Transcript DSP & Digital Filters

AGC
DSP


Digital Filter Specifications
Only the magnitude approximation problem
Four basic types of ideal filters with magnitude
responses as shown below (Piecewise flat)
Professor A G Constantinides
1
AGC
DSP

Digital Filter Specifications
These filters are unealisable because (one of
the following is sufficient)
 their impulse responses infinitely long noncausal
 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.
Professor A G Constantinides
2
AGC
DSP

Digital Filter Specifications
Consider the ideal LP response squared
(same as actual LP response)
Professor A G Constantinides
3
AGC
DSP



Digital Filter Specifications
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
Professor A G Constantinides
4
AGC
DSP


Digital Filter Specifications
A realisable response would effectively
need to have an approximation of the
delta functions in the differential
This is a necessary condition
Professor A G Constantinides
5
AGC
DSP

Digital Filter Specifications
For example the magnitude response
of a digital lowpass filter may be given as
indicated below
Professor A G Constantinides
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AGC
DSP

Digital Filter Specifications
In the passband 0     p we require
that
with a deviation   p
G (e j )  1
j
1   p  G (e )  1   p ,    p

In the stopband  s     we require
j
that G (e )  0
with a deviation  s
j
G (e )   s ,
s    
Professor A G Constantinides
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AGC
DSP
Digital Filter Specifications
Filter specification parameters
 
- passband edge frequency
p
 s
- stopband edge frequency
 p
- peak ripple value in the
passband
 s - peak ripple value in the

stopband
Professor A G Constantinides
8
AGC
Digital Filter Specifications
DSP




Practical specifications are often given
in terms of loss function (in dB)
G ( )   20 log10 G (e j )
Peak passband ripple
 p   20 log10 (1   p )
dB
Minimum stopband attenuation
 s   20 log10 ( s )
dB
Professor A G Constantinides
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Digital Filter Specifications
AGC
DSP


In practice, passband edge frequency F
p
and stopband edge frequency Fs are
specified in Hz
For digital filter design, normalized bandedge
frequencies need to be computed from
specifications in Hz using
 p 2 Fp
p 

 2 FpT
FT
FT
 s 2 Fs
s 

 2 Fs T
FT
FT
Professor A G Constantinides
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AGC
Digital Filter Specifications
DSP


Example - Let Fp  7 kHz, Fs  3
kHz, and FT  25
kHz
Then
2 (7 103 )
p 
 0.56
3
25 10
2 (3 103 )
s 
 0.24
3
25 10
Professor A G Constantinides
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Selection of Filter Type
AGC
DSP


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 z 1
p0  p1z 1  p2 z 2    pM z  M
H ( z) 
d0  d1z 1  d 2 z 2    d N z  N

H(z) must be stable and of lowest order N or
M for reduced computational complexity
Professor A G Constantinides
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AGC
DSP

Selection of Filter Type
FIR real digital filter transfer function is a
polynomial in z 1 (order N) with real
coefficients
N
H ( z )   h[n] z
n
n 0



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:
h[n]   h[ N  n]
Professor A G Constantinides
(More on this later)
13
AGC
DSP


Selection of Filter Type
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
Professor A G Constantinides
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AGC
DSP
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
Professor A G Constantinides
15
Finite Impulse Response
Filters
AGC
DSP

The transfer function is given by
N 1
H ( z )   h(n).z n
n 0



The length of Impulse Response is N
All poles are at z  0 .
Zeros can be placed anywhere on the zplane
Professor A G Constantinides
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AGC
DSP
FIR: Linear phase
For phase linearity the FIR transfer
function must have zeros outside
the unit circle
Professor A G Constantinides
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AGC
FIR: Linear phase
DSP

To develop expression for phase
response set transfer function (order n)
H ( z )  h0  h1z 1  h2 z 2  ...  hn z n

In factored form
n1
1
n2
1
H ( z )  K  (1  i z ). (1  i z )
i 1

i 1
Where i  1, i  1 K
,
real & zeros occur in conjugates
is
Professor A G Constantinides
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AGC
FIR: Linear phase
DSP

Let H ( z)  KN1 ( z) N2 ( z)
where
n1
N1 ( z )   (1  i z 1 )
i 1

Thus
n1
n2
N 2 ( z )   (1  i z 1 )
i 1
1
n2
1
ln( H ( z ))  ln( K )   ln(1   i z )   ln(1   i z )
i 1
i 1
Professor A G Constantinides
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AGC
FIR: Linear phase
DSP


Expand in a Laurent Series
convergent within the unit circle
To do so modify the second sum as
n2
1
n2
1
n2
1
i 1
i
 ln(1 i z )   ln(i z )   ln(1 
i 1
i 1
z)
Professor A G Constantinides
20
AGC
FIR: Linear phase
DSP

So that
n1
n2
1
ln(H ( z ))  ln(K )  n2 ln(z )   ln(1   i z )   ln(1  z )
i
i 1
i 1

1
Thus

N1
sm
m1
m
ln(H ( z ))  ln(K )  n2 ln(z )  

where
smN1

n1
m

 i
i 1
sNm2
z
m

N2
sm
m
z
m
n1
   i m
i 1
Professor A G Constantinides
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AGC
FIR: Linear phase
DSP



smN1
are the root moments of the
minimum phase component
N2
s m
are the inverse root moments of
the maximum phase component
Now on the unit circle we have z  e j
and
H (e j )  A( )e j ( )
Professor A G Constantinides
22
AGC
Fundamental Relationships
DSP
N1
N2
s
s
ln(H (e j ))  ln(K )  jn2   m e jm  m e jm
m
m 1 m

ln( H (e j ))  ln( A( )e j ( ) )  ln( A( ))  j ( )

hence (note Fourier form)
smN1 sNm2
ln( A( ))  ln(K )   (

) cos m
m
m 1 m

smN1 sNm2
 ( )  n2   (

) sin m
m
m 1 m

Professor A G Constantinides
23
AGC
FIR: Linear phase
DSP




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 Professor A G Constantinides
24
AGC
FIR: Linear phase
DSP

It follows that zeros of linear phase FIR
trasfer functions not on the
circumference of the unit circle occur in
the form
 ji 1
i e


Professor A G Constantinides
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AGC
FIR: Linear phase
DSP

For Linear Phase t.f. (order N-1)
h( n)   h( N  1  n)


so that
for
N
even:
N
2
1
H ( z )   h(n).z
n 0
n
N 1
  h(n).z n
n N
2
N 1
2
N 1
2
n 0
n 0
  h(n).z  n   h( N  1  n).z ( N 1 n )
N 1
2

  h( n) z  n  z  m
n 0

m  NProfessor
 1AGnConstantinides
26
AGC
FIR: Linear phase
DSP

for N odd:
N 1
1
2

H ( z )   h(n). z
n 0

n
z
m

N  1

 h
z
 2 
N 1 


 2 
I) On C : z  1 we have for N even,
and +ve sign
H (e
j T
)e
N 1  N 1
 jT 
 2
 2 
N  1 
 
.  2h(n).cos T  n 

2 
n 0
 
Professor A G Constantinides
27
AGC
FIR: Linear phase
DSP

II) While for –ve sign
H ( e jT )  e


N 1  N 1
 jT 

2
 2 
N  1 

.  j 2h( n).sin  T  n 

2 
n 0


[Note: antisymmetric case adds  / 2 rads
to phase, with discontinuity at   0 ]
III) For N odd with +ve sign
H (e
jT
)e
N 1
 j T 

 2 

N  1


 h
2



N 3
2

N  1  

  2h( n).cosT 
n


 
2
n 0

 

Professor A G Constantinides

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AGC
FIR: Linear phase
DSP

IV) While with a –ve sign
H (e jT )  e

N 3
N 1 

 jT 
2
 2  

N  1  
 
 
  2 j.h(n).sin T  n 
2  
n 0
 



[Notice that for the antisymmetric case to
have linear phase we require
N  1
h

  0.
 2 
The phase discontinuity is as for
N even]
Professor A G Constantinides
29
AGC
FIR: Linear phase
DSP

The cases most commonly used in filter
design are (I) and (III), for which the
amplitude characteristic can be written
as a polynomial in
T
cos
2
Professor A G Constantinides
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AGC
DSP
Design of FIR filters: Windows
(i) Start with ideal infinite duration h(n)
(ii) Truncate to finite length. (This
produces unwanted ripples increasing in
height near discontinuity.)
~
(iii) Modify to h (n)  h(n).w(n)
Weight w(n) is the window
Professor A G Constantinides
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AGC
Windows
DSP




Commonly used windows
Rectangular 1  2 n
n
N
2n 

Bartlett
1  cos

 N 
Hann
2n 

Hamming 0.54  0.46 cos N 


Blackman


Kaiser
N 1

2
2n 
 4n 
0.42  0.5 cos

0
.
08
cos



 N 
 N 
2

2
n

J 0  1  

  J 0 ( )
 N  1 



Professor A G Constantinides
32
AGC
Kaiser window
DSP

Kaiser window
β
2.12
Transition Min. stop
width (Hz) attn dB
1.5/N
30
4.54
2.9/N
50
6.76
4.3/N
70
8.96
5.7/N
90
Professor A G Constantinides
33
AGC
Example
DSP
• Lowpass filter of length 51 and
Lowpass Filter Designed Using Hamming window
0
Gain, dB
-50
-100
-100
0
0.2
0.4
0.6
0.8
0
1
/
0.2
0.4
Lowpass Filter Designed Using Blackman window
0
Gain, dB
Gain, dB
Lowpass Filter Designed Using Hann window
0
-50
c   / 2
0.6
0.8
1
/
-50
-100
0
0.2
0.4
0.6
/
0.8
1
Professor A G Constantinides
34
AGC
DSP
Frequency Sampling Method
• In this approach we are given H (k ) and
need to find H (z )
• This is an interpolation problem and the
solution is given in the DFT part of the
course
N
1 N 1
1 z
H ( z )   H (k ).
2
N k 0
j k
1  e N .z 1
• It has similar problems to the windowing
Professor A G Constantinides
approach
35
Linear-Phase FIR Filter
Design by Optimisation
AGC
DSP

Amplitude response for all 4 types of
linear-phase FIR filters can be

expressed as H ( )  Q( ) A( )
1,
for Type 1

where
 cos(/2), for Type 2

Q( )  
 sin( ), for Type 3
sin( / 2), for Type 4
Professor A G Constantinides
36
Linear-Phase FIR Filter
Design by Optimisation
AGC
DSP

Modified form of weighted error function
E ( )  W ( )[Q( ) A( )  D( )]
D ( )
 W ( )Q( )[ A( )  Q ( ) ]
~
~
 W ( )[ A( )  D( )]
where
~
W ( )  W ( )Q( )
~
D( )  D( ) / Q( )
Professor A G Constantinides
37
AGC
DSP

Linear-Phase FIR Filter
Design by Optimisation
Optimisation Problem - Determine a~[k ]
which minimise the peak absolute value
L
~
~
of E ( )  W
~
( )[  a [k ] cos( k)  D( )]
k 0

over the specified frequency bands   R
~
a
After [k ] has been determined,
j
construct the original A(e ) and
hence h[n]
Professor A G Constantinides
38
AGC
DSP
Linear-Phase FIR Filter
Design by Optimisation
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.
Professor A G Constantinides
39
AGC
DSP
FIR Digital Filter Order
Estimation
Kaiser’s Formula:
N

 20 log10 (  p s )
14.6( s   p ) / 2
ie N is inversely proportional to
transition band width and not on
transition band location
Professor A G Constantinides
40
AGC
DSP

FIR Digital Filter Order
Estimation
Hermann-Rabiner-Chan’s Formula:
D ( p ,  s )  F ( p ,  s )[( s   p ) / 2 ]2
N
( s   p ) / 2
where
D ( p , s )  [a1 (log10  p )2  a2 (log10  p )  a3 ] log10  s
 [a4 (log10  p )2  a5 (log10  p )  a6 ]
F ( p ,  s )  b1  b2 [log10  p  log10  s ]
with a1  0.005309, a2  0.07114, a3  0.4761
a4  0.00266, a5  0.5941, a6  0.4278
b1  11.01217, b2  0.51244
Professor A G Constantinides
41
AGC
DSP




FIR Digital Filter Order
Estimation
Formula valid for  p   s
For  p   s , formula to be used is
obtained by interchanging  p and  s
Both formulae provide only an estimate
of the required filter order N
If specifications are not met, increase
filter order until they are met
Professor A G Constantinides
42
AGC
DSP

FIR Digital Filter Order
Estimation
Fred Harris’ guide:
A
N
20(s   p ) / 2

where A is the attenuation in dB
Then add about 10% to it
Professor A G Constantinides
43