**THE OPTIMUM DETECTOR**

Our goal is to design a signal detector that makes a decision on the transmitted signal in each signal interval based on the observation of the vector r in each interval such that the probability of a correct decision is maximized. We assume that there is no memory in signals transmitted in successive signal intervals. We consider a decision rule based on the computation of the posterior probabilities defined as

P(sm|r)≣P(signal sm was transmitted|r), m=1,2,…,M. The decision criterion is based on selecting the signal

corresponding to the maximum of the set of posterior probabilities { P(sm|r)}. This decision criterion is called the maximum a posterior probability (MAP) criterion.

Using Bayes’ rule, the posterior probabilities may be expressed

As

$$P(s_m|r)=\dfrac{p(r|s_m)P(s_m)}{p(r)} \ \ \ \ \ \ ----(A)$$

where P(sm) is the a priori probability of the mth signal being transmitted. The denominator of (A), which is independent of which signal is transmitted, may be expressed as

$$p(r)=\sum^M_{m=1} p(r|s_m)P(s_m)$$

Some simplification occurs in the MAP criterion when the M signal are equally probable a priori, i.e., P(sm)=1/M. The decision rule based on finding the signal that maximizes P(sm|r) is equivalent to finding the signal that maximizes P(r|sm). The conditional PDF P(r|sm) or any monotonic function of it is usually called the likelihood function. The decision criterion based on the maximum of P(r|sm) over the M signals is called maximum-likelihood (ML) criterion. We observe that a detector based on the MAP criterion and one that is based on the ML criterion make the same decisions as long as a priori probabilities P(sm) are all equal.

The maximum of ln p(r|sm) over sm is equivalent to finding the signal sm that minimizes the -*Euclidean distance:*

$$D(r,s_m)=\sum^N_{k=1}(r_k-s_{mk})^2$$

We called D(r,sm), m=1,2,…,M, the distance metrics. Hence, for the AWGN channel, the decision rule based on the ML criterion reduces to finding the signal sm that is closest in distance to the receiver signal vector r. We shall refer to this decision rule as *minimum distance detection*.
Expanding the distance metrics:

$$D(r,s_m)=\sum^N_{n=1}r^2_n-2\sum^N_{n=1}r_ns_{mn}+\sum^N_{n=1}s^2_{mn} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =||r||^2 - 2r.s_m+||s_m||^2,m=1,2,......,M$$

The term || r||2 is common to all distance metrics, and, hence, it may be ignored in the computations of the metrics. The result is a set of modified distance metrics

$$D'(r,s_m)=- 2r.s_m+||s_m||^2$$

Note that selecting the signal sm that minimizes D'(r, sm ) is equivalent to selecting the signal that maximizes the metrics C(r, sm)= –D'(r, sm )

$$C(r,s_m)=2r.s_m-||s_m||^2$$

The term r⋅ sm represents the projection of the signal vector onto each of the M possible transmitted signal vectors. The value of each of these projection is a measure of the correlation between the receiver vector and the mth signal. For this reason, we call C(r, sm), m=1,2,…,M, the correlation metrics for deciding which of the M signals was transmitted. Finally, the terms || sm||2 =εm, m=1,2,…,M, may be viewed as bias terms that serve as compensation for signal sets that have unequal energies.

If all signals have the same energy, || sm||2 may also be ignored. These metrics can be generated by a demodulator that crosscorrelates the received signal r(t) with each of the M possible transmitted signals and adjusts each correlator output for the bias in the case of unequal signal energies.

We have demonstrated that the optimum ML detector computes a set of M distances D(r,sm) or D'(r,sm) and selects the signal corresponding to the smallest (distance) metric. Equivalently, the optimum ML detector computes a set of M correlation metrics C(r, sm) and selects the signal corresponding to the largest correlation metric.

The above development for the optimum detector treated the important case in which all signals are equal probable. In this case, the MAP criterion is equivalent to the ML criterion. When the signals are not equally probable, the optimum MAP detector bases its decision on the probabilities given by:

$$P(s_m|r)=\dfrac{p(r|s_m)P(s_m)}{p(r)} \ \ or \ \ PM(r,s_m)=p(r|s_m)P(s_M)$$

**CORRELATION TYPE RECEIVER**

When the whitening filter g.t/ has been determined we can construct the optimum receiver. Note that we actually place the whitening filter at the channel output r .t/ as in the left part of figure 17.5 but this is equivalent to the circuit containing two filters g.t/ as in the right part of figure 17.5. Therefore the effect of applying the whitening filter g.t/ is that we have transformed the channel into an additive white Gaussian noise waveform channel and that the signals sm.t/ are changed by the whitening filter g.t/ into the signals

$$s^0_m(t)=s_m(t)*g(t)=\int^\infty _{-\infty} s_m(\alpha)g(t- \alpha)d \alpha \ \ for m \in M$$

This observation leads to the optimum receiver shown in figure 17.6. This receiver correlates the output r o.t/ of the whitening filter g.t/ with the signals som .t/ for all m 2 M. The signals Som .t/ are obtained at the output of filters with impulse response g.t/ when sm.t/ is input. Note that the constants com for m 2Mare now

$$c^0_m(t)=\dfrac{N_0}{2} \ \ In \ \ Pr \{M=m\} - \dfrac12 \int^\infty _{-\infty} [s^0_m(t)]^2dt,$$

since the signal sm.t/ has changed into som .t/.