Wavelet Channel Coding: An Algebraic Approach

In this paper, the wavelet channel coding (WCC) is revisited in a more general analysis embracing flat real wavelet matrices derived from a Haar matrix and complex input symbols. WCC encoding and decoding are algebraically described and a probability distribution of wavelet symbols is formulated. Signal constellations for transmission of wavelet symbols are proposed and the constellation average energy is deduced from probability generating functions of the wavelet symbols. System performance over a flat Rayleigh channel is analyzed and compared with symbol-by-symbol detecting systems and diversity two space-time block coding (STBC) systems. Simulation results show that WCC presents better performance than ordinary symbol-by-symbol detecting systems, particularly at higher signal-to-noise ratios for higher spectral efficiencies, and STBC systems for spectral efficiency of 1 bit/s/Hz.

provide any coding gain over an additive white Gaussian noise channel.
Articles on the performance of systems that employ WCC can be found at [5], [6], [17], [20]. In general, WCC provides an increase in system performance at the cost of a large increase in the dimensionality of the output code. In order to counteract this drawback, some works propose constellations and increase spectral efficiency to wavelet symbols transmission, as in [8], [12], [13], [15].
WCC encoding is performed through a finite impulse response filter-like state machine composed by shift registers to store input symbols, coefficient taps from a wavelet coding matrix and adders. Information from each input symbol is spread over several wavelet symbols [1]. The consistent orthogonality between wavelet matrix rows, which remains even after overlapping and addition between them, makes possible to decode simply by using a bank of correlators formed from the rows of the matrix itself. This computational simplicity in decoding, with the same low complexity of the encoding process, and the ability to protect information from communication channel fading are the main advantages in employing WCC [2]. Furthermore, a coding rate change in WCC may be achieved by simple rearrangement of the coefficient taps and adders in the encoder, which suggests its use in adaptive systems to channel conditions. Alternative encoding and decoding approaches for WCC-based systems are explored in [9], [10], [16].
A WCC encoder maps a sequence of equiprobable input symbols to a sequence of not equiprobable wavelet symbols. The maximum number of wavelet symbols that carry part of the information from an input symbol is known as the constraint length of the wavelet code and is given by the product of the coding rate with the number of columns of the wavelet matrix [1]. An increase in input symbols cardinality or in constraint length has a strong impact on the cardinality of the wavelet symbol set and consequently on the cardinality of the transmission signal constellation [7]. In contrast, encoding and decoding complexities vary just linearly with the number of columns of the wavelet matrix.
WCC is presented and analyzed here in a more general approach, with WCC encoding and decoding algebraically described. The proposed system consider flat real wavelet matrices derived from a Haar matrix and complex input symbols. These input symbols are provided by a prior data mapper, which lets to increase the spectral efficiency, ordinarily restricted to 1 bit/s/Hz 1 [1]. A probability distribution of wavelet symbols is formulated. Signal constellations for transmission of wavelet symbols are proposed and the constellation average energy is deduced from probability generating functions of the wavelet symbols.
In order to infer about the employability of this technique, its performance was analyzed and compared with ordinary symbol-by-symbol detecting systems. In addition, WCC system performance was compared with space-time block coding (STBC) systems with two transmitting and one receiving antennas proposed by Siavash Alamouti in [21].
Section II briefly reviews the concepts on wavelet matrices. Section III presents the proposed WCC system, an algebraic description of WCC encoding and decoding processes, as well as the probability distribution of the generated wavelet symbols. Section IV proposes signal constellations for transmission of the WCC symbols. In addition, it calculates probability generating functions for the generated wavelet symbols and employs them to deduce the average energy of these constellations. Section V evaluates the performance of the proposed WCC system. At last, Section VI presents the conclusions for this work.

II. WAVELET MATRIX
A matrix A = [a s k ] with m ≥ 2 rows and mg columns, such that 1 Excepting the approach proposed in [13], [14], where a constellation formed by pairs of consecutive integer wavelet symbols increased the spectral efficiency to 2 bits/s/Hz. and a s k ∈ R or C, is said to be a wavelet matrix or matrix of wavelet coefficients of rank m and genus g if the linear condition and the quadratic condition are met, where [k + mr] denotes the operation k + mr mod mg, a is the complex conjugate of a and δ x,y is the Kronecker delta [1]. Equation (2) ensures that the sum of components of the first wavelet matrix row is equal to the rank m, whereas the sum of the components of each other row is zero. Equation (3) establishes that vectors represented by the m wavelet matrix rows are orthogonal to each other, even when shifted by an arbitrary multiple of m. Equation (3) also indicates that each row in this matrix is orthogonal to itself when shifted by a multiple step of m.
A wavelet matrix is said to be flat when all elements have a same absolute value. If they are real numbers it is said to be a flat real wavelet matrix. If they are complex numbers it is said to be a flat complex wavelet matrix [1].
A flat real wavelet matrix with normalized elements satisfy the modified wavelet linear and quadratic conditions [1], [2]  Example 1 (Wavelet coding matrix with rank m = 2 and genus g = 4): A 2×8 is given by In simulations, the wavelet matrices and A 4×256 (m = 4, g = 64) were used.

A. System model
We consider the communication system illustrated in Fig. 1.
As WCC works over any subfield of the complex field, it is natural to think about a more general representation of input information than in [2]. It is additionally interesting to recall the OFDM system, where input information is firstly formatted into complex symbols, so we followed the same idea.
In order to provide complex input symbols to the WCC Encoder (see Fig. 1), a Data Mapper was employed prior to the WCC Encoder to map n b -bit sequences into 2 n b -ary complex symbol sequences x, employing a Gray code, as illustrated in Table I. This mapping may not be the best one, but simulations suggest that simple rotations or expansions in the signal lattice (e.g., for n b = 2, 00 → 1, 01 → j, 11 → −1 and 10 → −j) does not alter results. Notice that this mapping from {0, 1} n b to C is common for classical systems (e.g., QAM), but in those cases points are mapping to the signal space directly. In the wavelet channel coding approach this step is mediated by a wavelet matrix.
In Fig. 1, input symbols in x are encoded forming wavelet symbols y, according to some coding rate R. At last, wavelet symbols are modulated and transmitted through a flat Rayleigh fading channel. At the reception, a reverse process gives back an estimation of the source bits. For simplicity, throughout this work, the scope of analysis was limited to n b = 1 and even values of n b , as well as to R = 1 and R = 1/k, considering k an even integer. Odd values of n b and k result in non-regular WCC encoders and asymmetrical constellations, without loss of generality of the presented analysis.

B. Channel model
In Fig. 1, symbols of a sequence y = {y 0 , y 1 , y 2 , . . .} are transmitted serially. The channel is assumed to be a flat Rayleigh fading channel, where the path gain α n at instant n is modeled as samples of independent zero-mean complex Gaussian random variables with variance 1/2 per axis.
At instant n, when a signal s n is transmitted, the received signal r n is given by where noise samples η n are independent samples of a zero-mean complex Gaussian random variable with variance is the average energy of the transmission constellation and E b /N 0 is the energy per bit to noise power spectral density ratio. We assume that perfect channel state information is available at the receiver. So, the Demodulator module employs the decision metric over all possible transmitted signals {s k }. The MAP (maximum a posteriori probability) criterion is adopted as decision rule.
C. Algebraic WCC encoding Consider a binary source of i.i.d. equiprobable bits. Thus, the Data Mapper module yields a sequence of i.i.d. equiprobable symbols x = {x 0 , x 1 , x 2 , . . .}. In order to describe algebraically the encoding process, x may be expressed as a formal series where D represents a delay or shift operator, as in [3]. As m symbols enter the encoder at a time, consider a serial to parallel operation prior to the encoding process and define the m series which express the m input symbols {x im , x im+1 , . . ., In [2], the WCC encoding may be understood as a table operation, where m input symbols multiply the m rows of the wavelet coding matrix, a symbol for each line, and the m 2 g resulting values are stored in m lines of the table. Then, m/R wavelet symbols are obtained from the summations of the first m/R columns of the table. Afterwards, the entire table is shifted by m/R columns and the same operation is performed with the next m input symbols to produce the next m/R wavelet symbols. As a consequence of this process, 1/g < R < 1. Tables II, III and IV illustrate this process for the cases of the wavelet coding matrix in (6) with coding rates R = 1, 1/2 and 1/4, respectively.
According to this encoding process, in order to express algebraically the resulting wavelet symbols, consider the wavelet coding matrix A m×mg of rank m and genus g in (1) and define the matrix m/R×m of polynomials of degree gR − 1 So, a column vector of m/R polynomials which own the m/R wavelet symbols due to each m input symbols may be given by . . .
At last, considering a parallel to serial operation, the sequence of wavelet symbols y = {y 0 , y 1 , y 2 , . . .} may be obtained from (13) as Combining Equations (10), (11), (12), (13) and (14), (15) Thus, the n-th wavelet symbol can be given by By inspection in (16), one observe that each wavelet symbol y n is derived from a linear combination of mgR input symbols weighted by coefficients  (6), WITH RANK m = 2, GENUS g = 4 AND CODING RATE R = 1. from the wavelet coding matrix. Each one of the m 2 g matrix coefficients are employed just once in the equations of the m R wavelet symbols derived from a block of m input symbols. Also, real flat wavelet matrix coefficients, as considered here, have only values +1 or −1 (see (6) in Example 1). As a consequence, in the case of n b = 1, wavelet symbols are integer values y n ∈ {−mgR, −mgR + 2, . . . , −2, 0, 2, . . . , mgR}.
Example 2 (WCC 2 × 8 with R = 1): Consider the wavelet coding matrix in (6). Table II presents the encoding process and Fig. 2 shows the encoder diagram for a coding rate R = 1.
The matrix of polynomials A(D) in (12) in this case is Thus, the wavelet symbols may be algebraically  (6), WITH RANK m = 2, GENUS g = 4 AND CODING RATE R = 1/2.  (6), with rank m = 2, genus g = 4 and coding rate R = 1/2 (two input symbols in red and four output symbols at a time).
expressed as Example 3 (WCC 2 × 8 with R = 1/2): Consider the wavelet coding matrix in (6). Table III presents the encoding process and Fig. 3 shows the encoder diagram for a coding rate R = 1/2.
The wavelet symbols may be algebraically ex-   (6), with rank m = 2, genus g = 4 and coding rate R = 1/4 (two input symbols in red and eight output symbols at a time).
pressed as Consider the wavelet coding matrix in (6). Table IV presents the encoding process and Fig. 4 shows the encoder diagram for a coding rate R = 1/4.
The wavelet symbols may be algebraically expressed as An important remark about the WCC encoding process is that encoder diagrams for the same wavelet matrix and different coding rates just differ on the number of outputs (adders) and registers. Tap components stay the same, but in a different connection array. This feature suggests the employment of WCC where adaptability to channel conditions is desired.
(18) If n b > 1, formatted input symbols are complex, consequently wavelet symbols are as well. In the data mapping proposed (see Table I If input bits are i.i.d., formatted input symbols will also be i.i.d. Since wavelet coding is linear, wavelet symbols in (18) and (19) will be independent, but not equally distributed. In this case, when n b = 1, the wavelet symbols {y n } in (18) present a binomial probability distribution [2] given by where 0 ≤ k ≤ mgR. For n b > 1, real and imaginary parts of the wavelet symbols in (19) are multinomially distributed (see subsection IV-B on the methodology employed to derive wavelet symbol probabilities from a probability generation function).
Since Re {y n } and Im {y n } are independent of each other, then where j = √ −1 is the imaginary unit, 0 ≤ k ≤ mgR 2 n b /2 − 1 , 0 ≤ l ≤ mgR 2 n b /2 − 1 and the integer indexes k 1 , . . . , k 2 n b /2 and l 1 , . . . , l 2 n b /2 represent the yielded Re {y n } and Im {y n }, respectively. Each index in k 1 , . . . , k 2 n b /2 (similarly in l 1 , . . . , l 2 n b /2 ) is associated to one of the 2 n b 2 formatted input symbols and means how many times the respective input symbol appears in the coder's register to produce y n . As the coder has length mgR, so k 1 + · · · + k 2 n b /2 = mgR.

E. Correlation WCC decoding
Input symbols x may be estimated by correlation of the last mg received wavelet symbolsŷ with the rows of the wavelet coding matrix [2]. Each new m/R received wavelet symbolsŷ produce m symbolsx at the output of the WCC Decoder module.
Denote by z s r the correlation between the s-th line of the wavelet matrix and the last mg received wavelet symbolsŷ at instant r, such that From (16), Detailed descriptions of this decoding process for R = 1 and n b = 1 can be found in [2], [7], [8], [17]- [20].
Example 5 (WCC 2 × 8): Consider the wavelet coding matrix in (6). Fig. 5 shows the correlation decoder diagram for a coding rate R = 1. In fact, this decoder operates for any R, when it receives 2/R input symbols and yields two output symbols at a time.
In Fig. 5, blocks ≶ implement a decision logic over values of z s r in (24).

IV. CONSTELLATIONS FOR WCC SYSTEMS
System performance depends directly on the modulation scheme. This Section presents signal constellations for transmitting wavelet symbols produced by the proposed system in Fig. 1. In the next Section, its performance is evaluated over a flat Rayleigh fading channel. The proposed WCC system requires specially designed constellations since generated wavelet symbols are not equiprobable and the alphabet cardinality grows with the wavelet coding matrix order. The construction of special constellations was explored in [7], [8], [13], [14].

A. Constellation model
As introduced in Section I, an increase in the input symbols cardinality and in the code constraint length have a strong impact on the cardinality of the wavelet symbols set and consequently on the cardinality of the transmission signal constellation.
The WCC entropy rate is 1 bit/symbol when the coding rate is unitary and the spectral efficiency is limited to 1 bit/s/Hz [1]. In order to improve this efficiency, a method to increase it to 2 bits/s/Hz by constructing a signal constellation from consecutive pairs of integer wavelet symbols was proposed in [13]. These symbol pairs were directly associated to points in a signal space diagram forming a constellation for transmission. One observed that greater probability constellation points corresponded to lower energy signals. In [15], an optimizing effort suggested that these constellations were optimal. The method proposed in [13] could be generalized to blocks of length greater than two to achieve an arbitrary spectral efficiency, however the block-tosignal mapping would not be as simple as in the case of symbols pairs.
The data mapping proposed in Fig. 1 to generalize the WCC encoding and decoding algebraic formulations results in an alternative way to improve the spectral efficiency when n b > 1 (see Table I). The Modulator module directly associates a complex wavelet symbol a + jb to a constellation point (a, b). This method allows to achieve a spectral efficiency for a coding rate R. However, This approach entails an exponential growing of the constellation cardinality, since the number of possible wavelet symbols in (20) rises with 2 n b R 2 when n b > 1.

B. Probability generating function of WCC symbols
In [13], a probability generating function (PGF) for a wavelet symbol sequence was formulated. This alternative PGF permitted to predict integer symbol pairs generated by a WCC encoder and provided the joint probability distribution of these pairs. Following this methodology, a PGF that identifies the sequence of complex wavelet symbols generated in the system proposed here and their probability distribution are derived. From this PGF, one can also derive the average energy of the constellation.
Firstly, consider a binary source generating i.i.d. equiprobable bits. As a consequence, the Data Mapper module yields i.i.d. equiprobable input symbols x to the WCC Encoder (see Fig. 1). So, one can write the PGF of the n-th input symbol x n as where {α i } is the set of all possible input symbols. Example 6 (PGF of WCC Encoder input symbols from Table I A common PGF of a sequence of n wavelet symbols {y k+1 , . . . , y k+n } is a power series representation of the joint probability distribution P (y k+1 , . . . , y k+n ) of this sequence, given by G y k+1 ,...,y k+n (z 1 , . . . , z n ) = E z y k+1 1 · · · z y k+n n = y k+1 ,...,y k+n P (y k+1 , . . . , y k+n )z y k+1 1 · · · z y k+n n . This is equivalent to the definition of a characteristic function of a random vector, as in [22, p. 255-256]. So, an ordinary PGF of a sequence of n wavelet symbols can be viewed or interpreted as a mapping Consider now a sequence of wavelet symbols generated by the system expressed as a polynomial y(D) as in (15). One may consider, as in [13], an alternative definition of a PGF of a sequence of wavelet symbols as which employs a single complex variable z. In this last case, the PGF consists of a mapping where Z[D] is a ring of polynomials in D.
In the PGF in (26), as in the PGF of wavelet symbols deduced below, exponents of z indicate the generated wavelet symbols and constants multiplying terms in z inform the probability of each corresponding symbol. In (26), all symbols are equiprobable, but resultant wavelet symbols shall not be equiprobable.
If y(D) is the sequence of wavelet symbols generated by a WCC Encoder, from (15), As input symbols are independent, As input symbols are equiprobable and each wavelet symbol is a function of mgR input symbols weighted by wavelet coding matrix coefficients (values +1 and −1), one can find for the n-th wavelet symbol in the generated sequence that where {α i } is the set of all possible input symbols. Applying the multinomial theorem 2 to solve the power of sum in (32), 2 By the well-known multinomial theorem (generalization of the binomial theorem), In (33), as in (26), exponents of z indicate generated wavelet symbols and constants multiplying each term inform the probability of the corresponding symbol. For n b > 1, wavelet symbols are complex and (33) may be changed to treat separately real and imaginary parts of the symbols, as in subsection III-D. There, (22) was derived to provide wavelet symbol probabilities with lower complexity.
Example 7 (PGF of wavelet symbols for a coding matrix A 2×8 and R = 1/2): Consider the wavelet coding matrix in (6) and a coding rate R = 1/2.
Considering equally distributed input symbols, from (32), for n b = 1 in Table I, From (33), for n b = 2,

C. Constellation average energy
One can employ the wavelet symbols PGF to determine the average energy E s of the resulting constellation. The mean value of wavelet symbols, E [y n ], can be obtained evaluating the first derivative of the PGF in z = 1 and E [y 2 n ] − E [y n ] can be calculated evaluating the second derivative of the PGF in z = 1 [22]. One can remark that E s = E [y 2 n ]. Initially, consider the PGF in (32) when n b = 1, case of integer wavelet symbols. Its first derivative is So, E [y n ] = G y (z) z=1 = 0.
Thus, when n b = 1, the second derivative of the PGF in z = 1 directly yields the average energy of the constellation.
The second derivative of (32) is So, the constellation average energy when n b = 1 is When n b > 1, wavelet symbols are complex. Consider now a complex wavelet symbol y n = u n + jv n , where u n and v n are the real and imaginary parts of y n . Observe that u n and v n are independent because u n is due to the real part of the input symbols and v n is due to their imaginary part. In addition, the real and imaginary parts of the input symbols may be associated to half the input bits in data mapping (n b /2). So, u n and v n behave as y n when n b = 1 and u n , v n ∈ {−mgR2 n b /2 , −mgR2 n b /2 + 2, . . . , mgR2 n b /2 }. Then, the PGFs of u n and v n can be directly derived from (32) as In order to evaluate the mean of u n and v n , as in the case of n b = 1, the first derivative of (38) is evaluated in z = 1. So, and since the mean value of the input symbols is null. Thus, the second derivative of the PGF in z = 1 The second derivative of (38) is As u n and v n are independent, the constellation average energy when n b > 1 is  Fig. 7. Performances of the WCC system in Fig. 1 with a wavelet coding matrix 4 × 16 (rank m = 4 and genus g = 4) for a broad range of parameters. Curves for the same encoding rate have the same color and curves for the same value of n b employ the same point mark.

V. PERFORMANCE ANALYSIS
In this Section, the performance of the proposed WCC system is evaluated by simulation. Firstly the consequences of variation of wavelet code parameters on the system performance is examined. Then, its performance is compared with the performance of ordinary symbol-by-symbol detecting systems and space-time block coding (STBC) systems.

A. Performance of WCC systems
In order to obtain a broad view of performance of the WCC system proposed in Fig. 1, it was simulated considering a wide range of parameters. Wavelet coding matrices with ranks m = 2 and 4, genera g = 4, 16 and 64, coding rates R = 1, 1/2 and 1/4, and data mapping with n b = 1, 2, 4, 8 and 16 bits per input symbol were considered. Fig. 7 presents performance curves of a WCC system employing a wavelet coding matrix 4 × 16 (rank m = 4 and genus g = 4) for some parameter variations. The cases were limited to spectral efficiencies (see Equation (25)) of 1, 2 and 4 bits/s/Hz. In Fig. 7, curves for the same encoding rate have the same color and curves for the same value of n b employ the same point mark (e.g., blue curves have R = 1/2 and curves with n b = 2 employ as point mark).     7, as expected, displays that decreasing rates result in better performance while increasing n b , on the contrary, worsens that. Unfortunately, the first does not compensate the second for the same spectral efficiency and the employment of greater values of n b rapidly degrades system's performance. An exception occurs when n b goes from 1 to 2, case in which similar performances are obtained for the same encoding rate (modulation with n b = 2 corresponds to modulations with n b = 1 in quadrature). Furthermore, for the same spectral efficiency, the best WCC system performance is obtained employing n b = 2. Similar behavior is obtained for any other wavelet coding matrix m × mg. Fig. 8 presents performance curves of WCC systems with encoding rate R = 1/4 and data mapping with n b = 4 for wavelet coding matrices 2 × 8, 2 × 32, 2 × 128, 4 × 16, 4 × 64 and 4 × 256 (ranks m = 2 and 4 and genera g = 4, 16 and 64).
We observe in Fig. 8 that increases in rank or genus (increasing the constraint length) of the wavelet matrix yield better performances of the proposed WCC system over a fading channel, mainly at higher signal-to-noise ratios. It happens because the correlation decoder employed here takes advantage of the input information spread over the transmitted symbols. A sequential decoding must take greater advantage of that and yield a better performance for a longer constraint length. This fact was verified in [16] with the employment of a Viterbi decoder. Future works shall verify it for other sequential decoders, as the Fano algorithm and the M -algorithm.

B. Comparison with other systems
In order to evaluate the WCC ability to improve system performance over a flat Rayleigh fading channel, the WCC system performance is compared with the performance of ordinary symbol-by-symbol detecting systems and STBC systems proposed in [21] considering two transmitting and one receiving antennas.
As WCC symbol-by-symbol decoding are employed here, it is more natural to compare its performance with ordinary symbol-by-symbol detecting systems. The symbol-by-symbol detecting systems simulated were BPSK for a spectral efficiency of 1 bit/s/Hz, QPSK for 2 bits/s/Hz and 16-QAM for 4 bits/s/Hz. STBC systems' performance is taken here as a reference of a powerful technique employed in modern systems. A performance comparison between WCC and STBC is not quite fair, since WCC is as simple as systems employing symbol-by-symbol decoding and only one transmitting and one receiving antennas, without spatial diversity as in STBC systems. Thus, it is not expected that WCC provides in general better performance than STBC. In fact, it is expected that a spatial diversity system may be developed from WCC, which can be explored in future works.
STBC was simulated employing the generating matrix where s 1 and s 2 are two transmitted signals considering as modulation BPSK for a spectral efficiency of 1 bit/s/Hz, QPSK for 2 bits/s/Hz and 16-QAM for 4 bits/s/Hz. Fig. 9 shows performance curves of WCC, BPSK and STBC systems for a spectral efficiency of 1 bit/s/Hz. WCC systems with wavelet matrices 2×8, 2×32 and 2×128, considering R = 1/2 and n b = 2, exhibited better performance than BPSK and STBC systems. STBC presented performance better than WCC just for the smaller constraint length (wavelet matrix 2 × 8) for E b /N 0 > 23 dB. Fig. 10 presents performance curves of WCC, QPSK and STBC systems for a spectral efficiency of 2 bit/s/Hz. The WCC system presented better  Fig. 9. Comparison between WCC, BPSK and STBC system performances with a spectral efficiency of 1 bit/s/Hz. performance than QPSK. Differently than the former case, WCC performance was worse than STBC performance. The WCC performance was restrained by the rise of n b in order to attain greater spectral efficiency. Fig. 11 presents performance curves of WCC, 16-QAM and STBC systems for a spectral efficiency of 4 bit/s/Hz. A similar behavior may be seen in Fig. 10: higher values of n b degraded the WCC system performance. The WCC system presented worse performance than 16-QAM, except for the longer constraint length (wavelet matrix 2 × 128) in E b /N 0 > 19 dB.

VI. CONCLUSIONS
This paper presented a more general analysis of WCC considering flat real wavelet matrices derived from a Haar matrix and complex input symbols produced by a data mapper and representing n b source bits. WCC encoding and decoding processes were algebraically described and a probability distribution was formulated for the generated wavelet symbols. Also, signal constellations were proposed for transmission of these wavelet symbols and PGFs were calculated for them and used to deduce the average energy of the constellations.
The performance analysis of this system over a flat Rayleigh channel revealed that increasing rank or genus of the wavelet coding matrix yielded better performances of the proposed WCC system, specially for high signal-to-noise ratios. It happened because the correlation decoder employed here take advantage of the input information spread over the transmitted symbols. Future works should explore the employment of sequential decoders, as the Fano algorithm and the M -algorithm, instead of the correlation one. Sequential decoding for WCC may yield important performance increases, specially for large constraint lengths, at the expense of decoding complexity.
We also observed that the data mapping process employed prior to the WCC encoding degraded the system performance, whereas it allowed to increase the spectral efficiency. For a spectral efficiency of 1 bit/s/Hz, the WCC system presented a performance better than an ordinary symbol-by-symbol detecting system and even an STBC system with diversity 2 and the same spectral efficiency. For greater spectral efficiencies, when n b increased, the WCC system performance got worse. However, despite the observed degradation due to increases in n b , we found great potential possibilities of application of WCC, since it showed better performances than ordinary symbol-by-symbol detecting systems, particularly at higher signal-to-noise ratios for higher spectral efficiencies.
Many critics pointed to the low spectral efficiency provided by WCC as its main disadvantage. Further improvements in this regard are planned for future works. Still as future work, as successive wavelet symbols in WCC are generated from a matrix with consistent orthogonality properties, one may investigate its employment as a diversity system. Application of WCC in multi-band ultra wideband systems (UWB) may also be investigated, since subbands of UWB may be coded with Hadamard codes and real flat wavelet matrices of genus g = 1 are Hadamard matrices [24].