Generation of Walsh functions

program description

This applet allows to display the Hadamard matrices  $\mathbf{H}_J$  for the construction of the Walsh functions  $w_j$. The factor  $J$  of the band spreading as well as the selection of the individual Walsh functions (by blue bordering the rows of the matrix) can be changed.

Theoretical background

Application

The  Walsh functions  are a group of periodic orthogonal functions. Their application in digital signal processing is mainly in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.

• Due to their orthogonal properties and the favourable PKKF conditions (periodic KKF), the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system. If you take any two lines and form the correlation (averaging over the products), the PKKF value is always zero.
• In asynchronous operation (example:   uplink of a mobile radio system) or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading - see   [Tasks:5.4_Walsh Functions_(PKKF,_PAKF)|Task 5.4]].
• In terms of PAKF (periodic AKF) these sequences are less good:   Each individual Walsh function has a different PAKF and each individual PAKF is less good than a comparable PN sequence. That means:   The synchronization is more difficult with Walsh functions than with PN sequences.

= Construction

The construction of Walsh functions can be done recursively using the 'Hadamard matrices. A Hadamard matrix $\mathbf{H}_J$ of order $J$ is a $J\times J$ matrix, which contains line by line the $\pm 1$ weights of the Walsh sequences. The orders of the Hadamard matrices are fixed to powers of two, i.e. $J = 2^G$ applies to a natural number $G$. Starting from $\mathbf{H}_1 = [+1]$ and

$$\begin{equation} \mathbf{H}_2 = \left[ \begin{array}{rr} +1 & +1\\ +1 & -1 \\ \end [array}\right] \end{equation}$$ the following relationship applies to the generation of further Hadamard matrices: $$\begin{equation} \mathbf{H}_{2N} = \left[ \begin{array}{rr} +\mathbf{H}_N & +\mathbf{H}_N\\\\ +\mathbf{H}_N & -\mathbf{H}_N \\\\ \end [array}\right] \end{equation}$$
{TEXT= $\text{example:}$  The graphic shows the Hadamard matrix  $\mathbf H_8$  (right) and the spreading sequences which can be constructed with it  $J -1$ .

Datei:P ID1882 Mod T 5 3 S7 new.png

Walsh spreading sequences  $(J = 8)$  and Hadamard matrix  $\mathbf H_8$ 

• $J - 1$ because the unspread sequence  $w_0(t)$  is usually not used.
• Please note the color assignment between the lines of the Hadamard matrix and the spreading sequences  $w_j(t)$.
• The matrix  $\mathbf H_4$  is highlighted in yellow.

To use the applet

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    (A)     Selection of the factor for band spreading as power of two of $G$

    (B)     Selection of the respective Walsh function $w_j$

About the authors

This interactive calculation tool was desi

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