applications of dsp 1.imaging 2.medical imaging 3.bandwidth compression 4.sgraphic 5.spectrum...

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.


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


• The magnitude response specifications may be given in a normalized form.


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


15

• Example - Let kHz, kHz, and kHz

• Then

7pF 3sF25TF

56.01025

)107(23

3

p

24.01025

)103(23

3

s


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


• 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


• 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


• 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


• 3.HANN window• 4.Hamming window


• 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


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


• Side lobs– -44 dB

• Hamming window performs better

• Same complexity as Hamming

Hamming Window

else0

Mn0M

n2cos46.054.0nw


• 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


• 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


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


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


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


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)(


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

)()()(


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


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


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


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


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


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


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

THANK YOU

applications of dsp 1.imaging 2.medical imaging 3.bandwidth compression 4.sgraphic 5.spectrum...

Documents

hamming window slide

blackman window slide

fir digital filter

hamming slide

stopband slide

blackman slide

radar slide

hann window