noise in modulation systems - vlsi signal processing lab, ee, nctu

Principles of Communications I (Fall, 2007) Noise in Modulation Systems

NCTU EE 1

Noise in Modulation Systems SNR — Signal-to-Noise Ratio：(S/N)

One of the most important parameters in communication systems. (The other is B.W.)

point certaina at

powernoisepowerSignalSNR =

channel model：additive, white, Gaussian noise with power

(AWGN) (independent of signal)

spectral density 2

0N

N0 / 2

Receivermessage+

+ A B

n(t) noise

signal(containingmessage)

Compare different receiver (modulation)： Assume the same “SNR” at A, which one has better SNR at B.

【Note】：The power spectral density is 2

0N . ( some books use 0N )

Physical meaning： 0N is the average noise power per unit bandwidth

(single-sided psd) at the front end of the receiver.


NCTU EE 2

Noise in Linear Modulation Baseband model：a basis for comparison.

Message signal power = TP (watts) (transmitted power)

At the filter input

BNdfNB

B 0021

=∫− , BNP

SNR Ti

0

)( =

At the filter output

WNdfNW

W 0021

=∫− , WNP

SNR To

0

)( =

SNR enhancement WB

SNRSNR

i

o =)()(

This is a basic assumption in many communication system：

we, at least, filter out the out-of-band noise. (This filtering usually does not change signal (power)). 【Note】：The signal power is often referred to the “transmitted” signal.

(not message itself) (modulated)


NCTU EE 3

A. DSB-SC Systems：Coherent Detection

xr(t) = xc(t) + n(t)　Post det.　LF

Ａ

2 cos(wct + q )

ｙD (x)e3 (x)e2 (x) Predetection filter

( BPF )

f0 fc- fc

BPF

2W2W

N0 / 2

SXc(f)

0 f

LPF

fc 2 fc-2fc - fc2W

f

++= )cos()()( θtwtmAtx ccr n(t) white noise

BPF passing through BP noise

)sin()()cos()()cos()()(2 θθθ +−+++= twtntwtntwtmAte cscccc

Noise power： WNtntntn sc 0222

0 2)(21)(

21)( =+=

pass band

Signal power： )(2

22

tmAc

Predetection SNR：0

22

4)(

WNmASNR c

T =

)cos(2)()( 23 θ+⋅= twtete c

))(2sin()())}(2cos(1){())}(2cos(1){( θθθ +−+++++= twtntwtntwtmA cscccc

↓LPF

)()()(0 tntmAty cc +=


NCTU EE 4

Noise power： WNtnc 02 2)( =

Signal power： 22 mAc

Postdetection SNR：WN

mASNR cD

0

22

2)( =

)3(2)()(gain Detection dB

SNRSNR

T

D ===Δ

Does this detection (demodulation) provide a 3dB gain？ Not quite

(1) What is the “equivalent” baseband system?

22

0 21power dtransmitte mA

WNP

cT =←=γ

Dc SNR

WNmA )(

2 0

22

==

(2) Why? The baseband noise is )(0 WBwWN =

The passband noise is )2(2 0 WBwWN =

The detection gain cancels the noise BW increase.

【Note】：We assume coherent detection, i.e., the LO signal )cos( θ+twc

has the same cw and θ as the carrier.


NCTU EE 5

B. SSB System：Coherent Detection

m of ans.Hilbert trˆ where)()]sin()(ˆ)cos()([)( :Signal Received

：mtnwtmtwtmAtX cccr ++±+= θθ

Demodulation：

ffcfc-w0-fc+w-fc fw0-w

xr(t)

2 cos(wct + θ )

ｙD (x)e3 (x)e2 (x)BPF LPF

<Key>：Decompose noise into in-phase and guadrature components

(BP noise!)

noiseBP

NLet

)sin()()cos()()(

0222

0

↑

====

+−+=Δ

WNnnn

twtntwtntn

scT

cscc θθ

Thus,

term.noise)()()(LPF e

)cos(2* caret don'

)sin()]()(ˆ)cos()]()([)(

3

2

←+=→↓

+↑↓

+±++=

tntmAty

tw

twtntmAtwtntmAte

ccD

c

cscccc

θ

θθ m

NOISE

Lower-Sideband

ffcfc-w

N0 / 2

Upper-Sideband

coherent


NCTU EE 6

Assume m(t) is independent of ).(tnc

Post-detection signal power: )( 22 mAS cD =

Post-detection noise power: WNnN cD 02 )( ==

Post-detection WNmASNR c

D0

22

=

What is transmitted signal power (Predetection)

TDc

c

cc

ccc

cc

ccccc

cccT

PStmA

tmtmtmtmA

twtw

twtmtwtmA

twtwtmtmtwtwtmtmtwtmtwtmA

twtmtwtmAS

===

=+=

⟩+−⟨⟩++⟨

↓↓

⟩+⟨+⟩+⟨=

++−+++=

+−+=

!))((

))(ˆ)(( })(ˆ21)(

21{

)2cos(21

21 )22cos(

21

21

)(sin)(ˆ)(cos)({

.)orthogonal are )sin( and )cos( ( .)orthogonal are )(ˆ and )( ( )sin()cos()(ˆ)(2)]sin()(ˆ[)]cos()([{

)]}sin()(ˆ)cos()([{

22

22222

22222

222

2

θθ

θθ

θθθθ

θθ

∴ Detection gain：

TD

T

T

D

D

T

D SNRSNR

WNNS

NS

SNRSNR

==

=

= 1

)()()(

0

No gain! But DT NN = , no loss! (In DSB, )!2 DT NN =

Classnotes, p.3-16

ffc-fc+w

N0 / 2

-fc fc-w0

2)2

( 0 ⋅⋅WN


NCTU EE 7

C. AM System：Coherent Detection

Received Signal：

message normalized: index modulation: )cos()](1[)(

n

cncc

matwtamAtx θ++=

ffc-fc fc+w0

carrier

Mn(f)

fW0-W

X 2 cos(wct + θ )

e2 (x)

e3 (x)

BPF

xr(t) = xc(t) + n(t)

LPF

yD(t) = Ac a mn(t) + nc(t) + Ac Ac is removed

it. removesimply weThus, .)( of termdcrecover cannot wereality, In 2) term)(dc term;dc remove thuscan we,0)( assume )1 removed is

tmtmAc =Q

Post detection signal power: 2222))(( ncncD maAtamAS ⋅==

Post detection noise power: WNnN cD 02 2==

WNmaA

SNR ncD

0

222

2)( =

f

N0 / 2

fc-w0

WWN0

0 2N2)22

( =⋅⋅

fc+w


NCTU EE 8

) )cos(221)(cos , )((

)(cos)()(cos

)}cos()](1[{:power signalTx

222

222222

2

θθ

θθ

θ

++=+=

+⋅++⋅=

++=•

ccnn

cnccc

cncT

wwmtm

wtmaAwA

wtamAS

2222

21

21

nccTT maAASP +==

BP noise power: WNNT 02=

Pre-detection WN

maAASNR nccT

0

2222

221)( +

⋅=

∴ Detection gain：

1)(loss!1

2

)(21)(

)(22

22

2222

222

<+

=+

=n

n

ncc

nc

T

D

mama

maAA

maASNRSNR

Recall, (power) Efficiency: 22

22

1 n

nff

mamaE

+=

ffT

D ESNRSNR 2

)()(

=∴

【Note】WN

PEWN

PESNRESNR Tff

TffTffD

00

2

2)(2)( =⋅=⋅=

Ex： 2nm =0.1 a=0.5 0244.0

1.0)5.0(11.0)5.0(

2

2

=⋅+

⋅=ffE

Detection gain = 2Eff = 0.0488!

ffc-fc0


NCTU EE 9

D. AM System：Envelope Detection

xr(t) ｙD (x)e2 (x)BPF Envelope

detector

WNnn

twtntwtntwtamAtn

txtntxte

sc

cscccnc

cc

022

0

02

2)()( :Noise

)sin()()cos()()cos()](1[ noise band-narrow:)(

signal modulated:)( )()()(

==

+−++++=

+=

θθθ

f

N0 / 2

0 fc-fc

))()](1[

)((tan)(

)()}()](1[{)(

)](cos[)()(

1

22

2

{tntamA

tnt

tntntamAtr

ttwtrte

cnc

s

scnc

c

++=

+++=

++=→

−φ

φθ

Phasor diagram：

Letθ=0 ( θ does not affect envelope demod.)

Recall what we did forinterference φ(t)

ns(t)r(t)

nc(t)Ac(t) [ 1+aMn(t) ]

Ideally, average:)( )()()(

carrier) the(including dc remove (envelope)

amplitude )()('

trtrtrty

trty

D

D

−=

↓

=


NCTU EE 10

Case 1： TSNR)( is large. i.e., small noise.

)decreased.(neither increasednot is noise -detectioncoherent as same

)()()(mean zero

time theofmost )()](1[)()()()](1[

↑

+≅⇒

↑↑

++≅⇒

>>++

tntamAty

tntamAtrtntntamA

cncD

cnc

scnc

Case 2： TSNR)( is small. i.e., large noise.

φ(t)

rn(t)

φn Ac(t) [ 1+aMn(t) ]

r(t)

[ ] nncn φ(t)aMArr(t) cos1++≈

Bandpass noise: ))(cos()()(0 ttwtrtn ncn φ+=

Thus,

))(cos()( importantnot is ))(cos()()cos()](1[)(2

ttwtrttwtrtwtamAte

nc

ncncnc

φψθθφθ

++=+++++=

Now, assume timeofmost )()](1[ trtamA nnc <<+

)()(cos)](1[)()(

dc"" remove

)(cos)](1[)( )](sin)](1([)](cos)](1[)([)( 22

trttamAtrty

ttamAtrttamAttamAtrtr

nncnD

nncn

nncnncn

−++≅

↓

++≅

++++=

φ

φφφ

Remove “dc”


NCTU EE 11

Recall：

0 rn

fRn(rn)

r

n(t) =

nc(t)

ns(t)

rn(t)

where nc(t) and ns(t) are all independent zero mean Gaussian

lost! is message

random:)(cos )(cos)](1[)(~)( Now,distr.-Rayleigh)()()( 22

tttamAtrtrtntntr

nnncn

scn

φφ++

=+=

What we try to show here is that when noise > (message) signal, “noise” becomes the dominate output component. In the output signal, no message signal is proportional to the message signal.

(cf. Coherent detection: )()()( tamAtnty nccD += )

(Recall: The analysis in interference.) This is the threshold effect.

( ∵The envelope detector is nonlinear,

∴the message is “lost” when SNR< threshold )

Remark：It’s difficult to calculate the exact (SNR)D.

Def. of threshold：

A value of the carrier-to-noise ratio (or SNR) below which the noise

performance of a detector deteriorates much more rapidly than

proportionately to the carrier-to-noise ratio (or SNR).

Haykin, p.325


NCTU EE 12

E. AM System: Square-Law Detection.

──An example of simple nonlinear detector that we can calculate

SNR and thus the “threshold region” can be more precisely

determined.

xr(t) ｙD (x)e2 (x)BPF Remove

dc( )2ｙD'(x)

LPFe3 (x)

~2W

)sin()()cos()()cos()](1[ noise bandpass:)( )()( '

0'02

twtntwtntwtamAtntnxte

cscccnc

c

−++=+=

) freq. high : )2cos( ),2(sin (

)]2cos(21

21)[()2sin()()}( )](1[{

)]2cos(21

21[)}()](1[{

)(sin)()sin()cos()()}( )](1[{2

)(cos)}()](1[{ )()(

2

2

22

22223

twtw

twtntwtntntamA

twtntamA

twtntwtwtntntamA

twtntamAtete

cc

cscscnc

ccnc

csccscnc

ccnc

−+⋅++

−+++=

+⋅⋅++

−++==

LPF

)()()()(2)(2)()(2 )envelope( )(

)()}()](1[{)(

2222222

22

22'

tntntntamAtnAtmaAtamAAtr

tntntamAty

sccncccncncc

scncD

++++++=

==

+++∝


NCTU EE 13

)()()()()()(2

)(2])()([)(2)(

message

0)()( ,0)( )()(

2222

22222

dc remove'

tntntntntntamA

tnAtmtmaAtamAty

tntntmtyty

sscccnc

ccnncncD

scnDD

−+−+

++−+=

↓

===⎯⎯⎯ →⎯

)(4:power Signalion Postdetect 224ncD maAS =

224224

22222244

)()()()(

)](1[4]))([( :power Noise ionPostdetect

sscc

cncnncD

nnnn

ntamAmtmaAN

−+−

+⋅++−=

⎥⎦⎤

⎢⎣⎡ −=+⋅−=− )()()(2)()(

22422224222 xxxxxxxx

(Assume the cross terms are either zero or can be neglected)

Note: 224 )(3)( cc nn ⋅= , if cn is Gaussian.

This can be shown using moment generator )(2

22

tet

θσ

=

WNnn nsc 0222 2)()( === σ .

44222222244 22)1(4])[( nnnncnncD maAmmaAN σσσ +⋅+⋅++−=

4222222244

224

4)1(4)(

4)(

nnncnnc

nc

D

DD

maAmmaA

maANS

SNRσσ +++−

==


NCTU EE 14

Example：

81)]2cos(

21[

]21)([cos))((

signal randoma not )cos()(

2

2222n

2n

==

−=−

←=

tw

twmtm

twtm

m

m

mn

If we assume this part can be neglected, (to be discussed below)

4222 4)211(4 nncD aAN σσ ++=

)161(

81 :power Its

noise. random-not -.distortion harmonic called is ))cos(2 ( termThe )()()()()()]cos(1[2

)2cos(21)cos(2)(

)2cos21

21cos (

)()()()]cos(1[2 dc remove

)]([cos)cos(2)( )( and )( examine weIf :Note

244

m

2222

222

2

22

22222'

'

DcD

sscccmc

mcmcD

mm

sccmc

mcmccD

DD

SaaAD

twtntntntntntwaA

twaAtwaAty

twtw

tntntntwaA

twaAtwaAAtytyty

⋅==

−+−+⋅+

++=

+=

++⋅+↓

+++=

Now, 2424224 2

2144 aAaAmaAS ccncD =⋅==

)(2)2(4)2(2

2)(2

22

2

22

4222

24

c

n

n

c

nnc

cD

Aa

aA

aAaASNR

σσ

σσ ++=

++=

compare: “Baseband” system: WN

PT

0

=γ

)cos(w

)211(

21)1(

21)}cos()](1[{

m

222222

t

aAmaAtwtamAP cnccncT

↑

+=+=⋅+=


NCTU EE 15

Now, r

ra

a

PWNWN

P

aaSNR

T

T

D ++=

++=

1)

2(2

)(1)

2(2)( 2

20

022

( WNn 02 2=σ )

)2

(2)( ,1 If

0

0

22

0

WNP

E

WNP

aaSNR

WNP

T

TD

T

≅

⋅+

≅>>

( similar to the coherent case )

) small is when,(.) (

)1( )()2

(2)( ,1 If

00

002

0

22

0

WNP

WNP

PWN

PWN

WNP

aaSNR

WNP

TT

TT

TD

T

⋅<<

≈+⋅+

≅<< Q

This illustrates threshold effect.

Remarks﹔

1) The performance of envelope detector is better than that of square

law detector for high SNR and a=1. (by 8.1≈ dB)

2) The B.W. of )(2 tnc is wider than W and thus part of it can be

filtered out by LPF. =


NCTU EE 16

General form：(assume 4222 )( amm nn ⋅− is small)

T

T

n

nT

T

c

T

nn

c

T

nn

c

n

nnnc

n

nc

D

n

n

cncncT

nnnc

ncD

PWNWN

P

ma

maWN

PWNP

EA

P

EA

P

EA

mamaA

mamaA

SNR

ma

maE

twtamAmaAP

maA

maASNR

0

0

222

22

00

2

42

2

42

2

22

42222

22

224

22

22

2222

42222

224

1)1(21

41

21

21

21

)1(4)1(4

)1(4

)(

1

))}cos()](1[{( )1(21

4)1(4

4)(

+⋅

+=⋅

+

⋅=

+

⋅=

+

⋅=

+++

+=

+=

⋅+=+=

++=

σσσσσσ

σσ

WNP

mamaSNRWNP T

n

nDT

0222

22

0)1(

)( , If ⋅+

≈>>

2

0222

22

0 )()1(

)( , IfWN

Pma

maSNRWNP T

n

nDT ⋅

+≈<<

Special case: 212 =nm


NCTU EE 17

Noise and Phase Errors in Coherent Systems -- Random noise and carrier phase error combined effect.

General signal model: (recall: at the same carrier frequency, we can transmit two “separate” messages – quadrature representation).

ttmttmtx ccc ωω sin)(cos)()( 21 +=

(quadrature double-sideband QDSB)

Special cases:

(1) DSB: ⎩⎨⎧

==

0)()()(

2

1

tmtmtm

(2) SSB: ⎩⎨⎧

±==

)(ˆ)( )()(

2

1

tmtmtmtm

Narrowband noise: Tsc BNnn 022 ==

ttnttntn cscc ωω sin)(cos)()( −=

Receiver local carrier:

))(cos(2 ttc φω + )(tφ : Gaussian, mean = 0, variance = 2φσ

( phase error )

Note: A constant θ can be inserted to all the above three terms → same results


NCTU EE 18

Goal: compute (random) demodulation error statistics (t)(t)-ymε D1=

⎩⎨⎧

===

? 0

21

2 }(t))(t)-yE{(mε D

ε

(1) ttnttnttmt(t)mte cscccc ωωωω sin)(cos)(sin)(cos)( 21 −++=

(2) )(sin))()(()(cos))(()( 21 ttntmttn(t)mty scD φφ −−+=

(3) Error

)(sin))()(()(cos))()(()()()( 2111 ttntmttntm(t)mtytmt scD φφε −++−=−=

φφφ

φφφε22

221

2212111

21

2

sin)(cossin))((2

cos)(sin)(2cos)(2

ssc

csc

nmnmnm

nmnmmnmmm

−+−+−

++−++−=

Assume: m1, m2, nc, ns and φ are all uncorrelated.

φφφφ coscoscoscos)( 211

2111 ⋅=+=+∴ mnmmnmm cc

(4) 2222

221

21

21

2 sincoscos2 nmmmm +⋅+⋅+⋅−= φφφε

Tnsc BNnnn 02222 ==== σ


NCTU EE 19

Case A: QDSB and 222

21 mmm σ== (equal power message);

WBT 2=

222

22222222

cos22

)sincos(cos2

nmm

mnmmQ

σφσσ

φφσσφσσεε

+⋅−=

+++⋅−==

Assume )(211)(cos1)( 2 ttt φφφ −≈⇒<<

22

211)(

211cos φσφφ −=−≈⇒ t

2222nm σσσε φ +≈

multiplicative additive (phase error) (white channel noise)

After normalization

2

222

m

nNQ σ

σσε φ +≈


NCTU EE 20

Case B: SSB ⎩⎨⎧

±==

)(ˆ)( )()(

2

1

tmtmtmtm

, WBT =

222

21 mmm σ== , m1(t) and m2(t) are orthogonal

2222 cos22 nmms σφσσε +⋅−=

But WNn 02 =σ ( not 2N0W )

Thus 2

22 )cos1(2

m

nNS σ

σφε +−≈

For 1)( <<tφ , 2

222

m

nNS σ

σσε φ +≈

⇒ same result as QDSB

Case C: DSB ⎩⎨⎧

==

0)( )()(

2

1

tmtmtm

, WBT 2=

222

222222

)cos1(

coscos2

nm

nmmmD

+−⋅=

+⋅+⋅−=

φ

φφε

For 1)( <<tφ , 2

21cos1 φφ ≈−

2422422

43

41

nmnmD σσσσφσε φ +⋅=+⋅≈

( If φ zero-mean, real Gaussian, 224 )(3 φφ = )

2

242

43

m

nND σ

σσε φ +≈


NCTU EE 21

<Remarks>

(1) If 02 =φσ ( no carrier phase error )

SNR detectioncoherent 2

22

* ==m

nN σ

σε

(2) DSB is less sensitive to phase error.

For 1)( <<tφ , 24

43

φφ σσ <

Pilot: one technique to reduce phase error

Pilot signal: frequencycarrier cos ⋅=⋅ KtKA cP ω Pilot filter output: (assume AWGN)

)](cos[)(sin)(cos)(cos

ttKtrtKtntKtntKA

c

cscccP

γωωωω

+=⋅−⋅+⋅

phase error: )()(tan)( 1

tnAtntcP

s

+= −γ

where Pcs BNnn 022 ==


NCTU EE 22

(1) If AP >> nc , P

s

Atnt )()( ≈γ

TP

s

SNRAn

21

2

22 ==⇒ γσ ( Pilot power = 2

21

PA )

Thus, a strong pilot SNR ensures small phase error.

(2) Frequency divider:

])(cos[2)](cos[21

KttttKw c

frequencyK

cγωγ +⎯⎯⎯ →⎯+

Thus, 22

2 1γφ σσ

K= (noise reduction)

Small Noise in Angle Modulation Ideal angle demodulation system

BW WDBT )1(2 +≈ Hz (Carson’s rule) Received signal:

)()](cos[)()()( tnttAtntxtx cccr ++=+= φω

⎪⎩

⎪⎨⎧

=∫ for FM )(2

for PM )()( t

nd

nP

dmftmK

t ααπφ

n(t): AWGN )(W/Hz 21

0N


NCTU EE 23

Output of the predection filter

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

tttrttAttnttnttAte

ncncc

cscccc

φωφωωωφω

+++=−++=

Rayleigh dist uniform dist

where Tcs BNnn 022 ==

Reference: )cos( tA cc ω ( if Ac >> n(t) )

Let )()()( ttt n φφθ −=

))()(cos()( ))(sin())()(sin()(

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

ttttRtttttr

tttttrttAtttttrttAte

ec

cnn

cnncc

ncncc

φφωφωφφ

φωφφφωφφφωφω

++=+−−

+−++=+−+++=

noise

where ))()(cos()())()(sin()(tan)( 1

tttrAtttrt

nnc

nne φφ

φφφ−+

−= −


NCTU EE 24

Output of discriminator: (phase deviation)

)()()( ttt eφφψ +=

(If 0)( =teφ , perfect recovering of message)

Case A: High detection (SNR)T or )(trA nc >> most of the time

))()(sin()())()(cos()(

))()(sin()(tan)( 1 ttA

trtttrA

tttrt nc

n

nnc

nne φφ

φφφφφ −≈−+

−= −

Then, ⎪⎩

⎪⎨⎧ ⋅

= for FM 21

for PM )()(

dtdK

tKty

D

P

D ψπ

ψ

(A.1) PM: )()()()()( tntmKKtKtKty PnPDeDDDP +⋅=⋅+⋅= φφ

where )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP φφφ −=⋅≡

signal power (message): 222nPDDP mKKS ⋅=

noise power:

(i) Complete analysis --- 0)( ≠tφ . Difficult!

(note )(teφ depends on )(tφ )

Produce additional terms (frequency components) for f > W at the discriminator output. Can be filtered out. ⇒ Neglect!

(ii) Assume 0)( =tφ

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

tttrtttrtAte

cnn

cnncc

ωφωφω

−+=

Compare to

)sin()( )cos()()cos()(1 ttnttntAte cscccc ωωω −+=


NCTU EE 25

)(sin)()( ttrtn nns φ=⇒ c

sDP A

tnKtn )()( =⇒

02

2

2

2

)()( NAKfS

AKfS

c

Dn

c

DnP s

==⇒

(a) At discriminator output:

Tc

DB

Bc

DDP BN

AKdfN

AKN

T

T 02

22

202

2

== ∫−

(b) At postdetection filter output:

WNAKdfN

AKN

c

DW

Wc

DDP 02

2

02

2

2== ∫−

Transmitting power: 2

2c

TAP =

Post-detection SNRD

WNPmK

WNAmK

AWKNmKK

NSSNR T

nPcnP

cD

nPD

DP

DPD

0

22

0

222

220

222

2/2)( ====

Note: In order to assure unique demodulation.

π≤= deviation phase)(tmK nP

10222 ≈≤⇒ πnP mK

SNR baseband equivalent 0

≡WN

PTQ

⇒ “maximum” improvement = 10.


NCTU EE 26

(A.2) FM:

)()()()(21)( tntmfKtn

dttdKty FndDFDDF +=+=

φπ

where )]}()(sin[)(

{2

)(2

)( ttA

trdtdK

dttdK

tn nc

nDeDF φφ

πφ

π−=≡

signal power (message): 222ndDDF mfKS ⋅=

noise power: ( assume 0)( =tφ )

)sin()( )cos()()cos()(1 ttnttntAte cscccc ωωω −+=

)}({2

)}(sin)({2

)( tndtd

AKt

Atr

dtdKtn s

c

Dn

c

nDF π

φπ

==

202

2

02

22

2

)2()2(

)( fNAKNf

AKfS

c

D

c

DnF ==⇒ π

π

At postdetection filter output: 3

02

22

02

2

32 WN

AKdffN

AKN

c

DW

Wc

DDF == ∫−

Transmitting power: 2

2c

TAP =

Post-detection SNRD


NCTU EE 27

WNP

mD

WNP

mWf

WNAmf

WNAK

mfKSNR

Tn

Tn

dcnd

c

D

ndDD

0

22

0

223

0

222

302

2

222

3

)(323

32

)(

=

===

where D = deviation ratio Wf

Wdt

td

dt

=≡

)(max

21

φ

π

( )(2)( tmfdt

tdndπφ

≈Q and 1)( ≤tmn )

↑⇒↑⇒↑⇒

≈∴

BW (SNR) D

3t improvemen (SNR)

D

22D nmD

For D >> 1, BT ≈ 2DW (cost)

<Remark> So far, all the above analysis is based on the assumption that Ac >> rn(t) ( high (SNR)T or “above threshold” ).

Case B: Low predetection (SNR)T or )(trA nc << most of the

time


NCTU EE 28

Let )()()( ttt n φφθ −=

Output of the predetection filter

))(cos()()(1 tttRte c ψω +=

)()(cos)()( trtAtrtR ncn ≈+≈ θ

)]()()[()]()(sin[)()]()(sin[ tttrtttRttA nnnnc ψφψφφφ −≈−=−

)]()(sin[)(

)()( tttr

Att n

n

cn φφφψ −−=⇒

ψ(t) is dominated by noise; message is lost

Threshold Effect!

Preemphasis and deemphasis filter Preemphasis

filter FM

modulator

FM demodulator

Deemphasisfilter

Channel

Noise n(t)

m(t)

m’(t)

Let deemphasis filter = )( fH DE ;

Preemphasis filter = )(

1fH DE

.

Why pre- and de-emphasis? Change the shape of message spectrum before modulation to take the advantage of the nonlinear characheristics of FM noise spectrum.


NCTU EE 29

For example, HDE(f) is a 1st-order RC circuit.

Input Output

dB

DE

ffj

fH

3

1

1)(+

= ,RC

f dB π21

3 = 2

3

)(1

1)(

dB

DE

ff

fH+

=⇒

Output noise power

)tan(2

)(1

1)()(

3

1

3

3302

2

223

22

302

2

202

2

2

3

2

dBdBdB

c

DW

WdB

dBc

D

W

Wc

D

dB

W

W nFDED

fW

fW

fNA

Kdf

ff

ffN

A

K

dffNA

K

ff

dffSfHN

−

−

−−

−=+

=

+

==

∫

∫∫

Typically, f3dB << W ⇒ 2

)(tan3

1 π≈−

dBfW

and thus dBdB f

WfW

33

1 )(tan <<−

WfNAK

N dBc

DD

2302

2

2≈

Hence, WN

Pm

ff

WfNAK

mfKSNR T

ndB

d

dBc

D

ndDDF

0

22

32302

2

222

)(2

)( =≈

(cf. without preemphasis: WN

Pm

Wf

SNR Tn

dDF

0

22)(3)( = )

Thus, if f3dB << W, the improvement is significant.


NCTU EE 30

Example: FM: fd = 75kHz; W = 15kHz; D = 5; 1.02 =nm

(1) w/o deemphasis:

WNP

SNR TDF

0

5.7)( =

( Note: coherent DSB WN

PSNR T

D0

)( = )

(2) w deemphasis: f3dB = 2.1 kHz

WNP

SNR TDF

0

128)( =

<Remark> Disadvantage of deemphasis: may increase transmission

bandwidth. Deemphasis filter → lowpass filter Preemphasis filter → high filter After preemphasis filter: stronger high frequency components

increase the transmitter deviation wider transmission bandwidth

However, typical message sources have “little” high frequency components.


NCTU EE 31

Large Noise in FM --- Thresholds Threshold effect

(SNR)T

(SNR)D Linear model

FM

Probability distribution and PSD of spike noise Zero-crossing probability

A lowpass, zero-mean white Gaussian n(t) Bandwidth W, psd Sn(f) ↔ autocorrelation function Rn(τ)

Let

W21

<<Δ (so that more than one zero-crossing in unlikely)

)0'( )' and 0(

)0)'( and 0( in crossing zero plus-to-minus :

00

000

000

<<Δ−=Δ−><=

>Δ+<=Δ

−Δ

−Δ

nnpnnnpnnnpP

P

Look for all the probability of )',( 00 nn pairs that satisfy the

above conditions.

pdf of )',( 00 nn : ),()',(00 '00 zyfnnf nn=

∫ ∫∞

Δ−−Δ =0

0

' ]),([00

dzdyzyfPz nn


NCTU EE 32

0)(}'{0

00 ===ττ

τd

dRnnE nQ

20

2

20

2

0000

'2

2'0

2

20

''2

1

2

1)()(),( n

z

n

y

nnnn en

en

zfyfzyf−−

==ππ

∫

∫

∫ ∫

∞ −

∞ −

∞

Δ−

−−

−Δ

Δ=

ΔΔ≈

=

0

'2

2'0

20

0 20

'2

2'0

0

0 2

20

'2

2'0

2

1

2

small) is (if )(2

1

2

1

]2

1[2

1

20

2

20

2

20

2

20

2

dzzenn

dzzn

en

dzdyen

en

P

n

z

n

z

Z

n

y

n

z

ππ

ππ

ππ

Let 20

2

'2nz

=ζ , 20

20

0

20

20

20

'2

''2 n

ndennn

Pπ

ζπ

ζ Δ=

Δ≈ ∫

∞ −−Δ

(probability of a minus-to-plus zero crossing in Δ seconds )

Example: Lowpass white Gaussian noise

⎪⎩

⎪⎨⎧ ≤

=otherwise 0

21

)( 0 WfNfSn ⇔ )2sinc()( 0 ττ WWNRn =

0)(

0

==ττ

τd

dRnQ ⇒ uncorrelated ⇒ independent

WNRdffSnn nn 0020 )0()(}var{ ==== ∫

∞

∞−

)()2(31

21)2()(2}'var{' 0

20

220

20 WNWdfNfdffSfjnn

W

Wn πππ ==== ∫∫ −

∞

∞−

3)

)2(31

(2

21

0

02

WWN

WNWP Δ

=Δ

=−Δ

π

π

Probability of a zero crossing in Δ =3

2 WPPP Δ=+= +Δ−ΔΔ


NCTU EE 33

Average rate of origin encirclement Sinusoid + narrowband Gaussian

)](cos[)( sin)(cos)(cos)(cos)(

0

0000

tttRttnttntAtntAtz sc

θωωωωω

+=−+=+=

where

])(

)([tan)(

)()]([)(

1

22

tnAtnt

tntnAtR

c

s

sc

+=

++=

−θ

Q: How often does this spike happen (on the average)? Probability of R(t) cross the horizontal axis in the 2nd quadrant

in Δ seconds duration PccΔ ≡ counterclockwise encirclement probability = P{A+nc(t) < 0 and ns(t) makes + to – crossing} = P{nc(t) < -A} ⋅ PΔ-

Now, n(t) is a bandpass white Gaussian process Bandwidth = B; psd = N0 (single-side) or N0/2 (double-side)


NCTU EE 34

(1) 323

BWP Δ=

Δ=−Δ

(2) )(2

})({0

2

0

2 0

2

BNAQdn

BNeAtnP

A

c

BNn

c

c

==−< ∫−

∞−

−

π

)(32 0

2

BNAQBPcc

Δ=∴ Δ

Similar deduction for PcΔ = clockwise encirclement probability

)(3

)(1

secondper ntsencirclemeofnumber Expected

0

2

BNAQBPP ccc

Δ=+

Δ=

≡

ΔΔ

ν

BNA

BNPSNR T

T0

2

0 2)( == ⇒ ))(2(

3 TSNRQBΔ=ν

If (SNR)T << 1, BB 2887.021

3=

Δ≈ν


NCTU EE 35

PSD (pwer spectral density) psd of “periodic” pulsive noise: A random process

∑∞

−∞=

−=k

sk kTtntx )()( δ

and ,....2,1 0)( ±±== kkTR sn (orthogonal)

22 )0(})}({{1lim)( ks

s

nT

sTx nf

TR

txFET

fSs

s

===∞→

For FM spikes, ∑∞

−∞=

−±==k

kttdt

tdtx )(2)()( πδθ

∞<<∞=== fBN

AQBafS kx - )(3

4)2()(0

2222 ππνν

constant for fixed A, N0, and B.

General Case: with message )(])([2cos)( tnttftfAtz c +⋅+= δπ

The spike rate is increased by δν It can be shown (approximately)

BNA

ef 0

2

2−

⋅= δδν

Thus, the total psd is 2)2()()( πδνν ⋅+=fSx

Example: 0constant)( >== dftfδ

Average noise frequency < (fc+fd) signal frequency. ⇒ Noise phasor has more clockwise rotations relative to the

signal power. ⇒ More negative spikes than positive spikes

It can be shown that dff =δ in this case.


NCTU EE 36

FM system

dttdK

ty

tttRttnttnttAte

DD

ccscccc

)(2

)(

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

ψπ

ψωωωφω

=

+=−++=

Now, the spike noise psd = )()2()( 2' δννπψ +=fS

∴ the spike noise psd in yD :

T

c

BNA

DT

cTDDD efK

BNA

QB

KKS 0

2

22

0

222 )(

3)(

−

⋅+=+= δδννδ

( ≈ constant, another “white” noise ( ↑ pulses ))

Noise power:

message todue increase noise spike 2

message no noise, spike )(3

2

Gaussian small 32

0

2

22

0

22

302

2

T

c

BNA

D

T

cTD

c

DD

efWK

BNA

QWBK

WNAK

N

−

⋅+

+

=

δ


NCTU EE 37

Signal power: 222ndDD mfKS =

T

c

BNA

T

T

cTT

Tn

d

D

DD

eWN

PW

fBN

AQ

WNP

WB

WNP

mWf

NS

SNR0

2

2

00

2

0

0

22

)(6)()(321

)(3)(

−

++

==δ


NCTU EE 38

Key parameters: WN

PT

0

, modulation index DWf d = , fδ

(BT ≈ 2(D+1), Carson’s rule) 3 terms:

(1) small noise ~ 2

0

DWN

PT

(2) spike noise without message ~ DBN

PQ

T

T 1

0

)}2

({ −

(3) spike noise due to message ~ 21}{ 0 DeWf

T

T

BNP

−−δ

If WN

PT

0

is large, spike noise terms (2)(3) ↓ 0.

Example: Sinusoidal message (single tone): Wttmn π2sin)( =

Frequency deviation Wtftf d πδ 2sin)( =

WDWfdtWtfWf dW

d βπππ

πδ 2or 22 2sin2 21

0===⋅= ∫

upper term: 21)( 222 β=n

d mWf

lower term: WN

PBN

A T

T

c

00

2

)1(21

2 +=

β

WNP

TTT

T

D T

eWN

PWN

PQ

WNP

WNP

SNR0)1(2

1

000

0

2

12)1

1()1(341

5.1)(

+−

++

++

=ββ

πββ

β

Threshold: the WN

PT

0

value such that )5.1(21)(

0

2

WNP

SNR TD β=

Factor affecting “threshold”: β↑⇒ threshold↑


NCTU EE 39

Threshold Extension (spike suppression)

Goal: lower the threshold value (WN

PT

0

)

Methods: (1) Frequency-compression feedback (2) PLL Frequency-compressive feedback (or FM demodulator using feedback)

)](cos[)]()(cos[)( ttwAtttwAtx cceccr ψφφ +=++= message noise

παψ

α 2 )],(

11cos[)( VD

crKK

ttwKtx =+

+=


NCTU EE 40

The bandwidth of x(t) (= BP) is reduced by a factor of α+1

1 .

Hence, at the discriminator output:

(i) Signal (message) power reduced by 2)1

1(α+

(ii) Additive Gaussian noise power reduced by 2)1

1(α+

(iii) Spike noise power reduced “exponentially”.

P

c

BNA

P

CP efBN

AQB0

2

2

0

2

)(3

−

⋅+=+ δδνν

∴ Spike noise is significantly reduced. However, the (SNR)D at the post-detector output remains the same as before.

( Q Both SD and ND are reduced by 2)1

1(α+

. )

Noise in Pulse Code Modulation PCM (pulse code) Total error = quantization error “+” channel error (1) Quantization error:

Assume uniform quantizer with stepsize = S. Further assume the quantization error ε(t) is uniformly distributed.

2

2

2

2

22

2

22

1212

)(

121

Sm

SmSNR

SdxS

x

Q

S

S

==

== ∫−ε

(2) Channel error:

One PCM sample = n binary pulses (bit) Bit error probability = Pb Word error probability = any bit is wrong = 1 – (1-Pb)n=Pw


NCTU EE 41

Assume q levels, any bit error leads to

qSqS w 21

21

≤≤− ε

Assume wε is uniformly distributed over )21,

21( qSqS− .

2222

121

121 SSqw >>=ε

Total PCM noise power (quantization error is independent of channel error)

wwwD PPN 22 )1( εε +−=

Signal power: assume signal (message) is uniformly distributed over

)21,

21( qSqS− .

22

121 SqSD =

wwww

D PPqPSqPS

SqSNR

+−=

+−=⇒ − )1(

1

121)1(

121

121

)( 2222

22

If nq 2= (n bits)

)21(21)( 22 n

wnD P

SNR −− −+=⇒

Note: Pw is largely determined by WN

PT

0


NCTU EE 42

Companding

121 2

2

2

22 SdxS

xS

S == ∫−ε is independent of the signal magnitude

⇒ 2

2

2

2

1212

)(Sm

SmSNR Q ==

Hence, small amplitude signals suffer more from quantization effects than larger amplitude signals. Solutions: (1) The quantizing steps are small for small amplitudes and larger for

large amplitude portions of the signal – e.g., Max quantizer. (2) Nonlinear mapping – compressor + expander


NCTU EE 43

noise in modulation systems - vlsi signal processing lab, ee, nctu

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