Communication SystemsyLecture 25

D I KiDong In KimSchool of Info/Comm Engineering

Sungkyunkwan University

1

Outline

Noise in Angle Modulation Phase deviation

Large SNR Small SNR

O SNR Output SNR PM FM FM

2

R i f A l d l iReview of Angle Modulation

General form of angle modulated signal:( ) ( )cosx t A t tw fé ù= +ë û( ) ( )cosc c cx t A t tw fé ù= +ë û

( ) ( )PM: cosc c c p nx t A t k m twé ù= +ë û( ) ( ) : normalized message with max 1.n nm t m t =

( ) ( )FM: cos 2t

x t A t f m dw p a aé ù= +ê úò( ) ( )FM: cos 2c c c d nx t A t f m dw p a a= +ê úê úë ûò

Transmitted Power: 21P = ATransmitted Power: Demodulation output:

( )1 d tf

TP = .2 cA

3( ) ( ) ( )

( )1PM: , FM: 2D D D D

d ty t K t y t K

dtf

fp

= =

i i A l d l iNoise in Angle Modulation( )( )cx t +

N i (t)

PredetectionFilter

( )1e tDemodulation

FilterPostdetection

R d iNoise w(t):

AWGN N0/2

Reduce noise(explained later)

Bandwidth of the predetection filter: Carson’s rule: BT = 2(D+1)W.

( ) ( )From cos ,c c cx t A t tw fé ù= +ë û

To analyze the effect of the noise, we need to write

( ) ( )1( ) ( )cos ,c ee t R t t t tw q f fé ù= + + +ë û4

( ) ( )( ) ( )where is the noise added to .e t tf f

ë û

i i A l d l iNoise in Angle Modulation

Predetection filter output:( ) ( ) ( ) ( )cos ( )cos ( )sine t A t t n t t n t tw q f w q w qé ù= + + + + - +ë û( ) ( ) ( ) ( )1 cos ( )cos ( )sinc c c c s ce t A t t n t t n t tw q f w q w q= + + + + +ë û

( ) ( )( )cos ( )cosc c n c nA t t r t t tw q f w q fé ù= + + + + +ë û

( ) ( ) ( ) ( )( )cos ( )cosc c n c nA t t r t t t t tw q f w q f f fé ù= + + + + + - +ë û

( ) ( )( ) ( ) ( )( )( )( ) ( ) ( )( )

cos ( )cos cos

( )sin sinc c n c nA t t r t t t t t

r t t t t t

w q f w q f f f

w q f f f

é ù= + + + + + -ë û- + + -( )( ) ( ) ( )( ) ( )sin sinn c nr t t t t tw q f f f+ +

( ) ( )( )( ) ( )( )cos cosc n n cA r t t t t tf f w q fé ù= + - + +ë û

5( )( ) ( ) ( )( ) ( )sin sinn c nr t t t t tw q f f f- + + -

i i A l d l iNoise in Angle Modulation( ) ( )( )( ) ( )é ù( ) ( )( )( ) ( )

( )( ) ( ) ( )( )1( ) ( )cos cos

( )sin sin

c n n c

n c n

e t A r t t t t t

r t t t t t

f f w q f

w q f f f

é ù= + - + +ë û

- + + -

( ) ( )( )cos c eR t t t tw q f fé ù= + + +ë û

( )( ) ( ) ( )( )( )n c nf f f+ +

( )( ) ( )( )

( ) ( )( )1 ( )sin

tan( )cos

n ne

c n n

r t t tt

A r t t tf f

ff f

--

=+ -( ) ( )( )( )c n nf f

Phase deviation from the carrier:( ) ( )( ) ( )( ) et t ty f f= +

Only phase deviation affects the demodulation.6

y p Amplitude can be kept constant by a limiter (pp. 143).

Angle Modulation with High SNR

( )( ) ( )( ) ( ) ( )( )1 1( )sin ( )sin

t tn n n nr t t t r t t tt

f f f ff - -

- -

High SNR: ( )c nA r t most of the time.

( )( ) ( )( )

( ) ( )( )( ) ( )( )

( ) ( )( ) ( ) ( )( )

1 1tan tan( )cos

( )sin ( )sin

ecc n n

tAA r t t t

r t t t r t t t

ff f

f f f f

= »+ -

- -( ) ( )( ) ( ) ( )( )1 ( )sin ( )sin sin n n n n

c c

r t t t r t t tA Af f f f

-» »

( ) ( ) ( ) ( ) ( )( )( )( ) sinne n

c

r tt t t t t tA

y f f f f f= + » + -

increases then effect of ( ) reduces!A r t

Still dominated by the signal ( )tf

7

increases, then effect of ( ) reduces!c nA r t

A l d l i i h L S RAngle Modulation with Low SNRLow SNR: ( )A t t f th tiLow SNR: ( )c nA r t most of the time.

( ) : dominated by ( ).t ty f( ) : dominated by ( ).nt ty f

( ) ( ) ( ).nt t ty f a= -

( )ta

( ) ( ) ( )ny f

Need expression of ( )ta( )ta p ( )tain terms of ( ) and ( ).n t tf f

( ) is wrong in Fig. 6.13.e tfNote:

8

( ) g gef

A l d l i i h L S RAngle Modulation with Low SNRLow SNR: ( ) ( ) ( )y fLow SNR: ( ) ( ) ( ).nt t ty f a= -

( ) very small if ( ).c nt A r ta

( ) sin ( ) tan ( )t t ta a a» »

( )ta

sin ( )( )

c

n

A tr t

q»

( )ta ( )n

( ) ( )nt ty f»

( ) sin ( ) ( )( )c

nn

A t tr t

f f- -

9

( )n

A l d l i i h L S RAngle Modulation with Low SNR

SLow SNR:

( )( ) ( ) sin ( ) ( )cAt t t ty f f f» ( )( ) ( ) sin ( ) ( )( )n n

n

t t t tr t

y f f f» - -

Message signal is lost at low SNR! Threshold effect Threshold effect.

10

A l d l iPhase

Angle Modulation

( )é ù

deviation

High SNR:( )1( ) ( )cos ce t R t t tw q yé ù= + +ë û

High SNR:( ) ( )( )( )( ) ( ) sinn

nr tt t t ty f f f» + - (1)

Low SNR:

( ) ( )( )( ) ( ) ncA

y f f f+ ( )

Small most of the time

( )( ) ( ) sin ( ) ( )( )c

n nAt t t t

r ty f f f» - - (2)

( )nr t

Note: Eq (2) can be obtained from (1) by switching11( ) ( ), and ( ).n c nt t A r tf f« «

Outline

Noise in Angle Modulation Phase deviation

Large SNR Small SNR

O SNR Output SNR PM FM FM

12

O S RPM Output SNR( ) ( ) ( ) ( )( ) ( ) iq f q qé ù( ) ( ) ( ) ( )1 cos ( )cos ( )sinc c c c s ce t A t t n t t n t tw q f w q w qé ù= + + + + - +ë û

( ) ( ) ( )( )cos ( )cosc e cR t t t t R t t tw q f f w q yé ù é ù= + + + + +ë û ë û( ) ( ) ( )( ) ( )c e cf f y+ + + + +ë û ë û

( ) ( ) ( )( )( )( ) sinnn

r tt t t tA

y f f f» + -High SNR: ( ) ( ) ( )( )( ) ncA

y f f fg

PM Demodulation output:( ) ( ) ( ) ( )

( ) ( ) ( )( )( )D D D D ey t K t K t K t

r ty f f= = +

( ) ( ) ( )( )( )

( ) sin

P

nD D n

cn t

r tK t K t tA

f f f= + -

13( )( )

( )P

D p n PK k m t n t= +

O S RPM Output SNR

( ) ( ) ( ) ( )( )( ) sinnr ty t K k m t K t tf f= +

PM Demodulation output:( ) ( ) ( ) ( )( )

( )

sin

P

D D p n D nc

n t

y t K k m t K t tA

f f= + -

Demod Signal power: 2 2nDP D p mS K k P=

To compute noise power: let ( ) 0 (message is 0)tf =

( )( ) ( )( ) sinn sP D n D

c c

r t n tn t K t KA A

f= =( )t

( )sn t( )nr t

14

c c ( )cn t

O S RPM Output SNRS (f)R ll NB N i

ff f +B/2

Sn(f)N0/2

Recall: NB Noise

fc fc+B/2Sn(f-fc)

f

Sn(f+fc)

f

( ) ( ) ( )é ùB/2

( ) ( ) ( )Lpsn n c n cS f S f f S f fé ù= - + +ë û Sns(f)

N0

15fB/2

O S RPM Output SNR

( ) ( ) ( )D D p n Py t K k m t n t= + PM Demodulation output:( )( ) sn tK

2

( ) DKS f N( )( ) s

P Dc

n t KA

= 02( )P

Dn

c

S f NA

=

Spectrum of n (t) is in [ 1/2 B 1/2 B ] Spectrum of ns(t) is in [-1/2 BT, 1/2 BT] BT: Bandwidth of predetection filter (by Carson’s rule).

Output noise power: 2K 2

02D

DP Tc

KN N BA

=

16

O S RPM Output SNRP t d t ti filt Post-detection filter: Since BT= 2(D+1)W > 2W, the output noise power T ( ) , p p

can be reduced by applying a post-detection filter with bandwidth W

PredetectionFilter

( )1e tDemodulation

FilterPostdetection

Filter Filter

2DKN N B=

2

022 DDP

KN N WA

=

02DP Tc

N N BA

=

17

02DPcA

O S RTransmitted power

PM Output SNR1 2P A

Per unit message bandwidth

Transmitted Power: 2T

1P = .2 cA

O t t i

2T

0 0

P =N W 2N W

cA

1222 D TK PN N W K

-æ ö÷ç= = ÷ç

Output noise power:

020

2 .DP Dc

N N W KA N W

= = ÷ç ÷ç ÷è ø

Increasing transmitted power reduces output noise power!g p p pDifferent from linear modulation. PM D d O t t SNR

( )2 2

2nD p m TK k P PSNR k P

PM Demod Output SNR:

18

( )( )( ) 12

00/n

n

p Tp mDP

D T

SNR k PN WK P N W

-= =

O S RPM Output SNR( ) ( )é ù

( ) 2 TPSNR k P= PM Demod Output SNR:( ) ( )PM: cosc c c p nx t A t k m twé ù= +ë û

( )0

np mDPSNR k P

N W PM Demod Output SNR:

( ) sin or cosm mm t t tw w=Single-tone Modulation

: Modulation Indexpkb ( ) sin sinp m mt k t tf w b w=

P( ) 2

0

: increasing increases output SNR.n

TmDP

PSNR PN W

b b=

( )B t b d idth 2 1 i l i dB Wb+( )But bandwidth 2 1 is also increased.TB Wb= +

Input noise power will be increased.

19Eventually large input SNR assumption is invalid threshold effect.

Outline

Noise in Angle Modulation Phase deviation

Large SNR Small SNR

O SNR Output SNR PM FM FM

20

O S RFM Output SNR( ) ( ) ( ) ( )( ) ( ) it A t t t t t tq f q qé ù+ + + + +( ) ( ) ( ) ( )1 cos ( )cos ( )sinc c c c s ce t A t t n t t n t tw q f w q w qé ù= + + + + - +ë û

( ) ( ) ( )( )cos ( )cosc e cR t t t t R t t tw q f f w q yé ù é ù= + + + + +ë û ë û( ) ( ) ( )c e cë û ë û

( ) ( ) ( )( )( )( ) sinnn

r tt t t tA

y f f f» + -High SNR:

FM Demodulation output:( )

( )D d tK y

cA

( )( )

( )( ) ( )( )

2( )

DD

Ky tdt

d t r tK K d

yp

f

=

ì üï ïï ï( )( ) ( )( )

( )

( ) sin2 2

F

nD Dn

c

n t

d t r tK K d t tdt dt Af

f fp p

ï ï= + -í ýï ïï ïî þ

21( )( )

( )Fn t

D d n FK f m t n t= +

O S RFM Output SNR

FM Demodulation output:

( ) ( ) ( ) ( )( )( ) sinnD r tK dy t K f m t t tf fì üï ïï ï= + í ý( ) ( ) ( ) ( )( )

( )

sin2

F

D D d n nc

n t

y t K f m t t tdt A

f fp

= + -í ýï ïï ïî þ

Demod Signal power: 2 2nDF D d mS K f P=

To compute noise power: let ( ) 0 (message is 0)tf =

( )( )r t( )( )sin ( )r t t dn tK Kd fì üï ïï ï

( )cn t

( )sn t( )nr t( )( )sin ( )( )2 2

n n sD DF

c c

r t t dn tK Kdn tdt A A dt

fp p

ï ï= =í ýï ïï ïî þ( ) : quadrature component of noisen t

22How to find noise psd?( ) : quadrature component of noise.sn t

O S RFM Output SNR

( )( )sin ( )( )2 2

n n sD DF

c c

r t t dn tK Kdn tdt A A dt

fp p

ì üï ïï ï= =í ýï ïï ïî þc cî þ( ) 2 ( )dx t j fX f

dtp« d/dt is a linear filter

( )sn t ddt

( )u t

dt

dt2( ) ( ) ( )u nsS f H f S f=

( )2 2

2 20 02

1 1( ) 2 , f , .2 2 2

D DNF T T

c c

K KS f f N N f B BA A

pp

æ ö é ù÷ç= = Î -ê ú÷ç ÷ç ÷ ê úè ø ë û23

c cè ø ë û

O S RFM Output SNR2

202

1 1( ) , f , .2 2

DNF T T

c

KS f N f B BA

é ù= Î -ê ú

ê úë û

The noise has less effect on low-freq message signals.

After Post-detection filter with bandwidth W:2 2

2 32WD DK KN N f df N Wò

24

2 30 02 2 .

3D D

DF Wc c

N N f df N WA A-

= =ò

O S RFM Output SNR1 2P A

Transmitted Power: 2T

1P = .2 cA

Output noise power:

2T

0 0

P =N W 2N W

cA

Output noise power: 12 2 2

3 T2 PD DK K WN N W-æ ö÷ç= = ÷ç

Noise power is inversely proportional to TPN W

0203 3 N WDF

c

N N WA

= = ÷ç ÷ç ÷è ø

p y p p0N W

22 2K f P æ ö FM Demod Output SNR:

( )22 2

12 20T

3P

n

n

D d m d TmDF

D

K f P f PSNR PW N WK W

-

æ ö÷ç= = ÷ç ÷ç ÷æ ö è ø÷ç25

T

03 N WD

è ø÷ç ÷ç ÷ç ÷è ø

O S RFM Output SNR

( )2

3 d Tf PSæ ö÷ç

FM Demod Output SNR:

( )0

3n

d TmDF

f PSNR PW N W

÷ç= ÷ç ÷ç ÷è øR ll D i ti R ti f l (t)

max '( )peak frequency deviation tD

f= =

Recall: Deviation Ratio for general m(t):

bandwidth of m(t)D

W= =

( ) ( )FM: cos 2t

x t A t f m dw p a aé ù= +ê úò

( )maxd n df m t fD

( ) ( )FM: cos 2c c c d nx t A t f m dw p a a= +ê úê úë ûò

( ) 23 TPSNR D P26

( ) dfDW W

= = ( )0

3n

TmDF

SNR D PN W

=

O S RFM Output SNR

FM Demod Output SNR:

( ) 23 TPSNR D P( ) 2

0

3n

TmDF

SNR D PN W

=

F D 1 B 2(D+1)W 2DW TFor D 1, B =2(D+1)W 2DW:»

( )23 T TB PSNR P

æ ö÷ç= ÷( )04 nmDF

SNR PW N W

ç= ÷ç ÷çè ø

Increasing bandwidth can improve the output SNRInput noise power will be increased.

Increasing bandwidth can improve the output SNR.

27

Eventually large input SNR assumption is invalid threshold effect.

Pre-emphasis and De-emphasis for Noise

+ Pre DemoddetectionPost

detectionModPre

emphasisDe-

emphasis

( ) ( ) ( ) ( )( )( )D r tK d ì üï ïï ï

Noise f f

( ) ( ) ( ) ( )( )( )

( ) sin2

F

nDD D d n n

c

n t

r tK dy t K f m t t tdt A

f fp

ï ï= + -í ýï ïï ïî þDemod output:

Unmodulated carrier: ( )F

( ) 0 (message is 0)tf =

( )( )sin ( )r t t dn tK Kd fì üï ïï ï( )( )sin ( )( )2 2

n n sD DF

c c

r t t dn tK Kdn tdt A A dt

fp p

ï ï= =í ýï ïï ïî þ2 1 1K é ù( )d

28

202

1 1( ) , f , .2 2

DNF T T

c

KS f N f B BA

é ù= Î -ê ú

ê úë û( ) 2 ( )dx t j fX f

dtp«

Pre-emphasis and De-emphasis for Noise

+ Pre DemoddetectionPost

detectionModPre

emphasisDe-

emphasis

22 1 1( ) fDKS f N f B B

é ù= Î -ê ú

Noise f f

02( ) , f , .2 2

NF T Tc

S f N f B BA

= Î ê úê úë û

Assume de-emphasis filer is a 1st-order lowpass RC 1filter:( )2

3

1( )1 /

DEH ff f

=+

3f W

Noise power after de-emphasis filter:2( ) ( )

WN H f S f dfò

29

2( ) ( )DF DE NFWN H f S f df

-= ò

12 2

1 1 tan udua u a a

-=+ò

Pre-emphasis and De-emphasis for Noisea u a a+

+ Pre DemoddetectionPost

detectionModPre

emphasisDe-

emphasis

Noise power after de-emphasis filter:

Noise f f

Noise power after de emphasis filter:2( ) ( )

W

DF DE NFWN H f S f df

-= ò

( )

2 2 222

0 0 322 2 2 2331 /

W WD D

W Wc c

K K ffN df N f dfA A f ff f- -

= =++ò ò( ) 33

2 2 23 1 3 2

0 3 0 3 0 32 2 2

1 /

2 tan 2 22

c c

D D D

f ff f

K K KW W WN f N f N f WA f f A f A

p-

++

æ ö æ ö÷ ÷ç ç= - » - »÷ ÷ç ç÷ ÷ç ç÷ ÷30

0 3 0 3 0 32 2 23 3 3 2c c cA f f A f A÷ ÷ç ç÷ ÷è ø è ø

2T

0 0

P =N W 2N W

cA

Pre-emphasis and De-emphasis for Noise

+ Pre DemoddetectionPost

detectionModPre

emphasisDe-

emphasis

( ) ( ) ( )D D d n Fy t K f m t n t= +

Noise f f

2KDemod Signal power:

2 2nDF D d mS K f P=

( ) ( ) ( )D D d n Fy f

Noise power after de-emphasis: Output SNR:

22

0 322 DDF

c

KN N f WA

»

( ) ( )2 P Output SNR:( ) ( )2

30

/n

Td mDF

PSNR f f PN W

=

PSNR without deemphasis:

31( ) ( )2

0

3 /n

Td mDF

PSNR f W PN W

=

Pre-emphasis and De-emphasis for Noise

+ Pre DemoddetectionPost

detectionModPre

emphasisDe-

emphasis

Output SNR with de-emphasis:

Noise f f

Output SNR with de emphasis:( ) ( )2

3_0

/n

Td mDF DE

PSNR f f PN W

=0

Output SNR without de-emphasis:( ) ( )2

3 / TPSNR f W P=

SNR can be improved significantly3f W

( ) ( )0

3 /nd mDF

SNR f W PN W

=

32

p g yby de-emphasis filter.

Pre-emphasis and De-emphasis for Noise

( )2 P SNR with de-emphasis:

SNR ith t d h i

( ) ( )2

3_0

/n

Td mDF DE

PSNR f f PN W

=

SNR without de-emphasis:( ) ( )2

3 /n

Td mDF

PSNR f W PN W

=

Example: FM: 75 , W=15kHz, D = 5.df kHz=0N W

n3 m 2.1 , P =0.1.f kHz=

( ) ( )23 75/15 0 1 7 5T TP PSNR = =

( ) ( )275 / 2.1 0.1 128T TP PSNR = =

( ) ( )0 0

3 75/15 0.1 7.5DF

SNRN W N W

= =

33

( ) ( )_0 0

75 / 2.1 0.1 128DF DE

SNRN W N W

Transmitted power can be reduced by 17 times, with the same noise performance.