The Haar functions
The family of N Haar functions
on the interval . The shape of the specific function
of a given index depends on two parameters t and :
From the definition, it can be seen that determines the amplitude and width of the non-zero part of the function, while determines the position of the non-zero part of the function.
The Haar Transform Matrix
The N Haar functions can be sampled at , where
form an by matrix for discrete Haar transform. For example, when ,
We see that all Haar functions contains a single prototype shape composed of a square wave and its negative version, and the parameters
The Haar transform matrix is real and orthogonal:
Comparing this Haar transform matrix with all transform matrices previously discussed (e.g., Fourier transform, cosine transform, Walsh-Hadamard transform), we see an essential difference. The row vectors of all previous trnasform methods represent different frequency (or sequency) components, including zero frequency or the average or DC component (first row ), and the progressively higher frequencies (sequencies) in the subsequent rows ( ). However, the row vectors in Haar transform matrix represent progressively smaller scales (narrower width of the square waves) and their different positions. It is the capability to represent different positions as well as different scales (corresponding different frequencies) that distinguish Haar transform from the previous transforms. This capability is also the main advantage of wavelet transform over other orthogonal transforms.
A Haar Transform Example:
The Haar transform coefficients of a -point signal
can be found as
The inverse transform will express the signal as the linear combination of
the basis functions: