haar_transform, a Fortran90 code which computes the Haar transform of data.

In the simplest case, one is given a vector X whose length N is a power of 2. We now consider consecutive pairs of entries of X, and for I from 0 to (N/2)-1 we define:

        S[I] = ( X[2*I] + X[2*I+1] ) / sqrt ( 2 )
        D[I] = ( X[2*I] - X[2*I+1] ) / sqrt ( 2 )
We now replace X by the vector S concatenated with D. Assuming that (N/2) is greater than 1, we repeat the operation on the (N/2) entries of S, and so on, until we have reached a stage where our resultant S and D each contain one entry.

For data in the form of a 2D array, the transform is applied to the columns and then the rows.


The computer code and data files described and made available on this web page are distributed under the MIT license.


haar_transform is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:


cosine_transform, a Fortran90 code which demonstrates some simple properties of the discrete cosine transform (DCT).

fftpack5, a Fortran90 code which implements the Fast Fourier Transform by Paul Swarztrauber and Dick Valent;

sftpack, a Fortran90 code which implements the "slow" Fourier transform, intended as a teaching tool and comparison with the fast Fourier transform.

sine_transform, a Fortran90 code which demonstrates some simple properties of the discrete sine transform.

walsh_transform, a Fortran90 code which implements versions of the Walsh and Haar transforms.

wavelet, a Fortran90 code which does some simple wavelet calculations;


  1. Ken Beauchamp,
    Walsh functions and their applications,
    Academic Press, 1975,
    ISBN: 0-12-084050-2,
    LC: QA404.5.B33.

Source Code:

Last revised on 12 July 2020.