Probability of error in band limited spread spectrum binary communication systems



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The performance of a pseudonoise-modulated spread spectrum binary communication system is analyzed to determine the probability of error in the presence of band-limiting, gaussian noise, and synchronization errors. These degrading influences are considered individually and in combination. It is shown that the correlation receiver is an optimum receiver in the presence of white gaussian noise with no bandlimiting. Synchronization errors are introduced into the analysis in the form of a timing delay between the received code and local code. Bandlimiting is modeled by a causal low-pass filter at the input of the correlation receiver. Probability of error in detecting the binary message symbols is analyzed first by developing upper bounds based on the correlation properties of maximal length pseudonoise sequences and the decaying response typical of causal filters. Then, the average probability of error is computed using a series expansion of the characteristic function of the intersymbol interference. Results are presented for Butterworth filters which show that for large spreading ratios, the effects of intersymbol interference are negligible. Synchronization errors in addition to bandlimiting further degrade the performance, but less dramatically than in the case of the infinite-bandwidth system.