Bài giảng Digital Signal Processing - Chapter 6: Transfer Functions - Võ Trung Dũng

DSP-Chapter 6-Dr. Dung Trung Vo
Digital Signal Processing
Transfer Functions
Dr. Dung Trung Vo
Telecommunication Divisions
Department of Electrical and Electronics
September, 2013
DSP-Chapter 6-Dr. Dung Trung Vo
Equivalent Descriptions of Digital Filters
 Filter Descriptions: with the aid of z-transforms, we develop several 
mathematically equivalent ways to describe and characterize FIR and IIR filters.
• Transfer function H(z)
• Frequency response H(ω)
• Block diagram realization and sample processing algorithm
• I/O difference equation
• Pole/zero pattern
• Impulse response h(n)
• I/O convolutional equation
DSP-Chapter 6-Dr. Dung Trung Vo
Equivalent Descriptions of Digital Filters
 Relationships among these descriptions: the most important one is the 
transfer function description because from it we can easily obtain all the others
DSP-Chapter 6-Dr. Dung Trung Vo
Equivalent Descriptions of Digital Filters
 Steps to implement a filter: in practice, a typical usage of these descriptions 
contains:
 Specify a set of desired frequency response specifications H(ω).
 Obtain a transfer function H(z) that satisfies the given specifications through 
a filter design method.
 Derive a block diagram realization and the corresponding sample-by-sample 
processing algorithm that tells how to operate the designed filter in real time. 
DSP-Chapter 6-Dr. Dung Trung Vo
Equivalent Descriptions of Digital Filters
 Central role played of H(z): Given a transfer function H(z) one can obtain:
Impulse response h(n) 
Difference equation satisfied by the impulse response
I/O difference equation relating the output y(n) to the input x(n) 
Block diagram realization of the filter 
Sample-by-sample processing algorithm 
Pole/zero pattern
Frequency response H(ω)
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain the impulse response: use partial fraction expansion
where A0 and A1 are obtained by:
Assuming the filter is causal
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain the difference equation: The standard approach is to eliminate the 
denominator polynomial of H(z) and then transfer back to the time domain. 
Multiplying both sides by the denominator
Taking inverse z-transforms of both sides and using the linearity and delay 
properties
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain the general I/O convolutional equation: assume with causal 
solution, that is, the solution with the causal initial condition h(−1)= 0
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain the general I/O convolutional equation: assume with causal 
solution, that is, the solution with the causal initial condition h(−1)= 0
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain the difference equation for y(n): the standard procedure is to 
eliminate denominators and go back to the time domain
Taking inverse z-transforms of both sides, we have
Therefore
Given h(n), we could obtain H(z) by reversing all of the above steps
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 Many block diagram realization 
 Direct form realization: can be obtained by mechanize I/O difference equation
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain parallel form realization:
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain canonical form realization: by rearranging the I/O computations 
differently
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain transposed realization:
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: consider the transfer function:
 To obtain the frequency response: can be obtained by replacing z by ejω
into H(z)
The filter has a zero at z = −0.4 and a pole at z = 0.8
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions: IIR
 General transfer function of an IIR filter: is given as the ratio of two 
polynomials of degrees, say L and M
 Notes: 
 Note that by convention, the 0th coefficient of the denominator polynomial 
has been set to unity a0 = 1. 
 The filter H(z) will have L zeros and M poles. 
 Assuming that the numerator and denominator coefficients are real-valued, 
then if any of the zeros or poles are complex, they must come in conjugate 
pairs.
 To get a stable impulse response, we must pick the ROC that contains the 
unit circle
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions: IIR
 I/O difference equations: 
Multiplying by the denominator
Transforming back to the time domain
Or
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: Determine the transfer function of the following third-order FIR filter 
with impulse response
 Solution: The filter’s I/O equation is
z-transform of the finite impulse response
Frequency response
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Magnitude response:
 Block diagram and sample processing algorithm
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Example: Determine the transfer function and causal impulse response of the 
filter described by the difference equation
 Solution: the difference equation becomes in the z-domain
which can be solved for Y(z)/X(z) to give
with A1 = 0.5. The causal impulse response will be
DSP-Chapter 6-Dr. Dung Trung Vo
Transfer functions
 Magnitude response:
 Block diagram and sample processing algorithm
DSP-Chapter 6-Dr. Dung Trung Vo
Sinusoidal Response
 Steady-State Response:
 Using convolution in the time domain:
 Using the frequency-domain method:
Putting Y(ω) into the inverse DTFT formula
DSP-Chapter 6-Dr. Dung Trung Vo
Sinusoidal Response
 Steady-State Response: an infinite double-sided input sinusoid of frequency ω0 
reappears at the output unchanged in frequency but modified by the frequency 
response factor H(ω0)
In terms of its magnitude and phase
Taking real or imaginary parts of both sides of this result
DSP-Chapter 6-Dr. Dung Trung Vo
Sinusoidal Response
 Steady-State Response:
Magnitude and phase-shift modification introduced by filtering
DSP-Chapter 6-Dr. Dung Trung Vo
Sinusoidal Response
 Transient Response:
having ROC |z| > |ejω0| = 1. Assume a filter of the form
Output 
Apply the PF expansion
And take inverse DTFT
DSP-Chapter 6-Dr. Dung Trung Vo
Sinusoidal Response
 Homeworks: 6.1->6.5; 6.19; 6.26