Nowadays, multi-carrier modulations are widely used in many standards such as WIMAX (Worldwide Interoperability for Microwave Access) [1] , DAB (Digital Audio Broadcasting) [2] , DVB-T (Digital Video Broadcasting-Terrestrial) [3] . One of the main modulations used is CP-OFDM (Orthogonal Frequency Division Multiplexing with Cyclic Prefix) [4] [5] as it enables a simple structure of the transmitter and receiver implemented by a simple IFFT (Inverse Fast Fourier Transform) or FFT (Fast Fourier Transform). CP-OFDM is similar to OFDM/QAM (Orthogonal Frequency-Division Multiplexing with Quadrature Amplitude Modulation) when CP is not used. The use of Cyclic Prefic (CP) enables to cope with time-dispersive channels and Intersymbols Interference (ISI) that are produced by these channels. However, CP-OFDM has some drawbacks:

Since the Fourier transform of the rectangular window function is proportional to sinc(f) function. This bad PDS is tackled by inserting null sub-carriers at the boundary of the DFT (Discrete Fourier Transform) or IDFT (Inverse Discrete Fourier Transform) in order to avoid interference with neighbor systems. Null sub-carriers mean also loss of spectral efficiency. These drawbacks mainly the second one make OFDM/QAM not really suitable for application or system that requires a demanding frequency localization and for systems that will be designed in the range of cognitive radio. One alternative to OFDM/QAM is OFDM/OQAM (Orthogonal frequency-division multiplexing with Offset Quadrature Amplitude Modulation) [6] - [8] which allows the possibility to use different prototype filter such as the Square Root-Raised Cosine(SRRC) filter [9] - [11] , Extended Gaussian Functions (EGF) [12] [13] , other filters are designed using specific techniques such as the so-called frequency sampling technique [14] . These functions have the ability to provide a better PDS than the one obtained with the rectangular window function. This paper is about this specific OFDM/OQAM modulation, which relies on real orthogonality. However, [15] showed that when combining OFDM/OQAM with CDMA especially with Walsh-Hadamard codes one can obtain complex orthogonality. The current work is a continuation of the work in [15] where we use the two subsets of the Hadamard codes to create constructive interferences in order to improve performances of the decoding process at lower Signal to Noise Ratio (SNR). To reach this goal, firstly, we provide a structure of OFDM/OQAM modulation for the transmitter and the receiver without advance elements. Secondly, Code Division Multiplexing Access (CDMA) is introduced in time domain for real data transmission and later for complex data transmission. After that, our method which uses interference from two subsets is presented. This paper is divided in five sections. Section II named OFDM/OQAM develops and presents some modulator and demodulator structures. Then Section III describes CDMA and the method we are suggesting in order to use interferences of codes in a constructive manner. It can be noticed that a study pertaining to interference alignment can be found in [16] , however, the focus of our method is related to interferences using Walsh-Hadamard codes. This is followed by the section Simulations and Results, which gives the performances of several schemes in presence of AWGN channel. At the end a section entitled conclusion summarizes the main ideas and results of the article. As in [8] , we denote discrete filters with lower case letters for example h[n], and their Z-transform with upper case letters e.g. H(z). The real part of the complex-valued number c is designated respectively by $\Re \left\{c\right\}$. Superscript * denotes complex conjugation. Contrary to [8] H^*(Z) = (H(Z))^* and M = 2N will represents the total number of sub-carriers. δ_m,p represents the Kronecker function. ↑ N and ↓ N are respectively used for upsampling and downsampling processes with the factor N. For two continuous (Resp. discrete) functions g(t) (Resp. g[k]) and f(t) (Resp. f[k]) the scalar product between g(t) and f(t) (Resp. g[k] and f[k]) is given by:

$\langle g,f\rangle ={\displaystyle {\int }_{-\infty }^{+\infty }g\left(t\right)f^{*}\left(t\right)\text{d}t}\left(\text{Resp}.={\displaystyle \underset{k=-\infty }{\overset{+\infty }{\sum }}g\left[k\right]f^{*}\left[k\right]}\right).$ (1)

Finally, the superscript ^t represents the transpose operator. Let us start with OFDM/OQAM modulation.

This section is on OFDM/OQAM modulation. Where we give the main elements of OFDM/OQAM modulator and OFDM/OQAM demodulator. As the OFDM/OQAM modulator is concerned, in continuous form, we can define the OFDM/OQAM transmitted signal as in [8] by:

$s\left(t\right)={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{a}_{m,n}{h}_{m,n}\left(t\right)}}$ (2)

with $h_{m,n}\left(t\right)=h\left(t-n{\tau }_0\right)\exp \left(j\left(2\pi m{F}_{0}t+{\varphi }_{m,n}\right)\right)$, ${\tau }_0=T_0/2$, ${\varphi }_{m,n}=\pi \left(m+n\right)/2+k_{0}mn\pi$ and $k_0\in \left\{0,1,-1\right\}$. h(t) represents a prototype filter which must obey to the real orthogonality principle i.e. $\Re \langle h_{m,n},h_{p,q}\rangle ={\delta }_{m,p}{\delta }_{n,q}$. The symbols a_m,n here are real, that is why we have two more symbols than in OFDM/QAM, L_T is replaced by 2L_T to stress that the a_m,n symbols correspond to the real and imaginary part of complex QAM as described in [8] [17] . Using h[k] = h(kTs) and $h_{m,n}\left[k\right]=h\left[k-nN\right]\exp \left(j\left(2\pi mk/M+{\varphi }_{m,n}\right)\right)$, the discrete form s[k] = s(kTs) is obtained by sampling s(t) by the period interval Ts = T₀/M i.e.

$s\left[k\right]={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{a}_{m,n}{h}_{m,n}\left[k\right]}}$ (3)

Since h[k] can have an anticausal part of length D₀, we delay the signal s[k] by D₀ samples in order to obtain a structure with only causal filters. By using ${\theta }_{m,n}={\varphi }_{m,n}+\pi mn-2\pi m{D}_0/M$, M = 2N, $b_{m,n}=a_{m,n}\exp \left(j{\theta }_{m,n}\right)$, F₀T₀ = 1, $g\left[k\right]=h\left[k-D_0\right]$ we get:

$\begin{array}{c}{s}_1\left[k\right]=s\left[k-D_0\right]={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{a}_{m,n}{h}_{m,n}\left[k-D_0\right]}}\\ ={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{b}_{m,n}\underset{f_m\left[k-nN\right]}{\underset{︸}{g\left[k-nN\right]{\text{e}}^{j2\pi m\frac{k-nN}{M}}}}}}\\ ={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{b}_{m,n}{f}_m\left[k-nN\right]}}\end{array}$ (4)

The process of obtaining b_m,n from a_m,n will be referred as Pre-Processing as presented in [17] . We will also use the following notation $u_{m,n}\left[k\right]=h_{m,n}\left[k-D_0\right]$. s₁[k] is the sum of M sub-bands where the m^th branch is an N-fold expander follows by the filter f_m[n] [18, page 117] as depicted in Figure 1 .

This is actually a synthesis uniform filter-bank. An efficient structure is obtained by using: polyphase decomposition type I [18] [19] as described in ( [20] page 766) [21] , noble identity ( [18] page 119) [19] ( [20] page 769), and by taking into account that interpolating by a factor N then, delaying the m^th sub-band by m samples and summing all the sub-bands can be represented by two switches and an N-samples delay and an adder. All these elements lead to a complete OFDM/OQAM modulator given in Figure 2 . Some details to obtain such a structure can be found in [8] and [22] .

Now, let us talk about the demodulator and its structures while considering a perfect channel environment (the received signal is the same as the transmitted one). The demodulation process is based on the real orthogonality property, which is presented in continuous form as given in Equation (2). Indeed, let us denote by:

$c_{p,q}=\langle s,h_{p,q}\rangle ={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{a}_{m,n}}}\langle h_{m,n},h_{p,q}\rangle .$

By taking the real part we obtain:

$a_{p,q}=\Re \left\{c_{p,q}\right\}$

This last result is obtained following the real orthogonality principle. The design of the prototype filter [8] [12] [13] [23] - [25] is not the interest of this paper. However, is worth mentioning that the design in continuous time of this prototype while applying their discretized version as far as digital implementation is concerned will lead to loss of orthogonality in discrete mode due to:

These two problems were already mentioned in [8] . We assumed, we have a perfect orthogonality or a nearly-perfect orthogonality in discrete mode. Since we are dealing with s₁[k] instead of s[k] let us first rewrite the appropriate equation for s₁[k]. From Equation (4)

$s_1\left[k\right]={\displaystyle \underset{m=0}{\overset{M-1}{\sum }}{\displaystyle \underset{n=0}{\overset{2L_T-1}{\sum }}{a}_{m,n}{u}_{m,n}\left[k\right]}}$.

Using v = k − D₀ we have:

$\begin{array}{c}\langle u_{m,n},u_{p,q}\rangle ={\displaystyle \underset{k=-\infty }{\overset{+\infty }{\sum }}{u}_{m,n}\left[k\right]u_{p,q}^{*}\left[k\right]}\\ ={\displaystyle \underset{k=-\infty }{\overset{+\infty }{\sum }}{h}_{m,n}\left[k-D_0\right]h_{p,q}^{*}\left[k-D_0\right]}\\ ={\displaystyle \underset{k=-\infty }{\overset{+\infty }{\sum }}{h}_{m,n}\left[v\right]h_{p,q}^{*}\left[v\right]}=\langle h_{m,n},h_{p,q}\rangle \end{array}$ (5)

Thus for s₁[k] everything looks as for s[k] as long as we replace h_p,q by u_p,q i.e. $c_{p,q}=\langle s_1,u_{p,q}\rangle$. Still in discrete mode and setting $l_p\left[k\right]=g{\left[-k\right]}^{*}\exp \left(j2\pi pk/M\right)$, we obtain:

$c_{p,q}={\displaystyle \underset{k=-\infty }{\overset{+\infty }{\sum }}{s}_1\left[k\right]u_{p,q}^{*}\left[k\right]}={\text{e}}^{-j{\theta }_{p,q}}\underset{y_{p,q}}{\underset{︸}{{\displaystyle \underset{k=-\infty }{\overset{+\infty }{\sum }}{s}_1\left[k\right]l_p\left[qN-k\right]}}}$ (6)

Equation (6) means that y_p,q is obtained by filtering s₁[k] with the filter l_p[n] then down-sampling the output by a factor N[18, page 117]. Multiplying y_p,q = y_p[q] by exp(−jθ_p,q) (It is called Post-Processing) will generate c_p,q and taking the real part of it, we will obtain a_p,q as previously shown. This is actually an analysis uniform filter-bank. Figure 3 illustrates this demodulator structure.

An efficient structure can be obtained by using polyphase decomposition type I and by taking into accounts the delays elements and downsampling. Figure 4 summarizes a structure that we can obtain by these processes, where the polyphase decomposition type I of G(Z) is given by:

$G\left(Z\right)={\displaystyle \underset{l=0}{\overset{M-1}{\sum }}{Z}^{-l}{E}_l\left(Z^M\right)}$

where E_l(Z) is the Z-transform of the filter $e_l\left[k\right]=g\left[l+kM\right]$. Moreover n₀ and n₁ are integer obtained in the process of having only causal elements in the receiver structure. See [8] and [22] for some details.

Having given the details of OFDM/OQAM modulation, let us apply this OFDM/OQAM modulation with Code Division Multiplexing Access (CDMA).

This section is divided into three main parts. The first and second parts recall some elements pertaining to the combination of Walsh-Hadamard codes with OFDM/OQAM firstly for real data transmission and secondly for complex data transmission. Then the last part is about our proposed method in other to improve the bit error rate (BER) at low signal to noise ratio (SNR). We are going to start with CDMA and real data transmission in OFDM/OQAM.

In this section we focus on the performance of the different schemes presented in this paper in presence of AWGN (Additive White Gaussian Noise) channel in order to assess the basic performances and come up with the first findings concerning the studies carried out in this work. Performance will be evaluated:

We recall that single carrier (SC) transmission in presence of AWGN channel is modelled as: $y_k=a_k^{(c)}+n_k$ where y_k is the received signal at time k, $a_k^{(c)}$ is the symbol(QPSK modulation) sent at time k and n_k is the noise process which is considered to be: zero-mean, white, with a Gaussian distribution and independent of the data. We will denote by SC the performance of SC, by CDMA-R the performance of OFDM/OQAM with CDMA and real data at full load i.e. all codes are used, by CDMA-C the performance of OFDM/OQAM with CDMA and complex data. We apply our suggestion firstly for W = 16. The two subsets are: S2 = {2, 3, 5, 8, 9, 12, 14, 15} and S1 = {1, 4, 6, 7, 10, 11, 13, 16} where we use S2 for transmission and add some codes of S1 as follows:

1) data spread in half between code 2 ∈ S2 and code 1 ∈ S1. Data in code 1 are multiplied by j and data in frequency m are transmitted in frequency m + 1 for m ≠ M − 1, whereas data at frequency M − 1 are transmitted at frequency 0.

2) data spread in half between code 3 ∈ S2 and code 4 ∈ S1. Data in code 4 are multiplied by j and no change in position in frequency.

The performance of this scheme is denoted by CDMA-CM. Figure 8 shows the results in AWGN channel. Not surprising SC, CDMA-R, CDMA-C perform almost the same. Our proposed method CDMA-CM has a gain of about 0.3 dB (which corresponds in linear scale to an improvement of 7%) below 7 dB but this gain vanishes as the SNR increases. Above 9 dB its performance is worse than SC which reflects the presence interference.

In order to investigate if the number of subcarriers influences the results we decided to run simulation for M = 32 and for the previous proposed manner of spreading data. We denote its performance by CDMA-CM2 when M = 32. In addition, we also assess the performance of a scheme where we use W = 32 associated with the following two sets S2 = {2, 3, 5, 8, 9, 12, 14, 15, 17, 20, 22, 23, 26, 27, 29, 32} and S1 = {1, 4, 6, 7, 10, 11, 13, 16, 18, 19, 21, 24, 25, 28, 30, 31} and the following spreading rules:

1) data spread in half between code 2 ∈ S2 and code 1 ∈ S1. Data in code 1 are multiplied by j and data in frequency m are transmitted in frequency m + 1 for m ≠ M − 1, whereas data at frequency M − 1 are transmitted at frequency 0.

2) data spread in half between code 3 ∈ S2 and 4 ∈ S1. Data in code 4 are multiplied by j and no change in position in frequency.

3) data spread in half between code 17 ∈ S2 and code 18 ∈ S1. Data in code 18 are multiplied by −j and data in frequency m are transmitted in frequency m − 1 for m ≠ 0, whereas data at frequency 0 are transmitted at frequency M − 1.

4) data spread in half between code 20 ∈ S2 and 19 ∈ S1. Data in code 19 are multiplied by −j and no change in position in frequency.

Again we use S2 for transmission and with this spreading approach, for M = 16 the performance will be denoted by CDMA-CM3 and for M = 32 by CDMA-CM4. Figure 9 presents the results of CDMA-CM, CDMA-CM2, CDMA-CM3 and CDMA-CM4. CDMA-CM and CDMA-CM2 perform almost the same this conclusion holds for CDMA-CM3 and CDMA-CM4. Meaning that for these cases M has no influences on the performances. CDMA-CM3 performs almost the same as CDMA-CM for SNR less than 7 dB but it deteriorates faster after 7 dB. Meaning that the size W and the choice of adequate codes can influence the performances.

Since some data are received with a strong SNR that others, we also evaluate the performance of the main schemes using a binary convolutional code:

Figure 10 presents the performances of SC, CDMA-C and CDMA-CM with this coding scheme. SC and CDMA-C perform quite similarly whereas CDMA-CM provides a gain between 0.5 and 0.7 dB (which corresponds in linear scale to an improvement of around 12.5%). This gain is mainly due to the fact that reliability information (SNR) is taking into account by the decoder.

According to Equation (9) and Equation (10) some data introduce in the Viterbi decoder have better reliability than others thus they help the decoder to perform better at low SNR.

This paper was about combining OFDM/OQAM with CDMA for complex data transmission and the possibility to improve the performance at low SNR. For this goal, firstly as in [8] and [22] we presented some structures of OFDM/OQAM modulator and demodulator. Secondly, we recalled and stressed the main differences between real data transmission and complex data transmission in OFDM/OQAM with CDMA when using Walsh-Hadamard codes. To this aim CDMA with real data transmission and complex data with spreading in time domain were developed. After that, we suggested a method which combines two subsets in order to improve the performance at lower SNR of complex data transmission in OFDM/OQAM with CDMA (Walsh Hadamard codes). This method involves selecting some adequate codes to add to the set of codes of interest, then splitting the data to be transmitted between some codes of interest and the corresponding adequate codes. The splitting was done on half power basis. Simulations in presence of AWGN channel were done to evaluate the improvement one may have. A prospect from this work may be to combine the proposed method with turbo-codes in order to have a better use of the reliability introduce by some codes.