SignalProcessing

1. ECE 8443 – Pattern Recognition
EE 3512 – Signals: Continuous and Discrete
• Objectives:
FIR Filters
Design of Ideal Lowpass Filters
Filter Design Example
Digital Controller Design
Step-Response Matching
• Resources:
ISIP: Filter Transformations
Wiki: Finite Impulse Response
CNX: FIR Filters
Wiki: Window Function
EAW: Window Functions
LECTURE 37: DESIGN OF FIR FILTERS
AND CONTROLLERS
URL:

2. EE 3512: Lecture 37, Slide 1
Finite Impulse Response Filters
• Consider a filter with only
feedforward components:
• The transfer function is:
• Since the impulse response of this filter, h[n], has a finite number of nonzero
terms, this filter is referred to as a finite impulse response (FIR) filter.
• Observe that this filter only has zeroes (the roots of H(z)).


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

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l
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(
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3. EE 3512: Lecture 37, Slide 2
• The simplest method for designing an FIR filter is to construct an ideal
frequency response, perform an inverse Fourier transform to obtain the
impulse response, and then truncate the impulse response:
• As we discussed previously, we can view this truncation operation as
multiplication of a signal by a window:
• Since multiplication in the time domain implies convolution in the frequency
domain, the spectrum of the windowed signal is given by:
• In the case of the rectangular window:
Design of FIR Filters


 



otherwise
,
0
1
0
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4. EE 3512: Lecture 37, Slide 3
• Consider the design of an ideal
lowpass filter. The ideal filter is
noncausal for the response shown.
• Introduce a phase delay which will
help us create a causal approximation:
• We can truncate the impulse response to obtain our FIR filter:
• Examples of the frequency response:
FIR Design Example
   
 
 
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sin
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,
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,
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[
N
n
m
n
n
h
c
c
d 


 N = 21
N = 21 N = 41 Wc = 0.4;
N = 21;
m = (N-1)/2;
n = 0:2*m + 10;
h = Wc/pi*sinc(Wc*(n-m)/pi);
w = [ones(1,N) zeros(1,length(n)-N)];
hd = h.*w;

5. EE 3512: Lecture 37, Slide 4
Digital Controllers
• There are many strategies for
approximating an analog controller
with a digital controller.
• This is analogous to the filter design
problem, and many of the same
techniques (e.g., bilinear transform)
can be applied.
• This process is referred to as
analog emulation.
• In this section, we will explore a technique known as step-response
matching in which we design the digital system to approximate the step
response of the analog system.
• The approach involves taking the inverse Laplace transform of the output, y(t),
corresponding to the response of the system to a step function:
• We will need to make sure the sampling interval is chosen to be small enough
that the discrete system matches the analog system.
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(
]
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)
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)
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1
)
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)
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L

6. EE 3512: Lecture 37, Slide 5
Design Example
• Consider a CT system with transfer function:
• The transform of the step response is:
• The inverse Laplace transform is:
• The discrete version is:
• We can work backwards to derive the equivalent transfer function:
• We can compare the frequency response of the discrete system to the analog
system, and also compare impulse responses.
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7. EE 3512: Lecture 37, Slide 6
Design of a Feedback Controller
• Consider a feedback system:
• The overall transfer function is:
• Using our step-response design method, with , we have:
• The step response of this system
can be plotted using MATLAB.
• Note how the overshoot varies as
a function of the sample interval.
• There are a variety of ways we can
approximate an analog control system
using a digital controller. For example,
we can match the impulse response
rather than the step response.
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   
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rad/sec
41
.
1
2
:
Note
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

8. EE 3512: Lecture 37, Slide 7
Summary
• Reviewed FIR filters.
• Introduced a design methodology based on truncating the impulse response
of an ideal lowpass filter.
• Demonstrated an application of this design approach.
• Introduced a methodology for design digital controllers that can approximate
an equivalent analog controller.
• Demonstrated an approach involving matching the step response.
• Applied this to two control examples.