functions H(z) one can obtain:
((a)) the
th impulse
i
l response h(n)
h( )
(b) the difference equation satisfied the impulse response
( ) the
(c)
h I/O
/ difference
diff
equation
i relating
l i the
h output y(n)
( ) to the
h input
i
x(n).
(d) the block diagram realization of the filter
( ) the sample-by-sample
(e)

p y
p p
processingg algorithm
g
(f) the pole/zero pattern
(g) the frequency
q
y response
p
H(w)
( )

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Impulse response
™ Taking the inverse z-transform of H(z) yields the impulse response
h(n)
Example:
p 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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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:
p consider the transfer function
Multiplying both sides by denominator, we find

Taking inverse z-transform
z transform of both sides and using the linearity and
delay properties, we obtain the difference equation for h(n):

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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
byy block diagram
g
Example: consider the transfer function
We have the I/O difference equation


The direct form realization is given by

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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-delayy at time n:
Similarly,
y w1((n)) is the content of the y-delay
y
y 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
we obtain
b i an expression
i ffor the
h magnitude
i d response

‰ Drawing peaks when
passing near poles
‰ Drawing dips when
passingg near zeros
p
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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
q
y response
p
and plot
p the magnitude
g
response
p
of the filter.
d)) Plot the block diagram
g
of the system
y
and write the sample
p
processing algorithm

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2. Digital filter realizations
™ Construction of block diagram of the filter is called a realization of
the filter.
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
g 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)
a)) Draw
D
the
th direct
di t form
f
realization
li ti off this
thi digital
di it l filter
filt ?
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

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

1
X ( z)
D( z )

‰ 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
second-order
order sections
(SOS):

™ Each of SOS mayy 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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