# fftn#

## Purpose#

Computes a complex 1- or 2-D FFT.

## Format#

y = fftn(x)#
Parameters:

x (NxK matrix) – The data used to compute the FFT.

Returns:

y (LxM matrix) – where L and M are the smallest prime factor products greater than or equal to N and K, respectively.

## Remarks#

fftn() uses the Temperton prime factor FFT algorithm. This algorithm can compute the FFT of any vector or matrix whose dimensions can be expressed as the product of selected prime number factors. GAUSS implements the Temperton algorithm for any power of 2, 3, and 5, and one factor of 7. Thus, fftn() can handle any matrix whose dimensions can be expressed as

$2^p \times 3^q \times 5^r \times 7^s$

where p, q and r are nonnegative integers and s is equal to 0 or 1.

If a dimension of x does not meet this requirement, the fftn() pads matrices to the next allowable dimensions. However, it generally runs faster for matrices whose dimensions are highly composite numbers. Highly composite numbers are products of several factors (to various powers), rather than powers of a single factor.

For example, even though it is bigger, a 33600x1 vector can compute as much as 20% faster than a 32768x1 vector, because 33600 is a highly composite number, $$2^6 \times 3 \times 5^2 \times 7$$, whereas 32768 is a simple power of 2, $$2^15$$.

For this reason, you may want to hand-pad matrices to optimum dimensions before passing them to fftn(). The Run-Time Library includes a routine, optn(), for determining optimum dimensions.

The Run-Time Library also includes the nextn() routine, for determining allowable dimensions for a matrix. (You can use this to see the dimensions to which fftn() would pad a matrix.)

fftn() scales the computed FFT by $$1/(L*M)$$.