EEE207 Tutorial Solutions.doc

8/16/2019 EEE207 Tutorial Solutions.doc

1/42

Communications

Module Code: EEE207

Tutorial No 1 Solutions


2/42

Communications Tutorial 1 – Modulation – Solutions

1)

a) )102cos(5)( 4 t t m π = , i.e. V m = 5 Volts, f m = 10!"

t

V

t

V

t V t v mcm

mc

m

c DC s )cos(2)cos(2cos)( ω ω ω ω ω −+++=

V DC

= 0

5/2 5/2

m = ∞

90 110100 kHz


3/42

#)

+5

-5

d (t )

Condition Waveform = Type

Input

+5

-5

t

t

Input

PSK/PRK

Carrier

2kHz

V DC = 0

1 mse

t t d V c DC ω cos))(( +

+10

-10

t

!SK/""KV DC = 5

+15

-15

t

!SKV DC = 10+5

-5

2.

a) Modulation $e%t& 5'010

5===

DC

m

V

V m

#) T&e total aerae side#and %o*er +a #e deter+ined # one o- t*o +ain

*as:

A. . a%%lication o- t&e e/uation

+=

21

2m P P C T

i.e.

Side#andTotal

2

Carrier 2

m P P P cC T +=

i.e. Total side#and %o*er =2

2m P c


4/42

*&ere ( )

≡

=2

22

RMS DC

L

DC

c

V

R

V

P

!ence,att1

50

2

10 2

=

=c P

∴Total side#and %o*er =( )

att

1

2

5'0

21

22

==

m

B. 3lternatie +et&od is to consider diara+e/uation, i.e.

+#

0#

V DC

m(t )

( )

( )

( )t V

t V

t V

t t V V t v

mcm

mcm

c DC

cmm DC s

ω ω

ω ω ω

ω ω

−+

++=

+=

cos2

cos2

cos

coscos)(

t cω cost V mm ω cos

∴Total side#and %o*er = ( )

L

m

L

m

L

m

R

V

R

V

R

V

222222

22

=

+

!ence, total side#and %o*er = att

1

50

)5(2 2

=×

c) Total aerae %o*er = carrier %o*er total side#and %o*er

= 1 att 1 att = 11 atts

3. 6o*er out%ut = 24 *&en +odulation de%t& m = 1

a) - t&e carrier is un+odulated, 6o*er out = P carrier onl since P USB and P LSB = 0

i.e. )1(since18

2

11

24

21

242

1

22

2

==

+

=

+

=∴

=

+=

mm

P P

m P P

T c

cT

Power output = 16 kW (unmodulated carrier)

#) - m is reduced to 0'9 t&en

2

)9'0(118

21

22

+=

+= m P P cT

Total output power P T = 16.7 kW


5/42

c) #sere -ro+ a#oe t&at total side#and %o*er = 0'72 or -ro+

44

22m P m P

P P cccT ++=

∴ 6o*er in one side#and = 98'0

4

)9'0)(18(

4

22

==m P c

∴ ;atio = 0215'072'18

98'0= , (i.e. 2'15< o- total %o*er in one side#and *&en

m=0'9)

d) SS. di+inis&ed carrier %roduced *it& SS. %o*er = 0'98 and P c = 18

reduced # 28d. (%o*er)' To deter+ine P c (reduced)

Met!od 1"

P c = 18 × 109 atts

P c d. = d.42 att1

atts)(lo10 10 =

c P (relatie to 1 att)

;educed # 28 d. ies P c reduced = 18 d.

∴ P c reduced = '9=101018

= atts

∴ Total 6o*er = carrier side#and = 9' atts

Met!od "

=

reduced

10lo1028c

c

P

P dB , 11'9=10

8'2

reduced

==∴c

c

P

P

∴ P c reduced = atts2'4011'9=

000,18=

P T ≈ 400 atts

4.

V DC

t cω cos

t V t V t mmm

ω ω coscos)( 11 +=

( ) t t V t V V t v cmm DC s ω ω ω coscoscos)( 2211 ++=


6/42


7/42

φ φ ω

ω φ ω ω φ ω

φ ω ω

cos2

)()2cos(

2

)(

))cos((2

)())cos((

2

)(

)cos(cos)(

t mt

t m

t t t m

t t t m

t t t mV

c

cccc

cc x

++=

−++++=

+=

>6@ re+oes 2ωc co+%onents, &ence φ cos2

)(t mV UT

= B -or $S.SC

Note, i- t V t m mm ω cos)( = , t V

V mm

UT ω φ coscos2

=

cos φ a--ects t&e a+%litude o- t&e out%ut -or $S.SC'

i) - φ = 0, cos φ = 1, i.e. carrier %&ase o--set = 0

∴ 2

)(t mV UT =

ii) - φ = π2 (00), 02cos =π

0cos2

)(== φ

t mV UT B i.e. "ero out%ut

n eneral, as t&e local oscillator %&ase aries (assu+in t&e -re/uenc is o) t&e

amplitude o- t&e out%ut aries' 3s φ increases -ro+ 0 → π2 t&e out%ut a+%litude

decreases to "ero' (no*n as #adin$)' &en φ = ± !π2, *it& ! odd, t&e out%ut *ill #e

"ero'

". T&e in%ut to t&e snc&ronous de+odulator no* is SS.SC'

a)

t cω cos

P&

#*V OUT

SSSC

t V

V

t V

t V

t t V

t t V

V

mm

UT

m

m

mc

m

Differe!ce

mcc

S#m

mccm

cmcm

x

c

ω

ω ω ω

ω ω ω ω ω ω

ω ω ω

ω

cos4

cos4

)2cos(4

))(cos())(cos(2

1

2

cos')cos(

2

sinalMessaere+oes>[email protected]

=∴

+−=

−−+−+=

−=

∆Σ


8/42

6arts #) and c) can #e soled # considerin ∆ω and t&en φ' @or a eneral

solution consider a eneral local oscillator ))cos(('' φ ω ω +∆+= t L c , *&ere∆ω and φ +a #e %ositie or neatie'

[ ]

φ ω ω

φ ω ω φ ω ω ω

ω ω φ ω ω ω ω φ ω ω

φ ω ω ω ω

+∆+=∴

+∆+++−∆+=

−−+∆++−++∆+=∴

+∆+−=∴

t V

V

t V

t t t V

t t t t V

V

t t V

V

m

m

UT

m

m

mc

m

mccmcc

m

x

cmc

m

x

)cos((4

)cos((4

)2cos(4

)()cos(()()cos((2

1

2

))cos((')cos(

2

re+oes>6@

Note t&is is -or SS.SC (co+%are $S.SC in D8)' n t&is case ∆ω and φ o--sets

cause t&e out%ut to s&i-t in -re/uenc and %&ase'

#) @re/uenc o--set ∆ω, #ut %&ase o--set φ = 0'

t V

V mm

UT )cos(4

ω ω ∆+=∴

n t&is case t&e de+odulated out%ut is s&i-ted in -re/uenc # ∆ω

co+%ared to t&e in%ut t V mm ω cos '

@or ea+%le, i- ω+ = 1!" and ∆ω *as 200!" t&en t&e out%ut -re/uenc

*ould #e a sinal at 1200!"' @or s%eec& sinals, i- ∆ω is s+all #ut sta#le,

t&e -re/uenc s&i-t at t&e out%ut +a #e tolera#le'

c) No* consider ∆ω = 0 *it& a %&ase o--set φ

)cos(4

φ ω +=∴ t V

V mm

UT

n t&is case, i.e. -or SS., t&e out%ut is o--set # φ (Note -or $S., a %&ase

o--set *ill a--ect t&e a+%litude and cause -adin' @or s%eec& sinals a

%&ase o--set +a not #e er serious #ecause t&e ear is relatiel

insensitie to %&ase'

Note: n snc&ronous de+odulation *it& SS.SC in%ut, -re/uenc and

%&ase o--sets in t&e >'' can #e tolerated +uc& +ore t&an in $S.SC'

SS.SC is a %o%ular -or+ o- +odulation'


9/42

Communications

Module Code: EEE207



10/42

1) a)

T&e s%ectru+ at eac& %oint is s&o*n #elo*:

T&e out%ut, , is a sinle side#and su%%ressed carrier, SS.SC sinal, in t&is

case t&e lo*er side#and' T&e sinal is -re/uenc inerted'

%)

&' F &'( )t m

Carrier t C$s cω

f c=4G!"

ut%ut

( )t v s & C

( )t m

fre%

&

fre%

.and li+ited( )t m

fre%

f c

4 G!"

C

>S. HS.

*

f c

fre%

ut%ut ( )t v s

F &'

t C$s cω

4 G!"

( )t v sSS&SC

Vout

(t)V

( )t v s


11/42

T&e s%ectru+ at eac& %oint is s&o*n #elo*'

T&e out%ut sinal is t&e oriinal #andli+ited +essae sinal, '

a)

$iara+ to ie ( ) ( ) t C$st C$sV t V cm DC s ω ω '01+=

i'e' ( ) ( ) t C$st C$sV V t V cm DC DC s ω ω '0+=

%)

@ro+ a#oe , ( ) t C$sV t C$sV t m m DC mm ω ω '0==

i'e' ( )( ) v$&tsV V DC m 10'0'0 ===

Note also t&at +odulation de%t& = '0= DC

m

V

V

i'e' ( ) ( ) '0,1 =+= mt C$st C$smV t V cm DC s ω ω

f c

fre%

( )t v s

f c

fre%

V

fre%

Carrier

f c

fre%

fre%

Vout

(t)

F

t C$sV m DC ω '0 t C$s cω

( )t v s DC V

( )t m


12/42

c)

+ormali,ed a-era$e power"

1) 3%%l E/uation

21

2m

C T +Ρ =Ρ

&ere,

( )

2

2

DC C

V =Ρ

( ) ( ) ( ) ( )

( ) $'mi!(atts

mV

T

DC T

18892'150284'01

2100

2

'01

2

10

21

2

222

2

== +=Ρ

+=+=Ρ

Total 6o*er = 88 *atts (Nor+ali"ed 3erae 6o*er)

2)

6T = Carrier 6o*er HS. %o*er >S. %o*er

i'e'

( ) ( ) ( )

)(8850

84

84

2100

22222

22222

a)$veas(atts

V V V

t C$sV

t C$sV

t C$sV t V

T

mm DC T

mc

m

mc

m

c DC s

=++=Ρ

++= + + =Ρ

−+++= ω ω ω ω ω

d)

G!"10

G!"100

=

=

m

c

f

f

G!"0=

B mc f f

G!"110

mc f f +

G!"100

c f

vV m 42

=v4

vV DC

10=

fre%#e!c*


13/42

9'

a)

&m1(t )

!1

f 1 = 100kHz

&m2(t )1

f 2 = 110kHz

&m,(t )C1

f , = 120kHz

!2

2

C2

S".

S%ectru+ at eac& %oint

m1(t )

5kHz f ,kHz f

!1

10, f f c1

100

9

!2

5kHz f

m2(t )

,kHz f

1

11, f f c2

110

10

2

m,(t )

5kHz f ,kHz f

C1

12, f f c3

120

11

C2


14/42

10,f c1

100kHz

9 11,f c2

110kHz

10 12, f f c1120kHz

11

S".

Note t&is is si+ilar to >on ae (>) or Mediu+ ae (M) radio'

#) ;eceier

P&

"

0-,kHz

#".

SI

n t&e receier, *e +a tune t&e -re/uenc o- t&e local oscillator to select

*&ic& +essae *e *is& to receie'

>et >'' = t L ''cosω , i.e. -re/uenc = f L..

i) Tune t&e >'' to f L.. = f c2, i.e. f L.. = 110!"ii)

&

0-,kHz

#".

SI

t L ''cosω

#*

( )

t t t mt t t mt t t m

t t t mt t mt t m

t S V

Lc Lc Lc

Lccc

L +, x

''99''22''11

''992211

''

coscos)(coscos)(coscos)(

coscos)(cos)(cos)(

cos

ω ω ω ω ω ω

ω ω ω ω

ω

++=++=

=


15/42

t t m

t t m

t t m

t t m

t t m

t t m

V

Lc Lc

c Lc L

c Lc L x

)cos(2

)()cos(

2

)(

)cos(2

)()cos(

2

)(

)cos(2

)()cos(

2

)(

!"10110120

''99

!"290110120

''99

!"0110110

2''2

!"220110110

2''2

!"10100110

1''1

!"210100110

1''1

=−=+

=+=+

=−=+

−+++

−+++

−++=

ω ω ω ω

ω ω ω ω

ω ω ω ω

it& >'' set to 100!" s%ectru+ at V

P&

m2(t )

m1(t ) m,(t )

10

#*m1(t ) m2(t ) m,(t )

210 220 2,0 kHz

3-ter t&e >6@

2

)(2 t mV UT =

f

4'

a) Sste+

SS

t cω cos

#*

t 0cosω

#

m(t )

#C S".

t t t mt t V

t t t mV V

t t mV V

cc DC

c DC *

c DC x

00

0

coscos)(coscos

coscos))((

cos))((

ω ω ω ω

ω ω

ω

+=

+=

+=


16/42

t t m

t t m

t V

t V

V ccc DC

c DC

* )cos(2

)()cos(

2

)()cos(

2)cos(

2 0000 ω ω ω ω ω ω ω ω −+++−++=

.e-ore considerin t&e s%ectru+ ien # t&is e/uation, consider t&e sinals

#elo*'

50Hz 15kHzf

#C + m(t )

f c

f

f

Carrier f c

#*

"sf 0

#

SS

&i%ter

S".

f c

103Hzf c 415kHz f

c +15kHz

f 0

1003Hz

f 0

+ f c

1103Hz

f

f

f

f

f 0

1003Hz

f 0 4 f

c

903Hz

f 0

+ f c

1103Hz

2

DC V

2

DC V

T&e e/uation -or V * a#oe +a#e seen to #e consistent *it& t&e s%ectru+ -or

V *' T&e SS. %asses t&e su+ -re/uencies to t&e out%ut sinal S UT as s&o*n,

*&ere

t t m

t V

S cc DC

UT )cos(2

)()cos(

2 00 ω ω ω ω +++=

Note t&at t&e #ase#and sinal &as #een +odulated # a 10M!" carrier to

%roduce t&e $S.3M sinal at V x, t&en u%Bconerted (anot&er +odulation

stae) # a 100M!" oscillator to %roduce t&e dou#le side#and sinal centred

on 100 M!" at V *, t&en -iltered to %ass t&e HS., *&ic& in t&is case is t&e u%B

conerted $S.3M sinal, centred on 110M!"'

#) ;eceier$e+odulator si+%le'


17/42

P&

0-,kHz

#".

SI

t L1cosω t cω cos

#*

$a%

"si%%at$r f "1 = 1003Hz

4))(( DC V t m +

+++++=

+

++=

+=

==+=+=

t t t t mV

t t t mV

t t t mV V

t t f f

t t t t t mV V

t t t mV S

cc DC

c DC

c DC x

L L

c Lc DC x

c DC +,

)(2cos2

1

4

12cos

4

12cos

4

1

4

1))((

2cos2

1

2

12cos

2

1

2

1))((

cos'cos))((

)cos(cos*it&

cos'cos'cos'cos))((

cos'cos))((

00

0

0

22

1001

10

0

ω ω ω ω

ω ω

ω ω

ω ω

ω ω ω ω

ω ω

>6@ -ilter re+oes all co+%onents at 2ω0, 2ωc, etc. to ie

4

)(

4

t mV V DC UT +=

n t&is case >1 do*nBconerts t&e receied sinal to 10M!" and cos ωct and>6@ de+odulate to recoer m(t )'


18/42

Communications

Module Code: EEE207



19/42

1)

3udio sinal ( ) t C$st m mω 10= , V+ = 10 olts@re/uenc +odulator, α = 10 -/ %er olt'

a) 6ea deriation

I- c = α V+ = 10olt

G!"' 10 olts = 100 G!"

6ea deriation I- c = 100 G!"

#) Modulation inde, J =+-

-c∆=

+-

V+ α

+- 2π ω =m i'e' +- = 104 G!" = 10 G!", J = 10

G!"10

G!"100=

Modulation inde, J = 10

)

I- c = 1G!" *&en +- = 1 G!" , t&ere-ore Mod' nde, J =+-

-c∆= 1

Modulation inde, J = 1

a) Co+%onents in t&e @M s%ectru+ are -ound -ro+:

( ) ( ) ( )∑∞

−∞=

+=!

mc!c s t !C$s 0 V t v ω ω β

&ere Vc = 10 olts, J =+-

-c∆= 1

G!"1

G!"1=

T&e nt& %air o- t&e co+%onent is ( ) ( )t !C$s 0 V mc!c ω ω β + (n = e)

and ( ) ( )t !C$s 0 V mc!c ω ω β −− (n = B e)

@ro+ t&e ta#le o- t&e .essel -unctions and in t&is case usin t&e identit

( )β ! 0 − = ( ) ( ) 11 =− β β f$r 0 !!

n ( )β

!

0

Amp =( )β

!c

0 V

Frequency Hz


20/42

0 0'7852 7'852 fc

1 0'4400 4'40 mc f f +

B1 B0'4400 B4'40 mc f f −

2 0'114 1'14 mc f f 2+

B2 0'114 1'14 mc f f 2−

9 0'018 0'18 mc f f 9+

B9 B0'018 B0'18 mc f f 9−

Co+%onent -or n a#oe 9 &ae ( )β ! 0 K 0'01 and are considered insini-icant, andinored' T&e (B1) sin in t&e a+%litude indicates a %&ase o- 100'

#)

CarlsonLs rule a%%roi+ation . = 2(I- c - +) = 2 (1 G!" 1 G!")

CarlsonLs rule ies .= 4 G!" (Note a%%roi+ation)

c) >oad resistance ; > = 50 o&+s, Vc = 10 olts'


21/42

i) C&annel #and*idt& sini-icant @M s%ectru+, 8 G!" co+%onents outside

t&is #and*idt& are cutBo--L' To -ind t&e aerae %o*er receied need to -ind t&e

%o*er in t&e sini-icant s%ectru+'

Eac& co+%onent in t&e sinal &as a %ea a+%litude ( )β !c 0 V or ( )β !c 0 V −

3erae %o*er = ( ) ( )>

2

%ea

2

>

2

;MS

; 2

V2

;

V=

= L

1ea2

R

V

i'e 3erae %o*er =( )( )

>

2

; 2

β !c 0 V

-or n = e or e since O ( )( ) 2β ! 0 = ( )( ) 2

β ! 0 − P

Total %o*er in s%ectru+ 6T =( )( )∑

∞

∞−=!

!c 0 V

>

2

; 2

β

@or sini-icant side#and to n = a±

6o*er in s%ectru+ 6s =( )( )∑

−=

a

>

2

; 2

a!

!c 0 V β

6s =

( ) ( ) ( ) ( )

9210

)2((50)2

1=8'0)2(

(50)2

14='1)2(

(50)2

4'4

(50)2

852'72222

±=±=±==+++

!!!!

= 0'55591 0'972 0'028404 7'892 10B4

= 0'099 *atts

6o*er receied (i'e in t&e sini-icant s%ectru+) = 0'099 *atts

ii) &en t&e carrier is not +odulated'


22/42

6o*er = ( )*att1

)(502

102 2

2

==

L

1ea2

R

V

6o*er in un+odulated carrier = 1 *att

.ut Note in @M t&e carrier a+%litude is constant at Vc, onl - cL c&anes (i'e' - c ±

I- c ) and t&e %o*er is inde%endent o- -re/uenc'

T&ere-ore

( )( ) ( )*att1

2; 2

2

>

2

=== ∑∞

∞−= L

c

!

!cT

R

V 0 V P

β

)

Qien Vc= 10 olts, J =2 and also since *e are to -ind t&e %o*er, *e +a use( ) ( )β β !! 0 0 =− '

@ro+ .essel ta#les -or ( ) 01'02 ≥! 0

( ) 29='220 = 0 V c

( ) 878'521 = 0 V c

( ) 52'922 = 0 V c

( ) 2='129 = 0 V

c

( ) 940'024 = 0 V c

!ence s%ectru+ s&o*in +odulus o- a+%litudes is:

T&ere-ore, %o*er in s%ectru+ -or n u% to ± 4, -or ( ) 01'0≥β ! 0


23/42

( )( )∑−=

=4

4 >

2

; 2

!

!c s

0 V P

β

( ) ( ) ( ) ( ) ( )[ ]22222 94'022='1252'92878'5229='22

1×+×+×+×+=

L

s

R

P

*atts=47442'4

2

=49'=7

L L

s R R

P ==

Total %o*er in @M sinal

( ) ( )( )∑∞

∞−=

=

==!

!c

L

12

L

RMS T

0 V

R

V

R

V P

>

2

2

2

;

2 β

( ) *atts502100

210

2

L L L

T R R R

P ===

!ence,

( )

50

=47442'4

%o*er Total

01'0R-ors%ectru+in6o*er n =≥β

= 0'74

!ence, %ro%ortion o- total %o*er in s%ectru+ -or *&ic& onl sini-icant co+%onent

included 0'7 (7'


24/42

#) Vc= 10 olts' @ro+ .essel ta#les -or β =2

n Amp = ( )β !c 0 V FrequencyK Hz

0 2'29 fc = 100

1 5'787 mc f f + = 110

B1 5'787 mc f f − =0

2 9'52 mc f f 2+ =120

B2 9'52 mc f f 2− =0

9 1'2 mc f f 9+ =190

B9 1'2 mc f f 9− =70

4 0'940 mc f f 4+ =140

B4 0'940 mc f f 4− =80

c) Since Vc= 10olts %ea, 3erae %o*er = ( )>

2

;MS

; V

@M Sinal 6o*er =*att1

E502

0012

2

==

L

c

R

V

'

T&e %o*er in t&e s%ectru+ dra*n a#oe, *it& 4 side#and %airs *ill #e less t&an 1

*att, and is ien #


25/42

( )( )∑−=

=4

4 >

2

; 2

!

!c s

0 V P

β (See Duestion 9)

0)

a) Since Modulation nde β =+-

-c∆ and β = 5 is re/uired *it&

f m = 15 G!" (-ro+ ( ) t C$st m 9

101525 ×= π )

6ea $eiation I f c = β f m = (5) (15) = 75 G!"

#) Since I f c = α V m and V m = 5 olts,

@re/uenc conersion -actor α =+V-c

∆

=olts5G!"75 = 15 G!" %er olt'

c) @ro+ .essel ta#les, -or β =5 , ( ) 01'05 ≥is 0 ! -or n = 0 to , i'e t&ere *ill #e %airs o- sini-icant side#ands'

i'e'


26/42

CommunicationsModule Code: EEE207


1)

a)


27/42

P$er

3eter m = 13Hz

1 = 1kHz

2 = 10kHz

, = 103Hz

p0= 10-6

7atts/Hz

6o*er = ! B 10 , *&ere B! is t&e noise #and*idt&' e &ae to tae note *&ic&

is t&e s+allest #and*idt&'

6o*er +easured at (i) 101 B 1 , = (since B Bm)98

1 1010 ×= −

, = 1+att (0d.+)

6o*er +easured at (ii) 202 B 1 , = (since B2 Bm)98

2 101010 ××= −

, = 10+att (10d.+)

6o*er +easured at (iii) m B 1 , 09 = (since Bm B3)88

2 1010 ×= −

, = 1att (90d.+ or 0d.)

3ctual noise %o*er at (iii)88

90 1010 ×= − B 1 = 10atts (10d.)

Note 1: t is i+%ortant to #e a*are o- *&at #and*idt& noise is +easured in

(e.. *it& %o*er +eter or s%ectru+ analser), i.e. t&e sste+ #and*idt& or t&e

test #and*idt&'

Note 2: #sere &o* control o- t&e #and*idt& can reduce t&e noise (10atts

at 10M!" → 1 +att at 1!")' t is i+%ortant to &ae a #and*idt& Uust *ide

enou& -or t&e sinal, #ut no *ider in order to +ini+ise t&e noise %o*er and

+ai+ise t&e (SN),

#) .ot& -ilters are ideal, noise -ree, *it& power ains and 2, and #and*idt& B and B2'

811

822

!

p0 I ".

1 = 13Hz 2 = 1kHz

110 B5 1 , +, = = noise at


28/42

2210 )( B55 1 ,

UT = , since B2 B and since 1 is t&e sa+e an*&ere in t&e %at&,

allo*in -or ains'

.ut11

0 B5

, 1 +, =

=

=∴

1

22

11

221

B

B5 , ,

B5

B55 , ,

+, UT

+, UT

Note: , UT is not si+%l 2 , +, t&e #and*idt&s +ust #e taen into account'

Since( )

UT

UT UT

R

V ,

2

≡ ,( )

+,

+, +,

R

V ,

2

=

( ) ( )2

1

2

22

'' 5 B

B

R

V

R

V

+,

+,

UT

UT =

Matc&ed Sste+, i.e. RUT = R +,

2

1

2 5 B

BV V +, UT =

2)

8ain

8

(S/)".(S/)I

+,

UT

+,

UT

+,

+,

UT

UT

+,

+,

5,

,

5S

,

,

S

S

,

,

S 7 === ''

I ". = 8I + a8

a

, a is added noise *&ic& a%%ears at t&e out%ut'

I ". = 8(I + e)

8

e

, e is t&e e/uialent noise re-erred to t&e in%ut, *&ic& ies , a at out%ut, i.e. , a = , e


29/42

)(, e +, UT +,

UT , , 5 , 5,

, 7 +≡=

i.e.

+=

+=

+,

e

+,

e +,

,

,

5,

, , 5 7 1

)(

i.e. +, e , 7 , )1( −=

Since , = TB, assu+in sa+e #and*idt& B

B2T 7 B2T +, e )1( −=

+, e T 7 T )1( −= (*&ere T +, is t&e source te+%erature T s)i.e. se T 7 T )1( −= , *&ere T +, = T s is usuall taen to #e 20G'

9)a)

!

1

$ss = ,

1 = 2

81 = :&1 = 2

.e1 = 290K

28 = 10

& = 6

82 = 10

&2 = ;

.e2 =


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-

55

T

5

T T T T T

c s2*

eeee s2* s*s

1=501450500

947701455004

1'10

70

10

7070

2

12=0

2

11000

;e

92

4

2

921

=+=++++=

+++

+

=

++++=

;e-erred to & (sste+ e/uialent noise te+%erature)

Note: as *e *ould e%ect t&is is eactl &al- t&e sste+ te+%erature

re-erred to t&e ca#le &as ain o- (9d. loss) and noise %o*er is

%ro%ortional to T s*s, i.e. %o*er at & = %o*er at '

#) >o* noise %rea+% installed:

%tion (a) note reBnu+#er t&e ele+ents

!

1

&1 = 2

81 = 10

.e1 = 290K

2

&2 = 2

82 = :

.e2 = 290K

,

&, = ;

8, = 10.e, =


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-

5555

T

555

T

55

T

5

T T T T

c s2*

eeeee s2* s*s

219111911000

8'8=4'17174502=01000

4

1'10'10'

2

1

70

10'10'2

1

70

10'2

1

70

2

1

2=0

2=01000

;e

4921

5

921

4

21

9

1

21

=+=+++++=

+

+

+

++=

+++++=

;e-erred to , o%tion (#)' T&is is #etter t&an no %reBa+% #ut not as ood as

o%tion (a)

%tion (a) T s*s = 150G

%tion (#) T s*s = 2191G

No %rea+% T s*s = 900G

!ence, -or #est noise %er-or+ance, t&e +ast &eadL location is t&e #est

solution'

T&is solution can also #e in-erred -ro+ t&e e/uation

+++=21

9

1

21;e

55

T

5

T T T eeec

To ee% T Rec s+all, t&e ain o- t&e -irst stae s&ould #e W 1 (i.e. an a+%li-ier rat&er t&an a ca#le)' Successie noise contri#utions are t&en reduced'

Note: >o* noise (receiers) is not t&e onl consideration' Too +uc& ain at

t&e -ront end, *&ic& is wide open (a *ide #and*idt&) to noise and

inter-erence can oerdrie or saturate later staes, e.. t&e +ier, and cause

%ro#le+s due to nonBlinear distortion and inter+odulation %roducts' n so+e

receiers t&e aerial is connected strai&t to t&e -irst +ier' T&e %ri+e

considerations are t&e /ualit o- t&e sinal at t&e out%ut in ter+s o- (SN) and

distortion'

4)

a) n eneral eac& T e is re-erred to in%ut'


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81.e1

82.e2

8,.e,

8;.e;

÷ 81

÷ 8182

÷ 81828,

A

Noise %o*er is %ro%ortional to T e ( , = T e B)' T&ere-ore, to re-er to T e earlier

staes, diide # ain o- %recedin staes as s&o*n'

i.e. T Rec re-erred to is ++++=921

4

21

9

1

21;ec

555

T

55

T

5

T T T eeee

#)

i) Note: conert all d.L to ratios'

A

= ;& = 28 = 10 =6 & = 6n = 250kHz

1&1 = 2512

1 = 2512

.e1 = ;,


33/42

iii) T&e receier, T Rec, at includes t&e ca#le'

A

R>C

Since T e = (1 7 )T +, , +,

e

T

T 7 +=1

@actor)(Noise47'20100

1=4711

;ec

;ec =+=+= +,

e

T

T 7

Noise -iure 7 d. = dB 7 1'19)47'20(lo10lo10 1010 ==

c)

i)

A

R>C

.sk = 100K

7 = ,03Hz = ae

! = 250kHz

(S/)".

3ctual (SN) at = 5=4'1

1090100101'9

atts108'8829B

14

=

××××

×=

−

ae s2* B2T

S

3ctual (SN) at is 1'5, i.e. sinal Uust a#oe t&e noise'

(SN) at +easured *it& an instru+ent *it& a 250!" #and*idt& is:

9'1=1000,250100109'1

108'829

14

=×××

×−

−

i.e. in t&e sa+e #and*idt& (SN)N WW (SN)HT, as *ill #e s&o*n'

ii)( )

)7'=(948'=000,2502047109'1

108'8

3tore-erred

3at+easured

29

14

dB

T

S

B2T

S , S

s*s s*s

UT

≡=×××

×=

=

−

−

iii) Noise %o*er s%ectral densit re-erred to in%ut, -ro+ , = T s*s B,

1 = T s*s = 1'9×10B29×2047 = 2'25×10B20 atts!"

3ctual %o*er s%ectral densit at out%ut *ill #e 1 HT = 1×ain o- receier


34/42

atts!"1011'2)1000(4

1)10(

512'2

110525'2 1720

−− ×=

××=

T&e ratio (S 1) re-erred to is (also sa+e at out%ut)

)8'89(109989'21025'2

108'8 820

14

dB≡×=××

−

−


35/42


36/42


37/42

ii)

Sinle #it error correction i'e success-ul trans-er i- no errors or 1 error in

+essae' t&er*ise -alse trans-er since -urt&er error detection not carried out'

6ro#a#ilit o- success-ul trans-er = %(0) %(1) = 6sucess = 0'4904872 0'928975

6ro#a#ilit o- Success-ul trans-er = 0'191047

3ll ot&er +essaes are acce%ted, i'e' none reUected

6ro#' o- -alse trans-er = 18=59'01)(1

=−=∑=

s#ccess

,

R

P R 1

ONote 1< o- t&e in-or+ation acce%ted is *ron t&is is #etter t&an (i) #ut

still not oodP

iii)

Code *&ic& can correct sinle error in #loc and detect 2 errors'

i'e' No errors +essae acce%ted correct

1 error +essae correctedacce%ted correct

2 errors errors detected +essae reUected lost

9 or +ore errors errors not detected +essae acce%ted -alse

6ro#a#ilit o- success-ul trans-er = %(0) %(1) = 0'19047

6ro#a#ilit o- lost trans-er = 1(2) (2 errors detected, +essae reUected)

1(2) = 0'14094

@alse trans-er occurs i- t&ere are 9 or +ore errors'

6ro#a#ilit o- -alse trans-er = )P2()1()0(O1)()9(9

1 1 1 R 1 1 ,

R

++−==≥ ∑=

6ro#a#ilit o- -alse trans-er = 0'0901

ONote in t&is case 9'< o- t&e +essaes trans+itted are acce%ted and are

-alse t&is is a#out 4'47< o- t&e +essae acce%tedP

)

T&e +ini+u+ distance o- t&e code is t&e +ini+u+ no' o- #its c&ane, to

conert one alid code*ord in t&e code to anot&er alid code*ord'

@or ea+%le Code 1 Code 2 Code 9

Valid Code*ord 3 01101 1111 010101

Valid Code*ord . 00001 0000 101010

!a++in $istance 2 4 8


38/42

T&e distance #et*een alid code*ord in a code is called t&e &a++in

distance' 3ll code *ords in a code are not se%arated # t&e sa+e !a++in

distance T&e +ini+u+ alue o- t&e &a++in distance in a code is called t&e

+ini+u+ distance (d+in or d )'

@or a code *it& d+in=5, usin d+in = t 8 & 8 1, t K & *&ere t = no o- #itscorrected, &= no o- #it error detected'

d+in = t 8 & 8 1

5 = 0 4 1 detect u%to 4 errors (dB1)

5 = 1 9 1 detect u%to 9 errors , correct 1 error

5 = 2 2 1 detect and correct u%to 2 errors

n eneral a code can

$etect u% to (d+in B1) errors,

Correct u% to NT

−

2

1+ind errors'

)

% = 10B2 =0'01

SXNCL N@ and C!ECGL

a)

Snc&roni"ation #its are not included in t&e error detection correction %rocedures, i'e' all snc #its are to #e receied error -ree -or sncL'

6ro#' o- Success-ul snc= 6ro# o- no errors in #its = %(0)

&ere %(0) = (1B%)S = (1B%) = (1B0'01)

6ro#' o- Success-ul snc = 0'227447

#)

Success-ul %acet trans-er re/uires success-ul snc and a correct %acet'

@or correct %acet, re/uire 24 #its *it& no errors, or 1 error (*&ic& can #e

corrected)'

6ro#' o- correct %acet = %(0) %(1)

= (1B%)24 124C %1 (1B%)24B1

= (0')24 24 (0'01) (0')29

= 0'781455

6ro#a#ilit o- success-ul %acet trans-er = 6ro#' o- success-ul snc and 6ro#'

o- correct %acet= %(succ' snc ) ' %(correct %acet)

= / %itsS= 2 %its


39/42

= 0'227447 Y 0'781455

= 0'007991

6ro#a#ilit o- success-ul %acet trans-er = 0'007991

/)

TV sets, -ailure rate = 10B2

a)

50 TV sets %roduced, i'e N=50 , % = 10B2

6ro#a#ilit t&at all 50 are ood in t&e %ro#a#ilit o- no -ault ones, i'e'

6(0) = (1B%) N

6(0) = (1B10B2)50 = 0'805008

T&e %ro#a#ilit o- #ein a#le to delier an order -or 50 V sets i- onl 50 are+ade is onl 0'805 (80'5


40/42

Communications

Module Code: EEE207


1) $iscussion &o* sinle %arit #it codes +a #e used -or error detection see

notes'


41/42

2) $iscussion on re%etition codes +aUorit ote decodin and a%%lication to

error detectioncorrection see notes'

9) Messae, #its M 9 M 2 9 M 3 9 :9 M " 9 P trans-erred ia a c&annel *it& error

rate 1 = 10B2 = 0'01

a) n t&is case, success-ul trans-er occurs i- t&e ei&t #it +essae is

receied *it& no errors, i.e. =227448=5'0)1()0( =−== 1 P P s

#) @alse trans-er occurs i- errors are not detected, i.e. t&e +essae is

acce%ted, #ut it contains undetected errors' n t&is case, -or a sinle

%arit #it:

∑=

=+++=eve! R

7 R P P P P P P )()()8()4()2(

Since t&e %arit code cannot detect een errors, i.e. 2, 4, 8,

180

111928

8

7=44

4

9582

2

10)01'0()1()(

107442'21001'=2)1()8(

1072417207'810805=801'=70)1()4(

1089814441'210414'=2)1()2(

−

−−

−−

−−

===−=

×=××=−=

×=××=−=

×=××=−=

1 1 1C P

1 1C P

1 1C P

1 1C P

i.e. 9

108981889'2)( −

=

×== ∑eve! R

7 R P P

c) Messaes are lost or reUected i- errors are detected' n t&is case a %arit

code can detect all odd errors

)7()5()9()1()( P P P P R P P $dd R

L +++== ∑=

e could calculate t&ese as -or P 7 , #ut since

( ))()8()4()2()0(1)7()5()9()1(

1)()7()8()5()4()9()2()1()0(

P P P P P P P P P

P P P P P P P P P

++++−=+++

=++++++++

i.e. P L = 1 ( P S P 7 ) = 1 (0'2274485 2'898189×10B9)

P L = 0'074815

4)

a) @or a ;e%B5 code, *it& 1 = 0'1

9554

5429

1058'1010=5101'10

)1(5)1(10)5()4()9(

−−−− ×=+××+××=

+−+−=++=

UT

UT

P

1 1 1 1 1 P P P P

n t&is case, all +essaes are no* acce%ted, eit&er correct or -alse


42/42

i.e. P S = P (0), %ro#a#ilit o- no errors in an ei&t #it +essae su#Uect to

an error rate P UT .

6ro#a#ilit o- success P S = P (0) = (1 P UT ) = 0'995980

#) 6ro#a#ilit o- -alse trans-er = P (1) P (2) P (9) Z P ()

.ut P s P 7 = 1, i.e. P 7 = 1 P S = 0'0884890

EEE207 Tutorial Solutions.doc

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