response of bank of correlators to noisy input

8/10/2019 Response of Bank of Correlators to Noisy Input

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Response of bank of correlators

to noisy input


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Analyzer for generating the set of signal

vectors si.

0

i=1,2,....,M( ) ( ) , (5.6)

j=1,2,....,M

T

ij i js s t t dt


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Conversion of the Continuous AWGN

Channel into a Vector Channel Suppose that the si(t) is

not any signal, but

specifically the signal at

the receiver side,

defined in accordance

with an AWGN channel:

So the output of thecorrelator can be

defined as:

( ) ( ) ( ),

0 t T (5.28)

i=1,2,....,M

ix t s t w t

i0

x ( ) ( )

= ,

j 1, 2,....., (5.29)

T

j

ij i

x t t dt

s w

N


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deterministic quantity random quantity

contributed by the

transmitted signal si(t)

sample value of the

variable Wi due to noise

0 ( ) ( ) (5.30)

T

ij i is s t t dt

0( ) ( ) (5.31)

T

i iw w t t dt


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Now,

Consider a randomprocess x1(t),with x1(t),asample function which isrelated to the received

signalx(t)as follows: Here we get:

1

( ) ( ) ( ) (5.32)N

j i

j

x t x t x t

1

1

( ) ( ) ( ) ( )

= ( ) ( )

= ( ) (5.33)

N

ij j j

j

N

j j

j

x t x t s w t

w t w t

w t

which means that the sample function x1(t) depends only on

the channel noise!


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The received signal can be expressed as:

1

1

( ) ( ) ( )

( ) ( ) (5.34)

N

j i

j

N

j i

j

x t x t x t

x t w t


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Statistical Characterization

The received signal (output of the correlator ) is arandom signal. To describe it we need to usestatistical methodsmean and variance.

The assumption is : We have assumed AWGN, so the noise is Gaussian, so

X(t) is a Gaussian process and being a Gaussian RV, X jisdescribed fully by its mean value and variance.


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Mean Value

Let Wj, denote a random variable, represented by

its sample value wj, produced by the jth correlator

in response to the Gaussian noise component w(t).

So it has zero mean (by definition of the AWGNmodel)

=

= [ ]

= (5.35)

j

j

x j

ij j

ij j

x ij

E X

E s W

s E W

s

then the mean of

Xjdepends only onsij:


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Variance

Starting from the definition,we substitute using 5.29and 5.31

2

2

2

var[ ]

= ( )

= (5.36)

ix j

j ij

j

X

E X s

E W

0

( ) ( ) (5.31)T

i iw w t t dt 2

0 0

0

= ( ) ( ) ( ) ( )

= ( ) ( ) ( ) ( ) (5.37)

i

T T

x j j

T T

j i

o

E W t t dt W u u du

E t u W t W u dtdu

2

0

0

= ( ) ( ) [ ( ) ( )]

= ( ) ( ) ( , ) (5.38)

i

T T

x i j

o

T T

j i w

o

t u E W t W u dtdu

E t u R t u dtdu

Autocorrelation function ofthe noise process


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It can be expressed as:

(because the noise is

stationary and with a

constant power

spectral density)

0R ( , ) ( ) (5.39)

2

w

Nt u t u

After substitution for

the variance we get:

2 0

0

20

0

= ( ) ( ) ( )2

= ( ) (5.40)2

i

T T

x i j

o

T

j

Nt u t u dtdu

Nt dt

And since j(t) has

unit energy for thevariance we finally

have:

2 0= for all j (5.41)

2ix

N

Correlator outputs, denoted by Xjhave varianceequal to the power spectral density N0/2 of thenoise process W(t).


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Properties

Xjare mutually uncorrelated

Xjare statistically independent (follows from above

because Xjare Gaussian) and for a memory less

channel the following equation is true:

1

( / ) ( / ), i=1,2,....,M (5.44)j

N

x i x j i

j

f x m f x m


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Xjare mutually uncorrelated


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Define (construct) a vector X of N random variables, X1, X2,XN, whose elements are independent Gaussian RVwith

mean values sij, (output of the correlator, deterministic partof the signal defined by the signal transmitted) and varianceequal to N0/2 (output of the correlator, random part,calculated noise added by the channel).

then the X1

, X2

, XN

, elements of X are statisticallyindependent

So, we can express the conditional probability of X, givensi(t)(correspondingly symbol mi) as a product of theconditional density functions (fx) of its individual elementsfxj.

NOTE: This is equal to finding an expression of the probabilityof a received symbolgiven a specific symbol was sent,assuming a memoryless channel


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that is:

1

( / ) ( / ), i=1,2,....,M (5.44)j

N

x i x j i

j

f x m f x m

where, the vectorxand thescalar xj, are sample

values of the random vector Xand the random

variable Xj.


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Vector xandscalar xj

are sample values of

the random vector X

and the randomvariable Xj

Vector x is called

observation vector

Scalar xjis called

observable element

1

( / ) ( / ), i=1,2,....,M (5.44)j

N

x i x j i

j

f x m f x m


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Since, each Xjis Gaussian with mean sjand

variance N0/2

/ 2 2

0

0

j=1,2,....,N1( / ) ( ) exp ( ) , (5.45)

i=1,2,....,MjN

x i j ijf x m N x sN

we can substitute in 5.44 to get 5.46:

/ 2 2

0

10

1

( / ) ( ) exp ( ) , i=1,2,....,M (5.46)

NN

x i j ij

jf x m N x sN


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If we go back to the formulation of the received

signal through a AWGN channel

1

1

( ) ( ) ( )

( ) ( ) (5.34)

N

j i

j

N

j i

j

x t x t x t

x t w t

The vector that wehave constructed fully

defines this part

Only projections of the noise onto

the basis functions of the signal set

{si(t)M

i=1affect the significant

statistics of the detection problem


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Finally,

The AWGN channel, is equivalent to an N-

dimensional vector channel, described by the

observation vector

, 1,2,....., (5.48)ix s w i M


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response of bank of correlators to noisy input

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