Tel Basics

7/31/2019 Tel Basics

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S-72.245 Transmission Methods inTelecommunication Systems (4 cr)

Noise in analog carrier wave (CW) modulationsystems


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2 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Noise in analog CW modulation systems

Understanding noise

Lowpass presentation of bandpass noise and its conversionto baseband noise

Noise statistics of quadrature presentation in rectangular and

polar coordinates Modeling detectors for linear and exponential modulation

Analysis of post-detection SNR

Synchronous detector

PM-detector

FM-detector


3/22


Noise in carrier wave modulation

systems:basic definitions

Objectives: Define bandpass noise and use it to analyze postdetection SNR of analog CW systems

Assume signal is ergodic, e.g., all ensemble averages E[] equalthe corresponding time averages . Then, for instance

where the time average is defined by

( ) E[ ( )]v t v t 2 2( ) E[ ( )]v t v t

( ) ( ) E[ ( ) ( )]v t v t v t v t

/ 2

/ 2

1( ) lim ( )

T

Ti iTv t v t dt

T

/ 2

/ 2

1( ) ( )

T

Ti iv t v t dt

T (for a known period)

or

average valueaverage power

autocorrelation


4/22


The system model

We consider normalized ergodic analog message whoseamplitude and power are normalized2 2

( ) 1, ( ) , 1x x

x t S x x t S Channel loss

Modulated signal

Transmitted powerReceived power

Received signal (not altered by HR)

Detector

2/R T c

S S L x

Post-detection filter

Pre-detection filter

Pre-detection noise (after HR)


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Detection models

Pre-detection signal v(t) is presented in quadrature-carrier form:

Detection models:

( ) ( )cos[ ( )]

( )cos( ) ( )cos( )

v c v

i c q c

v t A t t t

v t t v t t

( ) ( )cos ( )

( ) ( )sin ( )

i v

q v

v t A t t

v t A t t

v

( ) Synchronous detector

( ) Envelope detector( )

( ) Phase detector

( ) / Frequency detector

i

v v

v

v t

A t Ay t

t

d t dt

0 0( ) cos[ 2 ( ) ], )

t

tC C Cx t A t f x d t t

(Remember that FM was defined by:


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Pre-detection noise in bandpass channel

Signal and noise are statistically independent and therefore theirpower can be added to form the total pre-detection power:

The pre-detection (bandpass) noise power is filtered from thechannel noise:

2 2 2 c R R

v x n S N

2

0( / 2) ( ) 2 ( / 2)T

B

R R TN H f df df B

fromchannel

to detector


7/227 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Pre-detection SNR Pre-detection signal-to-noise ratio for bandpass channels is

defined by

Note that above BT is the transmission bandwidth passingchannel noise power to the detector

For comparison, we can write the received signal-to-noise interms of baseband system (BW = W) SNR defined by

and therefore also

Note that always (limiting case is the SSB with BT= W)

(We will see, however, that post detection SNR can be muchlarger than !)

/ /( )R R R T

S N S B

2

( / 2) ( )

R T RN B H f df

/( )R

S W

/ /( ) ( / )( / ) /R R R T R T T

S N S B S W W B W B

TB W /

R RS N



Bandpass noise:

We assume stationary, zero mean Gaussian noise process forwhich

Bandpass noise in terms of lowpass equivalent signals

The in-phase and quadrature componentsare independent and hence

Their average is zero and their average power is thesame:

2 20,n R

n n N

( ) ( )cos( ) ( )sin( )i c q cn t n t t n t t

2 2 2

i q Rn n n N

( ) ( ) 0i qn t n t

0i q

n n

in

qn

nA

n

( ) ( )cos[ ( )]n c n

n t A t t t



Bandpass noise has Rayleigh distributed

envelope and evenly distributed phase

I-Q components of the bandpass noise can be presented inenvelope - phase format:

The PDF of envelope is Rayleighdistributed defined by

Therefore mean and variance for thebandpass noise are (integrate from above, how?)

2 2 2 , arctanq

n i q n

i

nA n n

n

2 2

( ) exp ( )2n

n n

A n n

R R

A Ap A u A

N N

2 2

2

2 2 2 2

2

/ 2, 2

/ 2 ( ) / 2i q

n R n R

R n i qn n n

A N A N

N A n n n

in

qn

nA

n

Two independentr.v.:s - sum of theirvariances equals varianceof the envelope



Post detection noise in synchronous detection

Signal component of synchronous detector:

Noise component of synchronous detector:

Detector extracts i-components and removes double frequencycomponents

( ) ( )exp( )

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

( )( )cos( )

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

2 2

DSB c

DSB c c c c

c

DSB c

c c

c c

v t x t j t

v t A x t t jA x t t

A x tv t t

A x t A x tt j t

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

cos(2 ) / 2 ( )si( ) 1 n(2 ) / 2

c c i c q c

ci q c

n t t t n t t n t t tn t tt n

( ) ( ) ( )D c i

y t A x t n t cos( )cos( ) 1/2 cos(2 )/ 2

cos( )sin( ) sin(2 )/ 2

x x x

x x x

detected message

received DSB signal



Post-detection SNR for DSB

Obtain signal and noise power after detection from:

where average noise and signal power are

Received average signal power is

and therefore SNR after DSB detector is

( ) ( ) ( )D c i

y t A x t n t

2 2 2 2

( ) , ( ) D i D c c x

N n t S A x t A S

2 2 2 2 2

0

1/ 2

( ) cos ( ) / 2 2 / x

R c c x x R c

S

S A x t t A S S S A

2

2

/ /D D c D

c

xSS N A N

A

2

2R

c

S

A2 (DSB)

2 2(DSB)

T

R R R

D B WD T

S S S

WN BN



Comparing SNR for DSB and AM

It can be show, that for AM the post detection SNR is

Comparison of this to the SNR of DSB can done by noting that

in practice

Hence AM performs usually much worse than DSB

It can be shown that for SSB performance is the same as for

DSB, e.g.

/1

x

D D AM

x

SS N

S

x

0.5 tone modulation

S 0.1 speech signal

xS

,/ R

D D USB DSB

SS N

W



Exponential modulation and channel noise

Both PM and FM have constant envelopes so the receivedpower is constant

Received SNR is yielding for wideband FM

where for wideband modulation

2 2

( ) cos[ ( )]

/ 2

c c c

R c c

x t A t t

S x A

22

22

R c c

R R T

A

N N B

S A

1,2( 2) 2

2

mT m f W

B f W

DW

2 2

R R

R T

RS S

N B DW D

S



Detection of exponential modulation

assuming small noise power

( )sin[ ( )]( ) arctan

( )cos[ ( )]

n n

v

c n n

A t tt

A A t t

carrier+noise

noise

( ) cos[ ( )] ( )cos[ ( )]c c n c n

Signal Noise

v t A t t A t t t

( )sin[ ( )]( ) n n

v

c

A t tt

A

small compared to Ac

Detected noise component



Post-detection noise spectra for PM

The channel noise is bandpass noise filtered at the transmissionbandwidth and therefore the respective post-detection noise powerspectral density GPM(f) and the total noise power NDare

( )sin ( )( )

( )

2

n nv

c

q

R

A t tt A

n t

S

2

2 2

/ 2

sinR c

q n n

i q R LP

S A

n A

n n N B

2

( )( )

2

2

q

PMTR

R T

n t fG f

BS

f

S B

Note that afterdetection signalbandwidth is Wandthus a post detectionfilter is required to removeout-of-band channel noise

( ) /W

WD PM R

N G f df W S

LPB



Post-detection SNR for FM

Recall the definition of FM-signal

Frequency discriminator (detector) differentiates theinstantaneous phase to cancel out the inherent integration inphase. Now

Inspection in frequency domain (In order to find the respectivePSDs) yields after detector

and the signal PSD is

( ) 2 ( ) ( ) / v

t x t d t dt

( )( ) 1 ( ) ( ) 1 ( )( )2 2 2 2

qv S N S

R

Signal Noise

d n td t d t d t d t x tdt dt dt dt S dt

( ) 2 ( )v

X f j f f

2

( ) ( ) ( 2FM

G f X f 22 2

) ( ) / (2 )v

f f 22

( )v

f f

0( ) cos ( ) , ( ) 2 ( )

t

c C C v v tx t A t t t f x d

( )

( 2 ) ( )

n

n

n

d x t

j f X fdt

( )( )sin ( )( )

2

qn n

vc R

n tA t tt

A S



Post-detection SNR in FM (cont.)

Therefore, the post-detection noise PSD can be written as

and now the PSD for FM post detection noise is

and the respective total noise power is

2

( )2

FM

R T

f fG f

S B

3( ) / 3WWD FM RN G f df W S

22( ) ( )

FM vG f f f

( )( )

2

q

v

R

N ff

S

with



For PM we have

For FM we have

Under wideband condition and

Destination S/N for PM and FM

2

2 2 2/ ,where

/

x

D D x x

R

SS N S S

W S

2

2

3

2

// 3

3 3

x

D D

R

R

x x

D

f SS NW S

f SS D S

W W

2

3/

4

T

D D x

BS N S

W

1D Note that S

D/N

Dcan

be increased just byincreasing deviation!

2 3

( )

2 3

W

D FMW

R R

f WN df

S S

( )

2

W

WD PM

R R

WN df

S S

1,2( 2) 2

2 /(2 )

mT m f W

T

B f W

DW D B W



FM post-detection S/N with deemphases

Deemphases filter (that is a lowpass filter connected afterdetector) can suppress noise further. FM post-detection noisePSD and total noise power without deemphases:

With deemphases filter (for simplification assume W/Bde>>1):

where

2

( )2

FM

R T

f fG f

S B

3

( )3

W

WD FM

R

WN G f df

S

3

2 2

/

( ) ( ) arctan /

de

W de

WD FM de de R

R de de

W B

B W WN G f H f df B W S

S B B2

21( )1 ( / )

de

de

H ff B

2 22

2/

/

x x R

D D x

de R de de

f S S S f fS N S

B W S W B B



Example

FM radio

Without deemphases

With deemphases

Therefore if DSB or SSB system could be exchanged to FMsystem 640 fold transmission power savings could be achieved.Note, however that the required transmission bandwidth is nowabout 220 kHz /15 kHz = 15 times larger! Also, a problem is theFM threshold effect that we discuss next.

75 kHz, 15 kHz, 5, 1/ 2, 2.1kHz

x de

f W D S B

2

21 2

/ 3

(3 5 ) 38

D D xS N D S

2

/ ( / ) 640

D D de x

S N f B S

R

S

W


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22

Comparison of carrier wave

modulation systems

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