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In this paper a novel multicarrier modulation system called Complex Wavelet Packet Modulation (CWPM) has been proposed. It is based on using the Complex Wavelet Transform (CWT) together with the Wavelet Packet Modulation (WPM). The proposed system has been tested for communication over flat and frequency selective Rayleigh fading channels and its performance has been compared with some other multicarrier systems. The simulation results show that the performance of the proposed CWPM system has the best performance in all types of channel considered as compared with OFDM, Slantlet based OFDM, FRAT based OFDM and WPM systems. Furthermore, the proposed scheme has less PAPR as compared with the traditional WPM multicarrier system.

Multicarrier modulation (MCM) [

In recent years different types of multicarrier modulation are produced like slantlet based OFDM [

In this paper, the principles of CWT are applied to WPM in order to improve its performance introducing a novel multicarrier communication scheme called complex wavelet packet modulation (CWPM). The rest of the paper is arranged as follows: the next section reviews the discrete wavelet packet transform. Section 3 presents the proposed complex wavelet packet modulation system. Section 4 shows the simulation results while the conclusions deduced through the work are given in Section 5.

Wavelet packets are a class of generalized Fourier transforms with basis functions localizing well in both time and frequency domains. They are constructed using Quadrature Mirror Filter (QMF) pairs h(n) and g(n), satisfying the following conditions [

where usually h(n) and g(n) are low-pass and high-pass filters, respectively, and L is the span of the filters. The QMFs h(n) and g(n) are recursively used to define the sequence of basis functions φ_{n}(t), called wavelet packets as follows:

Wavelet packets have the following orthogonality properties:

where is the inner product of functions and δ(.) is the delta function. Based on h(n) and g(n), and the corresponding reversed filters h(−n) and g(−n), four operators (H^{−1}, G^{−1}, H and G) are defined that can be used to construct a wavelet packet tree. H and G are the downsampling convolution operators and H^{−1} and G^{−1} are upsampling deconvolution operators. The four operators acting on the sequence of samples x(n) are defined as follows [

_{2}(z) into two orthogonal subspaces l_{2}(z). Each decomposition (H or G) step results in two coefficient vectors each half the length of the input vector keeping the total length of data unchanged. This operation can be iterated by cascading the operators for multiple numbers of steps. In this iterative decomposition procedure, the output coefficient vectors have size reduced at each step by 2 so that eventually these output vectors become scalars. This decomposition process using G and H is called Discrete Wavelet Packet Transform (DWPT). The decomposition is a reversible process and the Inverse Discrete Wavelet Packet Transform (IDWPT) can be used to reconstruct the original input vector from the coefficients vectors. The IDWPT is a series of upsampling filtering processes defined by the operators H^{−1} and G^{−1}.

^{th} complex element is formed from the i^{th} real elements of the two input vectors. The