CHAPTER 7

Filter Design
DIGITAL FILTER SPECIFICATION
 Digital Filter designed to pass signal
components of certain frequencies without
distortion.
 The frequency response should be equal to the
signal’s frequencies to pass the signal.
(passband)
 The frequency response should be equal to zero
to block the signal. (stopband)
DIGITAL FILTER SPECIFICATION
4 Types
DIGITAL FILTER SPECIFICATION
 Themagnitude response specifications are given some
acceptable tolerances.
DIGITAL FILTER SPECIFICATION
 Transitionband is specified between the passband and
the stopband to permit the magnitude to drop off
smoothly.
 In Passband

1   p  G ( e j )  1   p , for    p

 In Stopband
G ( e j )   s , for  s    
 Where δp and δs are peak ripple values, ωp are passband
edge frequency and ωs are stopband edge frequency
DIGITAL FILTER SPECIFICATION
 Digitalfilter specification are often given in
terms of loss function,
A(ω) = -20 log10 |G(ejω)|
 Loss specification of a digital filter
 Peak passband ripple, αp = -20 log10 (1 – δp) dB
 Minimum stopband attenuation, αs = -20 log10 (δs) dB
DIGITAL FILTER SPECIFICATION
 Themagnitude response specifications may be
given in a normalized form.
DIGITAL FILTER SPECIFICATION
 Example 1
The peak passband ripple αp and the minimum
stopband attenuation αs of a digital filter are,
respectively 0.1 dB and 35 dB. Determine their
corresponding peak ripples values δp and δs.
DIGITAL FILTER SPECIFICATION
 The
passband and stopband edge frequencies, in
most applications are specified in Hz
p 2Fp
p    2FpT
FT FT
 s 2Fs
s    2FsT
FT FT

 Where FT denote the sampling frequency in Hz, Fp

and Fs denote, respectively, the passband and
stopband edge frequencies in Hz
SELECTION OF FILTER TYPE
 Objective of digital filter design is to develop a
causal transfer function meeting the frequency
response specification.
 For IIR digital filter design

p0  p1 z 1  p2 z 2    pM z  M
H ( z) 
d 0  d1 z 1  d 2 z  2    d N z  N
 H(z) must be a stable function, N must be of lowest
order.
SELECTION OF FILTER TYPE
 For FIR digital filter design
N
H ( z )   h[n]z  n
n 0

 The degree N of H(z) must be small, for a linear

phase, FIR filter coefficient must satisfy the
constraint
h[n]   h[ N  n]
IIR DIGITAL FILTER DESIGN
 Convert the digital filter specifications into analog
lowpass prototype filter specifications
 Determine the analog lowpass filter transfer function
to meet these specifications
 Then transform it into the desired digital filter
transfer function
 Why used this approach?
 Analog approximation techniques are highly advanced
 Usually yield closed-form solutions
 Extensive tables are available for analog filter design.
 Many applications require the digital simulation of analog
filters.
IIR DIGITAL FILTER DESIGN
Pa ( s ) P( z )
H a ( s)   G( z) 
Da ( s ) D( z )
A mapping from s-domain to z-domain
 The imaginary (jΩ) azis in s-plane be mapped
onto the unit circle of z-plane
 A stable analog tranfer function be transform into
a stable digital transfer function
 The most widely used transformation is the
bilinear transformation.
IIR ANALOG FILTER ORDER
ESTIMATION
 Selectivityparameter
p
k
s
Ωp and Ωs passband and stopband edge frequencies
 Discrimination parameter
d
1    p
2
1
or

 s  2
1 A2  1
Where δp and δs are peak ripple values
IIR ANALOG FILTER ORDER
ESTIMATION
3 dB cutoff frequency, Ωc
1 1
 p [(1   p )  1]
2 2N
  c   s [( s )  1]
2 2N
IIR ANALOG FILTER ORDER
ESTIMATION
 Butterworth Filter
 Magnitude response
2 1
H a ( j )  2N

1   
 c 
 The Filter order, N

N

1 log10  A2  1  2 log10 d


2 log10   s  p  log10 k
N
 p 
with    
 c 
IIR ANALOG FILTER ORDER
ESTIMATION
 The magnitude of Butterworth Low-Pass Filter can
also be defined as:
|Ha(jΩ)|2 = Ha(s)Ha(-s)|s=jΩ
Thus,
Ga(s) = Ha(s)Ha(-s)|s=jΩ = 1 / [1+(s/jΩc)2N]
= |Ha(Ω)|2 = 1 / [1+(s/Ωc)2N]
 The poles of Butterworth Low-Pass Filter can also
be defined as:
sk = Ωc ej[/2 + (2k + 1)/2N], k = 0,1,…N-1
Thus, the Transfer Function is defined as:
Ha(s) = 1 / [sN + a1sN-1 + … + aN-1s + aN]
IIR ANALOG FILTER ORDER
ESTIMATION
 Example 2:
Design the Butterworth Low-Pass Filter to meet
the following specification:
fp = 6 kHz, fs = 10 kHz, δp = δs = 0.1
IIR ANALOG FILTER ORDER
ESTIMATION
 Chebyshev Type I
 Magnitude response
2 1
H a ( j ) 
1   2TN2 (  p )

 Where the parameter that control the passband ripple can be

defined as
1
 
2
1
1   p  2
IIR ANALOG FILTER ORDER
ESTIMATION
 Chebyshev Type I
 Where the chebyshev polynomial of order N, TN(Ω) is :
 cos( N cos 1 ),   1
TN ()   1
cosh( N cosh ),   1

 Can also be derived via a recurrence relation

Tr ()  2Tr 1 ()  Tr  2 (), r  2
with T0 ()  1 and T1 ()  

 Filter order, N
cosh 1 (1 d )
N
cosh 1 (1 k )
IIR ANALOG FILTER ORDER
ESTIMATION
 Example 3
Design a lowpass Chebyshev Type 1 filter to meet
the following spec:
fp = 6 kHz, fs = 10 kHz, δp = δs = 0.1
IIR ANALOG FILTER ORDER
ESTIMATION
 Solution
1. Calculate d & k,
d = sqrt[((1-δp)-2 -1)/ (δs-2 – 1)] = 0.0487
k = Ωp / Ωs = 0.6
2. Calculate the Filter Order, N
N  cosh-1 (1/d) / cosh-1 (1/k)  3.38 = 4
3. Calculate passband ripple controller, 
 = [(1-δp)-2 – 1]1/2 = 0.4843
4. Calculate Nth order Chebyshev Polynomials,
T4(Ω) = 8 Ω 4 – 8 Ω 2 -1, -1 ≤ x ≤ 1
5. The magnitude of the Frequency response of the filter:
|Ha(Ω)|2 = 1 / [1 + 2TN2(Ω/ΩP)] =
= 1 / [1 + (0.4843)2 Ω/(12000)]
= 1 / [1 + 6.22x10-6 Ω]
IIR ANALOG FILTER ORDER
ESTIMATION
 Chebyshev Type II
 Magnitude response
2 1
H a ( j )  2
2  TN ( s  p ) 
1   
T
 N s (   ) 
 The order is determine as Type I
BILINEAR TRANSFORMATION
 A method to map the left half s-plane to inside
unit circle of z-plane
 The design of the digital IIR Filter, H(z) from
analog filter, Ha(s) require a mapping of s-plane
to z-plane.
 The point in the left half s-plane should map to
points inside the unit circle to preserve the
stability of the analog filter
BILINEAR TRANSFORMATION
 Bilinear transformation is given by
2  1  z 1 
s   
1 
T 1 z 
 Maps a single point in the s-plane to a unique point in
z-plane
 The relationship
G ( z )  H a ( s ) s  2  1 z1 
T  1 z 1 
BILINEAR TRANSFORMATION
 For s = σo + jΩo,
1  T2 ( o  j o ) (1  T2  o )  j T2  o
z 
1  2 ( o  j o ) (1  T2  o )  j T2  o
T

 Mapping of s-plane to z-plane

BILINEAR TRANSFORMATION
 A point on the jΩ-axis in the s-plane is mapped
onto a point on the unit circle in the z-plane
 A point in the left-half s-plane with σ < 0 is
mapped inside the unit circle in the z-plane
 A point in the right-half s-plane with σ > 0 is
mapped outside the unit circle in the z-plane
 Frequency wrapping,

2  
 tan 
T 2
SIMPLIFIED BILINEAR
TRANSFORMATION
 Choose T = 2 to simplify the design procedure.
 So the parameter T has no effect on G(z)

1 s
z
1 s
BILINEAR TRANSFORMATION
 Example 4
Design the Digital Low-Pass Filter with 3-dB Cut-Off
Frequency, ωc = 0.25 by using Bilinear Transformation
method to analog Butterworth Low-Pass Filter defined below:
Ha(s) = [1 / 1 + (s/Ωs)]

 Example 5
Determine the Low-Pass Digital Filter with 3-dB Cut-Off
Frequency of 0.2 and its Frequency
Response from the analog filter given below:
Ha(s) = Ωc / s + Ωc
where Ωc is the 3-dB Cut-Off Frequency.
DESIGN OF LOW-ORDER
DIGITAL FILTERS
 First-order Butterworth lowpass digital filters
 Analog transfer function
c
H LP ( s ) 
s  c
 Applying bilinear transformation,
 cT
c (1  z 1 )
GLP ( z )   2
s  c (1  z )  2cT (1  z 1 )
1
 1 
s  2  1 z 
T  1 z 1 

1    1  z 1  1  tan(c T 2)
   where  
1 
2  1  z  1  tan(c T 2)
DESIGN OF LOW-ORDER
DIGITAL FILTERS
 First-order Butterworth highpass digital filters
 Analog transfer function
s
H HP ( s ) 
s  c

 Applying bilinear transformation,

s 1    1  z 1 
GHP ( z )    
1 
s  c 2  1  z 
 1 
s  2  1 z 
T  1 z 1 
DESIGN OF LOW-ORDER
DIGITAL FILTERS
 Second-order bandpass digital filter
Bs 1  1  z 2 
H BP ( s )  2  G BP ( z )  
 
2 
s  Bs   o
2
2  1   (1   ) z  z 
1

where
1  tan( B T 2)
 ,   cos(oT )
1  tan( B T 2)

 Second-order bandstop digital filter

s2   o2 1    1  2  z 1  z 2 
H BS ( s )   GBS ( z )   
2 
s2  Bs   o2
2  1   (1   ) z  z 
1
DESIGN OF LOWPASS IIR
DIGITAL FILTERS
 Designa lowpass IIR digital filter with
following parameters
Passband edge frequency, ωp = 0.25π
Passband ripple, αp not exceeding 0.5dB
Minimum stopband attenuation, αs = 15dB
Stopband edge frequency, ωs = 0.55π
SPECTRAL TRANSFORMATION
 Used to modify the characteristic of a filter to
meet the new specifications without repeating
the filter design procedure
 For a lowpass filter with a passband edge at 2
kHz, the passband edge can be move to 2.1 kHz.
 A digital filter with highpass, bandpass or
bandstop characteristic can also be design by
transforming a given lowpass digital lowpass
filter.
FIR DIGITAL FILTER DESIGN.
 Causal FIR transfer function of length M+1
M
H ( z )   h[n]z 1
n 0

 Corresponding frequency response

M
H (e )   h[n]e  jn
j

n 0

 FIR Lowpass Filter Impulse response

sin c n
hLP [n] 
n
FIR FILTER WITH WINDOW
METHOD
 AnIdeal Frequency Response of Low-Pass Filter
and Impulse Response are shown below:
FIR FILTER WITH WINDOW METHOD
 How the Impulse
Response of the ideal
Low-Pass Filter is
windowed is shown
below:
FIR FILTER WITH WINDOW
METHOD
 TheFilter is designed by windowing the impulse
response:

h(n) = hd(n)w(n)

 w(n)is a finite-length window that is equal to

zero outside the interval of 0 ≤ n ≤ M.

M is the Filter order

FIR FILTER WITH WINDOW
METHOD
 Basically,
there are 4 type of window :
 Rectangular

w(n) = 1, 0 ≤ n ≤ M
0, elsewhere
 Hanning

w(n) = 0.5 -0.5cos(2n/M), 0 ≤ n ≤ M

0, elsewhere
FIR FILTER WITH WINDOW
METHOD
 Hamming
w(n) = 0.54 – 0.46cos(2n/M), 0 ≤ n ≤ M
0, elsewhere

 Blackman
w(n) = 0.42 - 0.5cos(2n/M) +
0.08cos(4n/M), 0≤n≤M
0, elsewhere
FIR FILTER WITH WINDOW
METHOD
 The
relationship between the length of window,
M and Filter Transition Band is shown below:

M∆ω
M∆ = c

c is a parameter of the window.

 Transition Band width

   s   p
FIR FILTER WITH WINDOW
METHOD
 The window parameter, c is shown below:

1. Rectangular
M∆ω = 0.9 , s = -21 dB

2. Hanning
M∆ω = 3.1 , s = -44 dB

3. Hamming
M∆ω = 3.3 , s = -53 dB

4. Blackman
M∆ω = 5.5 , s = -74 dB
FIR FILTER WITH WINDOW
METHOD
 The
diagram of Rectangular, Hanning, Hamming
& Blackman is shown below:
KAISER WINDOW METHOD
 The Kaiser window is defined as

 I  1  [ ( n   )  ]2
 0

, 0nM
w[n]   I 0 ( )
0, otherwise

Where α = M/2, M is the filter order, β is the shape
parameter and I0(µ) represents the zeroth-order
modified Bessel Function of the first kind.

   2 
r 2

I 0 ( )  1    
r 1  r! 
KAISER WINDOW METHOD
 Shape parameter,
0.1102 ( s  8.7),  s  50

   0.5842( s  21) 0.4  0.07886( s  21), 21   s  50
0,  s  21

 Transition Band width
   s   p
 The filter order
s  8
M
2.285( )
KAISER WINDOW METHOD
 Example 6
 Design a FIR Lowpass with the desired
specification as follows using Kaiser window:
Passband edge, ωp = 0.3π
Stopband edge, ωs = 0.5π
Minimum stopband attenuation, αs = 40 dB