Applications of DSP
1. Imaging2. Medical Imaging3. Bandwidth compression4. sGraphic5. Spectrum Analysis6. Array Processors7. Control and Guidance8. Radar
Reason for Processing of signals
• Signals are carriers of information– Useful and unwanted– Extracting, enhancing, storing and transmitting
the useful information• How signals are being processed?---
– Analog Signal Processing– Digital Signal Processing
DSP
PrF ADC DSP DAC PoFAnalog Analog
Equivalent analog signal processor
PrF: antialiasing filtering
PoF: smooth out the staircase waveform
Comparison of DSP over ASP
-Advantages • Developed Using Software on Computer;• Working Extremely Stable;• Easily Modified in Real Time ;• Low Cost and Portable;• Flexible
Comparison of DSP over ASP Contd…
-Disdvantages• Lower Speed and Lower Frequency• Can not be used at Higher frequency• Skilled manpower is required• Weak Signals can not be able to process
The two categories of DSP Tasks
• Signal Analysis:– Measurement of signal properties– Spectrum(frequency/phase) analysis– Target detection, verification, recognition
• Signal Filtering– Signal-in-signal-out, filter– Removal of noise/interference– Separation of frequency bands
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)
Basic Filter Types
• Low pass filters• High Pass filters• Band pass filters• Band reject filters
Digital Filter Specification
• 4 Types
Digital Filter Specification Contd…
• The magnitude response specifications are given some acceptable tolerances.
Digital Filter Specification Contd…
Transition band is specified between the passband and the stopband to permit the magnitude to drop off smoothly.
In Passband
In Stopband
Where δp and δs are peak ripple values, ωp are passband edge frequency and ωs are stopband edge frequency
ppj
p foreG ,1)(1
ssj foreG ,)(
• Digital filter 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 Contd…
• The magnitude response specifications may be given in a normalized form.
Digital Filter Specification Contd…
14
• 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
TFF
F
F pT
p
T
pp
2
2
TFFF
F sT
s
T
ss 2
2
sFpF
Digital Filter Specification Contd…
15
• Example - Let kHz, kHz, and kHz
• Then
7pF 3sF25TF
56.01025
)107(23
3
p
24.01025
)103(23
3
s
Digital Filter Specification Contd…
Digital Filter Type
n
nznhzH ][)(
• Objective of digital filter design is to develop a causal transfer function meeting the frequency response specification.
• For IIR digital filter design
• For FIR digital filter design
– The degree N of H(z) must be small, for a linear phase, FIR filter coefficient must satisfy the constraint
N
n
nznhzH0
][)(
][][ nNhnh
Digital Filter Type Contd…
FIR FILTERS
FIR Filter Design by Window function technique
• Simplest FIR the filter design is window function technique
• An ideal frequency response may express
where
( ) [ ]j j nd d
n
H e h n e
1[ ] ( )
2j j n
d dh n H e e d
FIR Filter Design by Window function technique Contd…
• To get this kind of systematic causal FIR to be approximate, the most direct method intercepts its ideal impulse response!
[ ] [ ] [ ]dh n w n h n
( ) ( ) ( )dH W H
FIR Filter Design by Window function technique Contd…
• 1.Rectangular window
• 2.Triangular window (Bartett window)
1, 0[ ]
0,
n Mw n
otherwise
2 , 0 22[ ] 2 , 2
0,
n MnMn Mw n n MM
otherwise
FIR Filter Design by Window function technique Contd…
• 1.Rectangular window • 2.Triangular window (Bartett window)
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Rectangular window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Bartlett window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi unitsF
requ
ency
res
pons
e T
(jw)(
dB) Rectangular window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi units
Fre
quen
cy r
espo
nse
T(jw
)(dB
) Bartlett window
FIR Filter Design by Window function technique Contd…
• 3.HANN window
• 4.Hamming window
1 21 cos , 0
[ ] 2
0,
nn M
w n M
otherwise
20.54 0.46cos , 0
[ ]0,
nn M
w n Motherwise
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi unitsF
requ
ency
res
pons
e T
(jw)(
dB) Hanning window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi units
Fre
quen
cy r
espo
nse
T(jw
)(dB
) Hamming window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n)
Hanning window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n)
Hamming window
FIR Filter Design by Window function technique Contd…
• 3.HANN window• 4.Hamming window
FIR Filter Design by Window function technique Contd…
• 5.Kaiser’s window
• 6.Blackman window
20
0
2[ 1 (1 ) ]
[ ] , 0,1,...,[ ]
nI
Mw n n MI
2 40.42 0.5cos 0.08cos , 0
[ ]0,
n nn M
w n M Motherwise
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi unitsF
requ
ency
res
pons
e T
(jw)(
dB) Blackman window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150
-100
-50
0
50
100
pi units
Fre
quen
cy r
espo
nse
T(jw
)(dB
) Kaiser window
• 5.Kaiser’s window• 6.Blackman window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Blackman window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Kaiser window
FIR Filter Design by Window function technique Contd…
Type of the window Transition Bandwidth Minor Lobe attenuation in dB
Rectangular 4π/M -21
Triangu;ar 8π/M -26
Hanning 8π/M -44
Hamming 8π/M -53
Blackmann 12π/M -74
Kaiser variable variable
Window Table
351M Digital Signal Processing
Filter Design by Windowing• Simplest way of designing FIR filters• Method is all discrete-time no continuous-time involved• Start with ideal frequency response
• Choose ideal frequency response as desired response• Most ideal impulse responses are of infinite length• The easiest way to obtain a causal FIR filter from ideal is
• More generally
n
njd
jd enheH
deeH21
nh njjdd
else0
Mn0nhnh d
else0
Mn01nw where nwnhnh d
Rectangular Window
else0
Mn01nw
• Narrowest main lob– 4/(M+1)– Sharpest transitions at
discontinuities in frequency
• Large side lobs– -21 dB– Large oscillation around
discontinuities
• Simplest window possible
Bartlett (Triangular) Window
else0
Mn2/MM/n22
2/Mn0M/n2
nw
• Medium main lob– 8/M
• Side lobs– -25 dB
• Hamming window performs better
• Simple equation
Hanning Window
else0
Mn0M
n2cos1
21
nw
• Medium main lob– 8/M
• Side lobs– -44 dB
• Hamming window performs better
• Same complexity as Hamming
Hamming Window
else0
Mn0M
n2cos46.054.0nw
• Medium main lob– 8/M
• Good side lobs– -53 dB
• Simpler than Blackman
Blackman Window
else0
Mn0M
n4cos08.0
Mn2
cos5.042.0nw
• Large main lob– 12/M
• Very good side lobs– -73 dB
• Complex equation
Lowpass filter• Desired frequency response
• Corresponding impulse response
c
c2/Mj
jlp 0
eeH
)(
sin)(
n
nnh c
Highpass filter
• Corresponding impulse response
)(
sin)(sin)(
n
nnnh c
Bandpass Filter
• The Impulse Response is
)(
sin)(sin)( 12
n
nnnh cc
• The Impulse Response is
)(
sin)(sin)(sin)( 12
n
nnnnh cc
Bandreject Filter
FIR Filter Design Procedure
• Step1:Draw the response of the given problem.
• Step2:Convert the Analog frequencies in to the Digital frequencies
• Step3:Calculate the Transition Band width.
• Step4:Calculate the order of the filter by equating the calculated Transitions band width to the transition band width in the table.
• Step5:Calculate the Ʈ parameter Ʈ =(M-1)/2
• Step6:Choose the Window to be used by considering the attenuation.
• Step7:Calculate ht(n)
• Step8:Calculate w(n) for the choosen window.
FIR Filter Design Procedure Contd…
FIR Filter Design Procedure Contd…
• Step9:Then calculate h(n)=ht(n) x w(n)
• Step10: For verifying the design use the equation for calculating the magnitude response and the frequency response.
1
0
)(cos)(*2)()(
n
jj nnhheeH
Table: Frequency Response
Ѡ Ø
Attenuation20log
0
0.2π
0.4π
0.6π
0.8π
π
HH
42
Kaiser Window Filter Design Method• Parameterized equation forming a set of
windows– Parameter to change main-lob width
and side-lob area trade-off
– I0(.) represents zeroth-order modified Bessel function of 1st kind
else0
Mn0I
2/M2/Mn
1Inw
0
2
0
Determining Kaiser Window Parameters• Given filter specifications Kaiser developed empirical equations
– Given the peak approximation error or in dB as A=-20log10
– and transition band width • The shape parameter should be
• The filter order M is determined approximately by
21A0
50A2121A07886.021A5842.0
50A7.8A1102.04.0
ps
285.2
8AM
• After the kaiser window design follow the same procedure for the filter design
IIR FILTER
n
nznhzH ][)(
• The transfer function of the IIR Filters will be of the form
IIR Filter Design
Commonly used analog IIR filters
• Butterworth filter • Chebyshev filters
Butterworth filters
• It is governed by the magnitude squared response
n
c
jH 2
1
1
• The response is maximally flat at the origin• Magnitude square is having a value of 0.5 at
the cutoff frequency• It is a monotonically decreasing function
beyond the cutoff frequency.
Butterworth filters-Properties
Butterworth Polynomial
Order Butterworth polynomial
1 S+1
2 S2+√2 S+1
3 (S2+S+1)(S+1)
This Polynomial may be obtained by finding the roots for n is odd and evenThen by considering the left half side poles the butterworth polynomial may be
constructed
Butterworth filter design
Step1:Find the order of the filtern=log[(10(k1/10)-1)/ (10(k2/10)-1)]/2log(Ω1/Ω2)
Step2:Obtain the normalised transfer function Hn(s)=1/Bn(s)
Step3:By substituting the value of s from the analog transformation Table the actual filter transfer function may be obtained
Analog Transformation
Filter Type
Normalised Response
Analog Transformation
Actual Response
Backward Equation
Low pass filter S=S/ΩC ΩS=Ω2/Ω1
High Pass Filter S=ΩC /S ΩS=Ω2/Ω1
• Relative linear scale– The lowpass filter specifications on the magnitude-squared
response are given by
||,1
|)(|0
||,1|)(|1
1
2
2
sa
pa
AjH
jH
Where epsilon is a passband ripple parameter, Omega_p is the passband cutoff frequency in rad/sec, A is a stopband attenuation parameter, and Omega_s is the stopband cutoff in rad/sec.
Chebyshev filter design- Some Prelimnaries
sa atA
jH 2
2 1|)(|
pa atjH
2
2
1
1|)(|
Analog Filter response
n=log[(g+(g2-1)1/2 ]/log(Ωr/(Ωr2-1)1/2
Design Procedure for chebyshev filters
Step1: Calculate the order of the filter
\Where
g-[(A2-1)/ϵ2]1/2
Ωr= Ω2/Ω1
A=10-K2/20
Step2:Obtained the normalised transfer function Hn(s)=k/(Sn+bn-I Sn-1+…+b1S+b0)
Analog to Digital Conversion
• Impulse Invariance Transformation• Bilinear Transformation
Impulse Invariance method
• The most straightforward of these is the impulse invariance transformation
• Let be the impulse response corresponding to , and define the continuous to discrete time transformation by setting
• We sample the continuous time impulse response to produce the discrete time filter
( )ch t
( )cH s
( ) ( )ch n h nT
Impulse Invariance method contd…
• The impulse invariance transformation does map the -axis and the left-half s plane into the unit circle and its interior, respectively
j
Re(Z)
Im(Z)
1
S domain Z domain
sTe
j
• is expanded a partial fraction expansion to produce
• We have assumed that there are no multiple poles
• And thus
( )cH s
1
( )N
kc
k k
AH s
s s
1
( ) ( )k
Ns t
c kk
h t A e u t
1
( ) ( )k
Ns nT
kk
h n A e u n
11
( )1 k
Nks T
k
AH z
e z
Impulse invariance method contd…
11)( Zeas aT
• Hence it is sufficient if we substitute
Impulse invariance method contd…
• Example:
Expanding in a partial fractionexpansion, it produce
The impulse invariant transformation yields a discrete time design with thesystem function
2 2( )
( )c
s aH s
s a b
1/ 2 1/ 2( )cH s
s a jb s a jb
( ) 1 ( ) 1
1/ 2 1/ 2( )
1 1a jb T a jb TH z
e z e z
Impulse invariance method contd…
Bilinear transformation method
• The most generally useful is the bilinear transformation.• To avoid aliasing of the frequency response as
encountered with the impulse invariance transformation.
• We need a one-to-one mapping from the s plane to the z plane.
• The problem with the transformation is many-to-one. sTz e
Bilinear transformation method Contd…
• We could first use a one-to-one transformation from to , which compresses the entire s plane into the strip
• Then could be transformed to z by with no effect from aliasing.
s 's
Im( ')sT T
's's Tz e
j
'
j
/T
/T
s domain s’ domain
TZ
Zs
2
1
11
1
• Hence by using this equation a digital transfer function may be obtained
Bilinear transformation method Contd…
Feb.2008 DISP Lab 65
• The discrete-time filter design is obtained from the continuous-time design by means of the bilinear transformation
• Unlike the impulse invariant transformation, the bilinear transformation is one-to-one, and invertible.
1 1(2/ )(1 )/(1 )( ) ( ) |c s T z z
H z H s
Bilinear transformation method Contd…
Filter Realization
Filter Structures
•Direct form I
•Direct form II
•Cascaded form
•Parallel form
Copyright © 2005. Shi Ping CUC
Basic elements of digital filter structures
Adder has two inputs and one output. Multiplier (gain) has single-input, single-output. Delay element delays the signal passing through it by
one sample. It is implemented by using a shift register.
z-1
a z-1
a
Copyright © 2005. Shi Ping CUC
a1
z-1
z-1a2
b0)(nx )(ny
)(nx )(ny
a1
a2
b0
z-1
z-1
1 2
3
4
5
)2()1()()()()()2()1()()()(
)2()1()()1()1()(
)()(
210501
2142315
34
23
2
nyanyanxbnwnxbnwnyanyanwanwanw
nynwnwnynwnw
nynw)2()1()()( 210 nyanyanxbny
Introduction
Computational complexity refers to the number of arithmetic operations (multiplications,
divisions, and additions) required to compute an output value y(n) for the system.
Memory requirements refers to the number of memory locations required to store the
system parameters, past inputs, past outputs, and any intermediate computed values.
Finite-word-length effects in the computations refers to the quantization effects that are inherent in any digital
implementation of the system, either in hardware or in software.
The major factors that influence the choice of a specific structure
IIR Filter Structures
The characteristics of the IIR filter IIR filters have Infinite-duration Impulse Responses The system function H(z) has poles in ||0 z
)(11)()(
)(1
1
110
1
0N
N
MM
N
k
kk
M
k
kk
zaza
zbzbb
za
zb
zXzY
zH
The order of such an IIR filter is called N if aN≠0
M
kk
N
kk knxbknyany
01
)()()(
Direct form
In this form the difference equation is implemented directly as given. There are two parts to this filter, namely the moving average part and the recursive part (or the numerator and denominator parts). Therefore this implementation leads to two versions: direct form I and direct form II structures
N
kk
M
kk knyaknxbny
10
)()()(
N
k
kk
M
k
kk
za
zbzH
1
0
1)(
Copyright © 2005. Shi Ping CUC
Direct form I
)(2 ny)(nx
b1
b2
b0
z-1
z-1
bM-1
z-1bM
)1( nx
)2( nx
)1( Mnx
)( Mnx
)(ny
a1
a2
z-1
z-1
aN-1
z-1aN
)1( ny
)2( ny
)1( Nny
)( Nny
)(1 ny
N
kk
M
kk knyaknxbny
10
)()()(
Copyright © 2005. Shi Ping CUC
Direct form II
b1
b2
b0
z-1
z-1
bM-1
z-1bM
)(nx )(nyz-1
z-1
z-1
a1
a2
aN-1
aN
For an LTI cascade system, we can change the order of the systems without changing the overall system response
Copyright © 2005. Shi Ping CUC
Cascade form
In this form the system function H(z) is written as a product of second-order sections with real coefficients
21
21
1
11
1
1
1
11
1
1
1
0
)1)(1()1(
)1)(1()1(
1)(
N
kkk
N
kk
M
kkk
M
kk
N
k
kk
M
k
kk
zdzdzc
zqzqzpA
za
zbzH
21
21
1
22
11
1
1
1
22
11
1
1
)1()1(
)1()1()(
N
kkk
N
kk
M
kkk
M
kk
zazazc
zbzbzpAzH 21
21
22NNNMMM
Copyright © 2005. Shi Ping CUC
Parallel form Structures
Parallel form
In this form the system function H(z) is written as a sum of sections using partial fraction expansion. Each section is implemented in a direct form. The entire system function is implemented as a parallel of every section.
21
12
21
1
110
110
1
0
111
)(N
k kk
kkN
k k
kN
k
kk
M
k
kk
zaza
zbb
zc
AG
za
zbzH
21 2NNN
Suppose M=N
Copyright © 2005. Shi Ping CUC
Example
1111
111
)21
21(1)
21
21(1)
811)(
431(
)21)(321)(
211(10
)(zjzjzz
zzzzH
1
4
1
3
1
2
1
1
)21
21(1)
21
21(1)
811()
431(
)(
zj
A
zj
A
z
A
z
AzH
57.1425.12 ,57.1425.12 ,68.17 ,93.2 4321 jAjAAA
21
1
21
1
211
82.2650.24
323
871
90.1275.14)(
zz
z
zz
zzH
Copyright © 2005. Shi Ping CUC
21
1
21
1
211
82.2650.24
323
871
90.1275.14)(
zz
z
zz
zzH
-12.9z-17/8
-3/32 z-1
-14.75
26.82z-11
-1/2 z-1
24. 5
)(nx )(ny
Conclusions
• Discussed about the FIR filter design• IIR Filter design• Realization of structures
Feb.2008 DISP Lab 80
References
• [1]B. Jackson, Digital Filters and Signal Processing, Kluwer
Academic Publishers 1986 • [2]Dr. DePiero, Filter Design by Frequency Sampling,
CalPoly State University • [3]W.James MacLean, FIR Filter Design Using Frequency
Sampling • [5]Maurice G.Bellanger, Adaptive Digital Filters second
edition, Marcel dekker 2001
Feb.2008 DISP Lab 81
References
• [6] Lawrence R. Rabiner, Linear Program Design of Finite Impulse Response Digital Filters, IEEE 1972
• [7] Terrence J mc Creary, On Frequency Sampling Digital Filters, IEEE 1972
• WWW.GOOGLE.COM
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