Computes the convolution, and optionally, the correlation of two real vectors.

The type double function is imsl_d_convolution.

A pointer to an array of length nz containing the convolution of x and y. To release this space, use imsl_free. If no zeros are computed, then NULL is returned.

The function imsl_f_convolution, by default, computes the discrete convolution of two sequences x and y. More precisely, let nx be the length of x, and ny denote the length of y. If a circular convolution is desired, the optional argument IMSL_PERIODIC must be selected. We set

and we pad out the shorter vector with zeros. Then, we compute

where the index on x is interpreted as a positive number between 1 and nz, modulo nz.

The technique used to compute the zi’s is based on the fact that the (complex discrete) Fourier transform maps convolution into multiplication. Thus, the Fourier transform of z is given by

where the following equation is true.

The technique used here to compute the convolution is to take the discrete Fourier transform of x and y, multiply the results together component-wise, and then take the inverse transform of this product. It is very important to make sure that nz is the product of small primes if option IMSL_PERIODIC is selected. If nz is a product of small primes, then the computational effort will be proportional to nzlog(nz). If option IMSL_PERIODIC is not selected, then a good value is chosen for nz so that the Fourier transforms are efficient and nz ≥ nx + ny − 1. This will mean that both vectors will be padded with zeros.

We point out that no complex transforms of x or y are taken since both sequences are real, and real transforms can simulate the complex transform above. Such a strategy is six times faster and requires less space than when using the complex transform.

Optionally, the function imsl_f_convolution computes the discrete correlation of two sequences x and y. More precisely, let n be the length of x and y. If a circular correlation is desired, then option IMSL_PERIODIC must be selected. We set (on output)

where α, β, and γ are nonnegative integers yielding the smallest number of the type 2α3β5γ satisfying the inequality. Once nz is determined, we pad out the vectors with zeros. Then, we compute

where the index on x is interpreted as a positive number between one and nz, modulo nz. Note that this means that

contains the correlation of x(k − 1) with y as k = 0, 1, …, nz/2. Thus, if x(k − 1) = y(k) for all k, then we would expect

to be the largest component of z. The technique used to compute the zi’s is based on the fact that the (complex discrete) Fourier transform maps correlation into multiplication. Thus, the Fourier transform of z is given by

where the following equation is true.

Thus, the technique used here to compute the correlation is to take the discrete Fourier transform of x and the conjugate of the discrete Fourier transform of y, multiply the results together component-wise, and then take the inverse transform of this product. It is very important to make sure that nz is the product of small primes if IMSL_PERIODIC is selected. If nz is the product of small primes, then the computational effort will be proportional to nzlog (nz). If IMSL_PERIODIC is not chosen, then a good value is chosen for nz so that the Fourier transforms are efficient and nz ≥ 2n − 1. This will mean that both vectors will be padded with zeros.

We point out that no complex transforms of x or y are taken since both sequences are real, and real transforms can simulate the complex transform above. Such a strategy is six times faster and requires less space than when using the complex transform.

This example computes a nonperiodic convolution. The idea here is that you can compute a moving average of the type found in digital filtering using this function. The averaging operator in this case is especially simple and is given by averaging five consecutive points in the sequence. We try to recover the values of an exponential function contaminated by noise. The large error for the last value has to do with the fact that the convolution is averaging the zeros in the “pad” rather than the function values. Notice that the signal size is 100, but only reports the errors at 10 points.

This example computes both a periodic correlation between two distinct signals x and y. There are 100 equally spaced points on the interval [0, 2π] and f1(x) = sin (x). Define x and y as follows:

Note that the maximum value of z (the correlation of x with) occurs at i = 25, which corresponds to the offset.