ece651 digital signal processing i digital iir filter design
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ECE651 Digital Signal Processing I Digital IIR Filter Design. Introduction Some Preliminaries on Analog Filters Digital IIR Filter Design (s – z) Impulse Invariance Transformation Bilinear Transformation Frequency Band Transformations Analog Domain (s – s ) - PowerPoint PPT PresentationTRANSCRIPT
ECE651 Digital Signal Processing IDigital IIR Filter Design
Introduction
Some Preliminaries on Analog Filters
Digital IIR Filter Design (s – z)
Impulse Invariance Transformation
Bilinear Transformation
Frequency Band Transformations
Analog Domain (s – s )
Digital Domain (z – z)
Introduction
Analog filter : Infinitely long impulse response
Digital IIR filter : Infinitely long impulse response
S – Z (complex-valued mapping)
Introduction
Introduction
Advantages• Analog filter design tables available
• Filter transformation (s – z) tables available
• Frequency band transformation (s – s / z – z) available
Disadvantages
• No control over the phase characteristics of the IIR filter
• Magnitude – only design
Introduction
Other Design Approaches• Simultaneously approximate both the magnitude and the phase response
• Require advanced optimization tools
• Not covered in the class
Preliminaries On Analog Filters
Analog lowpass filter specifications
: passband ripple parameter
A: stopband attenuation parameter
2
2
11)(
paH
22 1)(
AH sa
p : passband cutoff frequency (rad/sec)
s : stopband cutoff frequency (rad/sec)
Preliminaries On Analog Filters
Analog lowpass filter specifications
pR : passband ripple in dB
sA : stopband attenuation in dB
AAs 10log20 20/10 sAA
)1(log10 210 pR 110 10/ pR
Preliminaries On Analog Filters
Analog lowpass filter system function )(sH a
jsaaa HsHsH /2)()()(
• Poles and zeros of magnitude-squared function are distributed in a mirror-image symmetry with respect to the imaginary axis
• For real filters, poles and zeros occur in complex conjugate pairs ( mirror symmetry with respect to real axis)
Preliminaries On Analog Filters
Analog lowpass filter system function
1. Pick up poles On LHP
2. Pick up zeros on LHP or Imaginary axis
)(sH a
)(sH a
Stable Causal
Preliminaries On Analog Filters
Prototype analog filters1. Butterworth
2. Chebyshev (Type I and II)
3. Elliptic
Preliminaries On Analog Filters
Butterworth lowpass filters (Magnitude-Squared Response)
N
c
aH 22
1
1)(
c The Cutoff frequency (rand/sec)
N The order of the filter
Preliminaries On Analog Filters
Butterworth lowpass filters (System Function)
12,1,0 ,2)12(
Nkep NNk
j
ck
poles LHP
)()(
k
Nc
a pssH
Preliminaries On Analog Filters
Butterworth lowpass filters (Design equations)
)/(log2
)]110/()110[(log
10
10/10/10
sp
AR sp
N
NR
pc
p 21
10/ )110(
NA
pc
s 21
10/ )110(
Digital IIR Filter Design
S - Z transformation
• Complex-valued mappings
• Derived by preserving different aspects of analog filters and digital filters
Digital IIR Filter Design
Impulse Invariance transformation
• Preserve the shape of impulse response
sTez
Digital IIR Filter Design
Impulse Invariance transformation (Design Procedure) (MATLAB function: impinvar)
1. Choose T and determine the analog frequencies
2. Design an analog filter using specifications
3. Partial fraction expansion
4. Transform analog poles into digital poles to obtain
Tp
p
Ts
s
)(sH a spsp AR and ,,,
N
k k
ka ps
RsH1
)(
}{ kp }{ Tpke
N
kTpk
zeRzH
k1
11)(
Digital IIR Filter Design
Impulse Invariance transformation (Aliasing)
>> f=0:0.01:5;T=0.1;>> z=exp(j*2*pi*f*T);>> zH=(1-0.8966./z)./(1-1.5595./z+0.6065./z./z);>> s=j*2*pi*f;>> sH=(1+s)./(s.^2+5*s+6);>> plot(f,abs(zH),f,abs(sH)/T);legend('Digitital','Analog')>>title('Magnitude Response of Analog and Digital IIR Filters')
Digital IIR Filter Design
Impulse Invariance transformation
Advantages:
• Stable design
• Analog frequency and digital frequency are linearly related
Disadvantage
• Aliasing
• Useful only when the analog filter is band-limited (LPF and BPF)
Digital IIR Filter Design
Bilinear transformation
• Preserve the system function representation
2/)(12/)(1
STsTz
1
1
112
zz
Ts
Digital IIR Filter Design
Bilinear transformation (Design Procedure) (MATLAB function: bilinear)
1. Choose T (1)and determine the analog frequencies
2. Design an analog filter using specifications
3. Bilinear transformation
)2
tan(2 pp T
)(sH a spsp AR and ,,,
)2
tan(2 ss T
)112()( 1
1
zz
THzH a
Digital IIR Filter Design
Bilinear transformation
Advantages
• Stable design
• No aliasing
• No restriction on the type of filters that can be transformed
Frequency DomainTransformations
Analog Domain
Frequency DomainTransformations
Digital Domain