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SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGYCOIMBATORE-10

DEPARTMENT OF ECE

NAME OF THE STAFF : G.SATHYA & L.MALATHI

SUB NAME : SIGNALS AND SYSTEMSYEAR / SEM / SEC : II / III / A & B

UNIT ICLASSIFICATION OF SIGNALS AND SYSTEMS

PART A

1. Def!e "#e$ f%!#'! (!) )e*#( f%!#'!.

A!" : CT unit step function u(t) = 1 for (t 0)0 for (t < 0)

DT unit step function u(n) = 1 for (n 0) 0 for (n < 0)

CT delta function (t) = 1 (t) = 1 for t=0

0 for t0 DT delta function (n) = 1 for n=0

0 for n0

+. ,(# " #e #'#(* e!e 'f #e )"e#e #e "!(* 23!4 5 #(6e" #e7(*%e 'f %!# (# !8 -19091

A!" : Energy of the signal is gien as! " 1E = # $%(n)$& = # $%(n)$&

n = '" n = '1

= $%('1)$& $%(0)$& $%(1)$& =

+. C*(""f #e "!(*" ("9 $e') ' !'!$e') "!(* e(! 9 (;1

A!" : *ince it is e%ponential the gien signal is non periodic+


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>. Def!e $'5e "!(*.A!" : , signal is said to -e po.er signal if tis nor/alied po.er is noneroand finite+ i+e++ (0


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. Ce6 5e#e #e ""#e (7! !$%#-'%#$%# e*(#'!3#48 - 23#4 )# " *!e( #e !7((!# ' !'#.A!":

This is an integration of input+ ,n integration is al.ays independent of ti/e

shift+ 4ence this is ti/e inariant syste/+

,(# )' '% e(! (! e7e! "!(* (!) (! ')) "!(*A!":dd signal %('t) = '%(t) CT

%('n) = '%(n) DTEen signal %('t) = %(t) CT

%('n) = %(n) DT

10.. D(5 #e "!(* 23!4 8 %3!4 %3!-


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1+. ,(# (e #e *(""f(#'!" 'f #e "!(*A!":1+ Deter/inistic and >ando/ *ignal&+ Energy and 5o.er *ignal+ 5eriodic and nonperidic *ignal

?+ ,nalog and Digital signal

1 @! / - e + @! / ?

A!":

1 = ? 2 @f1= & 2 @ = 812 A1

& = & 2 B f&= 1 2 B = 8&2 A&A1= @ and A&= B+ The funda/ental period .ill -e least co//on /ultiple

of A1 and A& (i+e+!) B+1>. F!) #e f%!)(e!#(* $e') 'f #e "!(* 23!4 8 J < e


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6(t) depends upon %(t10) (i+e+!) future input+ 4ence the syste/ isnoncausal+

+


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>. (i) Test .hether the syste/ descri-ed -y the e7uation y(n) = n %(n) is linear(? /ar8s)

(ii) Qerify the linearity ! causality and ti/e inariance of the syste/+ 6(n&) = a %(n1) - %(n)+ (3 /ar8s)

?. (i) Test .hether the syste/ descri-ed -y the e7uation y(n) = n %(n) is *hiftinariant+ (? /ar8s)(ii) Deter/ine .hether or not each of the follo.ing signals is periodic+ f the signal is periodic! deter/ine the funda/ental period+

(i) %(t) = : cos((&t I (2)) ;&(ii) %(n) = #(n) I ?8 ' (n'1'?8) (G Nar8s)

H+ (i) Descri-e the classification of the syste/+ (10 Nar8s) (ii) Chec8 for the linearity and ti/e inariance (H Nar8s)

@+ (i) Deter/ine .hether the syste/ are linear! ti/e inariant ! causal andsta-le+

a+ y(n) = n%(n)-+ y(t) = %(t) %(t'&) for t0 and 0 for t


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UNIT I IA!(*"" 'f '!#!%'%" Te S!(*"

PART A

1. ,(# )' #e f'%e "ee" 'effe!#" e$e"e!#A!":Lourier series coefficients represent arious fre7uencies present in the signal+t is nothing -ut spectru/ of the signal+

+. Def!e f'%e "ee".A!":

"(t) = # (8) e 80 t

8 = '"

.here ! (8) = (12T) %(f) e 8

0 t dt


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(t) cos(& fct V) < ''' U ( e V2& ) (f'fc) ( e' V2& ) (ffc)

&+ Conolution property MLT

%1(t) W %&(t) < '''' U 1(f) + &(f)

. ,#e #e f'%e #(!"f' $( f' 23#4.A!":

LT LT%(t) < ''' U (f) (or) %(t) < ''' U ()

"

() = %(t) e't dt '"

" %(t) = ( 1 2 & ) () et d '"

. De#e!e #e *($*(e #(!"f' 'f 23#4 8 e-(# "!3

#4 %3#4A!":

XT e'at sin(t) = XT e'at :(et - e't) / 2j ] }

= (1 / 2j) LT (e'(a') t - e'(a)t)

= (1 / 2j) (1/ (! " (a-j) } - (1/ (! " (a"j) }

= / # ($"a)2 " 2 ] % &' % &($)*-a

. ,(# " #e*($*(e #(!"f' 'f e-(# "!3#4 %3#4A!": XT

e'at sin(t) < '''''' U / # ($"a)2 " 2 ]

. A "!(* 23#4 8 '"3+@f #4 " $(""e) #'% ( )e7e 5'"e !$%# '%#$%# " e*(#e) 3#482+3#4. ,(# (e #e fe%e! '$'!e!#" !#e '%#$%#

A!":

*ince an input is s7uared!6(t) = ( cos(&ft) )&

= (1cos(?ft) ) 2 &

= (12&) (12&) cos:&(&f)t;

The output present in the output is Y &f Y

9


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10.Def!e #e f'%e #(!"f' $( f' '!#!%'%" #e "!(*.A!":

Lourier Transfor/ M " () = %(t) e't dt '"

nerse Lourier Transfor/ M "(f) = ( 1 2 & ) () et d '"

11. F!) #e *($*(e #(!"f' 'f 23#4 8 #e-(# %3#4 9 5ee 3(;04A!":

XT

e'at u(t) < '''' U ( 1 2 (saZ) ) ! >CM >e(s) U ('a)

Differentiation in *'Do/ain property gies! XT

't %(t) < ''' U (d (s) 2 ds )

XT

't e'at u(t) < ''' U (d(12(sa)) 2 ds )

XT

t e'at u(t) < ''' U 12(sa)& >CM >e(s) U ('a)

1+. F!) #e F'%e #(!"f' 'f 23#4 8 #e-(# %3#4 9 5ee 3(;04A!": " (f) = %(t) e'&ft dt '"

" = e'at e'&ft dt 0

"

= e

'(a&f)t dt 0

= (1 2 (a&f) )

1


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1. ,(# " #e e*(#'!"$ e#5ee! f'%e #(!"f' (!) *($*(e#(!"f' A!":

(s) = () .hen s=

+0. Def!e f'%e "ee" (!) f'%e #(!"f'A!":

Lourier series "(t) = # (8) e 80 t

8 = '"

.here ! (8) = (12T) %(f) e 80 t dt

Lourier Transfor/ M

" () = %(t) e't dt '"

nerse Lourier Transfor/ M "(f) = ( 1 2 & ) () et d '"

+1. ,(# " )ffee!e e#5ee! f'%e #(!"f' (!) *($*(e #(!"f'A!":

' Xaplcae transfor/ is ealuated oer co/plte s'plane ! -ut fouriertransfor/ is ealuated oer a%is in s'plane+

' Xaplcae transfor/ is -roader co/pared to fourier transfor/+ nother .ords! fourier transfor/ is the special case of laplacetransfor/+

++. o-tain the fourier transfor/ of %(t)= e + f # .

A!": "

(f) = %(t) e' &fc t dt '"

"

= e &f te' &fc t dt '"

"

= e &(f' fc) t dt '"

=(f ' fc)

12


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+


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(H Nar8s)

&+ (i) Deter/ine the trigono/etric fourier series representation of the half.ae rectifier output+ (10 Nar8s)

(ii) *tate and proe parsaalZs theore/ for co/ple% e%ponential fourierseries+ (H Nar8s)

+ Lind the fourier transfor/ of the signal %(t) and plot the a/plitudespectru/+ (1H Nar8s)

(t) = 1 ! (' 2\& ) StS( 2\& )0 ! other.ise

&?+ Lind the fourier series of the signal %(t) = sin(&f0 /)t + cos(&f0 n)t dt!

.here0

f0 is the funda/ental fre7uency and / and n are any positie integer+ (1H Nar8s)

B+ (i) Deter/ine the trigono/etric fourier series representation of the full .ae rectified output+ (10 Nar8s)

(ii) Lind the nerse laplace transfor/ of(&s& 3s I ?@ ) 2 (s1)(s& Hs &B) F + (H Nar8s)

H (i) Lind the laplace transfor/ of %(t) = e'-$t$ for (-


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U!#-IIIL!e( Te I!7((!# C'!#!%'%" Te S"#e

PART A

1.G7e f'% "#e$" #' '$%#e '!7'*%#'! !#e(*.A!": a) LoldingM ne of the signal is first folded at t=0 -) *hiftingM The folded signal is shifted right or left depending upon ti/e at.hich output is to -e calculated+ c) NultiplicationM The shifted signal is /ultiplied other signal+ d) ntegrationM The /ultiplied signals are integrated to get the conolutionoutput+

+. ,(# " #e '7e(** $%*"e e"$'!"e 3#4 5e! #5' ""#e" 5#

$%*"e e"$'!"e 13#4 (!) +3#4 (e ! $((**e* (!) ! "ee"A!": Lor parallel connection h(t)= h1(t) h&(t) Lor series connection h(t)= h1(t)Wh&(t)


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. ,(# " #e e*(#'!"$ e#5ee! !$%# (!) '%#$%# 'f (! LTI ""#eA!":

nput and output of an XT syste/s are related -y!

6(t) = %(]) h(t']) d] (i+e+! ) conolution+

. ,(# " #e #(!"fe f%!#'! 'f ( ""#e 5'"e $'*e" (e (# -0.< 0.>(!) ( e' (# -0.+ A!":

51 = '0+ 0+?! p& = '0+ I 0+? ^ = '0+&

4(s) = ( s') 2(s'p1)(s'p&)

= (s0+&) 2 (s0+'0+?)(s0+0+?)

= (s0+&) 2 ( s& 0+Hs 0+&B)

10. F!) #e $%*"e e"$'!"e 'f #e ""#e 7e! H3"4 8 1/3"4A!":

e'atu(t)


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`````` 4 (_) = 1 2 9a& _&

,ngle 4(_) = 'tan'1(_2a)

1. ,(# " e(!# $%*"e e"$'!"eA!":

The i/pulse response is the output produced -y the syste/ .hen unit

i/pulse is applied at the input+ The i/pulse response is denoted -y h(t)+1?. ,#e #e '!7'*%#'! !#e(* 'f 23#4.A!":

The conolution integral of %(t) is gien -y

6(t) = %(] ) h(t ' ]) d]

1. ,(# (e #e (" "#e$" !7'*7e) ! '!7'*%#'! !#e(*A!":

a) LoldingM ne of the signal is first folded at t=0 -) *hiftingM The folded signal is shifted right or left depending upon ti/e at

.hich output is to -e calculated+ c) NultiplicationM The shifted signal is /ultiplied other signal+ d) ntegrationM The /ultiplied signals are integrated to get the conolution

output+

1. Te $%*"e e"$'!"e 'f #e LTI CT ""#e " 7e! (" 3#4 8e-#%3#4. De#e!e #(!"fe f%!#'! (!) e6 5e#e #e ""#e " (%"(*(!) "#(*e.

A!":

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h (t) = e't u(t)

Ta8ing Xaplace transfor/!

4(s) = 12(s1)

4ere pole at s='1! ie located in left half of splane+ 4ence this syste/ iscausal and sta-le+

1. Def!e ee! 7(*%e (!) ee! f%!#'! 'f LTI CT ""#e".A!":

6(t) = 4(s) e st

Thus the output is e7ual to input /ultiplied -y 4(s)+ 4ence est is called eigenfunction and 4(s) is called eigen alue+

1. ,(# " #e $%*"e e"$'!"e 'f #e ""#e 3#4 8 23#-#04A!":

Ta8e laplace transfor/ of gien e7uation

6(s) = e'st (s)

4(s) = 6(s)2(s)

= e'st0Ta8ing inerse laplace transfor/ of the a-oe e7uation

h(t) = (t't0)+

PART ---B

1+Lind the conolution of %(t) and h(t)

(t) =1 0StS&

0 other.ise h(t) = 1 0StS 0 other.ise

&+ a) 4o. do you represent any ar-itrary signal inter/s of delta function andits delayed function (G Nar8s) -) Deter/ine the response of the syste/ .ith i/pulse response h(t) = u(t)for the input %(t)= e'&t u(t) (G Nar8s)

+ a) Lind the output of an XT syste/ .ith i/pulse response h(t) =(t') for

the input %(t)= cos ?t cos @t+ (G Nar8s)

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-) sing laplace transfor/ find the i/pulse response of an XT syste/descri-ed -y the differential e7uation d&y(t)2dt&I dy(t)2dt '&y(t) =%(t) (G Nar8s)

?+ a) Lind and plot the /agnitude spectru/ of the transfer function! 4(_) = (e_ ) 2( e_12) (10 Nar8s)

-) Define linear ti/e inariant syste/+ (H Nar8s)

B+ a) Derie the conolution integral of a syste/ (G Nar8s)-)Lind the response of the syste/ %(t) = (t)'(t'1+B)+4ere h(t) is thei/pulse response of the syste/+ (G Nar8s)

H+ a) Ohat is /eant -y causality and sta-ilityP Derie conditions for causalityand sta-ility+ (G Nar8s)-) Deter/ine the i/pulse response of the CT syste/ descri-ed -ydifferential e7uation! d&y(t)2dt& ? dy(t)2dt y(t) = d%(t)2dt &%(t)+(G Nar8s)

@+ a) Derie the e%pression for conolution integral (H Nar8s)-) Descri-e ho. state e7uation are used to continuous ti/e syste/ andho. fre7uency response is o-tained fro/ the state e7uationP (10 Nar8s)

G+ a) E%plain any properties of conolution integral (H Nar8s)-) Lind the state e7uation of a continuous ti/e XT syste/ descri-ed -y d&y(t)2dt& dy(t)2dt &y(t) =%(t) (G Nar8s)

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UNIT - IANALYSIS OF DISCRETE TIME SIGNALS

PART A

1.,(# " #e e*(#'! e#5ee! V-#(!"f' (!) F'%e #(!"f' 'f)"e#e #e "!(*A!": () = () = e.

This /eans ^'transfor/ is sa/e as Lourier transfor/ .hen ealuated on unitcircle+

+. ,(# " #e V- #(!"f' 'f (!%3!4A!":

^ an u(n)F = 12(1'a'1) !>C M U a

egion of conergence(>C) is the area in ^' plane .here ^'transfor/conerges +n other .ords ! it is possi-le to calculate () in >C+

>. S#(#e #e !#(* 7(*%e #e'e 'f - #(!"f'.A!":

The initial alue of the se7uence is gien as!

(0) = Xt "()

?. ,(# " #e )ffee!e e#5ee! #e "$e#% 'f #e CT "!(* (!) #e"$e#% 'f #e 'e"$'!)! "($*e) "!(*.A!":

1+ The spectru/ of CT signal and sa/pled signal are related

as! (f) = fs # (f'nf s) n='"

&+ This /eans spectru/ of sa/pled signal is periodicrepetition of spectru/ of CT signal+

+ t repeats at sa/pling fre7uency and a/plitude is also/ultiplied -y fs+

. S#(#e #e f!(* 7(*%e #e'e f' -#(!"f'.A!":

%(") = Xt 1(1''1)()

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. Def!e DTFT $(.A!":

DTLT ! () = # %(n) e'n !

DTLT %(n) = 12& () e n d+

. Def!e ROC 'f - #(!"f'.A!":

The alue of for .hich the 'transfor/ conerges is called theregion of conergence+

. O#(! - #(!"f' 'f 23!4 8 J19+9A!":

() = # %(n) 'n

= 1&'1'&?'

10.Def!e !#(* 7(*%e #e'e (!) )ffee!#(#'! #e'e 'f -#(!"f'.

A!": The initial alue of the se7uence is gien as!

(0) = Xt"

()The Differentiation theore/

n%(n)U ' d2d ()

11. Def!e P("e7(*Q" #e'e.A!": E = # %(n) & = 12& ()&d+

The a-oe e7uation gies the energy of the signal+

1+.,#e #e $'$e#e" 'f ROC 'f -#(!"f'.A!":

1+ >C of causal se7uence is e%terior of so/e circle of radius r=a+&+ >C of non causal se7uence is interior of so/e circle of radius r=-++ f the se7uence is -oth sided then its >C is a disc lying -et.een

a


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-) Lind the inerse ' transfor/ of %() = (?)2(&'?) 10 Nar8s)

&+ a) 4o. .ill you ealuate fourier transfor/ fro/ pole ero plot of 'transfor/+ (H Nar8s) -) Lind the inerse ' transfor/ of () = 12(1'1+B'10+B'&) for >C M

0+B


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UNIT

LINEAR TIME INARIANT DISCRETE TIME SYSTEMS

PART A

1. I" #e '%#$%# "e%e!e 'f (! LTI ""#e f!#e 5e! #e !$%# 23!0 "f!#e X%"#f #e (!"5e.

A!":

f the i/pulse response of the syste/ is infinite! then output se7uence isinfinite een though inout is finite+Lor e%M input ! %(n) = (n) finite length /pulse response h(n) = an u(n) nfinite length utput *e7uence y(n) = hn) W %(n)

= an u(n) W (n) = an u(n)

+. C'!")e (! LTI ""#e 5# $%*"e e"$'!"e 3!48 3!-!04 f' (!!$%# 23!4 9Y3e4.

A!":

4ere 6(ej) is the spectru/ of output+ ky conolution theore/ .e can .rite

6(ej) = 4(ej) (ej)

4(ej) = DTLT (n'n0) = ejno

6(ej) = e'jno (ej)

99-.A!":

3!4 --- ;

Y3!4

ZZ

1 + < > ?

+ & ? H G 10 1&

-> '? 'G '1& '1H '&0 '&?

H 1& 1G &? 0 H

- 'G '1H '&? '& '?0 '?G

6(n)= + 0 ? 0 '? 'G '&B '? '?G F

>. De#e!e #e ""#e f%!#'! 'f #e )"e#e #e ""#e )e"e) #e )ffee!e e%(#'!.

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Y3!4 - J 31/+4 3!-14 J 31/>4 3!-+4 8 23!4 23!-14A!":

6() ' (12&) '1y() F (12?) '& y() F = %() I '1%()

4() = 6() 2 () = (1''1) 2 (1 I :(12&) '1; :(12?) '&;

?. ,(# " #e *!e( '!7'*%#'! 'f #5' "!(*". [+9\ (!) [19-+91\A!":

6:n; = + '1 0 'B ?F

. ,(# " #e e"$'!"e 'f (! LSI ""#e 5# $%*"e e"$'!"e 3!48 3!4+ 3!-14 f' #e !$%# 23!48J19+9


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. D(5 #e )e# f'-II e(*(#'! 'f #e ""#e )ee) #e)ffee!#(* e%(#'!9

[)+3#4 / )#+\ ? [)3#4 / )#\ >3#4 8 [)23#4 / )#\

A!":

10. De#e!e #e #(!"fe f%!#'! 'f #e ""#e )e"e) 3!48 ( 3!-14 23!4A!":

4() = 6() 2 () = 1 2 (1 ' a'1)

11. S#(#e #5' ()7(!#(e" 'f FFT '$%#(#'!".A!":

1+ LLT algorith/s are e%tre/ely fast+4ence they are co/putationallyefficient+

&+ LLT algorith/s re7uire less /e/ory+

1+.D(5 )e# f'-II e$e"e!#(#'! 'f H34 8 31-1


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1


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1 1 jhd fD(5 #e *'6 )(( f' #e ""#e "$efe) #e )ffee!ee%(#'! [!\([-+\802[!\12[!-1\

A!":

1.F' ( "#(#e "$(e e$e"e!#(#'! 'f #e ""#e. F!) #e #(!"fef%!#'! 'f #e ""#e.

A 8 0 1 B 8 0 C 8 1 + -< -+ 1

A!":

-1

1

1

2

(n)X(n)

-1

-1

-1

-1

-1

-1

4

X(n)

-a

1

0 (n)

-1

-1

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4()= ( &'1 '& ) 2 (1 &'1 '&)

1. D(5 #e *'6 )(( 'f "#(#e 7((*e e%(#'!.A!":

1. ,(# (e #e $'$e#e" 'f '!7'*%#'!A!":1+ Co//utatie property of conolution&+ ,ssociatie property of conolution

+ Distri-utie property of conolution+0.,(# (e I$%*"e e"$'!"e (!) $'$e#e" 'f LTI ""#e"A!":1+ Causality&+ *ta-ility

PART B

1+ Lind the output of the syste/ .hose input'output is related -y!y(n) = @ y(n'1) I 1& y(n'&) & %(n) I %(n'&) for the input %(n) = u(n)+

(1H Nar8s)

&+ Lind the i/pulse response of the sta-le syste/ .hose input'output relationis gien -y the e7uation y(n) ' ? y(n'1) y(n'&) = %(n) & %(n'1) (1H Nar8s)

+ (i) Lind the linear conolution of %(n) = 1!&!!?F and h(n)=&!!?!1F(H Nar8s)

(ii) Co/pute the conolution of the t.o se7uences gien and plot theoutput+ (10 Nar8s)

?+ Lind the output se7uence y(n) of the syste/ descri-ed -y the e7uation6(n) = 0+@ y(n'1) I 0+1 y(n'&) & %(n) I %(n'&)+Lor the input se7uence %(n) = u(n)+ (1H Nar8s)

-a2

-a1

;1(n"1)

;1(n)

1

;2(n"1)

;2(n)

2

X(n) (n)

-1

-1

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B+ (i) Ohat is the i/pulse response %(n) of the syste/ if the poles and erosareradially /oed 8 ti/es their original locationP ( Nar8s)

(ii) Lind the oerall i/pulse response of the causal syste/ in fig+ h1(n) = (12)n u(n) ! h&(n) = (12&)n u(n) and h(n) = (12B)n u(n) (1& Nar8s)

H+ >ealie direct for/' ! direct for/' ! cascade and parallel realiation of the discrete ti/e syste/ haing syste/ function

4() = &(&) 2 ('0+1) (0+B)0 (0+?)F (1HNar8s)

@+ (i) The difference e7uation of the syste/ isy(n) I (2?) y(n'1) (1G) y(n'&) = %(n) (12&) %(n'1)+Dra. the direct for/' and structures+ (10

Nar8s) (ii) Lind the conolution of %(n) = 1!&!!?!BF .ith h(n) = 1!&!!!&!1F

(H Nar8s)

G+ (i) Lind the i/pulse of the discrete ti/e syste/ descri-ed -y the differencee7uation+ 6(n'&) I y(n'1) & y(n) = %(n'1) (H Nar8s)(ii) Descri-e radi%'& DT LLT algorith/ (10 Nar8s)

3+ (i) E%plain the state aria-le description of discrete ti/e syste/+ (G Nar8s) (ii) Co/pute the linear conolution of %(n) = 1!1!0!1!1F and h(n) =1!'&!'!?F (G Nar8s)

10+ ien 4() = (0+ '1I 0+?@ ''&) 2 (1'0+B '1 '& H ') + Dra. the -loc8diagra/ representation using DL and DL realiation+ (1H Nar8s)

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