## 3 Walsh-Hadamard Transform  
Due to the disruptive effects of the multipath propagation created by the wireless communication channel, very deep fading may cause distortion to the received signal. These deep fading is one of the biggest factors that reduce the system performance, especially in multi-carrier communication systems such as OFDM, due to the required high signal to noise ratio. As the frequency response of a wireless communication channel shown in Figure 2 shows that some sub-carriers become very weak at the receiver, these sub-carriers cannot be recovered and therefore due to this information loss, the system performance decreases. In such a channel, to improve performance, the missing information must be recovered in some way.  
Figure 1: The block scheme of the GFDM waveform transceiver structure.  
Without applying any correction or equalization, even if some sub-carriers are completely lost, what needs to be done is to load each bit of information to all sub-carriers at different rates to be sent to the receiver. Thus, even if some sub-carriers are completely lost, it will be possible to recover them as the information they carry is partially loaded on other carriers. One way to accomplish this is to use the Walsh-Hadamard Transform (WHT), which is an orthogonal transformation [20,21].  
Although there are different orthogonal transformations, the Walsh-Hadamard Transform was used in this study. There are some important reasons for using WHT:  
* It is an orthogonal transformation,
* The transformation matrix is a square matrix its dimensions are \(2^{\Delta}\) (A = 0, 1, 2,... ),
* The transformation matrix can be recursively defined in any size,
* Since the matrix elements consist of 1 and \(-1\), the transformation process can be performed only by addition and subtraction without the need for multiplication,
* Since there is no relationship between any two rows or any two columns of the transformation matrix, the information loaded on the sub-carriers is also different from each other,  
Because WHT matrix is square matrix and is the size of the powers of 2, it is easy to apply to multi-carrier systems.  
Walsh-Hadamard transformation matrix,  
\[W_{1\times 1}=[1] \tag{4}\]  
including,  
\[W_{2\times 2}=\left[\begin{array}{cc}W_{1\times 1}&W_{1\times 1}\\ W_{1\times 1}-W_{1\times 1}\end{array}\right]=\left[\begin{array}{cc}1 &1\\ 1&-1\end{array}\right] \tag{5}\]  
and  
\[W_{2K\times 2K}=\left[\begin{array}{cc}W_{K}&W_{K}\\ W_{K}-W_{K}\end{array}\right] \tag{6}\]  
can be acquired in desired size as.  
The Walsh-Hadamard Transform of a sequence \(\{v[n]\}\), consisting of \(N\) samples, can be given by the Equation (7).  
\[V(k)=\sum_{n=0}^{N-1}v[n]W(n,k),\ k=0,\ 1,\ \ldots,\ N-1 \tag{7}\]  
Figure 2: Frequency response of the wireless communication channel.  
where \(W(n,k)\) describes the Walsh-Hadamard transform. Similarly, the inverse Walsh-Hadamard Transform can also be given with Equation (8).  
\[v[n]=\frac{1}{N}\sum_{k=0}^{N-1}V(k)W(n,k),\;n=0,\;1,\;\ldots,\;N-1 \tag{8}\]  
Walsh-Hadamard Transform is applied in the frequency region in OFDM or multi-carrier communication system. The data to be sent is spread to all sub-carriers by WHT process before Inverse Fast Fourier Transform (IFFT) process. In the receiver, after Fast Fourier Transform (FFT) and channel equalization, the spread data over the sub-carriers will be recovered by the Inverse Walsh-Hadamard Transform (IWHT) process.  
Data string to be sent in an OFDM signal or multi-carrier communication signal is defined as Equation (9).  
\[D=[D_{0}D_{1}\cdots D_{62}D_{63}] \tag{9}\]  
Including the 64 \(\times\) 64 size WHT matrix in (10),  
\[W=\left[\begin{array}{cccc}w_{0,0}&w_{0,1}&\cdots&w_{0,