Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis

2.1.1 Discrete wavelet transform (DWT)

The computation cost for wavelet coefficients in the CWT is high since they are highly redundant data, which is not desirable for real application. Therefore, discrete wavelets offer as the alternative for practical applications. In discrete wavelets, the scalable and translatable wavelets are discrete. The process of discrete scaling and translation of the mother wavelet can be expressed as

where a₀ represents the fixed step of dilation and b₀ indicates the translation factor. The integer αand βsignify the indices scale and translation, respectively. The scaling in time domain correlates with an inverse scaling in frequency domain, therefore the product of Δta,bΔfa,bis independent of the dilation parameter a. If any time resolution gain is obtained, this inversely effect the cost of frequency resolution and vice versa. Therefore, this detains the Heisenberg uncertainty principle for the dilated and translated wavelet ψCWTa,btand the mother wavelet ψCWTt. The most natural choice for dilation step is 2 that results in octave bands or dyadic scales. The wavelet is compressed in frequency domain by a factor of 2 for each successive value of scale index. This produces the stretched in time domain by the same factor. The translation factor is set to the value of “1” to get the dyadic sampling fashion. The time-shift and scaling function are set as [16];

where Zis the set of all integer numbers, and L2Ris the vector space of square integrated function. The parameter v0is a space spanned by scaling function, which is defined as

In this subspace, if xt∈v0, it can be expressed as

One can increase the size of the subspace by changing the time scale of the scaling functions. The two-dimensional parameterization (time and scale) of scaling function φtfrom v0to vαcan be expressed as

Then, the new function for the expanded subspace v_αis given as

In the extended subspace, whenever xt∈va, then it can be expressed as

From (Eq. (7)), the span v_αis larger than v₀, for α>0and φα,βtable to represent the finer detail (due to its finer scale). For α<0this condition is true that represents for the coarse scale. Wavelet obeys to multi-resolution concept’s requirement, where every signal is decomposed into finer detail gradually as expressed as [17, 18].

where the terms v+∞=L2, and v−∞=0indicate that within the same vector space of L2, there exist both high resolution and low-resolution coefficients. Consequently, if xt∈va, then x2t∈va+1. Additionally, the ϕtterm is expressed as the weighted sum of the time-shifted scaling function

where the term h(n) represents the scaling function coefficients (sequence of real or imaginary numbers). The vα+1is the expanded space of vαand wαrepresents the corresponding orthogonal complement. Therefore, a new set of spaces is produced. Suppose that wα+1be the subspace spanned by the wavelet, the enlargement of v₁ and v₂ space are written as (Eq. (10)) below and as illustrated as in Figure 1 [19].

The definition of the wavelet function ψtis the same as the scaling space v0. Let the space spanned by the wavelet function ψβtbe w0, and the expanded space spanned by ψα,βtbe wαthat is obtained after using (Eq. (3)) to (Eq. (6)). The wαterm is orthogonal to vαand thus the orthogonality between φtand ψtis given as [19];

Due to these wavelets are in space spanned by the next finer scaling function, the wavelet function ψ(t) can be expressed by the sum of the weighted time-shifted wavelet function given as

where g(n) is called the wavelet function coefficient. The relationship between wavelet filter g(n) and scaling filter h(n) can be expressed as [19];

Both coefficients are restricted by the orthogonality condition. If h(n) has a finite even length N, then the (Eq. (13)) can be rewritten as

The wavelet function coefficients gnis normally required by the orthonormal perfect reconstruction (PR) process. In the communication system point of view, this PR process offers advantage to the receiver whereby the received signals can be reconstructed perfectly. For example, Haar wavelet below is analyzed with the wavelet function ψtcan be expressed as

and its scaling function is

Furthermore, the basic version of Haar wavelet for wavelet and scaling function is shown in Figure 2.

The Haar filter coefficients are obtained by applying (Eq. (9)) and (Eq. (12)).

Furthermore, the signal xt∈L2Rhas its discrete wavelet expansion given as [14].

where α,β,∈Zwhich Zis real integer. The α0is an arbitrary integer, and L2Ris the vector space of the square integrated function. The frequency (or scale) and time localizations are provided by the parameters αand βrespectively. The approximation coefficient and the detail coefficient have been deduced as cαβand dαβrespectively.

In the wavelet expansion, by manipulating (Eq. (9)) and (Eq. (19)), the higher scale (i.e. α+1) can also be obtained that results the approximation coefficient as

while the detail coefficient is expressed as

Both the terms of cαβand dαβin (Eq. (20) and (21)) are computed by taking the weighted sum of DWT coefficients of higher scale α+1. In order to obtain the scaling of the DWT coefficients cαβat scale α, the scaling function coefficient hnis convolved with the scaling DWT coefficients cα+1βat scale α+1, followed by subsampling with a factor of 2. Similarly, to obtain the wavelet DWT coefficients dαβat scale α, the wavelet function coefficient gnis convolved with the scaling DWT coefficients cα+1βat scale α+1, followed by subsampling with a factor of 2. Hence, as shown in Figure 3, that both of these expressions can be illustrated as 2-channel filter banks analysis [20].

The input signal to the 2-channel filter bank is split into two parts. The first portion of the signal goes to filter Hand second goes to filter G. Subsequently, both the filtered signals are subsampled by 2. Each filtered signal contains half of the number of original samples and spans half of the frequency band. However, the number of samples at the output of the filter bank is the same as the original signal since there are two filters used. The decomposition process starts at the largest scale of cβ. If there are three level of decompositions involved, this implies the term c3βexist and produces the terms c0β, d0β, d1βand d2βat the decomposition branches, as illustrated in Figure 4.

On the other hand, the reconstruction of the DWT coefficients process is expressed by (Eq. (19)). If (Eq. (9)) (for scaling refinement) and (Eq. (12)) (wavelet function) are substituted into (Eq. (19)) (reconstruction function), thus produces

By multiplying both sides of (Eq. (22)) with φ2α+1−βand taking the integral produces the lower scale of DWT coefficients [18], the scaling DWT coefficients of higher scale is given as

This implies that the scaling DWT coefficients at a certain value α+1can be computed by taking the weighted sum of wavelet DWT coefficients that are multiplied with the scaling DWT coefficients at scale α. Figure 5 illustrates this process which is known as a 2-channel synthesis filter bank. The scaling DWT coefficients cαβand wavelet DWT coefficients dαβat scale αare first up-sampled by factor 2. Then, the scaling DWT coefficients cαβare filtered with a LPF Ĥ, and the wavelet DWT coefficients dαβare filtered with a HPF Ĝrespectively. Finally, the two filtered signals are added together to form the scaling DWT coefficients at scale α+1i.e. cα+1β.

In short, the DWT decomposes signals into coefficients. The IDWT reconstructs the original signals from coefficients which can be implemented efficiently by iterating the 2-channel synthesis filter bank.

2.1.2 Wavelet packet transform (WPT)

In DWT decomposition, the direction of decomposition is heading towards the low pass branches, i.e. the sequence of iteration for the 2-channel filter bank is always taking the low pass filters. At the end of decomposition, the low frequencies portion contains fewer numbers of coefficients, hence occupying a narrow bandwidth. The high frequencies portion contains larger number of coefficients, hence occupying a wide bandwidth.

On the other hand, wavelet packet transform (WPT) executes the iteration of 2-channel filter bank on both sides, i.e. on the low pass and high pass filter branches for decomposition. Therefore, the WPT has evenly space frequency resolution and similar bandwidth size since both the high frequency and low frequencies components are decomposed. In WPT, the filter bank structure is expanded into a full binary tree. A set of WPT coefficients is labeled by ζand the level that corresponds to the depth a node in the tree structure is indicated by land parameter pindicates the position at current node. Every parent node is split by the WPT in two orthogonal subspaces WlPwhich is located at the next recursive level, and is given as [19];

In WPT, the scaling WPT coefficients are denoted as ζl+12pand wavelet WPT coefficients are labeled as ζ2p+1, given as in the following expressions, and are depicted as in Figure 6.

In WPT, the number of iterations by the 2-channel filter bank increases exponentially as the number of levels increased. Therefore, WPT has higher computational complexity than the regular DWT. The WPT requires O(Nlog(N)) operation (by using fast filter bank algorithm), while Fast Fourier Transform (FFT) requires only O(N) operations to complete DWT [16]. The reconstruction (inverse WPT) is executed by taking the reverse direction of the tree in Figure 6. The wavelet packet coefficients ζlpβat any level lcan be expressed as

3.1.1 System models descriptions

The three evaluated multicarrier modulation (MCM) system models are represented by a single general MCM model as illustrated in Figure 10. The information bits are generated based on the uniform random distribution binary number. The data are arranged (in every frame) in a horizontal matrix 1×ninit, and are converted into base16 number format. Subsequently they are encoded by Reed- Solomon (RS) codes, and converted into baseMnumber format, where Mcorresponds to the total mapping points in a particular QAM constellation. In this work, the Reed-Solomon of RS(n, k) is used, where nis the encoded data, and kis original data. In particular, the RS(15,11) is used throughout the work for protecting the original data. This channel coding scheme is compatible with hexadecimal number for encoding and decoding processes. In addition, RS encoded symbols are converted to the baseMsymbols to achieve the same configuration that adapts with modulation constellation mapping.

Table 1 shows four possible of baseMnumber format types associated with the number of bits per symbol, Nbpsas well as the corresponding constellation mapping modulation types. Then, the data frames are transformed into time-domain MCM signals by using a particular inverse transform prior transmission and they are retrieved back by the corresponding forward transform at the receiver. The particular inverse and forward transforms applied are labeled in block diagrams as shown in Figures 11 and 12.

In Figure 11, there are two types of wavelet-based MCM models to be considered i.e. the wavelet-based OFDM (WOFDM) and wavelet packet-based OFDM (WP-OFDM) systems. At the transmitter, either the inverse discrete wavelet transform (IDWT) or inverse discrete wavelet packet transform (IDWPT) is used. At the receiver, either the discrete wavelet transform (DWT) or discrete wavelet packet transform (DWPT) is used. These modulation techniques offer higher spectral efficiency since there is no for the system to use the cyclic pre-fix (CP) codes as in the conventional OFDM.

Figure 12 shows the conventional OFDM (C-OFDM) which is included for comparison system model. This model utilizes the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT). Additional blocks are required for appending and re-moving the CP codes where 25 percent of the OFDM frames tail are copied and appended to OFDM frames head [25, 26].

Each frame must contain Psymbols and is always less than or equal to the total number of subcarriers N,i.e. P≤N. The specified number of base for every modulation type is fixed as in Table 1. The number of initial binary information ninitincreases as the number of bits per symbol Nbpsincreased with constant value of N. In this work, the RS(15, 11) is used, and the encoded data n=15, and the original data k=11respectively. This implies that each time a sequence of symbols to be encoded, the number of original symbols taken is eleven and this produces total fifteen encoded symbols afterwards. Therefore, during the encoding process, the raw binary data (base2) need to be converted to base16 symbols to suites the requirement of RS(15, 11) coding scheme where each encoded symbol should have a value between 0 to 15.

3.1.2 Determination transmission parameter

This section describes how the transmission parameters values of Pis obtained by manipulating the base number of the symbols. Figure 10 above shows the block diagram of the of S/S encodes where raw input data bits are converted into baseMsymbols. Figure 13 shows further details of this process.

Figure 13 denotes three conversion processes for the initial input bit ninitwhich are indicated as the α, βand γprocesses. The αprocess converts every four bits (v=4)of binary source data to a base16 symbol. For example, if ninit,base2=10010011001100101000000001110001011100101001then, the output from process αis nα=93328071729. This means the number of overall symbols is reduced to one-fourth.

Next, βprocess converts every eleven base16 symbols k=11into fifteen base16 encoded symbols n=15. For instance, when nα=93328071729then, the output of βprocess is nβ=93328071729152711which increases the number of symbols along the encoding processes with the additional redundancy required for channel error protection. This implies the number of overall symbols now has been increased by 15/11. After that, γprocess converts every single base16 symbol into four base2 symbols (w=4). Then this follows by converting every Nbps=log2Mnumber of base2 symbols back to a single baseMsymbol. Suppose that M=16, continuing the above example, when nβ=93328071729152711then, the output of γprocess is nγ,base16=93328071729152711.The rest of other mappings are as listed in Table 2. The number of transmission symbols P, thus can be expressed as

where P≤N.

Using (Eq. (32)), the number of transmission symbols Pand initial input bit ninitcan be obtained after defining number of subcarriers N. Thus, number of bits per symbol, Nbpscan be obtained and the quantitative relationships between these parameters are shown in Table 3.

Figure 14 shows the partitions between the occupied slot positions of the encoded data Pand the remaining slot positions Rfor three different values of total subcarriers N=64128256. It can be observed that the subcarriers are not fully occupied for the whole slot positions with frames, hence the remaining positions, Ris filled up with zeros. There are four zero frames N=64, eight zero frames N=128and six zero frames N=256respectively.