Читать книгу Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai - Страница 30

The detection problem of the binary waveform signal when M = 2 has been introduced above, and then we will analyze the detection problem of the M-ary signal.

It is assumed that in the M-ary communication, there is a set {s₁(t), s₂(t), . . . , s_M(t)}, 0 ≤ t < T consisting of M signal waveforms. At this time, the data transmission rate is bit/s. It can be seen that the data transmission rate increases as M increases, which means that the larger the M value is, the better. However, in general, the detection performance of the signal will be worse as the value of M increases.

In the case of M hypotheses, assuming that the received signal is R(t) = s_m(t) + N(t), 0 ≤ t < T. The likelihood function and log-likelihood function of the L samples are expressed as

If the common terms in all hypotheses are ignored, the log-likelihood function becomes

Fig. 2.4. Maximum likelihood detector for M-ary signal detection based on a set of correlation detectors.

Using r(t) to represent the observed value of R(t), when L tends to infinity, the following equation can be obtained:

where is the energy of the mth signal s_m(t) and is the correlation function of r(t) and s_m(t).

According to the expression of the log-likelihood function, the maximum likelihood judging criteria for accepting S_m should be log f_m(r(t)) ≥ logf_m′(r(t)) or . The symbol “\” is defined as A\B = {x|x ∈ A, x ∉ B}.

Based on the preceding conclusions, the process of connecting a series of correlation detectors to achieve the maximum likelihood judging criteria is shown in Fig. 2.4.