Digital signal processing By Er. Swapnil Kaware

A Seminar On,

“Design of Digital IIR filter”
Presented By,

Mr. Swapnil V. Kaware,
svkaware@yaoo.co.in

svkaware@yahoo.co.in

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.

Filter Design Methods

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.


Design Procedure
To fully design and implement a filter five steps are required:

(1)
(2)
(3)
(4)
(5)

Filter specification.
Coefficient calculation.
Structure selection.
Simulation (optional).
Implementation.


Filter Specification - Step 1
|H(f)|

pass-band

stop-band

1

f c : cut-off frequency

f(norm)

f s /2
(a)

|H(f)|
(dB)

pass-band

transition band

|H(f)|
(linear)

stop-band

∆
p

1+ p
δ

0

1
1− p
δ

pass-band
ripple

-3

stop-band
ripple

∆
s
f sb : stop-band frequency
f c : cut-off frequency
f pb : pass-band frequency
(b)


δs
f s /2

f(norm)

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 parameters
 Passband ripple :
 Stopband attenuation :
 Discrimination factor :

 Selectivity factor :
1

 (-3dB) cutoff frequency : 1 − p
δ

δ
s
ω
p

ω
s

IIR Filters
Frequency response
 Transfer function : Rational
 Asymptotic attenuation at high frequency

 Attenuation function:

(rational or polynomial function)
: Square magnitude frequency response

: reference frequency
 If
 If

is monotone, so is
is oscillatory,

exhibits ripple.


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 types
 Butterworth
 Chebyshev
 Elliptic


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.

Butterworth Lowpass Filters
• Passband is designed to be maximally flat.
• The magnitude-squared function is of the form
Hc ( jΩ)

Hc ( s )

2

2

=

1

1 + ( jΩ / jΩc )

Ωc

2N

N The order of the filter

1

=

1 + ( s / jΩc )

2N

ΩcN
∏ (s − s k )

H a (s) =

LHP poles

s k = Ω ce

j

π ( 2 k + N +1 )
2N

,

The Cutoff frequency

⋅
k = 0,1, ⋅ ⋅ 2 N − 1

Butterworth filters
Magnitude Squared Response

| H (ω ) | 2
1
(1 − δ p ) 2
0.5

δ

2
s

Properties of a LP Butterworth filter
 Magnitude response : monotonically decreasing
 Maximum gain : 0 at

: -3 dB point
 Asymptotic attenuation at high frequency :
 Maximally flat at DC (maximally flat filter)


ω

p

ω

0

ω

s

Butterworth filters
Transfer function

 2N poles:

: N poles are on the left side
of the complex plane
 All pole filter

Im{s}

Im{s}
N= 3

Re{s}

 Normalized transfer function : Nth-order LP Butterworth filter


N= 4

Re{s}

Butterworth filters
LP Butterworth filter design procedure
1. Set up filter spec :
2. Compute N, using
3. Choose
using

4. Compute the poles

5. Compute

, using

, 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.

Chebyshev Filters
2
1
• Equiripple in the
Hc (jΩ) =
2
1 + ε2 VN (Ω / Ωc )
passband and monotonic
in the stopband.
1
• Or equiripple in the
| H c ( jΩ ) | 2 =
2
1 + [ε 2VN (Ω / Ω c )]−1
stopband and monotonic
in the passband.

(

VN (x ) = cos N cos−1 x

Type II

Type I


)

Chebyshev filters

TN (x)

Chebyshev polynomial of degree

N = 1,2,3,4

Recursive formula:
If N is even(odd), so is

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

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

π /N

bω 0
aω 0

Transfer function

(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
oscillates between 1 and



for

For
oscillates between

and


for

Elliptic Filters
Example
N= 2

N= 3


Elliptic Filters
Equiripple both in stopband and in the passband

| H c ( jΩ) |2 =

1
2
1 + ε 2U N (Ω)
Jacobian Elliptic function

Frequency transformation
Analog filter design
1. 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
1. 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


Impulse Invariant Transformation
 Definition

 Procedure
1.
2.
3.
 High-pass filter cannot be
transformed !!
 Filter orders are not changed
After transformation

Example)


Bilinear Transform
 Definition and Properties

(Approximation of continuous-time
integration by discrete-time
trapezoidal integration)

 For

1) # of poles are preserved. => Preserve the filter order
2) # of zeros increase from q to p if p > q (p-1 zeros at z=-1)


Bilinear Transform

If
=> Preserve the stability
θ

Im{s}

Re{s}

π
ω

1
z-plane

−π

Bilinear Transform
For

1) Frequency warping : One-to-one mapping,
2)
3) Can be used for all filter types
θ

Im{s}

Re{s}

π
ω

1
z-plane


−π

Bilinear Transform
 Prewarping
 Prewarp the analog frequencies -> Bilinear transform -> desired digital
frequencies
(A)
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



Bilinear Transformation
Transformation is unaffected by scaling.
Consider inverse transformation with scale
factor equal to unity
For

z =1+ s
1− s

s = σo + jΩo
2
(1 + σ o ) + jΩo
(1 + σ o ) 2 + Ωo
2
z=
⇒z =
and so
2
(1 − σ o ) − jΩo
(1 − σ o ) 2 + Ωo
σo = 0 → z =1

σo < 0 → z <1
σo > 0 → z >1

Mapping of s-plane into the z-plane


For

or

z =e

jω

with unity scalar we have

1 − e − jω = j tan(ω / 2)
jΩ =
− jω
1+ e
Ω = tan(ω / 2)


Mapping is highly nonlinear
Complete negative imaginary axis in the s-plane from
Ω = −∞ toΩ = 0 is mapped into the lower half of the unit
circle in the z-plane from = −1 to = 1
z
z
Complete positive imaginary axis in the s-plane from
Ω = 0 toΩ = ∞ is mapped into the upper half of the unit
z to −1
=
circle in the z-plane from 1
z=


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 0-07027389-8


Thank You!!

Digital signal processing By Er. Swapnil Kaware

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