Digital Communication Systems 2012 R.Sokullu

CHAPTER 5SIGNAL SPACE ANALYSIS Digital Communication Systems 2012 R.Sokullu 1/26

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 Digital Communication Systems 2012 R.Sokullu 2/26

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 Digital Communication Systems 2012 R.Sokullu 3/26

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) Digital Communication Systems 2012 R.Sokullu 4/26

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: Yes, but meaning is different… Looks very similar???? • Or log likelihood function ..l(mi)as: Digital Communication Systems 2012 R.Sokullu 5/26

Log-Likelihood Function of AWGN Channel Vector presentation of the 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. Digital Communication Systems 2012 R.Sokullu 6/26

So, which is the log likelihood function of the AWGN channel.. Digital Communication Systems 2012 R.Sokullu 7/26

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. Digital Communication Systems 2012 R.Sokullu 9/26

Figure 5.3(a) Synthesizer for generating the signal si(t). (b) Analyzer for generating the set of signal vectors si. Digital Communication Systems 2012 R.Sokullu 10/26

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). Digital Communication Systems 2012 R.Sokullu 11/26

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. Digital Communication Systems 2012 R.Sokullu 12/26

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: Digital Communication Systems 2012 R.Sokullu 13/26

The same rule can be more explicitly expressed using the a priori probabilities of the transmitted signals as: Conditional pdf of observation vector X given mk was transmitted Unconditional pdf of observation vector X a priori probability of transmitting mk Digital Communication Systems 2012 R.Sokullu 14/26

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: • 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 Digital Communication Systems 2012 R.Sokullu 15/26

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. • Decision rule for AWGN: • Or using Euclidian space notation Digital Communication Systems 2012 R.Sokullu 16/26

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) Energy of the transmitted signal sk(t) Digital Communication Systems 2012 R.Sokullu 17/26

Figure 5.8Illustrating 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. Digital Communication Systems 2012 R.Sokullu 18/26

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 theobservation 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. Digital Communication Systems 2012 R.Sokullu 20/26

Figure 5.9(a) Detector or demodulator. (b) Signal transmission decoder. Digital Communication Systems 2012 R.Sokullu 21/26

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 Digital Communication Systems 2012 R.Sokullu 22/26

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: Digital Communication Systems 2012 R.Sokullu 24/26

Using the likelihood function this can be re-written as: • 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. Digital Communication Systems 2012 R.Sokullu 25/26

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 26/26

Digital Communication Systems 2012 R.Sokullu