VI Sem ECE Assignments

8/19/2019 VI Sem ECE Assignments

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VI sem ECE

Digital Communication

Assignment for Jan-May 2016.

Q. o Questions

1. Consider the set of signals

≤≤−

=

elsewhere

T t it f T

E

t s ci

,0

0),4

2cos(2

)(

π π

where i = 0, 1, 2, 3 and f c is an integer multiple of 1/

(a) !etermine the dimensionalit", # of the signal set.

($) !etermine set of orthonormal $asis functions to represent this set of signals.

(c) !etermine the coefficients si% of the signals si(t). &lso gi'e the signal

constellation diagram.2. $tain the orthonormal $asis set for the gi'en s1(t) and s2(t).

≤≤

=Otherwise

t t s

,0

20,2)(1

≤≤

=Otherwise

t t s

,0

10,4)(2 .

3. nerg" signals s1(t) and s2(t) are represented with two $asis functions and their 'ector

representations respecti'el" are, s1=*2 1+, s2=*2 1+. -ind the distance $etween s1 and

s2, and also their signal energies.

4. Consider a signal s!t" # Asinc!t$%" defined o'er the inter'al ∞ t∞. !etermine and

mae a neat plot of the output of a filter ha'ing impulse response &!t" # sinc!t$%) from

∞ t ∞ if the signal s(t) is gi'en as input. Can we sa" that h(t) is a matched filter for

s(t) i'e reasons for "our answer.

. he two signals s1(t) = s2(t) = et

for t 0, and= 0 elsewhere.

&re transmitted with eual pro$a$ilit" o'er a channel with additi'e white aussian

noise of 5ero mean and power spectral densit" #0/2. he recei'er $ases the decision on

the recei'ed signal o'er the inter'al 0 6 t 6 2. !etermine the minimum attaina$le

pro$a$ilit" of error at the recei'er output.


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7. he signals g1(t) = 10 cos(1008t) and g2(t) = 10 cos(08t) are $oth sampled at times t n

= n/f s, where n = 0, 91, 92, :.., and f s = ; samples per second. (t)= 12 cos(?008t)cos2(1?008t) is ideall" sampled at 4700 samples per

second. hat is the minimum allowa$le sampling freuenc" hat is the range of

permissi$le cutoff freuencies for the ideal lowpass filter to $e used for reconstructing

the signal

?. -igure shows the spectrum of a message signal g(t). he signal is sampled at a rate

eual to !etermine and setch the spectrum of the resulting @&A signal.

B. -or the spectrum of $andpass signal shown $elow, chec the $andpass sampling

theorem for (i) f s = 4 5, (ii) f s = 0 5. &lso indicate if and how the signal can $e

reco'ered.

( f )

0 2 1 1 2 f , C 5

10.

& signal s(t) of duration DE sec is defined as,

≤≤−

≤≤+=

T t T a

T t at s

2/,2/

2/0,2/)(

(i) !etermine the impulse response of a filter matched to this signal and setch it asa function of time.

(ii) hat is the pea 'alue of output

11. wo signals s1(t) and s2(t) are defined o'er the inter'al 0 t . >press these signals

in terms of orthonormal $asis functions. @lot the orthonormal $asis functions.

40

0-400

G(f

)

f (Hz)

1


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s 1 ( t )s 2 ( t )

2

4

/ 2t t0 0

12.

Consider a signal s(t) defined $"

≤≤

=otherwise

T t t s

,0

0,1)( .

Ft is proposed to appro>imate the matched filter for this signal $" a lowpass GC filter

defined $" the transfer function)/(1

1)(

0 f f j

f H +

=

where RC

f π 2

10 = the cutoff freuenc" of the GC filter is.

!etermine the optimum 'alue of f 0 for which the GC filter $ecomes the $est

appro>imation for matched filter.

!etermine the pea output signal to noise ratio, assuming noise is &# of 5ero

mean and power spectral densit" #0/2.

!etermine $" how man" deci$els the transmitted energ" $e increased so that the

performance $ecomes same as that of perfectl" matched filer

13. Consider a pulse s(t) defined $"

0

1)t(s

elsewhere

t 0

where f 0 = 1/2πGC is the 3dH $andwidth of the filter.

(a) !etermine the optimum 'alue of f 0 for which the GC filter pro'ides the $estappro>imation to the matched filter.

($) &ssuming an additi'e white noise of 5ero mean and power spectral densit" #0/2,

what is the pea output signal to noise ratio

(c) H" how man" deci$els must the transmitted energ" $e increased so at to reali5e

the same performance as the perfectl" matched filter


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14. imum


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possi$le 'alue for the highest component of the analog signal

20. he $inar" data 011100101 are applied to the input of a modified duo$inar" s"stem.

(i) Construct the modified duo$inar" coder output and corresponding recei'er output

with and without a precoder at the transmitter.

(ii) plain with neat setch a complete precoded duo$inar" scheme. ncode and transmit

the $inar" seuence 1101001 in precoded duo$inar" scheme. &ssume 1 = J1K and 0 =

1K.

22. !etermine the power spectral densit" of the polar uaternar" format of #GL t"pe,

$ased on the natural code. &ssume statisticall" independent and euall" liel" message

$its.

23. &nswer the following.

M1.&. Fn $uilt s"nchroni5ation is achie'ed using t"pe wa'eform coding

techniue

H. ne $aud is eual to num$er of $its per second

M2. he discrete autocorrelation function G &(n) for a $ipolar signaling scheme is

M3. Construct the GL $ipolar signaling format for the $inar" seuence 011010110

M4. i'en the $inar" seuence 011011001, Construct the polar C&N #GL format

using ra" code

M. rite the time domain and freuenc" domain e>pressions for the $asic pulseused to achie'e 5ero F


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mae a neat plot of the output of a filter ha'ing impulse response h(t) = sinc(t/) from

∞ t∞ if the signal s(t) is gi'en as input. Can we sa" that h(t) is a matched filter for

s(t) i'e reasons for "our answer.

27. he suare wa'e >(t) of the -igure 1 of constant amplitude &, period 0, and dela" td,

represents the sample function of a random process O(t). he dela" is random,

descri$ed $" the pro$a$ilit" function

f (td) = , 0 6d 6 0 .

0, otherwise

a) !etermine the pro$a$ilit" densit" function of the random 'aria$le O(t ) o$tained

$" o$ser'ing the random process O(t) at time t . $) !etermine the mean and autocorrelation function of O(t) using ensem$le

a'eraging.c) !etermine the mean and autocorrelation function of O(t) using timea'eraging.d) sta$lish whether or not O(t) is stationar". Fn what case it is ergodic

2;. a). >plain the freuenc" di'ision multiple access(-!A&) and compare it with!A&.

$). & @CA s"stem uses a uniform uanti5er followed $" a ;$it $inar" encoder. he $it

rate of the s"stem is eual to 0× 107 $its/second.(i). hat is the ma>imum message $andwidth for which the s"stem operates

satisfactoril"

(ii). !etermine the output signal to uanti5ing noise ratio when the full load sinusoidal

modulating wa'e of freuenc" 1A5 is applied to the input

2?. & $ase$and $inar" data transmission s"stem uses a format nown as $inar" pulse

position modulation in which a $it D1E is represented $" placing a pulse of amplitude

DaE from time P /2 to 0 and a pulse of 5ero amplitude from 0 to /2 while $it D0E is

represented $" placing a pulse of amplitude 5ero from P/2 to 0 and a pulse of

amplitude DaE from time 0 to /2. !etermine the power spectral densit" of the data


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$eing transmitted and also e'aluate the pro$a$ilit" of $it error if &# of 5ero mean

and two sided @

2B. -or a full scale sinusoidal modulating signal with amplitude D&E, show that (imum message $andwidth for which the s"stem operates

satisfactoril"

(ii). !etermine the output signal to uanti5ing noise ratio when the full load sinusoidal

modulating wa'e of freuenc" 1A5 is applied to the input

33. !eri'e an e>pression for pro$a$ilit" of error for coherent detected $inar" @


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C4 = d1 d3

C = d1 d2 d3

C7 = d1 d2

i. rite down the generator matri> .

ii. Construct all possi$le code words.

iii. imum length seuence. i'e reasons for "our answer.

4. &n N!A s"stem operates with a sampling freuenc" of 30 5. Ff a sinusoidal signal

>(t), normali5ed so that >(t) 6 1 whose freuenc" is 3 5, is applied, what 'alue of Vǀ ǀ

minimi5es the slope o'erload

47. Fn a $inar" data transmission using duo$inar" con'ersion fliter without a precoder,

the recei'ed sample 'alue Wc X were found to $e 2 2 0 2 2


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0 2 2 0 2 2 0.

he starting $ was 1.

a) !o "ou feel there is an error in these 'alues

$) Ff there is an error, can "ou guess the correct W$ X seuence Fs this uniue

c) Ff the o$tained W$ X seuence is not uniue, write down all possi$le correct W$ X

seuences.

#oteR Aore than one error in Wc X seuence is e>tremel" unliel".

4;. & con'olutional encoder has a single shift register with two stages, three modulo2

adders, and an output multiple>er. he generator seuences of the encoder are as

followsR g(1)=(1, 0, 1), g(2)=(1, 1, 0), g(3)=(1, 1, 1). !raw the $loc diagram of the

encoder.

4?. >plain the principles of direct seuence spread spectrum and freuenc" hopping

($oth slow and fast freuenc" hopping) spread spectrum communication s"stems.

Fnclude suita$le diagrams and wa'eforms as reuired.

4B. he $inar" data stream 001101001 is applied to the input of !uo $inar" s"stem

a) Construct the !uo $inar" coder output and corresponding recei'er output without a

precoder.

$) plain how the" are minimi5ed

1. Consider the sinusoidal process O(t)=& cos(28f ct)

where the freuenc" f c is constant and the amplitude & is uniforml" distri$utedR

f & (S)= 1, 0 6 S 6 1

0, otherwise

!etermine whether or not this process is strictl" stationar".

2. >plain the transition pro$a$ilit" diagram of a $inar" s"mmetric channel.

Ff the input $inar" s"m$ols 0 and 1 occur with eual pro$a$ilit". -ind the pro$a$ilities

of the $inar" s"m$ols 0 and 1 appearing at the channel output.

Ff the input $inar" s"m$ols 0 and 1 occur with pro$a$ilit" 0.2 and 0.;.-ind the

pro$a$ilities of the $inar" s"m$ols 0 and 1 appearing at the channel output.

3. Consider the (;, 4) amming code. and parit" chec

matri> of the code satisf" the condition = 0 or = 0.

4. &n analog signal ha'ing 4 5 $andwidth is sampled at 1.2 times the #"uist rate,


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and each sample is uanti5ed into one of 27 euall" liel" le'els. &ssume that the

successi'e samples are statisticall" independent.

hat is the information rate of this source

Can the output of this source $e transmitted without error o'er an &# channel with

a $andwidth of 10Q5 and an (t) = w(t), where w(t) is

white aussian noise of 5ero mean and power spectral densit" #0/2. -or h"pothesis

1, a target is present, and >(t) = s(t) J w(t), s(t) is an echo produced $" the target.

&ssuming that s(t) is completel" nown, e'aluate the following pro$a$ilitiesR

a) he pro$a$ilit" of false alarm defined as the pro$a$ilit" that the recei'er decides a

target is present when it is not.

$) he pro$a$ilit" of detection defined as the pro$a$ilit" that the recei'er decides

a target is present when it is.

;. !etermine the power spectral densit" of the polar uaternar" format of #GL t"pe,

$ased on the natural code. &ssume statisticall" independent and euall" liel" message

$its.

?. !raw the $loc diagram and e>plain the $ase $and $inar" @&A s"stem and e>plain theinters"m$ol interference

B. hat is a pseudo P noise seuence ow is it generated >plain with an e>ample and

'erif" the properties.

70. he $andwidth of input signal to @CA is restricted to 4Q5. he input 'aries from

3.?K to J3.?K and has the a'erage power of 30m. he reuired signal to noise ratio


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is 20dH. &ssuming uniform uanti5ation, calculate the num$er of $its reuired per

sample.

71. >plain with neat setch a complete precoded duo$inar" scheme. ncode and transmit

the $inar" seuence 1101001 in this scheme. &ssume 1 = J1K and 0 = 1K.

72. & discrete memor"less source O has fi'e s"m$ols >1, >2, >3, >4 and > with pro$a$ilities p(>1)=0.17, p(>2)=0.1B, p(>3)=0.10, p(>4)=0.40 and p(>)=0.1. Construct

VI Sem ECE Assignments

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