CHAPTER 5 SIGNAL SPACE ANALYSIS

Chapter 5: Signal Space Analysis

Digital Communication Systems 2012R.Sokullu

1/45

CHAPTER 5

SIGNAL SPACE ANALYSIS


Digital Communication Systems 2012 R.Sokullu 2/45

Outline

• 5.1 Introduction• 5.2 Geometric Representation of Signals

– Gram-Schmidt Orthogonalization Procedure• 5.3 Conversion of the AWGN into a Vector Channel• 5.4 Maximum Likelihood Decoding• 5.5 Correlation Receiver• 5.6 Probability of Error



Introduction – the Model

• We consider the following model of a generic transmission system (digital source):– A message source transmits 1 symbol every T sec

– Symbols belong to an alphabet M (m1, m2, …mM)

• Binary – symbols are 0s and 1s

• Quaternary PCM – symbols are 00, 01, 10, 11



Transmitter Side• Symbol generation (message) is probabilistic, with

a priori probabilities p1, p2, .. pM. or

• Symbols are equally likely

• So, probability that symbol mi will be emitted:

( )

1 = i=1,2,....,M (5.1)

i iP m

forM



• Transmitter takes the symbol (data) mi (digital message source output) and encodes it into a distinct signal si(t).

• The signal si(t) occupies the whole slot T allotted to symbol mi.

• si(t) is a real valued energy signal (???)

2i

0

E ( ) , i=1,2,....,M (5.2) T

is t dt



• Transmitter takes the symbol (data) mi (digital message source output) and encodes it into a distinct signal si(t).

• The signal si(t) occupies the whole slot T allotted to symbol mi.

• si(t) is a real valued energy signal (signal with finite energy)

2i

0

E ( ) , i=1,2,....,M (5.2) T

is t dt



Channel Assumptions:

• Linear, wide enough to accommodate the signal si(t) with no or negligible distortion

• Channel noise is w(t) is a zero-mean white Gaussian noise process – AWGN– additive noise

– received signal may be expressed as:

0 t T( ) ( ) ( ), (5.3)

i=1,2,....,Mix t s t w t



Receiver Side• Observes the received signal x(t) for a duration of time T sec

• Makes an estimate of the transmitted signal si(t) (eq. symbol mi).

• Process is statistical – presence of noise

– errors

• So, receiver has to be designed for minimizing the average probability of error (Pe)

1

ˆ( / ) (5.4)M

i i ii

p P m m m

cond. error probability given ith symbol was

sent

Symbol sent

Pe =

What is this?



Outline





5.2. Geometric Representation of Signals

• Objective: To represent any set of M energy signals {si(t)} as linear combinations of N orthogonal basis functions, where N ≤ M

• Real value energy signals s1(t), s2(t),..sM(t), each of duration T sec

1

0 t T( ) ( ), (5.5)

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

N

i ij jj

s t s t

Orthogonal basis function

coefficient

Energy signal



• Coefficients:

• Real-valued basis functions:

0

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

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

T

ij i js s t t dt

T

i

0

1 if ( ) ( ) (5.7)

0 if j ij

i jt t dt

i j



• The set of coefficients can be viewed as a N-dimensional vector, denoted by si

• Bears a one-to-one relationship with the transmitted signal si(t)



Figure 5.3(a) Synthesizer for generating the signal si(t). (b) Analyzer for generating the set of signal vectors si.



So,

• Each signal in the set si(t) is completely determined by the vector of its coefficients

1

2

i

.s , 1,2,....,M (5.8)

.

.

i

i

iN

s

s

i

s



Finally,• The signal vector si concept can be extended to 2D, 3D etc. N-

dimensional Euclidian space• Provides mathematical basis for the geometric representation

of energy signals that is used in noise analysis• Allows definition of

– Length of vectors (absolute value)– Angles between vectors – Squared value (inner product of si with itself)

2

i i

2

1

s s

= , 1,2,....,M (5.9)

Ti

N

ijj

s

s i

Matrix Transposition



Figure 5.4Illustrating the geometric representation of signals for the case when N 2 and M 3.(two dimensional space, three signals)



Also,

• …start with the definition of average energy in a signal…(5.10)

• Where si(t) is as in (5.5):

2i

0

E ( ) (5.10) T

is t dt

1

( ) ( ), (5.5)N

i ij jj

s t s t

What is the relation between the vector representation of a signal and its energy value?



• After substitution:

• After regrouping:

• Φj(t) is orthogonal, so finally we have:

i1 10

E ( ) ( ) T N N

ij j ik kj k

s t s t dt

T

i j1 1 0

E ( ) ( ) (5.11) N N

ij ik kj k

s s t t dt

22i i

1

E = s (5.12)N

ijj

s

The energy of a signal

is equal to the squared length of its

vector



Formulas for two signals

• Assume we have a pair of signals: si(t) and sj(t), each represented by its vector,

• Then:

k0( ) ( ) s (5.13)

T Tij i k is s t s t dt s

Inner product of the signals is equal to the inner product

of their vector representations [0,T]

Inner product is invariant to the selection of basis

functions



Euclidian Distance• The Euclidean distance between two points

represented by vectors (signal vectors) is equal to

||si-sk|| and the squared value is given by:

2 2i

1

2

0

s s = ( - ) (5.14)

= ( ( ) ( ))

N

k ij kjj

T

i k

s s

s t s t dt



Angle between two signals• The cosine of the angle Θik between two signal vectors si and

sk is equal to the inner product of these two vectors, divided by the product of their norms:

• So the two signal vectors are orthogonal if their inner product si

Tsk is zero (cos Θik = 0)

kik

scos (5.15)

Ti

i k

s

s s



Schwartz Inequality• Defined as:

• accept without proof…

22 2

1 2 1 2( ) ( ) ( ) ( ) (5.16)s t s t dt s t dt s t dt



Outline





Gram-Schmidt Orthogonalization Procedure

1. Define the first basis function starting with s1 as: (where E is the energy of the signal) (based on 5.12)

2. Then express s1(t) using the basis function and an energy related coefficient s11 as:

3. Later using s2 define the coefficient s21 as:

11

1

( )( ) (5.19)

s tt

E

1 1 1 11 1( ) ( ) =s ( ) (5.20) s t E t t

21 2 10( ) ( ) (5.21)

Ts s t t dt

Assume a set of M energy signals denoted by s1(t), s2(t), .. , sM(t).



4. If we introduce the intermediate function g2 as:

5. We can define the second basis function φ2(t) as:

6. Which after substitution of g2(t) using s1(t) and s2(t) it becomes:

• Note that φ1(t) and φ2(t) are orthogonal that means:

22

220

( )( ) (5.23)

( ) T

g tt

g t dt

2 21 1

2 22 21

( ) ( ) ( ) (5.24)

s t s tt

E s

1 20( ) ( ) 0

Tt t dt

220

( ) 1 T

t dt

2 2 21 1g ( ) ( ) ( ) (5.22)t s t s t Orthogonal to φ1(t)

(Look at 5.23)



And so on for N dimensional space…,

• In general a basis function can be defined using the following formula:

0( ) ( ) , 1, 2,....., 1 (5.26)

T

ij i js s t t dt j i

• where the coefficients can be defined using:

1

i1

g ( ) ( ) - (t) (5.25)i

i ij jj

t s t s



Special case:

• For the special case of i = 1 gi(t) reduces to si(t).

i2

0

( )( ) , i 1, 2,....., (5.27)

( )

i

T

i

g tt N

g t dt

General case:

• Given a function gi(t) we can define a set of basis functions, which form an orthogonal set, as:



Outline





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 the correlator (Fig. 5.3b) can be defined as:

( ) ( ) ( ),

0 t T (5.28)

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

ix t s t w t

i 0x ( ) ( )

= ,

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

T

j

ij i

x t t dt

s w

N



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 dt0

( ) ( ) (5.31)T

i iw w t t dt



Now,• Consider a random

process X1(t), with x1(t), a sample function which is related to the received signal x(t) as follows:

• Using 5.28, 5.29 and 5.30 and the expansion 5.5 we get:

1

( ) ( ) ( ) (5.32)N

j ij

x t x t x t

1

1

( ) ( ) ( ) ( )

= ( ) ( )

= ( ) (5.33)

N

ij j jj

N

j jj

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!



• The received signal can be expressed as:

1

1

( ) ( ) ( )

( ) ( ) (5.34)

N

j ij

N

j ij

x t x t x t

x t w t

NOTE: This is an expansion similar to the one in 5.5 but it is random, due to the additive noise.



Statistical Characterization• The received signal (output of the correlator of

Fig.5.3b) is a random signal. To describe it we need to use statistical methods – mean and variance.

• The assumptions are:– X(t) denotes a random process, a sample function of which

is represented by the received signal x(t).– Xj(t) denotes a random variable whose sample value is

represented by the correlator output xj(t), j = 1, 2, …N.– We have assumed AWGN, so the noise is Gaussian, so X(t)

is a Gaussian process and being a Gaussian RV, X j is described fully by its mean value and variance.



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 AWGN model)

=

= [ ]

= (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 Xj depends only on sij:



Variance• Starting from the definition,

we substitute using 5.29 and 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 dt2

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 of the noise process



• It can be expressed as: (because the noise is stationary and with a constant power spectral density)

0R ( , ) ( ) (5.39) 2w

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 φj(t) has unit

energy for the variance we finally have:

2 0= for all j (5.41)2ix

N

• Correlator outputs, denoted by Xj have variance equal to the power spectral density N0/2 of the noise process W(t).



Properties (without proof)• Xj are mutually uncorrelated

• Xj are statistically independent (follows from above because Xj are Gaussian)

• and for a memoryless channel the following equation is true:

1

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

N

x i x j ij

f x m f x m



• Define (construct) a vector X of N random variables, X1, X2, …XN, whose elements are independent Gaussian RV with mean values sij, (output of the correlator, deterministic part of the signal defined by the signal transmitted) and variance equal to N0/2 (output of the correlator, random part, calculated noise added by the channel).

• then the X1, X2, …XN , elements of X are statistically independent.

• So, we can express the conditional probability of X, given si(t) (correspondingly symbol mi) as a product of the conditional density functions (fx) of its individual elements fxj.

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



• …that is:

1

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

N

x i x j ij

f x m f x m

• where, the vector x and the scalar xj, are sample values of the random vector X and the random variable Xj.



Vector x and scalar xj are sample values of the random vector X and the random variable Xj

Vector x is called observation vectorScalar xj is called observable element

1

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

N

x i x j ij

f x m f x m



• Since, each Xj is Gaussian with mean sj and variance N0/2

/ 2 20

0

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

i=1,2,....,Mj

Nx i j ijf x m N x s

N

• we can substitute in 5.44 to get 5.46:

/ 2 20

10

1( / ) ( ) exp ( ) ,

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

NN

x i j ijj

f x m N x sN



• If we go back to the formulation of the received signal through a AWGN channel 5.34

1

1

( ) ( ) ( )

( ) ( ) (5.34)

N

j ij

N

j ij

x t x t x t

x t w t

The vector that we have constructed fully defines this part

Only projections of the noise onto the basis functions of the signal set {si(t)M

i=1 affect the significant statistics of the detection problem



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



Outline





Maximum Likelihood Decoding

• to be continued….

CHAPTER 5 SIGNAL SPACE ANALYSIS

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