digital signal processing 2014 pptx.pptx
TRANSCRIPT
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Outcome Based Education
1Dr.P.Meena,Assoc.Prof., EEE
FocusLearning, not teachingStudents, not facultyOutcomes, not inputs or
capacity
INDIA HAS BECOME APERMANENT MEMBER OF THE
WASHINGTON ACCORD
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2
Academicabilities
CourseLectures/Demonstrations/
Videos/Animations /powerpoint presentations/hand outs
Problem solving
Teacher led Students in pairs/share
Industry Visits
pen ended e!periments
Components that contriute toAca!emic Ai"ities
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Components that contriute toTrans#era"e S$i""s
3
Trans"erable s#ills
StudentActivities
Pro$ectwor#/
pen endede!periments
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The components o# course !e"i%er& that contriute
to the !e'ne! attriutes o# the course (
Attributes
TechnicalSymposia
4Dr.P.Meena,Assoc.Prof.,EEE
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5 5
Di)ita" Si)na" Processin)
Intro!uction
Inception*+,-./ith the !e%e"opmento# Di)ita" Har!/are such as !i)ita"har!/are(
Persona" computer re%o"ution in+,01s an! +,,1s cause! DSP e2p"osion/ith ne/ app"ications.
Dr. P.Meena, Assoc.Prof(EE) BMSCE
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6 6
Advantages of DSP TechnologyHigh reliabilityReproducibility
Flexibility & PrograabilityAbsence of !oponent Drift proble
!opressed storage facility "especially inthe case of speech signals #hich has a lot
of redundancy$.DSP hard#are allo#s for prograable
operations.Signal Processing functions to beperfored by hard#are can be easilyodified through soft #are"efficient
algoriths$
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7 7
Advantages of DSP TechnologyHigh reliabilityReproducibility
Flexibility & PrograabilityAbsence of !oponent Drift proble
!opressed storage facility "especially inthe case of speech signals #hich has a lot
of redundancy$.DSP hard#are allo#s for prograable
operations.Signal Processing functions to beperfored by hard#are can be easilyodified through soft #are"efficient
algoriths$
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 8
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 9
Digital Signal Processing#ith overlapping borders
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1010
A T%P!A' D(TA' S()A'PR*!+SS)( S%ST+,
%&tx%&txa %&nx %&ns %&ts D3A
CON4ERTER
Ana"o)pre'"ter orAntia"iasin)'"ter
A3DCON4ERTER
Di)(Si)na"Processor
Reconstruction'"ter sameas the pre'"ter
5o/ pass'"tere!si)na"
Discretetimesi)na"
Discretetimesi)na"Samp"i
n)#re6uenc&
!B
!B
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 11
CO+* Ai"it& to app"& the $no/"e!)e o#
mathematics7 science an! #un!amenta"s o#si)na"s an! s&stems to ascertain the eha%ioro# comp"e2 en)ineerin) s&stems(
CO8*Ai"it& to I!enti#& techni6ues7 #ormu"ate
representations an! ana"&9e responses o#!i)ita" s&stems(
CO:*Ai"it& to Desi)n !i)ita" s&stemcomponents an! test their app"ication
usin) mo!ern en)ineerin) too"s7 asso"utions to en)ineerin) pro"ems.
Course Outcomes
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 12
Course Contents
Dierent operations on a signal in thedigital domain
Dierent forms of reali!ations of aDigital System.
Design Procedures for Digital "ilters
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Outcomes of this Course:By The 'nd " The Course (
Distinguish The Digital and Analog Domains)
Analyse Signals( and reconstruct)
Develop *loc# Diagrams +or Di""erent System,epresentations()
Design Analog And Digital +ilters)
,eady to Ta#e up Speciali-ed Courses in Audio( speech( image and ,eal.time Signal Processing( +urther n
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 14
Course Out"ine
Course Delivery:
Lecture(hand outs(videos(animations(discussions(activities
Course Assessment:
ar#s0
Tests0 12 &T3 4 T1%
5ui- 6 27
Tutorials0 32Lab0 37
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Revie# ofSignals
&Systes
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Signals
Audio Video (Represented as a function of 3 variables.) Speech-
Continuous-represented as a function of a single (time) variable).Discrete-as a one dimensional seuence !hich is a function of a
discrete variable. "mage#Represented as a function of t!o spatialvariables
$lectrical -Dr. P.Meena, Assoc.Prof(EE) BMSCE
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 17
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 19
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2020
Relation bet#een analog
fre-uency and digitalfre-uency
( )
s
1 unit is radians per second)
y8a sin ( a signal in the continuous time domain)
t8n9
-8a sin& %
sin& %
8 is the digital "re:uency in radians/sample
There"ore( given a 1 ( get
Ts
s
f
t is
n
z a n
where
f to
T
T
=
=
=
s
1 or &1 9 %T
s
ff
f
=
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 21
f !n "er#$(ana%o&)
' !n ra!anssa*+%eD!&!#a%
0
s2-s4-s2-s ss4 32s-32s
f0,'0
'/2,fs4
'/,fs2
' -/2,f-s4
'2/'/'0'/2'-/2
'-2/ '-/ '3/'-3/
!s# !n#era%
'-/,f-s2
Sl.)o.
Fre-uencyin Hert*f thesignal
SaplingFre-uencyFs in Hert
/ inradians0cycle
%. f&' s '
. f&s*+ s ,*
3 f&-s*+ s -,*
+. f&s* s ,
. f& -s* s -,
Diagrammatic ,epresentation o" relation between analog
"re:uency and digital "re:uency0;ve angle counter cloc# wise
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 22
amp"in) o# continuous time si)na"s
#he "ourier transform pair for continuous$timesignals is de%ned &y
( ) ( )
( ) ( )
d$1
3t
dtt$
e
e! > 8 = < $
Assignment
Dr. P.Meena, Assoc.Prof(EE) BMSCE
R l ti f th F t f
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 68
Resolution of the Fre-uency spectru forlonger DFT
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 69
Properties of the DFT
9inearit% : The D!T is a linear transform
D!T 0ax,0n1-x&0n112 a D!T0x,0n11 - D!T 0x&0n11
If x,0n1 and x&0n1 ha)e different durations that is3the% are #,point and #& se$uences3respecti)el%3 then choose #2max*#,3#&+ andproceed -% ta;ing # point D!Ts.
H*,*(+)T% *F TH+ DFT
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 70
H*,*(+)T% *F TH+ DFT
F f th i l
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 71
Fre-uency response of the ipulse response#ith padded eros.
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 72
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 73
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74P.Meena,Ass#.Prof(EE)BMSCE 74
> point DFT plots of a se-uence x9n:;9< < <
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7575
The +ight point DFT of 9< < <
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7676
+ffect of ero padding
1he 2ero padding gives a 5igh densit0Spectrum and a better displa0ed versionfor plotting but not a high resolutionspectrum.
6ore data points needs to be obtained inorder to get a high resolution spectrum
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7777
DFT plotindicating the syetry
about ? ;pi
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7878
,AT'A@ PR*(RA,n&input(7input the values of nin the form
8'#delta9#9-%:&7);
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7979
Ge#
con$ugate
comple!areG%.
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8080
123
4
5
6
78 9
10
1112
1314
Dr. P.Meena, Assoc.Prof(EE) BMSCE
D"# )(66$n779*F G66$>77
9F
8n:&8% 3 + B E %' %% % %3 %+:
8-n:mod9&8% %+ %3 % %% %' E B + 3 :
Circular Shift
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 81
Circular Shift
1
;13
2
4
5
6 78
9
10
1211
1314
;0
;2
;3
;4
;5 ;6;7
;9
;8
;10;11;12
;13
;1
;0
;2
;3
;4
;5 ;6
;7
;9
;8
;10;11;12
;13
141
2
3
45
6
7
8
9
1011
1213x0n1
x0n6,1 mod ,/
;1;0
;2;3
;4;5
;6;7
;9
;8
;10
;11;12
;13
23
4
6
7 8
9
10
11
12
13x0n,1 mod ,/
5
114
0right shift the se$uence1,/
0left shift the se$uence1,/
=3@(3B(31(33(32(F(D(E(A(7(@(B(1(3>=> =nx
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 82
In general3
mod
mod
> &2%( &3%()))))))) & B%( & 1%( & 3%=
> & 3%( &2%( &3%()))))))) & B%( & 1%=
> & 1%( & 3%( &2%( &3%()))))))) & B%=
> => 3=
> 1=N
N
x x x N x N x N
x N x x x N x N
x N x N x x x N
x nx n
x n
=
=
=
[ ]
> = >3 1 B @ 7 A E =
! >E 3 1 B @ 7 A=&n.1%x n =
=
[ ]
[ ]
[ ]
[ ]1(3(3@(3B(31(33(32(F(D(E(A(7(@(B=1>
3(3@(3B(31(33(32(F(D(E(A(7(@(B(1=3>
31(33(32(F(D(E(A(7(@(B(1(3(3@(3B=1>
3B(31(33(32(F(D(E(A(7(@(B(1(3(3@=3>
@mod
@mod
3@mod
3@mod
=+
=+
=
=
nx
nx
nx
nx
P i f Di F i
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8383
Properties of Discrete FourierTransfor
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8484
&.Circular !olding => modmod KXnx NNDFT =
mod@
mod@
mod@
>3 @ B 1=
32 32
.1 $1 .1 $1G= 6 > =
.1 .1
.1.$1 .1 $1
32
.1 $1
.1
.1 $1
> =
> =
> =
DFT
x n
x n
X K
= + = = +
= +
@=B1>3!>n= =let
@=B1>3!>n= =let10
-2H2
-2
-2-2
-2H2
10
-2-2
-2
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85
a.D!T of circular shifted se$uence
mod
D+T>!>n==8G=
then D+T G=!>n.m= K
N
!f
NW =
mod@!>n=8>3 1 1 2=) +ind D+T o" (mod@!>n=(!>n.3= !>n.1=!f
mod@>2 3 1 1=
3 3 3 3 2 7
3 .$ .3 ;$ 3 1
3 .3 3 .3 1 3
3 ;$ .3 .$ 1 1
!>n.3=
j
j
=
+ =
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86
a.D!T of circular shifted se$uence
mod
3
mod@
D+T G=
D+T G=@
!>n.m=!>n.3=
K
N
K
NWW
= =
@
&2% 3 3 3 3 3 7
&3% 3 .$ .3 ;$ 1 .3.$1&1% 3 .3 3 .3 1 3
&B% 3 ;$ .3 .$ 2 .3;$1
2 79@
3 .3.$19 @
H
H> 3=
X
XX
X
DFT x n
= =
=
793 7
&.3.$1%9&.$% .1;$
1 39&.3% .339@
&.3;$1%9&$% .1.$B .3;$19@
H
H
= =
1
67
-1
Dr. P.Meena, Assoc.Prof(EE) BMSCE
!n #e D: of
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 87
!n #e D: of
Circular Con)olution
!ind the circular con)olution of 30, 3& 3&3 41 and 0,3&33/1
/.If x0n1 is a )alued real se$uence,
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 88
;
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8989
-. ultiplication -% Exponentials or Cicular !re$uenc% Shift
> = & %
n
N NDFT x n X K W
=
If =*>+ is circularl% shifted3 the resulting in)erse transform ?ill -e themultiplication of the in)erse of =*>+ -% a complex exponential.
Dr. P.Meena, Assoc.Prof(EE) BMSCE
f &
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90
!onsider x9n:;9 < = 8 > 4 B C :>hat is
mod@!>n.@= =
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91
!ircular !onvolution
1
3 1
3 1 3 1
>3 1 1 2=6 > = >3 3 3 3=3
7 @
3 1 2> = > =
2 2
3 1 2
12
2> = > = 6 > = > =
2
2
122
2
2
> = n
jK K
j
K K !DFT K K
!DFT
n xx
X X
X X X X
= = = = +
=
3 3 3 3 12 73 ;$ .3 .$ 2 73
3 .3 3 .3 2 7@
3 .$ .3 ;$ 2 7
= =
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92
!ircular !onvolution
3 1
3 1 1 2 3 1 1 2 3 1 1 2 3 1 1 23
1 mod@
> = > = 3 1 1 2 3 3 3 3
> =
3333 3333 3333 3333> =
....... ....... ...... ......
n nx x
nx
n kx
= ==
7 7 7 7
3 1> = > =3 1> = > = !DFT K K n n X Xx x =
Dr. P.Meena, Assoc.Prof(EE) BMSCE
+valuating the DFT
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93
+valuating the DFT
Find the DFT of 94 "3
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 94
1
2
4
56
n0
n1n2
n3
n4 n5
12
45
6n0
n1n2
n3
n4 n5
x0n1x0n6,1mod@
3
x0 n1 x0# n1mod#
n0
n1n2
n3
n4 n5
2
3
34
5
6 1
n0
n1n2
n3
n4 n5
3
45
6
21
n0
n1n2
n3
n4 n5
4
56
1
2 3x0n,1 x0n&1 x0n1
n0
n1n2
n3
n4 n5
5
61
2
34
x0n/1
x0n6,12x0n51mod#
S**e#r +ro+er# for rea% a%e ;
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S**e#r +ro+er# for rea% a%e ;
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 96
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 97
S%mmetr% properties for real se$uences
If x0n1 is real and a # point se$uence. then x0n12x0n1. "sing the
a-o)e propert%3 =*>+2 =*6>+ #
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 98
;(e)
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 99
If x0n1 is imaginar% and e)en3 then its =0>1 is purel% imaginar%.
;(e)
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;
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 101
+
[ ] [ ] [ ][ ][ ] [ ] [ ][ ]n!n!
1
3n
n!n!13n
mododd
modev
!
!=
+=
!n #e een an o +ar#s of #e seence ;
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 102
Real and aginary Parts of G9:
[ ] [ ]
[ ] [ ][ ] [ ]
[ ] [ ]=>
=>
9
9
9
Im
9
=>13
=>1
3
=>=>1
3
=>=>1
3
KXX
KXX
xx
xx
Ni
Nre
re
KXK
KXK
nnxjn
nnxn
=
+=
=
+=
Co*+#a#!on of +o!n# D: of a rea% seence s!n& +o!n# D:
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 103
Co*+#a#!on of +o!n# D: of a rea% seence s!n& +o!n# D:.
&
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 104
,ultiplication2
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105
,ultiplication2
It is the dual of the circular con)olutionpropert%.
[ ] =>=>3=>=>1333
KKN
nnDFT XXxx =
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 106
ParsevalIs Relation
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107
Parseval s Relation1his Computes the energ0 in the freuenc0
domain3 31 1
2 2
1
1
3
is called the energy spectrum o" "inite
duration se:uences)Similarly ( "or periodic se:uences( the :uantity
called the power spectrum
> = > =
> =
& %
N N
xn kN
The "#antit$N
is
x n X KE
X K
X KN
= =
= =
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108
Soe -uic6 relations2
C.3.#n
C#82
C.3 C.32
C#82 #82
C.3
#82
3!>n=8 G=
C
3 3>2= G= G=
C C
G=8C >2=6
H
H
f
x
x
= =
3 31 1
2 2
3 31 1
2 2
3> = > =
> = > =
N N
xn k
N N
k n
N
%&
N
x n X KE
X K x n
= =
= =
= =
=
1.
2.
1C.3n
n82
!>n= be a 1C valued real se:uence(
C=8 !>n=&.3%
!f
3.
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109
Probles on Juic6 Relations
E E 1
#82 #82
> = >3 1 2 B .1 @ E 7=
with a point D+T
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110
0 1 2 3 4 5 6 70
50
100
w in radians
Magnitude
0 20 40 60 80 100 120 140 1600
50
100
k
magnitude
0 20 40 60 80 100 120 140 160-100
0
100
kangleindegrees
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
50
100
w/pi
magnitude
D!T O! A 54 H SI#E AE SAF9ED AT G>H
Dr. P.Meena, Assoc.Prof(EE) BMSCE
D!T Energ% Spectrum O! A 54 H SI#E AE ?ith SAJ
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111
0 500 1000 1500 2000 25000
5
10x 10
5
EnergySpectrum
DFT and energy spectrum of a wave form with sag
0 1 2 3 4 5 6 70
500
1000
mag
nitudeofDFT
0 500 1000 1500 2000 25000
500
1000
magofDFT
0 500 1000 1500 2000 2500-100
0
100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
500
1000
0 500 1000 1500 2000 2500-1
0
1
Dr. P.Meena, Assoc.Prof(EE) BMSCE
D!T Energ% Spectrum O! A 54 H #ormal SI#E AE and a Sine ?a)e ?iths
ag
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112Dr. P.Meena, Assoc.Prof(EE) BMSCE
DT,F T*)+ Allocation
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113
DT,F T*)+ Allocation
P.Meena, Ass#.Prof(EE) BMSCE 113Dr. P.Meena, Assoc.Prof(EE) BMSCE
D!T of DT!
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114Dr. P.Meena, Assoc.Prof(EE) BMSCE
"se of D!T in 9inear !ilterin
g
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 115
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 116
:erefore $ero +a!n& !s se #o *aFe#e s!&na%s of ea% %en.
'inear !onvolution & !ircular
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117
'inear !onvolution & !ircular!onvolution.
(!n!#e *+%se es+onse) f!%#ers are !*+%e*en#e s!n& %!nearcono%#!on.
I!en #o seences
[ ] [ ]
3Cn)convolutiolineartoidenticalisnconvolutiocircular
then the-eros(o"numbereappropriatanpadding
byClengthsametheo"madearese:uences
(Clengtho"and
13
1313
+= NN
'oth
Nandnn
xx
Dr. P.Meena, Assoc.Prof(EE) BMSCE
Ge#
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118
Ge#
[ ] [ ]
3=)33.1.3.33>0Ans
e:ual)arethat they
showandnconvolutiocirculartheCompute
n)convolutiolineartheirDetermine
3=6333>3=6113> 13 == nxnx
Dr. P.Meena, Assoc.Prof(EE) BMSCE
Error -et?een Circular con)olution 9inear
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 119
Error -et?een Circular con)olution 9inearcon)olution due to choice in #
hen #2max*#,3#&+ is chosen for circularcon)olution then the first 6, samples arein error ?here 2min*#,3#&+.
Hence this leads to different methods ofcon)olution in -loc; processing.
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 120
@'*! !*)K*'1T*)S
#ecessit% of Kloc; Con)olutions: To filter an input se$uence recei)ed continuousl% such as a speech
signal from a microphone and if this filtering operation is doneusing a !IR filter3 in ?hich the linear con)olution is computed usingthe D!T then there are some practical pro-lems .
A large D!T is to -e computed. Output samples are not a)aila-le until all input samples are
processed resulting in a large amount of dela%.
In Kloc; Con)olution: The speech signal is segmented into smaller sections CO#O9"TIO#S
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P.Meena, Ass#.Prof(EE) BMSCE 121
Errors in K9OC> CO#O9"TIO#S
f ;3 1= >3 1 B @ 7 =6
length o" h>n=86
h n =
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1
length o" h>n=86
length o" !>n=8L6
To evaluate the length o" the se:uences "or convolution(3(
1 3( @ 3 B) @
verlap and save method(Input se:uence overl
1
1
M N
N N N
!n
+
+ = =
3 1
aps by &.3% samples)
> = >3 1 2 2=6 ! >n=8>2 3 1 B=(! >n=8>B @ 7 =
3 1 2 2 3 1 2 2 3 1 2 2 3 1 2 2 J3 1 2 2 3 1 2 2 3 1 2 2 3 1 2 2
2 B 1 3 3 2 B 1
h n
=
1 3 2 B B 1 3 2 J B 7 @ @ B 7 7 @ B 7 @ B
.......... ........... ........... ........... J.......... ........... ........... ...........
3 @ E J 32 3B 3
......... ........... ........... ........... J.......... ..
......... ........... ...........
The linear convolution result y>n=8>3 @ E 32 3B 3=
122Dr. P.Meena, Assoc.Prof(EE) BMSCE
! % # # < = !f < =
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 123
s!n& oer%a+ an sae *e#o co*+#e =>7=B.1.3.3122>=>
1
3
=
==
nxnx
1=(3>=> =nhO)erlap and Add method of Sectional Con)olutionN
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 124
................................................
3133B32E@3
.................................................
3133B@
E@3
..................................................
@722@72@772@31B2231BB2311B23
221322132213221322132213221322132=7>@2=6B13>=>
=2213>=>
@array)in theconsideredsi-ebloc#theis(B
631@
631
nconvolutiocircularo"si-etheis
=(7@B13>!>n=
=(>=>
=
===
+=+=
=
nx
nh
NL
L
LM
311 +=== LMNM M
Ter!f oer%a+ an Sae Me#o
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 125
31=33B32E@3>0Ans
223133B3237
.........................................................
222222222222B@7B@77B@@7B
J22132213221322132213221322132213
........................................................
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231BB2311B2331B2
2213221322132213
=222>=>!
=7@B>=>
=B132>=>!
2=213>=>
136B6@
31(1
B
1
3
===
====
+===
n
nx
n
nh
MLN
LMNM
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 126
content)energy
the"indhence3)and.Cn2where
1
cos=>
se:uencetheo"D+Tbtain the
2
= NnK
nx
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 127
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 128
D!T of a S$uare a)e
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 129
Relationship of the DFT to *therT f
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Transfors
Relationship to F- transform#
kN
N
kj
Wez
j
N
n
n
kn
N
N
n
n
n
zXKXez
znx
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znxzX
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=
=
=
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=
=
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1%&%&(
=>n=sin
=>%&
=>%&
3
2
3
2
130Dr. P.Meena, Assoc.Prof(EE) BMSCE
2=)sing2)72>2)7o"trans"orm)theFind
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 131
2)27)27)27)27)2%B&2)37)27)27)27)2%1&
27)27)27)27)2%3&
2)37)27)27)27)2%2&
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@
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=
+=+++=
=
WXWX
WX
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zXkX
zzz
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kNWz
The 'iitations of the direct
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calculation of the DFT
"t reuires
paireach"or"ourtions(multiplicareal
@Cre:uires#each"or
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 133
TH+ FAST F*1R+R TRA)SF*R,
This is proposed by !ooley and Tu6ey @ased on decoposing 0brea6ing thetransfor into saller transfors andcobining the to give the totaltransfor. This can be done in both Tie and
Fre-uency doains.
D+!,AT*) ) T,+ FFT
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 134
:e n*er of +o!n#s !s ass*e as a +oer of 2,#a#
!s
obtained)ares"ormspoint tran.two
int@
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#ntil
transfors*o
transfors*o
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N +=
relation(theusing
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=
=
for
WWxWxKX kN
N
k
N
N
k
N
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 135
}{
}{
3).1
2(3()))))(#(=(>=>%
1=>=>
bins("re:uencyhal""irstthe
yieldsG=("ore!pressiontheintothisngsubstituti
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3).1
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sint1/%31&(=31>=>
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+
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=
=
forKHWK,N
K
WW
forKHWK,KX
forN
KHKH
forN
K,K,
*oNwithxDFTWxKH
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for
WxWWxKX
W
k
N
k
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KN
N
k
N
N
k
N
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k
N
N
n
k
N
k
N
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k
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 136
##+sNen&!neer!n&.+re.eTSEee438e*osf%asec!*a#!on.sf
&6FOI#T D!T
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 137
2x
3x
3
1W
2
1
W%2&X
%3&X
2x
1x 31W
2
1W
3x
Bx
2
1W
31W
2
@W3
@W
1
@W
B@W
(0)
(1)
(2)
(3)
/6FOI#T D!T
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 138
!n #e D: of ;
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 139
@T R+K+RS+D S+J1+)!+ +(HT P*)T DFT
;
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 140
2x
@x 31W
2
1W
1x
Ax
2
1W
3
1W
2
@W
3
@W
1
@W
B
@W
(1)
(2)
(3)
3
1W
2
1W
2
1W
3
1W
2
@W
3
@
W
1
@W
B
@W
(4)
(5)
(6)
(7)
2
W
3
W
1
W
B
W
@
W
7
W
A
W
E
W
3x
7x
Bx
Ex
< = < , , , , , , , =
1
-1
1
-1
1
-1
-1
2
0
0
-2
0
-2
-2
0
2
2
2
-2
-2
-2
2
-2
( )
3.69H1.96
2.82-0.78
1.53-0.39
4-0.0
1.53H0.392.82H0.78
3.69-1.96
0.0-0.0
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 141
363 @2 WW ==
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 142
1
1
1
1
1
1
1
1$
1
1
1
1
1
1
1
1
.363
E
B
1
7
3
jWjW
WjW
jWjW
WW
+====
+==
==
=!!!>!Let !>n= B132=
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 143
=!(>!=(!(>!
intodecimateor
=!!!>!Let !>n=
B312
B132
Di+ide ==!(>!=(!(>!
intodecimateor
B312
Di+ide
3!2 =
17)2!1=2
1W
2
1W7)2!3 =
317)2!B=
3
1W
3
1W
2
@W
B
@W
(0)1.875
(1)0.75-0.375
(2)0.625
(3)0.75H0.375
3
@W1
@W
2)317=2)172)73>=> =nx
17)3
E7)2
A17)2
BE7)2
!oputation of nverse DFT
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 144
( ) ( )=$13.3$1.3.>7
o"ID+Tthe
)=>
3!>n=
bygivenisD+Tinverse
3
2
+
=
=
Find
WKX
The
N
k
kn
N
T?iddle !actors are negati)e po?ers ofN
W
The output is scaled -% ,
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 145
2
1W
2
1W
3
1
W
3
1
W
2
@W
B
@
W
3@W
1
@
W1
@j
2
31 =x
133
jx =
13B jx +=
@
D
D
2
1
2
1
@
3
@
3
@
3
H
-
-1 1
32 =x
2W 2W3
3
2
1
0
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 146
12 =x
33 =x
3
1W
2
1W
31 =x
1B =x
21W
3
1W
2
@W
3
@W
1
@W
B
@W
6)B
3
B
3
1H
0
1-
33 =x
3
1
W
1W
3B =x
2
1W
3
1W
@W
3
@W1
@W
B
@W
2
1
31 =x
312H2
0
2H2
2-2
2W2
W2
2=x
2
2
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 147
2
1W
2
1W
3
1
W
3
1
W
2
@W
B
@
W
3@W
1
@
W1
@j
2
21
=x
@3
jx =
@B
jx =
@
D
D
@
3
@
3
@
3
2
1
2
1
D+!,AT*) ) FR+J1+)!% FFT "DF$ FFTor F0,1UU..-1,@ere !s #e #!%e fac#or an 2,4,8,16U can e e;+ane as,
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 148
< = ( ) ( ) ( )
A&a!n !f e s+%!# #e aoe ea#!on !n#o,
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 149
f ;
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Dr. P.Meena, Assoc.Prof(EE) BMSCE 150
2
1W
2@W
32 =x
11 =x
BB =x
@@=xD
4
3
3
6
-2
-2 M3@W
-2
-2
10
3
3
1
-2-2
2
1W
-2H2
0
2
1
3
=0>12 0,43 6&67&3 6&3 6&7&1
2o.of s#a&es N
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o.of s#a&es No.Qfco*+%e; *%#!+%!ca#!ons !n eac ##erf%2o.of B##erf%!es !n eac s#a&e 2n*er of co*+%e; *%#!+%!ca#!ons !n eac s#a&e.
no.of co*+%e; *%#!+%!ca#!ons N