User:Rishabh4112/sandbox

Introduction
Hadamard’s matrices are used widely at the forward links of communication channels to mix the information on connecting to and at the backward links of these channels to sift through this information is transmitted to reach the receivers this information in correct form, specially in the pilot channels, the Sync channels, the traffic channel and so much applications in the fields; Mod- ern communication and telecommunication systems, signal processing, optical multiplexing, error correction coding and design and analysis of statistics.

Formulate the Mathematics
n the modern world, Communication are most important aspect of our life, but their were some noise in the communication channel. As an engineer our work to remove this noise and solve this problem using some theory and formula.

Solve the Matrix
A square matrix whose entries are either +1 or 1 and whose rows are mutually orthogonal is called the Hadamard matrix. A Hadamard matrix has maximal determinant among matrices.Hadamard matrix H of order n satisfies HHT = nIn where In is the n x n identity matrix and HT is the transpose of H.

Example of a small order Hadamard's Matrix is:
 * $$\begin{bmatrix}1 & 1 \\1 & -1  \end{bmatrix}.$$

Walsh Functions
Walsh functions were initially introduced mathematically by Walsh in 1923.It forms an orthonormal set of rectangular waveforms with values of -1 or +1 on the interval [0,1). Hence the corresponding fast algorithms require only addition and subtraction of input values. Formula:

x2k-1m+1(t) = xkm(2t);                   0.5≤t≤1 (-1)k+1xkm(2t-1);           0.5≤t≤1 xkm+1(t) = xkm(2t);                      0.5≤t≤1 (-1)kxkm(2t-1);               0.5≤t≤1

Hadamard matrix to Walsh matrix
First we create the Hadamard matrix as we seen earlier. Then we note the number of sign change for each row. And then we arrange the row in the increasing order of sign change from the top to bottom.

H4 = $$\begin{bmatrix}1 & 1 & 1 & 1 \\1 & -1 & 1 & -1 \\1 & 1 & -1 & -1 \\1 & -1 & -1 & 1  \end{bmatrix}.$$

W4 = $$\begin{bmatrix}1 & 1 & 1 & 1 \\1 & 1 & -1 & -1 \\1 & -1 & -1 & 1 \\1 & -1 & 1 & -1  \end{bmatrix}.$$

Walsh-Hadamard Transform
The Walsh–Hadamard transform is a non-sinusoidal, orthogonal transformation technique that decomposes a signal into a set of basis functions. These basis functions are Walsh functions, which are rectangular or square waves with val- ues of +1 or –1. Walsh–Hadamard transforms are also known as Walsh, or Walsh-Fourier transforms. The Walsh–Hadamard transform returns sequency values. Sequency is a more generalised notion of frequency and is defined as one half of the average num- ber of zero-crossings per unit time interval. Each Walsh function has a unique sequency value. It can be used to estimate the signal frequencies in the original signal.

Formula for walsh Hadamard transformation W*n ___        N where W is the Walsh matrix, n is the sequence of even number, N is the size of Walsh function.

Formula for inverse walsh Hadamard transformation W'*n ____         N where W’ is the walsh Hadamard transformation and n is the sequence of even number.

Solution
1)First we take a normal signal which we want to send through the channel.

2) Then we get the output at the end of the channel with noise added to it.

3) Next we apply Walsh-Hadamard Transformation on the noisy signal.

4) Next we remove the higher coefficients from the signal.

5) Finally we reconstruct the signal using Walsh-Hadamard Transform