A Seminar On, “Design of Digital IIR filter” Presented By, Mr. Swapnil V. Kaware, [email protected] [email protected]
Digital signal processing By Er. Swapnil Kaware
Oct 31, 2014
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Introduction
(i). Infinite Impulse Response (IIR) filters are the first
choice when:Speed is paramount.Phase non-linearity is acceptable.
(ii). IIR filters are computationally more efficient than
FIR filters as they require fewer coefficients due to
the fact that they use feedback or poles.
(ii). However feedback can result in the filter
becoming unstable if the coefficients deviate from
their true values.
Butterworth :- Maximally Flat Amplitude.
Chepyshev type I :- Equiriple in the passband.
Chepyshev type II :- Equiriple in the stopband.
Elliptic :- Equiripple in both the passband and stopband.
Filter Design Methods
Design ProcedureTo fully design and implement a filter five steps are required:
(1) Filter specification.
(2) Coefficient calculation.
(3) Structure selection.
(4) Simulation (optional).
(5) Implementation.
Filter Specification - Step 1
(a)
1
f(norm)fc : cut-off frequency
pass-band stop-band
pass-band stop-bandtransition band
1
s
pass-bandripple
stop-bandripple
fpb : pass-band frequency
fsb : stop-band frequency
f(norm)
(b)
p1
s
p0
-3
p1
fs/2
fc : cut-off frequency
fs/2
|H(f)|(dB)
|H(f)|(linear)
|H(f)|
IIR Filters Better magnitude response (sharper transition and/or lower stopband
attenuation than FIR with the same number of parameters: HW efficient)
Established filter types and design methods.
IIR filter design procedure:-
1) Set up digital filter specification,2) Determine the corresponding analog filter specification,
(frequency translation involved)3) Design the analog filter,4) Transform the analog filter to digital filter using various
transformation methods, Impulse invariant method Bilinear transformation
IIR Filters
Important parametersPassband ripple : Stopband attenuation : Discrimination factor :
Selectivity factor :
(-3dB) cutoff frequency :
1
p1
s
IIR Filters
Frequency response Transfer function : Rational
Asymptotic attenuation at high frequency
Attenuation function: (rational or polynomial function)
If is monotone, so is If is oscillatory, exhibits ripple.
: Square magnitude frequency response
: reference frequency
IIR Filters
Frequency response For real rational transfer function
Stability requirement
must include all poles of
on the left half of the s plane and only those.
Analog filter typesButterworthChebyshevElliptic
Butterworth Filters
The Butterworth filter is a type of signal processing filter
designed to have as flat a frequency response as possible in the
passband. It is also referred to as a maximally flat magnitude
filter. It was first described in 1930 by the British engineer and
Physicist Stephen Butterworth in his paper entitled "On the Theory
of Filter Amplifiers"
Butterworth Filters
The frequency response of the Butterworth filter is maximally flat
(i.e. has no ripples) in the passband and rolls off towards zero in the stopband.
When viewed on a logarithmic Bode plot the response slopes off linearly towards negative infinity.
A first-order filter's response rolls off at −6 dB per octave (−20 dB
per decade) (all first-order lowpass filters have the same normalized frequency response).
A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on.
Butterworth filters have a monotonically changing magnitude function with ω, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband.
N2
c
2
cj/j1
1jH
N2
c
2
cj/s1
1sH
• Passband is designed to be maximally flat.• The magnitude-squared function is of the form
c The Cutoff frequency
N The order of the filter
12,1,0 ,2
)12(
Nkes N
Nkj
ck
poles LHP
)()(
k
Nc
ass
sH
Butterworth Lowpass Filters
Butterworth filters
Magnitude Squared Response
Properties of a LP Butterworth filterMagnitude response : monotonically decreasingMaximum gain : 0 at Asymptotic attenuation at high frequency : Maximally flat at DC (maximally flat filter)
12)1( p
2s
sp 0
5.0
2|)(| H
: -3 dB point
Butterworth filters
Transfer function
2N poles: : N poles are on the left side
of the complex plane All pole filter
Normalized transfer function : Nth-order LP Butterworth filter
3N
}Re{s
}Im{s
4N
}Re{s
}Im{s
Butterworth filters
LP Butterworth filter design procedure1. Set up filter spec :
2. Compute N, using
3. Choose using
4. Compute the poles , using
5. Compute , using
Chebyshev Filters
Chebyshev filters are analog or digital filters having a steeper roll-offand more passbandripple(type I) or stopband ripple (type II) than Butterworth filters.
Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband.
This type of filter is named after Pafnuty Chebyshv because its mathematical characteristics are derived from Chebyshev polynomials.
Because of the passband ripple inherent in Chebyshev filters, the ones that have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications.
• Equiripple in the passband and monotonic in the stopband.• Or equiripple in the stopband and monotonic in the passband.
Type IType II
xcosNcosxV /V1
1jH 1
Nc
2N
2
2
c
122
2
)]/([1
1|)(|
cN
cV
jH
Chebyshev Filters
Chebyshev filters
Chebyshev polynomial of degree
Monotone only in one band Chebyshev Type I : equiripple in the passband Chebyshev Type II : equiripple in the stopband
Sharper than Butterworth due to the ripples ! Why ? Sharpest if equiripple in both bands, pass- and stop-bands. Phase response : Better for maximally flat or monotonic mag
response filters
Recursive formula:
4,3,2,1N
)(xTN
If N is even(odd), so is
Chebyshev-I : Chebyshev Filter of the first kind
Properties All-pole filter For
For Monotonically decreasing because asymptotic attenuation :
Chebyshev-I : Chebyshev Filter of the first kind
Poles of a Nth-order LP Chebyshev-I filter
Transfer function0a
N/
0b
(N=3) case
Chebyshev-II : Chebyshev Filter of the second kind
Inverse Chebyshev filter or Chebyshev-II
Properties Passband : monotonic Stopband : equi-ripple Contains both the poles and zeros
for all
: monotonically decreasing
Elliptic filter(i). An elliptic filter (also known as a Cauer filter, named after
Wilhelm Cauer) is a signal processing filter with equalized ripple (equiripple) behavior in both thepassband and the stopband.
(ii). The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition ingain between the passbandand the stopband, for the given
values of ripple (whether the ripple is equalized or not).
(iii). Alternatively, one may give up the ability to independently adjust the passband and stopband ripple, and instead design a filter which is maximally insensitive to componenAn elliptic filter (also known as a Cauer filter,
Elliptic Filters
Overview Equiripple in both the passband and the stop band Minimum possible order for a given spec : Sharpest (optimum)
Magnitude Squared Response: LP elliptic filter
: Jacobian elliptic function of degree N Even(odd) function of for even(odd) For , oscillates between -1 and +1
For
oscillates between 1 and for
oscillates between and for
Elliptic Filters
Equiripple both in stopband and in the passband
Jacobian Elliptic function)(1
1|)(|
222
Nc U
jH
Frequency transformation
Analog filter design1. Design a LPF (Butterworth, Chebyshev, elliptic)
2. Frequency transformation to obtain HPF, BPF, BRF
Definitions : rational function ( , ) Transfer function of a LP filter : Transformed filter
: rational function of Class and stability of the filter is preserved after transformation.
Design domain : Target domain :
Frequency transformation
LP to LP transformation
LP to HP transformation
LP to BP transformation
LP to BS transformation
Digital IIR filter design
Digital IIR filter design1. Digital filter spec -> analog filter spec
2. Design analog filter
3. Transformation : Analog filter to digital filter
Transformation Goal
Requirements for Real, causal, stable, rational The order of should not be greater than that of if possible. should be close to where •transform should be simple, convenient to implement and applicable
to all analog filter types and classes
Digital IIR filter design
Impulse Invariant Transformation Definition
Procedure
1.
2.
3. High-pass filter cannot be
transformed !! Filter orders are not changed
After transformation
Example)
Digital IIR filter design
Bilinear Transform Definition and Properties
For
(Approximation of continuous-time integration by discrete-time trapezoidal integration)
1) # of poles are preserved. => Preserve the filter order2) # of zeros increase from q to p if p > q (p-1 zeros at z=-1)
Digital IIR filter design
Bilinear Transform Definition and Properties
If => Preserve the stability
}Re{s
}Im{s
z-plane
1
Digital IIR filter design
Bilinear Transform Definition and Properties
1) Frequency warping : One-to-one mapping, 2) 3) Can be used for all filter types
For
}Re{s
}Im{s
z-plane
1
Digital IIR filter design
Bilinear Transform Prewarping
Prewarp the analog frequencies -> Bilinear transform -> desired digital frequencies
For convenience, set => Prewarping -> BLT gives the same result.
IIR filter Design procedure using BLT
1. Convert each specified band-edge frequency of the digital filter to a corresponding band-edge freq of an analog filter, using (A)
- Leave the ripple values unchanged.
2. Design
3. using BLT
(A)
Bilinear TransformationBilinear Transformation
Transformation is unaffected by scaling. Consider inverse transformation with scale factor equal to unity
For
and so
ssz
11
oo js
22
222
)1()1(
)1()1(
oo
oo
oo
oo zjj
z
10 zo10 zo10 zo
Bilinear TransformationBilinear Transformation
Mapping of s-plane into the z-plane
Bilinear TransformationBilinear Transformation
For with unity scalar we have
or
)2/tan(11
j
eej j
j
jez
)2/tan(
Bilinear TransformationBilinear Transformation
Mapping is highly nonlinear
Complete negative imaginary axis in the s-plane from to is mapped into the lower half of
the unit circle in the z-plane from to
Complete positive imaginary axis in the s-plane from to is mapped into the upper half of the unit circle in the z-plane from to
0
0
1z 1z
1z 1z
Bilinear TransformationBilinear Transformation
Nonlinear mapping introduces a distortion in the
frequency axis called frequency warping
Effect of warping shown below,
References
(1). J.G. Proakis and D.G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Prentice Hall, 3rd Edition, 1996, ISBN 013373762- 4.
(2). S.S. Soliman and M.D. Srinath, Continuous and Discrete Signals and Systems, Prentice Hall, 1998, ISBN 013518473-8.
(3). A.V. Oppenheim and R.W. Schafer, Digital Signal Processing, Prentice Hall, 1975, ISBN 013214635-5.
(4). L.R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing, Prentice Hall, 1975, ISBN 013914101-4.
(5). E.O. Brigham, The Fast Fourier Transform and Its Applications, Prentice Hall, 1988, ISBN 013307505-2.
(6). M.H. Hayes, Digital Signal Processing , Schaum’s Outline Series, McGraw Hill, 1999, ISBN 0-07-027389-8
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