XL-FFT

A formula-based Fast Fourier Transform for Excel.

Calculate real and complex Fourier transforms of one-dimensional arrays in Excel without VBA macros.

Installation

The recommended installation method is Microsoft's Advanced Formula Environment for Excel (now part of Excel Labs). Create a new module (the suggested name is FFT, but any name will work) and paste the contents of the file XL-FFT.xlf in the text area, or import a new module directly from this gist.

Alternatively, it is also possible to copy the individual definition of LAMBDA formulas into the Excel Name manager, after removing all comments (and the trailing semicolons).

Usage

The Fourier Transform operates on real and complex vectors. Complex numbers are represented as a pair of horizontally adjacent real numberrs. A complex number $z=x+\mathrm{i}y$ is represented as:

  | A | B |
--|---|---|
1 | x | y |

Vectors are always column vectors. For example a real vector (x_1, x_2) is represented as:

  |  A  |
--|-----|
1 | x_1 |
2 | x_2 |

and a complex vector (z_1, z_2) = (x_1 + i*y_1, x_2 + i*y_2) as:

  |  A  |  B  |
--|-----|-----|
1 | x_1 | y_1 |
2 | x_2 | y_2 |

Complex Fourier Transform

Forward Fourier Transform

FFT(z, [n])

Compute the Fourier transform of z, where z is a complex vector and n is the length of the transform. If the length of z is smaller than n then z is padded with zeroes; if it is larger than n then z is truncated, if it is equal to n or if n is omitted, then z is not modified. The output is a complex vector of length n. The output is a complex vector of length n.

Inverse Fourier Transform

IFFT(z, [n])

Compute the inverse Fourier transform of z, where z is a complex vector and n is the length of the transform. If the length of z is smaller than n then z is padded with zeroes; if it is larger than n then z is truncated, if it is equal to n or if n is omitted, then z is not modified. The output is a complex vector of length n.

There are different conventions for the norm of the forward and inverse transform. Here the forward transform has norm n and the inverse transform has norm 1/n.

Real Fourier Transform

Forward Fourier Transform

RFFT(x, [n])

Compute the Fourier transform of x, where x is a real vector and n is the length of the real vector on which the transform is applied. If the length of x is smaller than n then z is padded with zeroes; if it is larger than n then x is truncated, if it is equal to n or if n is omitted, then x is not modified. The output is a complex vector of length (n + 2) / 2 if n is even and (n + 1) / 2 if n is odd.

Inverse Fourier Transform

IRFFT(z, [n])

Compute the inverse Fourier transform of z, where z is a complex vector and n is the length of the real transform of z. If the length of z is smaller than (n + 2) / 2 (if n is even) or (n + 1) / 2 (if n is odd) then z is padded with zeroes; if it is larger then z is truncated, if it is equal then z is not modified, if n is omitted then it is set to twice the length of z minus one and z is not modified. The output is a real vector of length n. Note that it is necessary to specify n in order to obtain an output of odd length.

The convention for the norm is the same as in the complex case.

License

Released under the MIT license. See the file LICENSE.