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Probability of bit-error (Pe) performance of asynchronous direct-sequence code-division multiple-access (DS-CDMA) systems is analyzed. In particular, the effects of pulse shaping, quadriphase (or direct-sequence quadriphase shift keying (DS-QPSK)) spreading, aperiodic spreading sequences and the coherent correlator or, equivalently, the matched filter (MF) receiver are considered. An exact Pe expression and several approximations: one using the characteristic function (CF) method, a simplified expression for the improved Gaussian approximation (IGA) and the simplified improved Gaussian approximation (SIGA) are derived. Two main results are presented. Under conditions typically satisfied in practice and even with a small number of interferers, the standard Gaussian approximation (SGA) for the multiple-access interference component of the MF statistic and for MF Pe performance is shown to be accurate. Moreover, the IGA is shown to reduce to the SGA for pulses with zero excess bandwidth. Second, the Pe performance of quadriphase DS-CDMA is shown to be superior or equivalent to that of biphase DS-CDMA. Numerical examples with Monte Carlo simulation are presented to illustrate Pe performance for square-root raised-cosine (Sqrt-RC) pulses and spreading factors of moderate to large values.