Optimal reception of digital signals

In a digital communication system, what the receiver receives is the sum of the transmitted signal and the channel noise.

The optimal reception of digital signals is based on the minimum error probability as the “best” criterion. Errors considered in this chapter are mainly caused by band-limited Gaussian white noise. Under this assumption, the binary digital modulated signal is divided into three types: sure signal, confidence signal and fluctuation signal, and the minimum error probability is quantitatively analyzed one by one. In addition, the error probability of receiving multi-base band signal is analyzed.

The basic principle of the analysis is to take the total sampling value of a receiving signal element as a vector in the K-dimensional receiving vector space, and divide the receiving vector space into two regions. Determine whether an error has occurred according to which region the received vector falls into. The block diagram of the optimal receiver can be obtained and the bit error rate can be calculated by the decision criterion. This bit error rate is theoretically optimal, that is, theoretically the least possible.

The optimal bit error rate of the binary determinate signal is determined by the correlation coefficient p and the signal-to-noise ratio E/n., but has no direct relationship with the signal waveform. The smaller the correlation coefficient p, the lower the bit error rate. 2PSK signal has the smallest correlation coefficient (p=-1) and the lowest bit error rate. The 2FSK signal can be regarded as an orthogonal signal with a correlation coefficient p=0.

For the signal with signal and fluctuation, only FSK signal is used as the representative analysis, because in this channel, the amplitude and phase of the signal are randomly changed due to the influence of noise, so FSK signal is mainly suitable for application. Incoherent demodulation is the best receiving method because of the random change of signal phase caused by the channel.

By comparing the bit error rate of the actual receiver and the best receiver, it can be seen that if the signal-to-noise power ratio r in the actual receiver is equal to the ratio E/n of the code energy and noise power spectral density in the best receiver, the bit error rate performance of the two is the same. However, because the actual receiver is always impossible to achieve this point. Therefore, the performance of the actual receiver is always inferior to that of the best receiver.

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