Chapter 9 Analysis and Design of Digital Filter
Chapter 9 Analysis and Design of Digital Filter
9-1 Introduction
What designs have we done in this course?
What do we mean by filters here?
What do we mean by filters design?
Given specifications (requirements) => H(z)
Let’s see how we can implement a digital filter (processor) if its H(z) is given?
9-2 Structures of Digital Processors
1. Direct-Form Realization
[pic]
The function is realized!
What’s the issue here?
Count how many memory elements we need!
Can we reduce this number?
If we can, what is the concern?
[pic]
[pic]
Denote [pic]
Implement H2(z) and then H1(z) ?
Why H2 is implemented?
(1)[pic](2)
[pic]
H2 is realized!
Can you tell why H1 is realized?
What can we see from this realization? Signals at [pic] and [pic]: always the same
( Direct Form II Realization
[pic]
Example [pic]
Solution: [pic]
[pic]
Important: H(z)=B(z)/A(z) (1) A: 1+…. (2) Coefficients in A: in the feedback channel
2. Cascade Realization
Factorize [pic]
[pic]
[pic]
General Form
[pic]
Apply Direct II for each!
3. Parallel Realization (Simple Poles)
[pic]
Example 9-1
[pic]
cascade and parallel realization!
Solution:
1) Cascade:
[pic]
[pic]
2) Parallel
[pic]
In order to make deg(num) [pic]
[pic]
[pic]
A discrete-time integrator: rectangular integrator
[pic]
[pic]
2. Trapezoidal Integration
[pic]
[pic]
or [pic]
[pic]
3. Frequency Characteristics
1. Rectangular Integrator
[pic]
Frequency Response [pic]
Or [pic] [pic]
Amplitude Response
[pic]
Phase Response
[pic]
2. Trapezoidal Integrator
[pic]
Frequency Response
[pic]
Amplitude: [pic]
Phase:
[pic]
3. Versus Ideal Integrator
Ideal (continuous-time ) Integrator
[pic]
when T=1 second (Different plots and relationships will result if T is different.)
[pic]
• Low Frequency Range [pic]
(Frequency of the input is much lower than the sampling frequency:
It should be!)
[pic]
[pic]
• High Frequency: Large error (should be)
Example 9-4 Differential equation (system)
[pic]
Determine a digital equivalent.
Solution
(1) Block Diagram of the original system
(2) An equivalent
(3) Transfer Function Derivation
[pic]
Design: 9-4 Find Equivalence of a given analog filter (IIR):
Including methods in Time Domain and Frequency Domain.
9-5 No analog prototype, from the desired frequency response: FIR
9-6 Computer-Aided Design
4. Infinite Impulse Response (IIR) Filter Design
(Given H(s) ( Hd(z) )
9-4A Synthesis in the Time-Domain: Invariant Design
1. Impulse – Invariant Design
1) Design Principle
[pic]
2) Illustration of Design Mechanism (Not General Case)
Assume:
1) Given analog filter (Transfer Function)
[pic] (a special case)
(2) Sampling Period T (sample ha(t) to generate ha(nT))
Derivation:
1) Impulse Response of analog filter
[pic]
2) ha(nT): sampled impulse response of analog filter
[pic]
3) z-transform of ha(nT)
Sampled impulse response of analog filter
[pic]
4) Impulse-Invariant Design Principle
[pic]
(
Digital filter is so designed that its impulse response h(nT)
equals the sampled impulse response of the analog filter ha(nT)
Hence, digital filter must be designed such that
[pic]
(5) [pic] (scaling)
=>[pic]
(3) Characteristics
(1) [pic] when T( 0
(
frequency response of digital filter
2) [pic]
3) Design: Optimized for T = 0
Not for T ( 0 (practical case) (due to the design principle)
(4) Realization: Parallel
[pic]
(5) Design Example [pic]
Solution: [pic]
[pic]
2. General Time – Invariant Synthesis
1) Design Principle
2) Derivation
Given: Ha(s) transfer function of analog filter
Xa(s) Lapalce transform of input signal of analog Filter
T sampling period
Find H(z) z-transfer function of digital filter
1) Response of analog filter xa(t)
[pic]
(2) ya(nT) sampled signal of analog filter output
[pic]
(3) z-transform of ya(nT)
[pic]
(4) Time – invariant Design Principle
[pic]
(
Digital filter is so designed
that its output equals the sampled
output of the analog filter
Incorporate the scaling :
[pic]
(
z-transfer function of digital filter
(5) Design Equation
[pic]
special case X(z)=1, Xa(s) = 1 (impulse)
=>[pic]
(6) Design procedure
A: Find [pic] (output of analog filter)
B: Find [pic]
C: Find [pic]
D: [pic]
Example 9-5 [pic]
Find digital filter H(z) by impulse - invariance.
Solution of design:
1) Find [pic]
[pic]
[pic]
2) Find [pic]
[pic]
3) Find [pic]
[pic]
4) Find z-transfer function of the digital filter
[pic]
use G = T
[pic]
5) Implementation
Characteristics
(1) Frequency Response equations: analog and digital
Analog : [pic]
Digital : [pic]
(2) dc response comparison ([pic])
Analog: [pic]
Digital: [pic]
Varying with T (should be)
[pic] [pic], [pic]
[pic]
[pic] for example [pic]
[pic], [pic]
[pic] good enough
(3) [pic] versus [pic] : [pic]
Using normalized frequency [pic]
[pic]
[pic]
(4) [pic] versus [pic]
[pic]
(5) Gain adjustment when [pic]
[pic] => frequency response inequality
adjust G => [pic] at a special [pic]
for example [pic]
If G = T = 0.3142 => [pic]
If selecting G = T/1.0745 => [pic]
3. Step – invariance synthesis
[pic] [pic]
[pic]
Example 9-6 [pic]. Find its step-invariant equivalent.
Solution of Design
[pic]
[pic]
[pic]
Comparison with impulse-invariant equivalent.
[pic]
9-4B Design in the Frequency Domain --- The Bilinear z-transform
1. Motivation (problem in Time Domain Design)
[pic] [pic]
Introduced by sampling, undesired!
x(t) bandlimited ([pic]
[pic]
[pic]
for [pic]
[pic]
for [pic]
Consider digital equivalent of an analogy filter Ha(f): [pic])
Ha(f): bandlimited => can find a Hd(z)
Ha(f): not bandlimited => can not find a Hd(z) Such that [pic]
2. Proposal: from [pic] axis to [pic] axis
[pic] ([pic]: given sampling frequency)
(s plane to s1 plane)
[pic]
Observations: (1) Good accuracy in low frequencies
(2) Poor accuracy in high frequencies
(3) 100% Accuracy at [pic]
a given specific number such that 0.01[pic]
Is it okay to have poor accuracy in high frequencies? Yes! Input is bandlimited!
What do we mean by good, poor and 100% accuracy?
Assume (1)[pic] (originally given analogy filter)
(2) The transform is [pic]
Then,[pic] is a function of [pic]. Denote [pic].
Good accuracy:
[pic]
Poor Accuracy
[pic]
100% Accuracy (Equal)
[pic]
Is [pic] bandlimited? That is, can we find a [pic] such that [pic]=0?
Yes, [pic]. ( We have no problem to find a digital equivalent
[pic] without aliasing!
Let’s use [pic]as a number (for example 0.2) representing any low frequency,
Then, because [pic] is a good approximation of [pic],
[pic][pic] should be a good approximation.
A digital filter can thus be designed for an analogy filter [pic] which is not bandlimted!
Two Step Design Procedure:
Given: analogy filter [pic]
(1) Find an bandlimited analogy approximation ([pic]) for [pic]
(2) Design a digital equivalent [pic] for the bandlimited filter [pic].
Because of the relationship between ([pic]) for [pic],
[pic] is also digital equivalent of [pic].
The overlapping (aliasing) problem is avoided!
The designed digital filter can approximate [pic] (for [pic] and [pic] take the
same value) at low frequency.
3. [pic] axis to [pic] axis (s plane to s1 plane) transformation
Requirement : [pic] ([pic] is given sampling frequency.)
Proposed transformation :
[pic]
Effect of C:
We want the transformation map
[pic] (for example, [pic]) to [pic] [pic]
=> [pic]
i.e. when the sampling period T is given, C is the only parameter
which determines what [pic] will be mapped into [pic] axis with the
same value.
Example: [pic]
[pic] not bandlimited
If we want to map [pic] to [pic]
1. [pic]
[pic]
Hence, for any given T or [pic]
[pic]
is bandlimited as a function of [pic] by [pic]
[pic] when [pic] at low frequencies.
Further [pic]
Exactly holds!
How to select [pic] or sampling frequency [pic] at which [pic]?
(1) [pic] should be small?
why? [pic] [pic],
(The accuracy should be good at low frequencies)
(2) When [pic] is given or determined by application, [pic] should be large
enough such that [pic] to ensure the accuracy in the frequency
range including [pic]
When [pic]
[pic] since [pic] for small x.
4. Design of Digital Filter using bilinear z-transform
A procedure: (1) [pic]
[pic] [pic]
(not bandlimited, (bandlimited,
original analog) analog)
or [pic]
(2) [pic]
[pic] [pic]
(Transfer replace [pic] by z
function of
digital filter)
* Question: Can we directly obtain Hd(z) from Ha(s) ? Yes! (But how?)
Bilinear z-transform
Preparation : (1) [pic]
(2) [pic]
Hence, [pic]
Replace [pic] by s , [pic] by s1 ([pic])
[pic]
Replace [pic] for digital filter
[pic] direct transformation from s to z (bypass s1)
Example [pic]
Digital Filter
[pic]
C: only undetermined parameter in the digital filter.
To determine C: (1) [pic]
(2) [pic] (related to the frequency range of interest)
Example 9-7 [pic]
[pic] : break frequency
Take [pic]
Consider [pic]
[pic]
( C determined => Hd(z) determined
( [pic]
( [pic], [pic], [pic]
To compare the frequency response with the original analog filter Ha :
[pic]
( replace s by [pic])
( [pic], [pic]
[pic]
Too low fs => poor accuracy in fc.
9-4C Bilinear z-Transform Bandpass Filter
1. Construction Mechanism
1) From an analog low-pass filter Ha(s)
to analog bandpass filter [pic]
i.e., replace s by [pic] to form a bandpass filter
For example [pic] low-pass
[pic] band-pass
Why? Original low-pass
[pic]
[pic]
Low => High Gain
High => Low Gain
After Replacement
[pic]
high [pic] => high => low gain
low [pic] => high => low gain
2) From analog to digital
Replace s in [pic] by [pic]
( [pic]
for example
[pic]
2. Bilinear z-transform equation
Analog Low-pass Bandpass (analog)
s [pic]
[pic]
[pic]
[pic]
with [pic]
3. How to select ( [pic]) for bandpass filter
(design)
Important parameters of bandpass
1) center frequency [pic]
2) [pic] upper critical frequency
3) [pic] low critical frequency
Selection of [pic] for bandpass: [pic]
Design of ( [pic])
We want [pic], Also want [pic]
one parameter C => impossible
solution [pic]
bandwidth [pic]
Hence, A and B can be determined to perform the transform.
4. Convenient design equation
[pic]
why no C?
[pic]
[pic]
[pic]
Example :[pic]
5. In the normalized frequency
Reference frequency: sampling frequency [pic]
[pic] [pic]
=> [pic], [pic]
=>[pic]
s => [pic]
Example 9-9 Lowpass [pic]
Transfer function of bandpass digital filter
[pic]
A and B? Determined by design requirements.
[pic] sampling frequency fs = 5000Hz
2. [pic]
3. [pic]
4. [pic]
[pic], [pic], [pic]
9-5 Design of Finite-Duration Impulse Response (FIR) Digital Filter
Direct Design of Digital Filters with no analog prototype.
Can we also do this for IIR? Yes!
Using computer program in next section.
9-5A A few questions
1. How are the specifications given?
By given [pic] and [pic]
2. What is the form of FIR digital filter?
Difference Equation [pic]
(What is T ? sampling period)
Transfer function [pic]
3. How to select T ?
4. After T is fixed, can we define the normalized frequency r and
[pic] and [pic] ? Yes!
Can we then find the desired frequency response
[pic] ? Yes!
5. Why must H( r ) be a periodic function for digital filter?
H( r ) = H ( n + r ) ? Why? What is its period?
6. Can H( r ) be expressed in Fourier Series ? Yes!
How?
See general formula :
[pic]
[pic]
In our case for H(r):
[pic] [pic]
What does this mean? Every desired frequency response H(r) of digital
filter can be expressed into Fourier Series ! Further, the coefficients of
the Fourier series can be calculated using H(r)!
9-6B Design principle
[pic]
Denote [pic]
Consider a filter with transfer function [pic]
What’s its frequency response ?
[pic]
given specification of digital filter’s frequency response!
9-6 C Design Procedure
1) Given H(r)
(2) Find H(r)’s Fourier series [pic]
where [pic]
(3) Designed filter’s transfer function
[pic]
What’s hd(nT) ? Impulse response!
Example 9-10: [pic]
Solution :
1) Given [pic]: done
2) Find [pic]’s Fourier series
[pic]
where [pic]
n = 0
[pic][pic]
[pic]
[pic]
( [pic]
(3) Digital Filter
[pic]
D Practical Issues : Infinite number of terms and non-causal
1) [pic]
Truncation => [pic]
Rectangular window function
[pic]
Truncation ( window
Effect of Truncation (windowing):
Time Domain: Multiplication ( h and w )
Frequency Domain: Convolution
[pic]
(After Truncation: The desired frequency Hr
1. [pic] frequency response of truncated filter )
The effect will be seen in examples!
(2) Causal Filters:
[pic]
k = n+M [pic]
Define [pic]( [pic]
Relationship: [pic]
Frequency Response [pic]
Design Examples
Hamming window: [pic]
Example 9-11 Design a digital differentiator
Step1 : Assign [pic]
[pic] should be the frequency response of the analog differentiator
H(s) = s
=> Desired [pic]
Step2 : Calculate hd(nT)
[pic]
[pic]
[pic]
i.e., [pic]
Step 3: Construct nc filter with hamming window (M=7)
[pic]
[pic]
Example 9-12: Desired low-pass FIR digital filter characteristic
[pic]
[pic]
[pic]
2. [pic]
[pic]
NC filter with 17 weight’s window: [pic], [pic]
[pic]
Example 9-13 (90o phase shifter)
[pic]
[pic]
n = 0 => [pic]
3. [pic]
[pic]
=> [pic]
Filter: [pic]
[pic]
[pic]
[pic]
Fig. 9-32 Amplitude response of digital 90 degree phase shifter
6. Computer-Aided Design of Digital Filters
9-6A Command yulewalk for IIR
Example 9-14
9-6B Command remez for FIR
Example 9-15
Chapter 9 Homework
2, 4, 6, 11, 13, 14, 25, 26, 27, 29, 30, 31, 44, 48, 53, 61, 62, 64, 66, 68, 69
-----------------------
Constant [pic]
Partial-Fraction Expansion for s
y(nT) : System output
x(nT) : System input , to be integrated
Constant
Constants
z-transfer function [pic] of the digital filter. Of course,
the z-transfer function of its impulse response.
T
Variable in [pic] domain
Variable in [pic] domain
Constant
Bilinear z – transformation
In the low pass filter
digital bandpass filter
Direct Transformation
s (in low-pass)
s (in low-pass)
[pic]
In low-pass
In low-pass
r is not time
[pic]
[pic]: sampling frequency? No! [pic] : period of [pic]
2M+1 terms
[pic]
0
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