DIGITAL FILTER SPECIFICATIONS AND MATHEMATICS.ppt

AGC
DSP
Professor A G Constantinides
1
Digital Filter Specifications
 Only the magnitude approximation problem
 Four basic types of ideal filters with magnitude
responses as shown below (Piecewise flat)

AGC
DSP
2
 These filters are unealisable because (one of
the following is sufficient)
 their impulse responses infinitely long non-
causal
 Their amplitude responses cannot be equal
to a constant over a band of frequencies
Another perspective that provides some
understanding can be obtained by looking
at the ideal amplitude squared.

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DSP
3
 Consider the ideal LP response squared
(same as actual LP response)

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DSP
4
 The realisable squared amplitude response
transfer function (and its differential) is
continuous in
 Such functions
 if IIR can be infinite at point but around
that point cannot be zero.
 if FIR cannot be infinite anywhere.
 Hence previous defferential of ideal response
is unrealisable


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DSP
5
 A realisable response would effectively
need to have an approximation of the
delta functions in the differential
 This is a necessary condition

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DSP
6
 For example the magnitude response
of a digital lowpass filter may be given as
indicated below

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DSP
7
 In the passband we require
that with a deviation
 In the stopband we require
that with a deviation
1
)
( 

j
e
G
0
)
( 

j
e
G s

p


p

 

0


 

s
p
p
j
p e
G 


 




 ,
1
)
(
1







 s
s
j
e
G ,
)
(

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DSP
8
Filter specification parameters
 - passband edge frequency
 - stopband edge frequency
 - peak ripple value in the
passband
 - peak ripple value in the
stopband
p

s

s

p


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DSP
9
 Practical specifications are often given
in terms of loss function (in dB)

 Peak passband ripple
dB
 Minimum stopband attenuation
dB
)
(
log
20
)
( 10

 j
e
G


G
)
1
(
log
20 10 p
p 
 


)
(
log
20 10 s
s 
 


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DSP
10
 In practice, passband edge frequency
and stopband edge frequency are
specified in Hz
 For digital filter design, normalized bandedge
frequencies need to be computed from
specifications in Hz using
T
F
F
F
F
p
T
p
T
p
p 

 2
2




T
F
F
F
F
s
T
s
T
s
s 

 2
2




s
F
p
F

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DSP
11
 Example - Let kHz,
kHz, and kHz
 Then
7

p
F 3

s
F
25

T
F


 56
.
0
10
25
)
10
7
(
2
3
3




p


 24
.
0
10
25
)
10
3
(
2
3
3




s

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DSP
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 The transfer function H(z) meeting the
specifications must be a causal transfer
function
 For IIR real digital filter the transfer function
is a real rational function of
 H(z) must be stable and of lowest order N or
M for reduced computational complexity
Selection of Filter Type
1

z
N
N
M
M
z
d
z
d
z
d
d
z
p
z
p
z
p
p
z
H 
















2
2
1
1
0
2
2
1
1
0
)
(

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DSP
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 FIR real digital filter transfer function is a
polynomial in (order N) with real
coefficients
 For reduced computational complexity,
degree N of H(z) must be as small as possible
 If a linear phase is desired then we must
have:
 (More on this later)




N
n
n
z
n
h
z
H
0
]
[
)
(
]
[
]
[ n
N
h
n
h 


1

z

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DSP
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 Advantages in using an FIR filter -
(1) Can be designed with exact linear phase
(2) Filter structure always stable with
quantised coefficients
 Disadvantages in using an FIR filter - Order of
an FIR filter is considerably higher than that
of an equivalent IIR filter meeting the same
specifications; this leads to higher
computational complexity for FIR

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DSP
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FIR Design
FIR Digital Filter Design
Three commonly used approaches to
FIR filter design -
(1) Windowed Fourier series approach
(2) Frequency sampling approach
(3) Computer-based optimization
methods

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DSP
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Finite Impulse Response
Filters
 The transfer function is given by
 The length of Impulse Response is N
 All poles are at .
 Zeros can be placed anywhere on the z-
plane





1
0
).
(
)
(
N
n
n
z
n
h
z
H
0

z

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DSP
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FIR: Linear phase
For phase linearity the FIR transfer
function must have zeros outside
the unit circle

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DSP
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FIR: Linear phase
 To develop expression for phase
response set transfer function (order n)
 In factored form
 Where , is
real & zeros occur in conjugates
n
nz
h
z
h
z
h
h
z
H 






 ...
)
( 2
2
1
1
0
)
1
(
).
1
(
)
( 1
2
1
1
1
1




 
 
 z
z
K
z
H i
n
i
i
n
i


1
,
1 
 i
i 
 K

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DSP
19
FIR: Linear phase
 Let
where
 Thus
)
(
)
(
)
( 2
1 z
N
z
KN
z
H 
)
1
ln(
)
1
ln(
)
ln(
))
(
ln(
2
1
1
1
1
1
 

 






n
i
i
n
i
i z
z
K
z
H 

)
1
(
)
( 1
1
1
1


 
 z
z
N i
n
i
 )
1
(
)
( 1
2
1
2


 
 z
z
N i
n
i


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DSP
20
FIR: Linear phase
 Expand in a Laurent Series
convergent within the unit circle
 To do so modify the second sum as
)
1
1
ln(
)
ln(
)
1
ln(
2
1
1
2
1
1
2
1
z
z
z
i
n
i
i
n
i
i
n
i 

  

 

 






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DSP
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FIR: Linear phase
 So that
 Thus
 where
)
1
1
ln(
)
1
ln(
)
ln(
)
ln(
))
(
ln(
2
1
1
1
1
2  

 






n
i i
n
i
i z
z
z
n
K
z
H


m
N
m
m
m
N
m
z
m
s
z
m
s
z
n
K
z
H
2
1
1
2 )
ln(
)
ln(
))
(
ln( 



 






1
1
1
n
i
m
i
N
m
s  




1
1
2
n
i
m
i
N
m
s 

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DSP
22
FIR: Linear phase
 are the root moments of the
minimum phase component
 are the inverse root moments of
the maximum phase component
 Now on the unit circle we have
and

j
e
z 
)
(
)
(
)
( 


 j
j
e
A
e
H 
1
N
m
s
2
N
m
s

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DSP
23
Fundamental Relationships
 hence (note Fourier form)



 jm
N
m
m
jm
N
m
j
e
m
s
e
m
s
jn
K
e
H
2
1
1
2
)
ln(
))
(
ln( 



 



)
(
))
(
ln(
)
)
(
ln(
))
(
ln( )
(



 


j
A
e
A
e
H j
j




 m
m
s
m
s
K
A
N
m
m
N
m
cos
)
(
)
ln(
))
(
ln(
2
1
1



 





 m
m
s
m
s
n
N
m
m
N
m
sin
)
(
)
(
2
1
1
2



 




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DSP
24
FIR: Linear phase
 Thus for linear phase the second term in the
fundamental phase relationship must be
identically zero for all index values.
 Hence
 1) the maximum phase factor has zeros
which are the inverses of the those of the
minimum phase factor
 2) the phase response is linear with group
delay (normalised) equal to the number of
zeros outside the unit circle

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DSP
25
FIR: Linear phase
 It follows that zeros of linear phase FIR
trasfer functions not on the
circumference of the unit circle occur in
the form
 1

 i
j
ie 


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DSP
26
FIR: Linear phase
 For Linear Phase t.f. (order N-1)

 so that for N even:
)
1
(
)
( n
N
h
n
h 













1
2
1
2
0
).
(
).
(
)
(
N
N
n
n
N
n
n
z
n
h
z
n
h
z
H
 












1
2
0
)
1
(
1
2
0
).
1
(
).
(
N
n
n
N
N
n
n
z
n
N
h
z
n
h
 
 





1
2
0
)
(
N
n
m
n
z
z
n
h n
N
m 

 1

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DSP
27
FIR: Linear phase
 for N odd:
 I) On we have for N even,
and +ve sign
 
 




 











 



1
2
1
0
2
1
2
1
).
(
)
(
N
n
N
m
n
z
N
h
z
z
n
h
z
H
1
: 
z
C
 










 









 
 1
2
0
2
1
2
1
cos
).
(
2
.
)
(
N
n
N
T
j
T
j N
n
T
n
h
e
e
H 



AGC
DSP
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FIR: Linear phase
 II) While for –ve sign
 [Note: antisymmetric case adds rads
to phase, with discontinuity at ]
 III) For N odd with +ve sign
 










 









 
 1
2
0
2
1
2
1
sin
).
(
2
.
)
(
N
n
N
T
j
T
j N
n
T
n
h
j
e
e
H 


2
/

0










 






 

2
1
)
( 2
1
N
h
e
e
H
N
T
j
T
j







 










 




2
3
0 2
1
cos
).
(
2
N
n
N
n
T
n
h 

AGC
DSP
29
FIR: Linear phase
 IV) While with a –ve sign
 [Notice that for the antisymmetric case to
have linear phase we require
The phase discontinuity is as for N even]










 










 









 
 2
3
0
2
1
2
1
sin
).
(
.
2
)
(
N
n
N
T
j
T
j N
n
T
n
h
j
e
e
H 


.
0
2
1






 
N
h

AGC
DSP
30
FIR: Linear phase
 The cases most commonly used in filter
design are (I) and (III), for which the
amplitude characteristic can be written
as a polynomial in
2
cos
T


AGC
DSP
31
Design of FIR filters: Windows
(i) Start with ideal infinite duration
(ii) Truncate to finite length. (This
produces unwanted ripples increasing in
height near discontinuity.)
(iii) Modify to
Weight w(n) is the window
 
)
(n
h
)
(
).
(
)
(
~
n
w
n
h
n
h 

AGC
DSP
32
Windows
Commonly used windows
 Rectangular
 Bartlett
 Hann
 Hamming

 Blackman

 Kaiser
2
1


N
n
N
n
2
1







N
n

2
cos
1







N
n

2
cos
46
.
0
54
.
0














N
n
N
n 
 4
cos
08
.
0
2
cos
5
.
0
42
.
0
)
(
1
2
1 0
2
0 
 J
N
n
J

















AGC
DSP
33
Kaiser window
 Kaiser window
β Transition
width (Hz)
Min. stop
attn dB
2.12 1.5/N 30
4.54 2.9/N 50
6.76 4.3/N 70
8.96 5.7/N 90

AGC
DSP
34
Example
• Lowpass filter of length 51 and 2
/

 
c
0 0.2 0.4 0.6 0.8 1
-100
-50
0
/
Gain,
dB
Lowpass Filter Designed Using Hann window
0 0.2 0.4 0.6 0.8 1
-100
-50
0
/
Gain,
dB
Lowpass Filter Designed Using Hamming window
0 0.2 0.4 0.6 0.8 1
-100
-50
0
/
Gain,
dB
Lowpass Filter Designed Using Blackman window

AGC
DSP
35
Frequency Sampling Method
• In this approach we are given and
need to find
• This is an interpolation problem and the
solution is given in the DFT part of the
course
• It has similar problems to the windowing
approach





 

1
0 1
2
.
1
1
).
(
1
)
(
N
k k
N
j
N
z
e
z
k
H
N
z
H 
)
(k
H
)
(z
H

AGC
DSP
36
Linear-Phase FIR Filter
Design by Optimisation
 Amplitude response for all 4 types of
linear-phase FIR filters can be
expressed as
where
)
(
)
(
)
( 

 A
Q
H 









4
Type
for
),
2
/
sin(
3
Type
for
),
sin(
2
Type
for
/2),
cos(
1
Type
for
,
1
)
(




Q

AGC
DSP
37
 Modified form of weighted error function
where
)]
(
)
(
)
(
)[
(
)
( 



 D
A
Q
W 

E
]
)
(
)[
(
)
( )
(
)
(




 Q
D
A
Q
W 

)]
(
~
)
(
)[
(
~


 D
A
W 

)
(
)
(
)
(
~


 Q
W
W 
)
(
/
)
(
)
(
~


 Q
D
D 

AGC
DSP
38
 Optimisation Problem - Determine
which minimise the peak absolute value
of
over the specified frequency bands
 After has been determined,
construct the original and
hence h[n]
)]
(
~
)
cos(
]
[
~
)[
(
~
)
(
0



 D
k
k
a
W
L
k




E
]
[
~ k
a
R


)
( 
j
e
A
]
[
~ k
a

AGC
DSP
39
Solution is obtained via the Alternation
Theorem
The optimal solution has equiripple
behaviour consistent with the total
number of available parameters.
Parks and McClellan used the Remez
algorithm to develop a procedure for
designing linear FIR digital filters.

AGC
DSP
40
FIR Digital Filter Order
Estimation
Kaiser’s Formula:
 ie N is inversely proportional to
transition band width and not on
transition band location





2
/
)
(
6
.
14
)
(
log
20 10
p
s
s
p
N




AGC
DSP
41
Estimation
 Hermann-Rabiner-Chan’s Formula:
where
with










2
/
)
(
]
2
/
)
)[(
,
(
)
,
( 2
p
s
p
s
s
p
s
p F
D
N



 
s
p
p
s
p a
a
a
D 



 10
3
10
2
2
10
1 log
]
)
(log
)
(log
[
)
,
( 



]
)
(log
)
(log
[ 6
10
5
2
10
4 a
a
a p
p 

 

]
log
[log
)
,
( 10
10
2
1 s
p
s
p b
b
F 


 


4761
.
0
,
07114
.
0
,
005309
.
0 3
2
1 


 a
a
a
4278
.
0
,
5941
.
0
,
00266
.
0 6
5
4 

 a
a
a
51244
.
0
,
01217
.
11 2
1 
 b
b

AGC
DSP
42
Estimation
 Formula valid for
 For , formula to be used is
obtained by interchanging and
 Both formulae provide only an estimate
of the required filter order N
 If specifications are not met, increase
filter order until they are met
s
p 
 
s
p 
 
p
 s


AGC
DSP
43
Estimation
 Fred Harris’ guide:
where A is the attenuation in dB
 Then add about 10% to it


 2
/
)
(
20 p
s
A
N



DIGITAL FILTER SPECIFICATIONS AND MATHEMATICS.ppt

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DIGITAL FILTER SPECIFICATIONS AND MATHEMATICS.ppt