Introduction to Discrete Time Signals & System
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apter 1
roduction to Discrete Time Signals & System:
Discrete–Time Signals representation and Manipulation,
Discrete–Time IIR and FIR Systems, Impulse Response,
(Infinite Impulse Response and Finite Impulse Response)
Transer Function,
Dierence !"uation,
Fre"uency Domain and Time Domain #nalysis o IIR ilter and FIR ilter,
Correlation,
$inear and Circular and Con%olution #lgorithm,
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pter 1 'uestions
ec( )1
i%e any i%e classiications o Discrete time systems +ith eamples
.t/ 0 sin.2π
t/ 3 4 sin .5)π
t/ is sampled +ith Fs 0 6 times per sec.
1/ 7hat are the re"uencies in radians in the resulting DT signal .n/8
)/ I .n/ is passed through an ideal interpolator,
+hat is the reconstructed signal8
une )11
.n/ 0 ), ;1, 4, , < o=tain ollo+ing:
.i/ .;n/ .ii/ .n;1/ .iii/ .n31/ .i%/ .;n3)/.%/ .)n/
or a discrete time system +hose impulse response h .n/ 0 1, ;), 1<
↑
ind the output or input .n/ 0 1, ), 4, <
↑
lassiy ollo+ing DT System on linearity> causality and time %ariance:;
i/ y .n/ 0 ).n/ 3 .n;1/ .ii/ y .n/ 0 .)n/ 3)
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1( .n/0 ?1, ), 1, ), 4, @ ind y .n/ 0 .1;n/ 3 .4;n/ 3 . –n/
)( !plain +hether the ollo+ing signals are po+er signal or energy signal(
a( (A n u .n/ =( # cos .ω
n/
4( Determine +hether the ollo+ing signals are periodic or non;periodic(
B.π
>/ n
a( .n/ 0 e =( .n/ 0 cos.A πt/ 3 cos.1 πt/
( Decompose the ollo+ing signals into e%en and odd parts
a( ?n@ 0 ? 1 ) 4 A 6 5 2 @
=( y ?n@ 0 ?2 2 2 2 @
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Signal
# signal is deined as any physical "uantity that %aries +ith:
1( Time,
)( Space or
4( #ny other independent %aria=le or %aria=les
; !amples o Signals
; Speech
; Music
; ictures; ideo
; !C*
; Mathematically, +e descri=e a signal as a
unction o one or more independent %aria=les
s1.t / 0 A t
s). t/ 0 ) t) Ene independent %aria=le t .time/
., y/ 0 4 3 ) y 3 1 y)
t+o independent %aria=les , and y
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!ample o speech signal
Signals can =e generated =y a system:
Speech ; ocal Cords
!C* ; !lectrocardiogram: olariation> depolariation o %entricles
!!* ; !lectroencephalogram
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; Speech, !C* and !!* signals are unction o a single %aria=le:
t, Time
; #n image signal is unction o t+o %aria=les:
, y .Coordinates o an image/
; These signals are generated =y some means:
; Speech signal =y %ocal cord and air lo+(
; Image signals =y eposing light; sensiti%e sensors to light
; The signals are generated =y a system
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Signal processing system
# signal processing system is deined as a de%ice
that perorm an operation on the signal
; #mpliier is a system that ampliies a signal
; Filter is a system that suppresses or allo+s
some re"uencies rom the signal
; Thus a system is an interconnection o components thatperorms an operation on
input signal and produces an output signal(
; The deinition o a system can =e =roadened to include not only
physical de%ices =ut also,
sot+are realiation o operations
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Digital signal processing system
; There are t+o types o signals:
; #nalog signals and
; Digital signals
#nalog signals:
; Most o the signals that occur in nature are analog signals(
; The analog signals are a unction o a continuous %aria=le
such as time or space,
taGes on %alues in continuous range
; #nalog signals can =e processed directly, in continuous orm =y analog systems .circuits/ such as:
#mpliiers, Filters, Fre"uency multipliers,
; #nalog processing:
Direct processing o signals in continuous orm .#nalog orm/
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!ample o analog processing o signals
oice ampliier:
; #mpliier is made up o resistors, capacitors, transistors etc(
; #mpliier taGes the analog signal .continuous/ rom microphone,
ampliies it , and produces analog output signal or speaGer
#nalog processing system:
; TaGes analog input, process it in analog orm and
produces analog output
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Digital Signal rocessing:
- Digital signal processing pro%ides an alternati%e method or
processing the analog signals
; First analog signals are con%ert to digital signals :
– #DC .#nalog to Digital con%erter/
#DC pro%ides the interace =et+een
the analog signal and the digital processor
; rocess the digital signals, using digital processor
; Digital processor can =e
- rogramma=le computer ,
; Small microcomputer , or
; DS chip(
; The digital output o the DS is con%erted =acG to analog
( DAC, Digital to Analog converter)
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!ample o digital processing o signals
- First, input analog signal rom microphone is con%erted to digital signal
=y #nalog to Digital Con%erter .#DC/
- Then, the digital signal is processed =y Digital Signal rocessor .DS/
- Finally, the digital output produced =y DS, is con%erted to #nalog signal,
=y Digital to #nalog con%erter .D#C/, or eeding it to speaGer
Digital
Output signal
Digital
Input signal
n is num=er(sample)
Analog
Output signal Analog
Input signal
t is time
(t) (n) ! (t)! (n)
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tages o Digital Signal rocessing o%er #nalog Signal processing:
S is more lei=le: !asily reconigura=le( .Sot+are/
(Reconfiguration of analog s!stem re"uires redesigning
#ard$are)eplication is easy, digital systems can =e easily replicated
ccuracy o design in digital systems is high
( %etting #ig# accurac! in analog s!stems is ver! difficult
&ecause of tolerances of #ard$are components)
torage o digital signals is %ery easy on magnetic media, memory, CD, DD
pen dri%e(
S allo+s implementation o more sophisticated signal processing algorithms(
.Comple mathematical algorithms can =e easily implemented/
he cost o implementation digital systems is lo+
due to the lo+er cost o the digital hard+are
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pplications o Digital Signal rocessing:
; Speech processing
; Signal transmission and reception in telephone> mo=ile systems
; Image processing
; Seismology . Study o seismic signals/
; Eil eploration .#nalysis o signals tra%eling through the layers o eart
imitations o DS:
1( The use o #DC and D#C may maGe the processing comparati%ely, slo+er (Analog s!stems are faster)
)( The po+er consumption o digital signal processors may =e higher
4( Hot suita=le or signals +ith huge =and+idth .#DC, Sampling rate has to =e t+o times the =and+idth/
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Sound signals
#udi=le range: 4 ert to 12 ert
For telephone "uality sound signals:
Range o re"uencies 0 J
thus sampling re"uency re"uired is 2 J
For CD "uality:Sampling re"uency is (1 J > channel
Range o re"uencies 0 )) J
#udi=le range o human =eings ) to ),
Hy"uist sampling theorem:
I a unction x .t / contains no re"uencies higher than B hert,
it is completely determined =y gi%ing its ordinates at a series
o points, spaced 1> .)B/ seconds apart(
.Er the signal +ith the maimum re"uency o K ert, can =e
completely represented =y sampling at a re"uency o ) K ert/
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Classiication o signals
; Signal rocessing techni"ue or any signal depends upon the
characteristics o the signal(
Multi;channel and Multidimensional Signals: 'C-'ultiple components of same signal
'D- 'ultiple inputs
Real %alued signal
s1.t/ 0 # sin 4 πt
Comple %alued signals) .t / 0 # e B4 π t 0 # cos4 πt 3 B # sin 4πt
ector representation
The signal generated =y:
multiple sources or
multiple sensors
can =e represented =y components o a %ector
s1.t/
s4.t / 0 s).t/
Three component
signal
!
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Three components o ground acceleration measured a e+ Gilometers
rom the epicenter o an earth"uaGe
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* + * + ***
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Ene dimensional> multidimensional signal
; one;dimensional signal
; I a signal is a unction o one independent %aria=le
It is one;dimensional signal
; Ether+ise it is multidimensional signal
; I it is dependent upon more than
one independent %aria=les
; #n still image can =e t+o dimensional signal
since I ., y/, intensity o light at any location
depends upon , y coordinates o the point in the picture
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; #n tele%ision picture can =e three dimensional
as I ., y, t/, intensity at any , y coordinates also
depends upon time t .rame time/
#n colour T image can ha%e three components:
Thus a colour T image is a three channel .red, green and =lue/,
three dimensional ., y and t/ signal
Ir ., y, t/
I ., y( t/ 0 Ig ., y, t/
I= ., y, t/
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lassiication o signals
( Continuous time and discrete time signals
( Continuous %alued and discrete %alued signals
( Deterministic and random signals
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Continuous;Time and Discrete Time Signals
1a( Continuous;time signals .#nalog signal/:
; # signal that eists all the time in a gi%en inter%al
; These signals are deined or e%ery %alue o time in the inter%al
; Most o the naturally occurring signals are continuous in nature
; These signal taGe on %alues in the continuous inter%al .a, =/
+here a can =e ; ∞ and = can =e 3 ∞
; Mathematically, these signal can =e descri=ed =y unctions o
a continuous %aria=le
1.t/ 0 cos π t
).t/ 0 e; L t L +here ;
∞
t ∞
(t)
t
(t)
t
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=( Discrete;time signals
The signals are deined only at speciic %alues o time
; The %alue o signal =et+een and 1, say at (A is not Gno+n
; Thus, discrete time signals eists only
at speciic %alues o time
; These time;instants usually are e"uidistant at e"ually
spaced inter%als((&ut need not &e e"uidistant)
; The discrete time signal is o=tained rom a continuous time signal
using the sampling at speciic %alues o time
* / + 0 1 2 n
(n)
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; # discrete time signal is an approimation o the continuous signal
; To impro%e the accuracy o the approimation
the sampling period .Ts / is reduced or
the re"uency o the sampling Fs is increased
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; # typical discrete time signal can =e represented =y:
.n/ 0 # cos ω(n (ω 3 * π f)
!ample o discrete;time signal:
; LtnL
.tn/ 0 e , n 0 , ± 1, ± ), ( ( ( 4 ((t), ((t), . . ((tn), 5
;I inde n o discrete;time instants is used as the independent
%aria=le .i(e(, a se"uence o num=ers/,
the signal %alue =ecomes a unction o an integer %aria=le
;Thus a discrete;time signal can =e represented mathematically
=y a se"uence o real or comple num=ers
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# discrete time signal is represented as .n/ ( instead of (t) )
; I the time instants tn are e"ually spaced, tn 0 nT .T 0 Time period/
; Signal can =e .nT/ 0 ? .T/, .1T/, ( ( .nT/ @
; There are some discrete time signals +hich are inherently discrete and
do not re"uire the sampling o continuous signal
i(e( #ccumulating a %aria=le o%er a period o time
.i(e( num=er o cars> hour/
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.n/ 0 (2n or n N and
.n/ 0 or n (negative values of n)
.n/
n
*raphical representation o the discrete time signal
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( Continuous;%alued and Discrete;alued signals
; The %alues o continuous time or discrete time signals
can =e:
continuous (ta6ing all possi&le values) or
discrete ( ta6ing onl! a finite set of values)
)a( Continuous;alued signal:
Signal taGes on all possi=le %alues ina range . +hich can =e inite or an ininite/
.y;ais can =e di%ided into ininite num=er o le%els/
. (A, (A), (A4, (6 ( ( (/
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)=( Discrete;alued signal:
Signal taGes on %alues only rom a inite set o possi=le %alues
(!-ais can &e divided into finite num&er of levels)
; Osually, these %alues are e"uidistant,
hence can =e epressed as an integer multiple o
distance =et+een t+o successi%e %alues(
.), 4, , A or 6, not )(4)), 4(1, (4A, 6(1)/
Digital signal:
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Digital signal:
# discrete;time signal ha%ing a set o discrete %alues(
For digital processing o a signal:
; It must =e discrete in time
; Its %alues also must =e discrete
; Digital signal is o=tained =y:
First - E=taining a discrete time signal =y
sampling an analog signal at discrete instants in time
Then – 'uantiing the %alues o discrete time toa set o discrete %alues
'uantiation:
It is the process o con%erting continuous;%alued signal into
discrete %alued signal =y simple rounding > truncation process
or =y mapping to a set o inite %alues
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Digital signal +ith dierent amplitude %alues: .1, ), 4 and /
1, 1, 1, ), 1, 4, ), , ), 1<
-* - * / + 0 1 n
7(n)
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Discrete time signal
It is deined or e%ery integer %alue o n or ;∞
n 3∞
.n/ is nth sample o the signal
i .n/ is o=tained =y sampling then . n/ ≡ .nT/
4 (8, (8), . . ,(n8)
+here T is sample period
i(e( the time =et+een t+o successi%e samples
#lternati%e representation o DT signals:to graphical representation
1( Functional representation,
1, or n 0 1, 4
.n/ 0 , or n0)
, other+ise
)( Ta=ular representation:
n ( ( ( ;) ;1 1 ) 4 A 6 ( ( (
;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
.n/ 1 1
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( Se"uential representation:
#n ininite duration signal or
se"uential +ith the time origin .n 0 /
indicated =y sym=ol↑
.n/ 0 ( ( ( , , 1, , 1, , , ( ( ( <
↑
# se"uence +hich is or n0 .n/ 0 , 1, , 1, , , ( ( ( <
↑
The time origin or a se"uence +hich is ero or n
; First letmost point is considered to =e the origin
inite duration se"uence
.n/ 0 4, ;1, ;), A, , , ;1< .se%en point se"uence/ ↑
# se"uence .n/ 0 or n .n/ 0 ,1, , 1< .;point se"uence
↑
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Some elementary Discrete;time Signals
1( The unit impulse se"uence
1, or n 0
Denoted as .n/ , or n
≠
; It is ero e%ery+here ecept n0,
+here it has a unit height
; #lso reerred as a unit impulse
. ero e%ery+here, ecept time t 0 /
; It has unit area
.n/ ≡
-* - * / +
)( The unit step signal
Denoted as u .n/ 1, or n ≥
, or n
; It is 1 at positi%e n . including at n0 / and
at negati%e n
u .n/≡
-* - * / +
%rap#ical representation
of unit impulse
%rap#ical representation
of step signal
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4( The unit ramp signal
denoted as u r .n / n, or n ≥
, or n u .n/ ≡
-* - * / +
( The eponential signal
; a se"uence o the orm .n/ 0 an
or all n 9 a 9
a
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;nit step signal
;nit ramp signal
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.n/ 0 a n
. /
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.n/ 0 a n
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I the parameter PaQ is real .n/ is a real signal
; i the parameter PaQ is comple %alued,
a≡
r e Bθ +here r andθ
are the parameters
.n / 0 r n e B θn
0 r n . cos θn 3 B sin θn/
Real part g .n/ 0 rn
cosθ
n
Imaginary part B .n/ 0 rn sin θn
I r 0 ( and θ 0 π >1
g .n/ 0 rn cos θn 0 .(/n cos π >1 ( H
i .n/ 0 rn sin θn 0 .(/n sin π >1 ( H
g .n/ and B .n/ are a damped, decaying . eponential/ cosine unctionand dam ed sine unction
.n/ 0 a n
.n/ 0 a n :
Real part
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Real part g
Imaginary part
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Imaginary part i
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.n / 0 r n e B θn can =e represented as amplitude and phase unction:
#mplitude unction L .n/L 0 # .n/ ≡ r n
hase unction∠
.n/ 0φ
.n / ≡
θ
n (π n
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Classiication o Discrete;time signals:
1( !%en and odd signals .Symmetric and asymmetric signals/
)( eriodic signals and aperiodic signals
4( !nergy signal and po+er signal
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1( !%en and odd signals .Symmetric and asymmetric signals/
# signal is e%en > or symmetric i .n/ 0 .;n/
# signal is odd > or asymmetric i .;n/ 0 ; .n/ or
.n/ 0 ; .;n/
For an e%en signal .1/ 0 .;1/, .)/ 0 .;)/ ( ( (
For an odd signal .1/ 0 ; .;1/, .)/ 0 ; .;)/ ( ( (
; I .n/ is odd then . / 0
!amples:
# sine +a%e is odd,
+hile a cosine +a%e is e%en signal
sin .A/ 0 ; sin.;A/
cos .A/ 0 cos . ;A/
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-+ -/ -* - * / + n e%en signal
* / + n odd signal
-+ -/ -* -
(n)
(n)
.n/ 0 .;n/
.n/ 0 ; .;n/
. / 0
(n)
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#ny ar=itrary signal can =e epressed as the sum o the t+o components:
; Ene e%en and the
; Ether odd
; The e%en signal component can =e ound out =y:
e.n/ 0 ?.n / 3 .;n/@
odd signal component
o .n/ 0 ? .n/ – .;n/@
#dding =oth:
.n/ 0 e.n/ 3 o.n/
Fi d d dd t th i di t i l
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Find e%en and odd components o the gi%en discrete signal:
.n/ 0 1, ), 4, , 1, ), )<
↑
.;n/ 0 ), ), 1, , 4 ), 1< mirror image o .n/
↑
a=out origin ./
.n/e 0 1(A, ), ), , ), ), 1(A< 0 ? .n/ 3 .;n/@ > )
↑
.n/o 0 ;(A, , 1, , ;1, , (A< 0 ? .n/ ; .;n/@ > )
↑
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)( eriodic signals and aperiodic signals
; # signal .n/ is periodic +ith period H .HN /
Enly i .n 3H / 0 .n/ or all %alues o n
Smallest %alue o H or +hich the a=o%e e"uation hold good
is called the undamental period
;The signal is non;periodic i there is no %alue o H
that satisies the a=o%e e"uation
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ChecG i .n/ 0 # cos .ωn 3 θ / is periodic
ereω
0 )π
.n/ 0 # cos .)π n 3 θ /
.n 3H/ 0 # cos .)π
.n 3H/ 3θ
/0 # cos .)π n 3 )π H 3 θ /
For periodicity:
.n 3 H/ 0 .n/
# cos .)π
n 3 )π
H 3θ
/ 0 # cos .)π
n 3θ
/
For this to =e true:
)π H 0 0 or 0 )πG +here G is an integer (, , *, / . . )
or 0 G > H +here =oth G and H are integers
Thus or periodicity, 0 G >H, +here =oth G and H should =e integers
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1( ChecG i cos.(1 πn/ is periodic8
(1π
0 )π
0 (1> ) 0 1>) cycles per sample,
0 G> H ratio o t+o integers
#s G 01 and H is ), =oth are integers, so cos .(1 πn/ is periodic
)( ChecG i .n/ 0 sin 4nere 4 0 )π or 0 4> ) π
since cannot =e epressed as raction o t+o integers ,
the signal .n/ 0 sin4n is not periodic
nergy and po+er signal
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nergy and po+er signal
!nergy o a signal or discrete time signal .n/
∞
is deined as ! ≡ L .n/ L )
n 0 ;∞ #s the magnitude s"uare is used or .n/,
The deinition is applies to real as +ell as comple;%alued signals
The energy o a signal can =e inite or ininite
!nergy signal
; I !nergy ! is inite then .n / is called an energy signal
; The inite !nergy can =e called ! o signal .n/
Many signals possess ininite energy,=ut ha%e a inite a%erage po+er
1 H
%erage o+er 0 lim ∑ L .n/L )
H → ∞ H 3 1 n 0 ; H
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! d it t i l . /
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!nergy and po+er o a unit step signal u.n/
P1Q or n 0 ≥
; Step signal u.n/ 0
PQ other+ise
∞ ∞
!nergy ! ≡ L .n/ L ) 0 Lu .n/ L ) 3 = = . . . 3 ∞
n 0 ; ∞
Since the ! is ininite, the unit step signal is not an energy signal
1 H
The a%erage o+er 0 lim ;;;;;;;;;;; ∑ u ) .n/
H → ∞ )H 31 n 0
( summation of u* (n) is > =)
1 .H31/ 1 3 1> H 1
0 lim ;;;;;;;;;;; 0 lim ;;;;;;;;;;; 0 ;;;;;;;
H → ∞ )H 31 H → ∞ ) 3 1>H )
Conse"uently, the unit step se"uence is po+er signal
and its energy is ininite
* / . .
Find !nergy or the signal .n/ 0 an u .n/ aL 1 ?u (n) step signal5
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Find !nergy or the signal .n/ 0 an u .n/ aL 1 ?u (n) step signal5
∝
; #s energy o D( T( Signal is: ! H ≡ ∑ L .n/L)
n 0 ;∝
0 ∑ Lan u .n/L) for n 0 to ∝ u (n) is a init step
Since u .n/ is 1, or to ∞ 0 ∑ Lan ( 1L) or n 0 to ∝
0?a)@ 3 ?a)@1 3 ?a)@) 3 . .
.*eometric series ∑ #n 0 1 3 # 3 #) 3 #4 ( ( ( 0 1> 1 –# i # 1/
Thus ! 0 1> 1; a) i La)L 1
! l i h th th ll i i l i l i l
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; First calculate energy o the signal
! o .n/ is gi%en =y:
n 0 ∞ n 0 ∞
! 0 L .n/L ) 0 L (An u .n/L )
n 0 ;∞
n 0 ;∞
-Since this signal is multiplied =y unit step: n is rom ; ∞
( signal is ero for n )
n 0∞
n 0∞
n 0∞! 0 L .1>)/ n L ) 0 .1>)/ ) n 0
.1>/ n
n 0 n 0 n0 n 3 ∞
?@tandard %eometric series Σ A n = A = A*= A / . . . < (-A) .# 0 /<
n 3 8#us ! 0 1 > .1 ;1> / 0 1> U 0 > 4 0 1(444
- Thus ! is inite
Since !nergy is inite it is energy signal
!plain +hether the ollo+ing signals are po+er signal or energy signal(
i (A n u .n/
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Eperations on discrete time signals:
The mathematical transormation rom one signal to another
is represented as:y .n/ 0 T ? .n/@
; Eperations in%ol%ing:
; Independent %aria=le . time/
; Dependent %aria=le .amplitude/
The =asic operations on DT signals are:
1( Time shiting
)( Time re%ersal Independent varia&le, time
4( Time scaling
( Scalar multiplication
A( Signal addition and
multiplication
Dependent varia&le,
Amplitude
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Eperations in%ol%ing independent %aria=le . time/
1( Time Shiting
; Shit n =y ; G in .n/ Delay.G is an integer/
y .n/ 0 .n ; G/ for 6 3 /, ! () 3 (-/)
; i G is negati%e the shiting results in ad%ancing =y LGL units
- .n / 0 ;1, , 1, ), 4 , , , , , <
↑
-.n;4/ 0 ;1, , 1, ), 4 , , , , , < shiting delay G 0 4 (-/ &ecomes origin)
↑
-. n 3)/ ;1, , 1, ), 4 , , , , , < ad%ancing G0 ;) (=* &ecomes origin)
↑
Hote: delay >ad%ance is easy in stored signal
.o+e%er, ad%ancing in real time, generated signal is not possi=le(/
Future
)( Time re%ersal –
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Folding> Relection a=out the time origin n 0 :
; Replacing n =y –n y .n/ 0 .;n/
.n / 0 ), ), ), , 1, ), 4, <
↑ .;n/ 0 , 4, ), 1, , ), ), )<
↑
.;n ; )/ 0 , 4, ), 1, , ), ), )< .Folding and delayed =y )/
↑ .;) =ecomes origin/
4( Time scaling:
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g
; Changing in time scale
; T+o types do+n scaling and up scaling
; Do+n scaling is represented as:
y .n/ 0 T ? .n/@
!ample y .n/ 0 .)n/
.n/ 0 ;4, ;), ;1, , 1, ), 4, , , , , , , <
↑
y .n/ is taGing e%ery other sample rom .n/
y./ 0 ./ 0 , y.;1/ 0.;)/0 )
y .1/ 0 .)/ 0 y.;)/ 0.;/0
y.)/ 0 ./ 0 ,
Thus y .n/ ;), , ) , , , , <
↑
.n/ 0 ;4, ;), ;1, , 1, ), 4, , , , , , , <
↑
y .n/ 0 .)n/ 0 ;), , ), , , , <
↑
Sampling is reduced to hal
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Op scaling
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Op scaling
y .n/ 0 .n > ) /
.n/ 0 A, , 4, ), 1, ), 4, , A<
↑
y . /0 .>)/ 0 ./ 0 1
y .1/0 .1>)/ 0 .(A/ 0 no sample
y .) /0 .)>)/ 0 .1/ 0 )
y .4 /0 .4>)/ 0 .1(A/ 0 no sample
y . /0 .>)/ 0 .)/ 0 4
y .6 /0 .6>)/ 0 .4/ 0 y .2 /0 .2>)/ 0 ./ 0 A
Thus y .n/ is epanded %ersion o .n/ +ith y .1/, y.4/, y.A/ ( ( Ho sample
- -1 - -0 -+ -/ -* - * / + 0 1
8#e sampling rate is increased from
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#ddition, multiplication, and scaling o se"uences
; #mplitude modiication includes addition, multiplication and scaling
o discrete;time signals
( #mplitude scaling:
; Multiplying =y a constant
y .n/ 0 # .n/ ; ∞ n ∞
A( Signal addition and multiplication
; The sum o t+o signals
y .n/ 0 1 .n/ 3 ) .n/ ; ∞ n ∞
.add corresponding terms)
; The product o t+o signals
y .n/ 0 1 .n/ ( ) .n/ ; ∞ n ∞
.multiply corresponding terms)
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At -*, - . 3
-, * . 3
, * . 3 *
, . * 3 *
*, . / 3
Discrete time Systems
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Sy
; DT system is a de%ice> algorithm that operates on
a discrete;time signal .input> ecitation/,according to some +ell;deined rule,
to produce another discrete;time signal .output> response/
; It is a set o operations perormed on input signal .n/ to produce
output signal y .n/
.n/ is said to =e transormed to y .n/
y .n/ ≡ T ? .n/ @ +here T is transormation
or .n /→ y .n/ .transormed to /
Determining the response:
LnL 4 n 4
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LnL, ; 4≤
n≤
4
.n/ 0 , other+ise? .n/0 , 4, ), 1, , 1 ,), 4, <
↑
V .n/ 0 .n ;1/ 0 , 4, ), 1, , 1 ,), 4, 4, ), 1, )>4 , 1 ,), A>6, 1, , <
↑
(n ) 3 ? , , /, *, , , , *, /, , ,
B() 3
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y . / . . / . / . /
.n/0 , 4, ), 1, , 1 ,), 4, <
↑
0 , 4, 4, 4, ), 1, ), 4, 4, 4, <
↑
n
y .n/ 0∑
.G/ 0 .n/ 3 .n;1/ 3 . n;)/ 3 ( ( B ( ) 3 () = (-) = (-*)
G 0 ; ∞ = (-/), = (-+) = (-0)
.n/0 , 4, ), 1, , 1 ,), 4, <
↑
y .n/ 0 , 4, A, 6, 6, 5, ,1),1)< . G 0 ; ∞/ sum o all past %alues 8
↑
y./ 0 ./ 3 .;1/ 3 .;)/ ( (
= /
= / =*
= / =*=
= / =*== =
Classiication o Discrete Time Systems
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y
Discrete time system is a de%ice or algorithm that operates on
a discrete time signal
; It is represented as y .n/ 0 T ? .n/@
1( Static and Dynamic systems (@tatic - current input)
)( Causal and #nti;causal systems .Causal - resent and past
no future
4( $inear and Hon;linear systems
( Time %ariant and Time in%ariant systems
A( Sta=le and unsta=le systems
6( FIR and IIR systems
(Finite Impulse Response< infinite impulse response)
Static %ersus Dynamic systems
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y y
Static system:
# system is static i the output o a system depends
only the current input andnot on the past or uture input
; !amples o static systems:
y .n/ 0 T ? .n/@
y .n/ 0 ( .n/, y .n/ 0 a .n/
y .n/ 0 log .n/
y .n/ 0 # cos .n/
y .n / 0 a .n / 3 = 4 .n/
; In each case y .n/ re"uires only present %alue o input .n/
; Static systems are also called memory less system
as no memory is re"uired to store pre%ious input %alues
e( in case o y .n/ 0 ( .n/, ! () 3 + ( (), ! (*) 3 + ( (*), . . .
Dynamic system
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; #ll the systems that are not static(
; In dynamic systems, the output may depends on the
present as +ell as
past input signals (ven future signals)
y .n/ 0 .n/ 3 .n;)/ is a dynamic system
; as or inding y ./ 0 ./ 3 .)/ . current input = past input)
past input is re"uired
; Thus the dynamic system re"uires memory to store the past inputs
Dynamic system is also called memory system
!amples:
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p
y .n/ 0 ? .n/ 3 .n;1)/@
for ! (0), (0) and (/) are re"uired
y .n/ 0 .n/ ( .n;1/
n
y .n/ 0 ∑ .n ;G/ Re"uires Finite memory
G0
ny .n/ 0 ∑ .n ; G/ Re"uires Ininite memory
G0 ; ∞
)( Causal and Hon;causal system
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; # system is causal i the output o the system y .n/ at any n
depends on present and past inputs =ut
not on uture inputs
; !ample:
y .n / 0 .n/ 3 .n;)/ 3 .n;4/
.present past past)
Hon;causal system:
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; I the output also depends upon the uture input
then it is Hon causal system
; !ample:y .n/ 0 .n/ 3 .n31 / Future
(present = future) y ./ 0 ./ 3 .A/
; ence or a non;causal system,
; Future input needs to =e predicted to ind the present
Ether eamples:
y .n/ 0 .)n/
y .n/ 0 .n/ – .n3)/
4( $inear and non; linear
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; $inear systems satisies superposition principle:
; The superposition principle states that the response to
a +eighted sum o input signals,
should =e e"ual to the corresponding
+eighted sum o the outputs o the system
to indi%idual input signals(
i( e(T? a11.n/ 3 a)).n/ @ 0 a1T ? 1.n/ @ 3 ? a) T ? ).n/ @
(response to t#e $eig#ted sum of inputs 3
t#e $eig#ted sum of responses to t#e individual inputs)
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1 .n/
) .n/
3 T y1 .n/ 0 T ?a1 1 .n/ 3 a) ) .n/@a1
a)
1 .n/
) .n/#HD
T
T 3 y) .n/ 0 a1 T ?1 .n/@ 3 a) T ? ) .n/@
a1
a)
ChecG +hether the y .n/ 0 n( .n/ is a linear or non linear system8
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#s y .n/ 0 T ? .n/@
ence, the system is n ( .n/
For linearity:
ChecG i T ?1 .n/ 3 ) .n/@ 0 T ? 1.n/ @ 3 T?) .n/@ or
! (n) 3 !* (n)
so y1
.n/ 0 T ?1
.n/ 3 )
.n/@
0 n( ?1 .n/ 3 ) .n/@ 0 n( 1 .n/ 3 n() .n/
y) .n/ 3 T ? 1.n/ @ 3 T?) .n/@ 0 n ( 1.n/ 3 n( ) .n/
ence y1 .n/ 0 y) .n/
The system y .n/ 0 n( .n/ is linear
similarly y .n/ 0 n ( ) .n/ is non;linear can =e pro%ed
Time;%ariant and time; in%ariant systems
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Time;in%ariant system:
In Time;in%ariant system
the input >output characteristic does not change +ith time
; suppose input .n / produces y .n/ at any stage,
y .n/ 0 T ? .n/@
; i the input signal is delayed =y G units,
the output y .n; G/ +ill =e same as y .n/ ecept that it is delayed =y G units
- First delay the input =y G samples , and o=ser%e the output → y .n, G/- Delay the output =y G samples → y .n ; G/
I y .n; G/ 0 y .n, G/ the system is time;in%ariant
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.n ;G/ y .n ;G/ .n/ y .n/T T
ChecG +hether the ollo+ing systems are time in%ariant8
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1( y .n/ 0 e .n/
; Delay input =y PGQ y .n, G/ 0 e .n; G/
; Delay the output y .n/ =y G, y .n ; G / 0 e .n;G/
Since =oth are e"ual the system is time;in%ariant
)( y .n/ 0 n ( .n/ is time %ariant as
1( Delaying input y .n, G/ 0 n( .n ; G/
)( Delaying output y .n ; G/
y .n; G/ 0 . n; G/ ( .n; G/
Koth are not e"ual the system is time;%ariant
Sta=le and unsta=le systems :
For a Sta=le system
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For a Sta=le system
a =ounded input, produces =ounded output
$et M is inite num=ers such that M ∞
,the input is said to =e =ounded i L .n /L ≤ M ∞
for all value of n
My is a inite num=er such that My ∞,
the output is said to =e =ounded i Ly .n/L≤
My ∞
for all value of n
; I or a =ounded input .n/, there is ∞ output
the system is unsta=le
ChecG +hether the ollo+ing systems are sta=le8
1( y .n/ 0 e .n/
)( y .n/ 0 .)n/
Koth are sta=le, as or =ounded input, there is =ounded output
#nalysis o $inear Time –In%ariant systems .$T I / systems:
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; $T I systems are characteried in time domain =y
their response to a unit sample se"uence
; #ny ar=itrary signal can =e represented as
+eighted sum o unit sample se"uences
Techni"ues or analysis o $inear systems:
1( Direct solution o the input;output e"uation
)( First decomposing the input signals into elementary signals, then
determining the response to each elementary signal and
adding all the responses
Direct solution o the input;output e"uation
I t t t ti h th
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Input –output e"uation has the orm
y .n/ 0 F ? y .n;1/, y .n;)/, ( ( V .n;H/, .n/, .n;1/, .n;1/, . . . - .n;M/@
F ?( ( ( ( @ denotes some unction o "uantities in =racGets
; Specially, or $TI .$inear Time;in%ariant/ systems
; The general orm o input;output relationship:
H M
y .n/ 0 ; ∑ aG y .n ; G/ 3 ∑ =G .n ; G/
G01 G 0aG and =G are constant parameters that speciy the system and
aG and =G are independent o .n / and y .n/
- The a=o%e e"uation is called a dierence e"uation- It represents one +ay to characterie the =eha%ior o a
discrete;time $TI system
Second method o analying the =eha%ior o a linear system
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; Resol%e the input signal .n/ into
+eighted sum o elementary signal components ?G .n/ @
- (8#e elementar! signals are selected so t#at t#e response
to eac# component is easil! determined)
- Ose the linearity property to add the responses to
indi%idual components to o=tain the total response .output/
.n / 0 ∑ cG G .n/ ()
G
G .n/ are the elementary signal component
cG are the set o amplitudes .+eighting coeicients/
in the decomposition o the signal .n/
I the y .n/ is the response o the system to
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I the yG .n/ is the response o the system to
the elementary component G .n/
then yG .n/≡
T ?G .n/@ (*)
#ssuming the system response to the cG G .n/ is cG ? yG .n/@
#s a conse"uence o scaling property o the linear systems (n) from ()
y .n/ 0 T ? .n/@ 0T ?∑
cG G .n/ @ 0∑
cG T ? G .n/@G G
Osing additi%e property o the linear system
y .n/ 0 ∑ cG yG .n/ ( ! 6 (n) from (*)
G
(8#e response to t#e input (n ) $#ose components are c6 (n)
e"uals to t#e $eig#ted sum of
t#e responses to t#e components)
Resolution o a discrete time signal into impulses
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- $et an ar=itrary signal =e .n/ ? , *, /, , *, , /
↑
; Resol%ing the signal .n/ into a sum o unit sample se"uences
let Gth component o .n/, G .n/ 0 .n ;G/ 4 (n- 6)t# impulse5
+here G is the delay o the unit sample se"uence
; Multiplying .n/ and .n ;G/ 6 3/
Since .n ; G/ is ero e%ery+here ecept at n 0 G .Impulse/
the result o the multiplication +ill =e a se"uence:
+hich is ero at e%ery+here ecept at n 0 G
Thereore .n/( .n; G/ 3 .G/( .n ; G/
-/ -* - * /
(n)
-/ -* - * /
(/)
i +e taGe dierent %alues o G +e +ill get:
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i +e taGe dierent %alues o G +e +ill get:
∞
.n/ 0∑
.G/ .n ;G/
G 0 ; ∞
0 summation o an ininite num=er o unit sample se"uence , .n ;G/
ha%ing amplitude %alues o .G/
Representation o a signal in terms o
+eighted sum o shited, discrete impulses :
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+eighted sum o shited, discrete impulses :
#ny ar=itrary signal can =e represented as
summation o shited and scaled impulses
.n/ 0 (A, ), 1, (A, 1, ), (A<
↑
; The sample ./ can =e o=tained =y
multiplying ./, the magnitude, +ith a unit unction .n/
./ W or n 0i( e( ./ ( .n/ 0 0 (A ( 1 0 (A . multipl!)
W or n≠
δ (n)
n n
() . δ (n) .0
-/ -* - * /
.n/ 0 (A, ), 1, (A, 1, ), (A
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↑
Similarly other components can =e o=tained:
:
.1/( .n;1/ 0 1 ( 1 0 1
.)/( .n;)/ 0 ) ( 1 0 )
.4/( .n;4/ 0 (A ( 1 0 (A
.;4/( .n 34/ 0 (A ( 1 0 (A
.;)/( .n 3)/ 0 ) ( 1 0 )
.;1/( .n 31/ 0 1 ( 1 0 1
Thus .n/ 0 (A( .n 34/ 3 ) ( .n 3)/ 3 1. .n 31/ 3
3 (A ( .n/ . 3 1 ( .n ;1 / 3 ) ( .n ;)/ 3 (A ( .n ;4/
The signal .n/ can =e %ie+ed as a summation o scaled andshited impulses
∞
in general, .n/ 0∑
.G/( .n ;G/
6 3 - ∞
$et .n / 0 ) 4
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$et .n / 0 ), , , 4<
↑
Resol%e into a sum o +eighted impulse se"uence
since .n / is non;ero at the time instant n 0 ;1, , )
+e need three impulses at G 0 ;1, , )
.n/ 0 ) (
.n31/ 3 ( .n/ 3 4 (
.n;)/
↑
Impulse response and con%olution
. / T ? . / @ O t t ( ) t f f ti ( )
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y .n/ 0 T ? .n/ @ Output ! (n) 3 transfer function (n)
I input .n/ is a unit impulse .n/ then the output o the system is
Gno+n as impulse response h .n/
; Thereore impulse response: h .n/ 0 T ? .n/@
.transfer function of unit impulse)
Impulse response completely characterie the system
.n/ y .n/8
.n/ h .n/8
Con%olution sum
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Since input signal .n/ can =e represented as
+eighted sum o discrete impulses
∞ i( e( .n/ 0 ∑ .G/( .n; G/ ($#ere n is t#e time inde,
G 0 ; ∞ 6 is a parameter s#o$ing
t#e location of input impulse
∞
Thus output signal y .n/ 0 T ? .n/@ 0 T ∑ .G/( .n ;G/
G 0 ;∞
; I the system is linear :
∞
y .n/ 0 ∑ T ? .G/( .n; G/ 5
G 0 ;∞
∞
y .n/ 0∑
.G/( T ? .n; G/ 5 () G 0 ;
∞
$et T ? .n G/@ 0 h .n G/ # (n 6) unit pulse response dela!ed
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$et T ? .n ;G/@ 0 h .n, G/ # (n, 6) unit pulse response dela!ed
∞
y .n/ 0 ∑ .G/( h .n, G/ (From ()
G 0 ;∞
I the system is time in%ariant: h .n, G/ 0 h .n ;G/ ( Response $it#
t#e input dela!ed 3
8#e response $it#
∞
t#e output dela!ed)
y .n/ 0∑
.G/ ( h .n ; G/
G 0 ;∞
. The output> response at PnQ is summation o
%alues o input at G, multiplied =y unit impulse response at n ; G,
or all %alues o G/
∞
y .n/ 0 ∑ .G/( h .n ; G/
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y . / . / . /
G 0 ;∞
The a=o%e e"uation is called con%olution sum:
For a $inear Time In%ariant .$TI/ system
i input se"uence .n/ and
the impulse response h .n/ is Gno+n
The response y .n/ can =e ound out =y the con%olution sum
∞
y .n/ 0 ∑ .G/( h .n ; G/
G 0 ;∞
(Output ! at n 3 sum of values of input (n) at n 3 6,multiplied &! unit impulse response at n -6,
for all values of 6
The con%olution sum is represented as
y .n/ 0 .n/ X h .n/
X
roperties o Con%olution
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1( Commutati%e $a+ : .n/ X h .n/ 0 h .n/ X .n/
G Gy .n/ 0 ∑ .G/( h .n ; G/ 0 ∑ h . G/( .n ;G/(
G 0 ; ∞ G 0 ; ∞
)( #ssociati%e $a+: ? .n/ X h1 .n/ @ X h) .n/ 3 .n/ X ? h) .n/ X h1 .n/ @
4( Distri=uted $a+: .n/ X ? h1 .n/ @ 3 h) .n/@ 3 .n/ X h1 .n/ 3 .n/ X h) .n/
Computation o $inear Con%olution
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1( *raphical Method
)( Ta=ular method
The linear con%olution is gi%en =y:
∞
y .n/ 0 ∑ .G/ ( h .n ; G/
G 0 ; ∞
( Response ! (n) at n, is summation of values of (n) at 6,multiplied &! unit impulse response at n - 6, for all values of 6)
Calculating the %alue y .n/ or time instant n 0
∞ ∞
y./ 0∑
.G/( h. ; G/ 0∑
.G/( h. ; G/ G 0 ; ∞ G 0 ; ∞
. h . ;G/ indicates olding /
For n 0 1
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∞
∞
y.1/ 0 ∑ .G/( h.1 ; G/ 0 ∑ .G/( h.; G 3 1/
G 0 ; ∞ G 0 ; ∞
h.; G 3 1/ indicates shiting o olded signal h
h .;G 31/ Indicates h .;G/, the olded signal is delayed =y P1Q sample
For n 0 )
∞ ∞
y.)/ 0∑
.G/( h.1 ; G/ 0∑
.G/( h.; G 3 )/ G 0 ; ∞
G 0 ; ∞
h.; G 3 )/ indicates shiting o olded signal h
h .;G 3)/ Indicates h .;G/, the olded signal is delayed =y P)Q sample
#nd so on
Steps or computing the con%olution =et+een .G/ and h .G/
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1 Folding: old h .G / # ( -6)
) Shiting: shit h . ;G/ # (-6 =)
4 Multiplication: Multiply .G/ =y h .n –G/,
ind all the product terms or all the %alues o G
Summation: Sum all the product terms to ind y .n/
Find y .n/ or all %alues o n, ollo+ing steps ) to or each
%alue o n
∞
Range o PnQ and PGQ , or calculating y .n/ 0∑ .G/ ( h .n ; G/
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g g y . / . / . /
G 0 ; ∞
Range o PnQ $o+est %alue o n in y. n/ 0 lo+est %alue o n in .n/ 3 lo+est %alue n in h h .n/
to
ighest %alue o n in y. n/ 0 highest %alue o n in .n/ 3
highest %alue n in h .n/
Range o PGQ
The range G +ill =e same as range o n in .n/
E=tain the impulse response o a system h .n/ 0 1, ), 1, ;1<
↑
I t Si l . / 1 ) 4 1
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Input Signal .n/ 0 1, ), 4, 1<
↑
Range o n in y .n/ 0 .$o+est %alue o n in .n/ 3 lo+est %alue o n in h .n//
to .highest %alue o n in .n/ 3 highest %alue o n in h .n//
n in y .n/ 0 .;1 3 0 ;1/ to .) 3 4 0 A / 0 ;1 to 3 A
Range o G 0 range o n in .n / 0 to 4
Determining the response> output y: For n 0 : y ./ h .n/ 0 1, ), 1, ;1
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y . / . / . / , , , <
y .1/ 0∑
.G/ ( h .;G 31/ G %aries rom to 4↑
.G/ ( h .;G 31 /
y .1/ 0∑
1, ), 4, 1< ( ;1, 1, ), 1
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Similarly y ./ 0 ;)
y. A/ 0 ;1
y . ;1/ 0 1
!ntire response o the system ( ( ( (, , , 1, , 2, 2, 4, ;), ;1, , , ( ( ( < ↑
.G/ ( h .;G 34 /
y .4/ 0 ∑ 1, ), 4, 1< ( , ;1, 1, ), 1
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.n/ 0 1, ), 1, )< h .n/ 0 1, 1, 1<
Since the se"uences are gi%en in terms o PnQ
o=tain .G/ and h .G / =y su=stituting =y G
.G/ 0 1, ), 1, )< h .G/ 0 1, 1, 1<
↑
↑
Con%olution is gi%en =y:
∞
y .n/ 0∑
.G/( h .n ; G/ G 0 ;
∞
Range o PnQ or y .n/ 0 $ 3 h$ 0 3 0 to h 3 hh 0 ) 3 4 0 A
0 to A
Range o PGQ : same as the %alue o n in .n/ 0 to 4
Ta=ulation method o linear con%olution
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$et .n/ 0 ./, .1/, .)/ < and h .n/ 0 h./, h.1/, h.)/<
↑ ↑
Step I: Form the matri as sho+n =elo+:.n/
./ .1/ .)/
h./
h .n/ 0 h.1/
h.)/
Step II Multiply the corresponding elements o .n/ and h .n/ and
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Step II Multiply the corresponding elements o .n/ and h .n/, and
Step III Separate out elements diagonally as sho+n =elo+:
.n/
./ .1/ .)/h./ h./(./ h./(.1/ h./(.)/
h.n/ 0 h.1/ h.1/(./ h.1/(.1/ h.1/(.)/
h.)/ h.)/(./ h.)/(.1/ h.)/(.)/
Step I : #dd the elements in each =locG(
Thi ill i di l . /
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This +ill gi%e corresponding %alues o y .n/
V./ 0 h./(./
V.1/ 0 h./(.1/ 3 h.1/(./ V.)/ 0 h./(.)/ 3 h.1/(.1/ 3 h.)/(./
V.4/ 0 h.1/(.)/ 3 h.)/(.1/
V./ 0 h.)/(.)/
V .n/ 0 y.1/, y.)/, y.4/, y./<
Range o n in y .n/ 0 $o+est %alue o n in .n/ 3 lo+est %alue o n in h .n/
to highest %alue o n in .n/ 3 highest %alue o n in h .n/
0 . 3 0 / to .) 3 ) 0 / 0 to 3
The range o PnQ or y .n/ 0 to
Compute con%olution y .n/ 0 .n/X h. n/
. / 1 1 1 1< h . / 1 ) 4
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.n/ 0 1, 1, , 1, 1< h .n/ 0 1( ;), ;4, <
↑
↑
.n/ 1 1 1 1
h .n/1 1 1 1 1 (
;) ;) ;) ;) ;) ( -*
;4 ;4 ;4 ;4 ;4 .-/
. +
y .n/ 0 1, 1;), ;);4, 13 ;43, 1 ;) 3 3, ;) ;4 3, ;43 , <
0 1, ;1, ;A, ), 4, ;A, 1, <
∑ .n/ ( ∑ h(.n/ 0 ∑ y .n/
.13133131/ ( .1 ;) ;4 3/ 0 . 1 ;1 ;A 3)34 ;A 31 3/
( 0
Erigin
Correlation:
; Correlation is used or the comparison o t+o signals
It i d t hi h t i l i il
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; It is measure o degree to +hich t+o signals are similar
; Cross correlation
Correlation o t+o separate signals is Gno+n ascross correlation
; #uto correlation
Correlation o a signal +ith itsel is Gno+n as
auto correlationCross Correlation:
Correlation =et+een t+o signals .n/ and y .n/
3 ∞
r y . l / 0∑
.n/ ( y .n ; l/ +here l 0 ,±
1,±
), ±
4 ( ( (n 0 ; ∞
Inde l is time shit .or lag/ parameter and E
the su=scripts , y on cross correlation se"uence r y . l /
indicates the se"uences =eing correlated
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#uto correlation
The correlation o a signal +ith itsel to determine time delay
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The correlation o a signal +ith itsel to determine time delay
=et+een the transmitted signal and the recei%ed signal
3∞
r .l/ 0∑
.n/ ( .n ; l/ +here l 0 ,±
1,±
), ±
4 ( ( (n 0 ; ∞
Compute the cross;correlation =et+een:
.n/ 0 1 1 1
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.n/ 0 1, 1, , 1<
↑
y .n/ 0 , ;4, ;), 1<
↑
; Cross correlation o .n/ and y .n/ is gi%en =y :
3 ∞
r y.l/ 0 ∑ .n/ ( y .n ; l/ +here l 0 , ±1, ±), ±4 ( ( (
n 0 ; ∞
; Range o n:#s y .n/ is delayed =y l, and .n/ is not changed
so the range o n in the summation is
same as .n/ 0 ;) to 31
1
So r y .l/ 0 ∑ .n/ ( y .n ; l/
n 0 ;)
Range o n in .n/ is ;) to 31
Range o n in y .n/ is ;) to 31
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Range o l:
since the e"uation y .n; I/ should ha%e maimum %alue at y .n/ 0 y.1/
n – l 0 1 maimum
Starting %alue o PnQ in summation is ;) in (n)
;) –l 0 1 or n 0 ;)
I 0 ;1 ;) 0 ;4
or starting %alue o l 0 ;4
The e"uation y .n – l / should ha%e minimum %alue at y n / 0 y .;)/
n; I 0 ;)
=ut the summation should stop at n 01 in (n)
1; I 0 ;)
or I 0 4
So the range o I 0 ;4 to 3 4
The range o n 0 ;) to 31
The range or l is ;4 to 4
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1
r y.l/ 0 ∑ .n/ ( y .n ; l/ +here l 0 ;4 to 3 4
n 0 ;)
Ho+ .n/ 0 1, 1, , 1< and y .n/ 0 , ;4, ;), 1<
↑
↑
R y .;4/ or I 0;4, n 0 ;) to 31
n 0 ;) ;1 1
.n/ ( y .n ; l/
r y.;4/ 0 .;)/ (y.;) 34 / 3 .;1/( y .;1 34/ 3 ./ ( y. 34/ 3 .1/( V.134/
.;)/( V.1/ 3 .;1/ y .3)/ 3 ./y.34/ 3 .1/( V./
1 ( 1 3 1 ( 3 ( 3 1 (
0 1 3 3 3 0 1
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r y . 1/ or I 01, n 0 ;) to 31
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r y.1/ 0 .;)/ y.;) ;1 / 3 .;1/ y .;1 ;1/ 3 ./y. ;1/ 3 .1/( V.1;1/
0 .;)/ y.;4 / 3 .;1/ y .;)/ 3 ./y.;1/ 3 .1/( V./
1 ( 3 1 ( 3 ( ;4 3 1 (;) 3 3 ;) 0 )
Similarly r y.)/ 0 ;4 and r y.4/ 0
Ry.l/ 0 1, ;1, ;A, ), ), ;4, < ↑
Simple method to calculate correlation
Solution using property o correlation
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r y .l / 0 .n/ X y .;n/ ( 3 convolution of (n) and ! (-n))
; So taGe .n/ as it is: .n/ 0 1, 1, , 1<↑
; Fold y .n/: y .n/ 0 , ;4, ;), 1<
↑
y .;n/ 0 1, ;), ;4, /
↑
E=tain the con%olution o .n/ and y .;n/ =y matri method
y.;n/ .n/ 1 1 1
1 1 1 1
;) ;) ;) ;)
;4 ;4 ;4 ;4
r y.l/ 0 .1, 1 ;), ;);4, 13;43, ;)33, ;43, <
r y.l/ 0 .1, ;1, ;A, ), ), ;4, <
↑
Determine the autocorrelation o the ollo+ing signal
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.n/ 0 .1, ), 1, 1< $et 1 .n/ =e 1, ), 1, 1<
↑ ↑
)
.n/ =e 1, ), 1, 1<
Folding ) .n/ ↑
).;n/0 1, 1, ), 1<
↑
So r .l/ 0 .n/ X .;n/
↓
1.n/ 1, ), 1, 1
).;n/ 1 1 ) 1 1
1 1 ) 1 1
) ), , ), ) → 1 1, ), 1, 1
r .I/ 0 1, ) 31, 13)3), 131331, 1 3)3), )31, 1<R .l/ 0 1, 3 4, 3A, 3 5, 3 A, ,3 4, 31 <
↑
roperties o Correlation
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1( The result o autocorrelation is maimum
+hen the signal matches +ith itsel and there is no phase shiting(
)( #uto;correlation is an e%en unction
r .l/ 0 r .; l/
4( The cross;correlation is not commutati%e(
That means r y.l/ ≠ r y .l/
FIR and IIR
From the con%olution sum:
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∞
y .n/ 0∑
.G/( h .n ; G/
G 0 ; ∞
h .G/ is the impulse response o the system
FIR
I h .G/ is o inite duration, the system is called
Finite Impulse Response .FIR/ system
IIR
I h .G/ is o ininite duration, the system is called
Ininite Impulse Response .IIR/ system
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!HD
)( .t/ 0 sin.2 πt/ 3 4 sin .5) πt/ is sampled +ith Fs 0 6 times per sec.
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.1/ 7hat are the re"uencies in radians in the resulting DT signal .n/8
.)/ I .n/ is passed through an ideal interpolator,
+hat is the reconstructed signal8
To ind DT signal re"uencies sample the CT signal
ut t 0 n Ts 0 n > Fs 0 n>6 Ts sampling time period
Fs sampling re"uency 0 1> Ts
. t/ 0 sin . 2π
t / 3 4 sin .5)π
t/
? n@ 0 sin . 2 π n > 6/ 3 4 sin .5) π n> 6/
0 sin . (2 π n / 3 4 sin . 1() π n /
.* π n 3 (* .) π n 3 -. π n
0 sin .(2π
n / 3 4 sin .;(2π
n/ 0 ; ) sin .(2π
n/
0 ; ) sin .+ n/
thus + 0 2 radians