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Approximations are obtained for the average probability of error in an asynchronous direct-sequence spread-spectrum multiple-access communications system. Both binary and quaternary systems are considered, and the chip waveforms are allowed to be arbitrary time-limited waveforms with time duration equal to the inverse chip rate. The approximation is based on the integration of the characteristic function of the multiple-access interference. The amount of computation required to evaluate this approximation grows only linearly with the product of the number of simultaneous transmitters and the number of chips per bit. The accuracy of the approximation is extremely good in most cases, but it can be improved, if necessary, by an application of a series expansion. Numerical results are presented for specific chip waveforms and signature sequences.