haar, a MATLAB 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.

Thanks to comments by Stephen Becker, the code has been modified so that the haar_1d() and haar_1d_inverse(), and haar_2d() and haar_2d_inverse() functions will be proper inverses of each other even in the case when the vector or matrix dimensions are NOT powers of 2.


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


haar is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:


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

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

walsh, a MATLAB code which implements versions of the Walsh and Haar transforms.

wavelet, a MATLAB 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 modified on 27 January 2019.