In this paper, we investigate the propagation of an ultra-wideband electromagnetic pulse in a homogeneous, cold plasma which is used to represent a simplified model of the atmosphere. The standard procedure for the computation of the corresponding transient field involves the application of a fast Fourier transform (FFT) to a well-known, analytical, frequency-domain solution. However, because of the long tails in both the time and frequency domains, a large number of sample points are required to compute the transient response using this FFT approach. In this paper, we introduce a new asymptotic extraction technique which dramatically reduces the number of sample points required by the FFT. First, we review the recently derived closed-form expression for a double-exponential pulse propagating in a homogeneous, collisionless, cold plasma. Since the high-frequency behavior does not depend on the electron collision frequency, an analytical frequency-domain expression, which is similar in form to the one encountered for the collisionless, cold plasma and encompasses this high-frequency behavior, can be subtracted from the exact expression for the plasma with a nonzero collision frequency. The extracted term is evaluated analytically. The remaining expression, which can be transformed to the time domain with a FFT, requires only a modest number of sample points. This dramatically improves the numerical efficiency of the approach. We find that the extracted analytical term provides a very good approximation for the early-time behavior of the transient pulse.