## 4 Lifting Wavelet Transform  
Though wavelet transform has been used in signal processing implementations for long time, it has begun to stand out in the area of communication in recent years. Because of the pliable construction of the wavelet transformation and the adaptive time-frequency intervals on the sub-carriers, the discrete wavelet transform (DWT)-based OFDM (DWT-OFDM) construction is an alternative to the FFT-based OFDM construction [22]. The DWT converts discrete data from time domain to time and frequency domain. On account of many interesting attributes, the discrete wavelet transformation is being used in many possible applications as for example voice compression, which gives quicker sending in mobile telecommunications. DWT is employed in medical applications for its real-time processing abilities. Besides the two applications already mentioned wavelets are used in other applications as well, as for example: in denoising, edge detection, echo cancellation, etc. An alternative method for constructing biorthogonal wavelet transform, the lifting scheme [23; 24], has some advantages over the classical standard wavelet transform which are given below. * it is a spatial domain method,
* it is easy to use for implementation,
* it allows the use of quicker and in-place computations,
* it allows the use of nonlinear, adaptable, disorderedly sampled and integer to integer wavelet transforms,  
Another advantage of the lifting scheme is that the LWT and the reverse-LWT shown in Figure 3 are completely symmetrical to each other. This ensures that LWT can be applied easily. Filters and subsampling are used in traditional wavelet transform processes. Lifting method was developed by [PERSON] in 1996 to reduce the complex mathematical operations used in filtering. Lifting method is the simplest and most efficient method for wavelet transformation [25; 26]. In the lifting method, the signal is divided into odd and even samples. Instead of filters, Split Predict and Update processes are applied. Complex calculations are not required for these operations as in traditional methods. In Figure 3, the Lifting wavelet transformation in the LWT unit is given in detail [27]. Figure 3: The transceiver structure of the proposed waveform. The original \(X[k]\) signal is primarily separated into odd and even components as in Equations (15) and (16). \[X_{\text{odd}}[k]=X[2k+1] \tag{15}\]  
\[X_{\text{even}}[k]=X[2k] \tag{16}\]  
There is a strong correlation between these two examples. In the predict step, odd samples are tried to be obtained approximately by making use of even samples. The second part, Predict, preserves the high frequency components by eliminating the low frequency components of the signal. In the prediction process, the subset \(X_{\text{odd}}[k]\) is predicted over the subset \(X_{\text{even}}[k]\) using the prediction operator \(P(.)\). The detail information of the \(X[k]\) signal is obtained as in Equation (17) by taking the difference between the subset \(X_{\text{odd}}[k]\) and the predicted \(P(X_{\text{even}}[k])\). The \(P(. )\) Prediction operator is a linear combination of a single neighboring subset [26]. \[P(X_{\text{even}}[k])=\sum_{i}(p_{i}X_{\text{even}}[k+i]) \tag{17}\]  
Here, \(p_{i}\) are the prediction coefficients. This step acts as a high pass filter and the high frequency components (\(d[k]\)) that give the detail part of the obtained signal. \[d[k]=X_{\text{odd}}[k]-P(X_{\text{even}}[k]) \tag{18}\]  
In the update step, the samples are scaled to fix and then added with even samples to provide low pass filtered values for transformation. In the third part, update, the even components are updated by using the detailed signal to reduce the frequency aliasing effect. In the update process, the approximate information about the signal is obtained as a result of collecting the detailed information entering the \(U(. )\) Update operator with the subset \(X_{\text{even}}[k]\). The \(U(. )\) Update operator is a linear combination of neighboring \(d[k]\) values. The formula for the \(U(.