digital communication systems 2012 r.sokullu

Chapter 5: Signal Space Analysis

Digital Communication Systems 2012R.Sokullu 1/26

CHAPTER 5

SIGNAL SPACE ANALYSIS



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 Likelihood Functions• 5.5 Maximum Likelihood Decoding• 5.6 Correlation Receiver• 5.7 Probability of Error



Likelihood Functions• In a sense, likelihood works backwards from probability: given parameter

B, we use the conditional probability P(A|B) to reason about outcome A, and given outcome A, we use the likelihood function L(B|A) to reason about parameter B. This mode of reasoning is formalized in Bayes' theorem:

• A likelihood function is a conditional probability function considered as a function of its second argument with its first argument held fixed, thus:

and also any other function proportional to such a function. That is, the likelihood

function for B is the equivalence class of functions



Likelihood Functions• As we discussed in the previous class, the

conditional probability density functions fX(x|mi), I = 1, 2, 3, …M are the very characterization of the AWGN channel.

• They express the functional dependence of the observation vector x on the transmitted message symbol mi. (known as the transmitted message symbol)



However,• If we have the observation vector given, and we

want to define the transmitted message signal, then we have the reverse situation

• We introduce the “likelihood function” L(mi) as:( ) ( / ), 1, 2,....., (5.49)i X iL m f x m i M

Looks very similar????

Yes, but meaning is different…

• Or log likelihood function ..l(mi) as:

( ) log ( ), 1, 2,....., (5.50)i il m L m i M



Log-Likelihood Function of AWGN Channel

• Substitute 5.46 into 5.50:

• where sij, j = 1, 2, 3, ..N are the elements of the signal vector si, representing the message symbol mi.

/ 2 20

10

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

NN

x i j ijj

f x m N x sN

( ) log ( ), 1, 2,....., (5.50)i il m L m i M

Vector presentation of the AWGN channel



So,

2

10

1( ) ( ) , 1, 2,....., (5.51)N

i j ijj

l m x s i MN

which is the log likelihood function of the AWGN channel..



5.5 Maximum Likelihood Decoding• Defining the problem

– Suppose that in each time slot duration of T seconds, one of M possible signals, s1(t), s2(t), …sM(t) is transmitted with equal probability, 1/M.

– As described in the previous part, for the vector representation, the signal si(t), i=1, 2, …M is applied to a bank of correlators, with a common input and supplied with a suitable set of N orthogonal basis functions, N. The resulting output defines the signal vector si.

– We represent each signal si(t) as a point in the Euclidian space, N ≤ M (referred to as transmitted signal point or message point).The set of message points corresponding to the set of transmitted signals si(t) {i = 1 to M} is called signal constellation.



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



– The received signal x(t) is applied to a bank of N correlators (Fig. 5.3b) and the correlator outputs define the observation vector x.

– On the receiving side the representation of the received signal x(t) is complicated by the additive noise w(t).

– As we discussed the previous class, the vector x differs from the vector si by the noise vector w.

– However only the portion of it which interferes with the detection process is of importance to us, and this is fully described by w(t).



• Based on the observation vector x we may represent the received signal signal x(t) by a point in the same Euclidian space used to represent the transmitted signal.



• For a given observation vector x we have to make a decision m' = mi

• The decision is based on the criterion to minimize the probability of error in mapping each observation vector into a decision.

• So the optimum decision rule is:

iˆSet m=m if( sent / ) ( sent/x) for all k i (5.53)i kP m x P m



• The same rule can be more explicitly expressed using the a priori probabilities of the transmitted signals as:

i

k

ˆSet m=m if( / ) is maximum for k=i (5.54)( )

X k

X

p f x mf x

a priori probability of transmitting mk

Unconditional pdf of observation

vector X

Conditional pdf of observation vector X

given mk was transmitted



• Thus we can conclude, according to the definition of likelihood functions, the likelihood function l(mk) will be maximum for k = i.

• So the decision rule using the likelihood function will be formulated as:

iˆSet m=m if( ) is maximum for k=i (5.55)kl m

• For a graphical representation of the maximum likelihood rule we introduce the following:– Observation space – Z, N-dimensional, consisting of all possible

observation vectors x– Z is partitioned into M decision regions, Z1, Z2, .. ZM

iObservation vector x lies region Z if( ) is maximum for k=i (5.56)kl m



For the AWGN channel..• Based on the log-likelihood

function, of the AWGN channel, l(mk) will be max when the term: is minimized by k = i.

2

1

( ) , N

j ijj

x s

• Decision rule for AWGN:

• Or using Euclidian space notation

i

2

1

Observation vector x lies region Z if

( ) , is minimum for k=i (5.57)N

j kjj

x s

iObservation vector x lies region Z ifthe Euclidean distance

is minimum for k=i (5.59)kx s



Finally,• (5.59) states that the maximum likelihood decision rule is

simply to choose the message point closest to the received signal point.

• After few re-organizations we get: (left as homework brain gymnastic exercise for you)

1

in1 is maximum for k=i (5.61)2

i

N

j kj kj

Observationvector x lies region Z if

x s E

2k

1

E (5.62)N

kjj

s

Energy of the

transmitted signal sk(t)



Figure 5.8

Illustrating the partitioning of the observation space into decision regions for the case when N 2 and M 4;it is assumed that the M transmitted symbols are equally likely.



5.6 Correlation Receiver• Based on the theoretical assumptions made in the previous

class we define the correlator at the receiver side.• It can be implemented as a optimum receiver that consists of

two parts:– Detector part – M product-integrators supplied with the

corresponding set of coherent reference signals (orthogonal basis functions), generated locally. It operates on the received signal s(t) to produce the observation vector x for 0≤ t ≤ T.

– Receiver part – signal transmission decoder – which is implemented in the form of a maximum likelihood decoder, operating on the observation vector x to produce the estimate m‘ of the transmitted symbol mi in a way to minimize the average probability of symbol error. According to (5.61) the N elements of the observation vector x are multiplied by the N elements of each of the M signal vectors s1, s2, ..sM and then summed up to produce the inner products [xTsk|k=1,2..M]. Largest of the resulting numbers is selected.



Figure 5.9(a) Detector or demodulator.

(b) Signal transmission decoder.



Note:• The detector shown

in Fig. 5.9a is based on correlators.

• Alternatively, matched filters, discussed in Chap. 4.2 may be used to produce the required observation vector x. Detector part of matched filter receiver;

the signal transmission decoder is as shown in Fig. 5.9b



5.7 Probability of Error• To complete the statistical characterization of the

correlation receiver (Fig. 5.9) we need to discuss its noise performance.

• Using the assumptions made before, we can define the average probability of error Pe as:

e 1

1

1

P = ( | ),

1= ( | ) (5.67)M

11 ( | )M

M

i i ij

M

i ii

M

i ii

p P x does not lie in Z given m

P x does not lie in Z given m

P xlies in Z given m



• Using the likelihood function this can be re-written as:

e1

1P 1 ( | ) (5.68)M

i

M

x ii Z

f x m dx

• The probability of error is invariant to rotation and

translation of the signal constellation.– In maximum likelihood detection the probability of

symbol error Pe depends solely on the Euclidian distances between the message points in the constellation

– The additive Gaussian noise is spherically symmetric in all directions in the signal space.



Conclusions:• This chapter presents a systematic procedure for the analysis

of signals in a vector space.• The basic idea of the approach is to represent each member of

a set of transmitted signals by an N-dimensional vector, where N is the number of orthogonal basis functions, needed for the unique representation of the transmitted signals.

• The set of signal vectors defines the signal constellation, the N-dimensional space defines the signal space.

• It is the theoretical basis for the design of a digital communication receiver in the presence of AWGN. The procedure is based on the theory of maximum likelihood detection.

• The average probability of symbol error is defined as Pe. It is dominated by the nearest neighbors to the transmitted signal.

digital communication systems 2012 r.sokullu

Documents