We numerically differentiate the integral of a time-domain diffraction operator, which has an integrable inverse-root singularity and an infinite tail, to get a truncated digital form of the operator. This truncated difference operator effectively simulates the singularity but is computationally inefficient and produces a convolutional truncation ghost. We thus use a least-squares method to model an equivalent ARMA filter on the difference operator. The recursive convolution of the ARMA filter with a wavelet has no truncation ghost and an error below 1% of the peak diffraction amplitude. Design and application of the ARMA filter reduces computer (CPU) time by 42% over the direct convolution. A combination of filter design at a coarse spatial sampling, angular interpolation of filter coefficients to a finer sampling, and recursive application reduces CPU time by 83% over the direct convolution or 80% over~the Fourier convolution, which also has truncation error.