Performance Evaluation of Complex Wavelet Packet Modulation (CWPM) System over Multipath Rayleigh Fading Channel ()
1. Introduction
Multicarrier modulation (MCM) [1] based on the discrete Fourier transform (DFT) has been adopted as the modulation/demodulation scheme of choice in several digital communications standards. These include wire line systems such as digital subscriber lines (DSL), wireless systems such as digital audio and terrestrial video broadcast (DAB/DVB-T), local area networks such as IEEE 802.11a/g/n, and metropolitan area networks such as IEEE802.16a, where it is commonly known as orthogonal frequency-division multiplexing (OFDM).
In recent years different types of multicarrier modulation are produced like slantlet based OFDM [2], FRAT based OFDM [3,4] and wavelet based OFDM [5-8]. Wavelet packet Modulation (WPM) [9] is a carrierless that uses a filtering and defiltering technique to convey orthogonal multi-sub-band information from transmitter to receiver. WPM shares all the benefits of multicarrier technique and exhibits further benefit such as higher efficiency due to elimination of guard interval (GI). It is considered as one of wavelet transforms which are well localized both in time and frequency domain, while sinusoid waveforms are only localized in frequency but not in time domain. The wavelet and subband transform applications in communications are viewed in [9]. Besides the features mentioned previously, WPM has additional attractive features inherited from the wavelet packet modulation: the Wavelet Transform (WT) is widely adopted in image/video and speech coding. Its use for modulation/demodulation on frequency-selective channels results in a better integrated system design and a reduced overall implementation cost. The WPT uses only real arithmetic, as opposed to the complex-valued DFT. This reduces the signal-processing complexity/power consumption, but it suffers from three major limitations: shift sensitivity, poor directionality, and absence of phase information. To solve these problems, Complex wavelet transform (CWT) [10] are proposed and applied in many signal processing applications especially in image processing applications [11-13].
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.
2. Discrete Wavelet Packet Transform (DWPT)
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 [14]:
(1)
(2)
(3)
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 φn(t), called wavelet packets as follows:
(4)
(5)
Wavelet packets have the following orthogonality properties:
(6)
(7)
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 [12]:
(8)
(9)
(10)
(11)
Figure 1 shows the construction of these operators. The operators H and G can be used to decompose (analyze) any discrete function x(n) on the space l2(z) into two orthogonal subspaces l2(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. Figure 2 shows a full DWPT tree (on the left side) and a full IDWPT tree (on the right side) which are used in the WPM system for demodulation and modulation, respectively.
3. Complex Wavelet Packet Modulation (CWPM)
Figure 3 illustrates the block diagram of proposed CWPM transceiver. CWPM system employs two filter banks performs Inverse Discrete Wavelet Packet Transform (IDWPT) placed at transmitter side, and Discrete Wavelet Packet Transform (DWPT) placed at the receiver side. The block “MAKE CMPLX” accepts two N-dimensional real vectors as inputs. Its output is an N-dimensional complex vector whose ith complex element is formed from the ith real elements of the two input vectors. The