Bài giảng Digital signal processing - Chapter 6: Transfer function and Digital Filter Realization - Nguyễn Thanh Tuấn
Chapter 6
Transfer function
and Digital Filter Realization
Nguyen Thanh Tuan,M.Eng.
Department of Telecommunications (113B3)
Ho Chi Minh City University of Technology
Email: [email protected]om
With the aid of z-transforms, we can describe the FIR and IIR filters
in several mathematically equivalent way
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Transfer function
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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
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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)
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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
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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):
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Transfer function
and Digital Filter Realization
I/O difference equation
Write
to the time domain.
then eliminate the denominators and go back
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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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
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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:
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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
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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
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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.
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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
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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 ?
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Transfer function
and Digital Filter Realization
Canonical form realization
1
1
Y(z) H(z)X (z) N(z)
X (z)
N(z)X (z)
Note that
D(z)
D(z)
The maximum number of
common delays: K=max(L,M)
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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.
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Transfer function
and Digital Filter Realization
Cascade form
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Transfer function
and Digital Filter Realization
Review
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Transfer function
and Digital Filter Realization
Homework 1
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Transfer function
and Digital Filter Realization
Homework 2
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Transfer function
and Digital Filter Realization
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