 Research Article
 Open Access
 Published:
The Alamouti Scheme with CDMAOFDM/OQAM
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 703513 (2010)
Abstract
This paper deals with the combination of OFDM/OQAM with the Alamouti scheme. After a brief presentation of the OFDM/OQAM modulation scheme, we introduce the fact that the wellknown Alamouti decoding scheme cannot be simply applied to this modulation. Indeed, the Alamouti coding scheme requires a complex orthogonality property; whereas OFDM/OQAM only provides real orthogonality. However, as we have recently shown, under some conditions, a transmission scheme combining CDMA and OFDM/OQAM can satisfy the complex orthogonality condition. Adding a CDMA component can thus be seen as a solution to apply the Alamouti scheme in combination with OFDM/OQAM. However, our analysis shows that the CDMAOFDM/OQAM combination has to be built taking into account particular features of the transmission channel. Our simulation results illustrate the Alamouti coding scheme for which CDMAOFDM/OQAM and CPOFDM are compared in two different scenarios: (i) CDMA is performed in the frequency domain, (ii) CDMA is performed in time domain.
1. Introduction
Increasing the transmission rate and/or providing robustness to channel conditions are nowadays two of the main research topics for wireless communications. Indeed, much effort is done in the area of multiantennas, where Space Time Codes (STCs) enable to exploit the spatial diversity when using several antennas either at the transmitting side or at the receiving side. One of the most known and used STC technique is Alamouti code [1]. Alamouti code has the nice property to be simple to implement while providing the maximum channel diversity. On the other hand, multicarrier modulation (MCM) is becoming, mainly with the popular Orthogonal Frequency Division Multiplexing (OFDM) scheme, the appropriate modulation for transmission over frequency selective channels. Furthermore, when appending the OFDM symbols with a Cyclic Prefix (CP) longer than the maximum delay spread of the channel to preserve the orthogonality, CPOFDM has the capacity to transform a frequency selective channel into a bunch of flat fading channels which naturally leads to various efficient combinations of the STC and CPOFDM schemes. However, the insertion of the CP yields spectral efficiency loss. In addition, the conventional OFDM modulation is based on a rectangular windowing in the time domain which leads to a poor () behavior in the frequency domain. Thus CPOFDM gives rise to two drawbacks: loss of spectral efficiency and sensitivity to frequency dispersion, for example, Doppler spread.
These two strong limitations may be overcome by some other OFDM variants that also use the exponential base of functions. But then, in any case, as it can be deduced from the BalianLow theorem, see, for example, [2], it is not possible to get at the same time (i) Complex orthogonality; (ii) Maximum spectral efficiency; (iii) A welllocalized pulse shape in time and frequency. With CPOFDM conditions (ii) and (iii) are not satisfied, while there are two main alternatives that satisfy two of these three requirements and can be implemented as filter bankbased multicarrier (FBMC) modulations. Relaxing condition (ii) we get a modulation scheme named Filtered MultiTone (FMT) [3], also named oversampled OFDM in [4], where the authors show that the baseband implementation scheme can be seen as the dual of an oversampled filter bank. But if one really wants to avoid the two drawbacks of CPOFDM the only solution is to relax the complex orthogonality constraint. The transmission system proposed in [5] is a pioneering work that illustrates this possibility. Later on an efficient Discrete Fourier Transform (DFT) implementation of the Saltzberg system [5], named Orthogonally Quadrature Amplitude Modulation (OQAM), has been proposed by Hirosaki [6]. To the best of our knowledge, the acronym OFDM/OQAM, where OQAM now corresponds to Offset QAM, appeared for the first time in [7]. In [7] the authors also present an invention of Alard, named Isotropic Orthogonal Transform Algorithm (IOTA), and explicitly use a real inner product to prove the orthogonality of the OFDM/OQAMIOTA modem. A formal link between these continuoustime modulation models and a precise filter bank implementation, the Modified Discrete Fourier Transform (MDFT) [8], is established in [9].
It is now recognized in a large number of applications, with cognitive radio being the most recent and important one [10], that appropriate OFDM/OQAM pulse shapes which satisfy conditions (ii) and (iii) can be designed, and these can lead to some advantages over the CPOFDM. However, most of these publications are related to a single user case and to SingleInputSingleOutput (SISO) systems. On the contrary, only a few results are available concerning more general requirements being related either to multiaccess techniques or multiantenna, that is, of Multiple Input Multiple Output (MIMO) type. In a recent publication [11], we have shown that, under certain conditions, a combination of Coded Division Multiple Access (CDMA) with OFDM/OQAM could be used to provide the complex orthogonal property. On the other hand, it has also been shown in [12] that spatial multiplexing MIMO could be directly applied to OFDM/OQAM. However, in the MIMO case there is still a problem which has not yet found a fully favorable issue: It concerns the combined use of the popular STBC Alamouti code together with OFDM/OQAM. Basically the problem is related to the fact that OFDM/OQAM by construction produces an imaginary interference term. Unfortunately, the processing that can be used in the SISO case, for cancelling it at the transmitter side (TX) [13] or estimating it at the receiver side (RX) [14], cannot be successfully extended to the Alamouti coding/decoding scheme. Indeed, the solutions proposed so far are not fully satisfactory. The Alamoutilike scheme for OFDM/OQAM proposed in [15] complicates the RX and introduces a processing delay. The pseudoAlamouti scheme recently introduced in [16] is less complex but requires the appending of a CP to the OFDM/OQAM signal which means that condition (ii) is no longer satisfied.
The aim of this paper is to take advantage of the orthogonality property resulting from the CDMAOFDM/OQAM combination introduced in [11] to get a new MIMO Alamouti scheme with OFDM/OQAM. The contents of our paper is as follows. In Section 2, after some general descriptions of the OFDM/OQAM modulation in Section 2.1 and the MIMO Alamouti scheme in Section 2.2, we will combine both techniques. However, as we will see in Section 2.3, the MIMO decoding process is very difficult because of the orthogonality mismatch between Alamouti and OFDM/OQAM. In Section 3, we propose to combine Alamouti and CDMAOFDM/OQAM in order to solve the problem. Indeed, in [11], we have shown that the combination of CDMA and OFDM/OQAM (CDMAOFDM/OQAM) can provide the complex orthogonality property; this interesting property is first recalled in Section 3.1. Then, two different approaches with Alamouti coding are proposed, by considering either a spreading in the frequency (in Section 3.2) or in the time domain (in Section 4.2). When spreading in time is considered, 2 strategies of implementing the Alamouti coding are proposed. Some simulation results finally show that, using particular channel assumptions, the Alamouti CDMAOFDM/OQAM technique achieves similar performance to the Alamouti CPOFDM system.
2. OFDM/OQAM and Alamouti
2.1. The OFDM/OQAM Transmultiplexer
The baseband equivalent of a continuoustime multicarrier OFDM/OQAM signal can be expressed as follows [7]:
with the set of integers, an even number of subcarriers, the subcarrier spacing, the prototype function assumed here to be a realvalued and even function of time, and an additional phase term such that where can be chosen arbitrarily. The transmitted data symbols are realvalued. They are obtained from a QAM constellation, taking the real and imaginary parts of these complexvalued symbols of duration , where denotes the time offset between the two parts [2, 6, 7, 9].
Assuming a distortionfree channel, the Perfect Reconstruction (PR) of the real data symbols is obtained owing to the following real orthogonality condition:
where denotes conjugation, denotes the inner product, and if and if . Otherwise said, for , is a pure imaginary number. For the sake of brevity, we set . The orthogonality condition for the prototype filter can also be conveniently expressed using its ambiguity function
It is wellknown [7] that to satisfy the orthogonality condition (2), the prototype filter should be chosen such that if and .
In practical implementations, the baseband signal is directly generated in discrete time, using the continuoustime signal samples at the critical frequency, that is, with . Then, based on [9], the discretetime baseband signal taking the causality constraint into account, is expressed as
The parallel between (1) and (4) shows that the overlapping of duration corresponds to discretetime samples. For the sake of simplicity, we will assume that the prototype filter length, denoted , is such that , with being a positive integer. With the discrete time formulation, the real orthogonality condition can also be expressed as:
As shown in [9], the OFDM/OQAM modem can be realized using the dual structure of the MDFT filter bank. A simplified description is provided in Figure 1, where it has to be noted that the premodulation corresponds to a single multiplication by an exponential whose argument depends on the phase term and on the prototype length. Note also that in this scheme, to transmit QAM symbols of a given duration, denoted , the IFFT block has to be run twice faster than for CPOFDM. The polyphase block contains the polyphase components of the prototype filter . At the RX side, the dual operations are carried out.
The prototype filter has to be PR, or nearly PR. In this paper, we use a nearly PR prototype filter, with length , resulting from the discretization of the continuous time function named Isotropic Orthogonal Transform Algorithm (IOTA) in [7].
Before being transmitted through a channel the baseband signal is converted to continuoustime. Thus, in the rest of this paper, we present an OFDM/OQAM modulator that delivers a signal denoted , but keeping in mind that this modulator corresponds to an FBMC modulator as shown in Figure 1.
The block diagram in Figure 2 illustrates our OFDM/OQAM transmission scheme. Note that compared to Figure 1, here a channel breaks the real orthogonality condition thus an equalization must be performed at the receiver side to restore this orthogonality.
Let us consider a timevarying channel, with maximum delay spread equal to . We denote it by in time, and it can also be represented by a complexvalued number for subcarrier at symbol time . At the receiver side, the received signal is the summation of the signal convolved with the channel impulse response and a noise component . For a locally invariant channel, we can define a neighborhood, denoted , around the position, with
and we also define .
Note also that and are chosen according to the time and bandwidth coherence of the channel, respectively. Then, assuming , for all , the demodulated signal can be expressed as [13, 14, 17]
with the noise component, , the interference created by the neighbor symbols, given by
and the interference created by the data symbols outside .
It can be shown that, even for small size neighborhoods, if the prototype function is well localized in time and frequency, becomes negligible when compared to the noise term . Indeed a good timefrequency localization [7] means that the ambiguity function of , which is directly related to the terms, is concentrated around its origin in the timefrequency plane, that is, only takes small values outside the region. Thus, the received signal can be approximated by
For the rest of our study, we consider (9) as the expression of the signal at the output of the OFDM/OQAM demodulator.
2.2. Alamouti Scheme: General Case
In order to describe the Alamouti scheme [1], let us consider the onetap channel model described as
where, at time instant , is the channel gain between the transmit antenna and the receive antenna and is an additive noise. We assume that is a complexvalued Gaussian random process with unitary variance. One transmit antenna and one receive antenna are generally referred as SISO model. We consider coherent detection, that is, we assume that the receiver has a perfect knowledge of .
The Alamouti scheme is implemented with 2 transmit and one receive antennas. Let us consider and to be the two symbols to transmit at time (time and frequency axis can be permuted in multicarrier modulation.) instants and , respectively. At time instant , the antenna transmits whereas the antenna transmits . At time instant , the antenna transmits whereas the antenna transmits . The factor is added to normalize the total transmitted power. The received signal samples at time instants and are given by
Assuming the channel to be constant between the time instants and , we get
Note that is an orthogonal matrix with , where is the identity matrix of size and stands for the transpose conjugate operation. Thus, using the Maximum Ratio Combining (MRC) equalization, the estimates and are obtained as
where,
Since the noise components and are uncorrelated, , where denotes the monolateral noise density. Thus, assuming a QPSK modulation, based on [18], the bit error probability, denoted , is given by
where denotes the SignaltoNoise Ratio (SNR) at the transmitter side. When the two channel coefficients are uncorrelated, we will have a diversity gain of two [18].
2.3. OFDM/OQAM with Alamouti Scheme
Equation (9) indicates that we can consider the transmission of OFDM/OQAM on each subcarrier as a flat fading transmission. Moreover, recalling that in OFDM/OQAM each complex data symbol, , is divided into two real symbols, and , transmitted at successive time instants, transmission of a pair of data symbols, according to Alamouti scheme, is organized as follows:
We also assume that in OFDM/OQAM the channel gain is a constant between the time instants and . Let us denote the channel gain between the transmit antenna and the receive antenna at subcarrier and time instant by . Therefore, at the single receive antenna we have
Setting
and using (16), we obtain
where,
This results in
We note that is an orthogonal matrix which is similar to the one found in (12) for the conventional Alamouti scheme. However, the term appears, which is an interference term due to the fact that OFDM/OQAM has only a real orthogonality. Therefore, even without noise and assuming a distortionfree channel, we cannot achieve a good error probability since is an inherent "noise interference" component that, differently from the one expressed in (9), cannot be easily removed. (in a particular case, where , one can nevertheless get rid of the interference terms.)
To tackle this drawback some research studies are being carried out. However, as mentioned in the introduction, the first one [15] significantly increases the RX complexity, while the second one [16] fails to reach the objective of theoretical maximum spectral efficiency, that is, does not satisfy condition (ii). The one we propose hereafter is based on a combination of CDMA with OFDM/OQAM and avoids these two shortcomings.
3. CDMAOFDM/OQAM and Alamouti
3.1. CDMAOFDM/OQAM
In this section we summarize the results obtained, assuming a distortionfree channel, in [19] and [11] for CDMAOFDM/OQAM schemes transmitting real and complex data symbols, respectively. Then, we show how this latter scheme can be used for transmission over a realistic channel model in conjunction with Alamouti coding.
3.1.1. Transmission of Real Data Symbols
We denote by the length of the CDMA code used and assume that is an integer number. Let us denote by , where stands for the transpose operation, the code used by the th user. When applying spreading in the frequency domain such as in pure MCCDMA (MultiCarrierCDMA) [20], for a user at a given time , different data are transmitted denoted by: . Then by spreading with the codes, we get the real symbol transmitted at frequency and time by
where is the number of users, / the modulo operator, and the floor operator. From the term, the reconstruction of (for ) is insured thanks to the orthogonality of the code, that is, ; see [21] for more details. Therefore, noise taken apart, the despreading operator leads to
In [19], it is shown that, since no CP is inserted, the transmission of these spread real data () can be insured at a symbol rate which is more than twice the one used for transmitting complex MCCDMA data. Figure 3 depicts the real CDMAOFDM/OQAM transmission scheme for real data and a maximum spreading length (limited by the number of subcarriers), where after the despreading operation, only the real part of the symbol is kept whereas the imaginary component is not detected. This scheme satisfies a real orthogonality condition and can work for a number of users up to .
3.1.2. Interference Cancellation
A closer examination of the interference term is proposed in [11] assuming that the CDMA codes are WalshHadamard (WH) codes of length , with an integer. The prototype filter being of length , its duration is also given by the indicating function , equal to 1 if and elsewhere. Then, the scalar product of the base functions can be expressed as
where is given by
For a maximum spreading length, that is, , based on [11, Equation (18)], the interference term when transmitting real data can be expressed as
It is shown in [11] that if spreading codes are properly selected then the interference is cancelled. The WH matrix being of size can be divided into two subsets of column indices, and , with cardinal equal to making a partition of all the index set. To guarantee the absence of interference between users, the construction rule for theses two subsets is as follows.
For , each subset is initialized by setting: and .
Let us now assume that, for a given integer , the two subsets contain the following list of indices:
These subsets are used to build two new subsets of identical size such that
Then, we get the subsets of higher size, , as follows:
Applying this rule one can check that for , as an example, we get
Hence, for a given user and at a given time, we get and and these equalities hold for a number of users up to . The complete proof given in [11] takes advantage of three properties of WH codes.
3.1.3. Transmission of Complex Data Symbols
As the imaginary component can be cancelled when transmitting real data through a distortionfree channel when using CDMAOFDM/OQAM, one can imagine to extend this scheme to the transmission of complex data. Indeed, the transmission system being linear, real and imaginary parts will not interfere if the previous rule is satisfied.
Then, denoting by the complex data to transmit, the OFDM/OQAM symbols transmitted at time over the carrier and for the code are complex numbers, that is, are complex symbols. The corresponding complex CDMAOQAM transmission scheme is depicted in Figure 4. The baseband equivalent of the transmitted signal, with a spreading in frequency, can be written as
In this expression, as in [11], we assume that the phase term is , that is, . Then, if the codes are all in , or , the interference terms are cancelled and we get
Otherwise said, this CDMAOFDM/OQAM scheme satisfies a complex orthogonality condition, that is, the backtoback transmultiplexer is a PR system for the transmission of complex data. Note also that, differently from what we saw for the transmission of real data symbols, as explained in Section 3.1.2, here the maximum number of users is instead of . In both cases the overall data rate is therefore the same.
In the presence of a channel, an equalization must be performed before the despreading since the signal at the output of the equalization block is supposed to be free from any channel distortion or attenuation. Then, the signal at the equalizer output is somewhat equivalent to the one obtained with a distortionfree channel. Then, despreading operation will recover the complex orthogonality.
Now, the question is: "Can we use this complex orthogonality for combining Alamouti coding scheme and CDMAOFDM/OQAM?''. Let us analyze this problem assuming a onetap equalization.
3.2. Alamouti with CDMAOFDM/OQAM with Spreading in the Frequency Domain
In a realistic transmission scheme the channel is no longer distortionfree. So, we assume now that we are in the case of a wireless DownLink (DL) transmission and perfectly synchronized.
3.2.1. Problem Statement
Before trying to apply Alamouti scheme to CDMAOFDM/OQAM, one must notice that the channel equalization process is replaced by the Alamouti decoding. When adapting Alamouti scheme to CDMAOFDM/OQAM, the equalizer component, depicted in Figure 4, must be replaced by the Alamouti decoding process and the despreading operation must be carried out just after the OFDM/OQAM modulator. Then, contrary to the DL conventional MCCDMA case, the despreading operation must be performed before the Alamouti decoding. Indeed, with OFDM/OQAM, we can only recover a complex orthogonality property at the output of the despreading block. This point is critical since it rises the question: does complex orthogonality hold in CDMAOFDM/OQAM if we perform despreading operation before equalization? and if yes, at which cost? The first point leads to the following problem: let us consider complex quantities , , . Does it sound possible to obtain (equalization + despreading) from (despreading)? Here, equalization is materialized by and the despreading operation by . The answer is in general (obviously) NO, except if all the are the same, that is, . That is the case if we are in the presence of a constant channel over frequencies. Indeed, only in this case the order of the equalization and despreading operations can be exchanged without impairing the transmission performance. Conversely, applying despreading before equalization should have an impact in terms of performance for a channel being nonconstant in frequency. So, let us consider at first a flat channel. Then the subset of subcarriers where a given spreading code is applied will be affected by the same channel coefficient.
3.2.2. Implementation Scheme
In a SISO configuration, if we denote by the single channel coefficient between the transmit antenna and the single receive antenna at time instant , the despreaded signal is given by:
where is the complex data of user being transmitted at time instant by antenna . Now, if we consider a system with 2 antennas with indexes 0 and 1, respectively, and if we apply Alamouti coding scheme to every user data, denoting by the main stream of complex data for user , we have
For a flat fading channel, ignoring noise, the despreaded signal for user is given by
Hence,
This is the same decoding equation as in the Alamouti scheme presented in Section 2.2. Hence, the decoding could be performed in the same way. Figures 5 and 6 present the Alamouti CDMAOFDM/OQAM transmitter and receiver, respectively.
3.2.3. Performance Evaluation
We compare the proposed Alamouti CDMAOFDM/OQAM scheme with the Alamouti OFDM using the following parameters:

(i)
QPSK modulation

(ii)
subcarriers

(iii)
maximum spreading length, implying that the WH spreading codes are of length ,

(iv)
flat fading channel (one single Rayleigh coefficient for all 128 subcarriers);

(v)
the IOTA prototype filter with length 512,

(vi)
zero forcing one tap equalization for both transmission schemes,

(vii)
no channel coding.
Figure 7 gives the performance results. As expected, both systems perform the same.
4. Alamouti and CDMAOFDM/OQAM with Time Domain Spreading
In this section, we keep the same assumptions as the ones used for the transmission of complex data with a spreading in frequency. Firstly, we again suppose that the prototype function is a realvalued symmetric function and also that the WH codes are selected using the procedure recalled in Section 3.1.2.
4.1. CDMAOFDM/OQAM with Spreading in the Time Domain
Let us first consider a CDMAOFDM/OQAM system carrying out a spreading in the time domain, that is, on each subcarrier the data are spread over the time duration frame length. Let us consider the length of the frame, that is, the frame is made of data in the frequency domain and data in time domain. is the length of the spreading code. We assume that is an integer number. Let us denote by: the code used by the th user. Then, for a user at a given frequency , different data are transmitted denoted by: . By spreading with the codes, we get the real symbol transmitted at frequency and time by
where is the number of users. From the term, the reconstruction of (for ) is insured thanks to the orthogonality of the code, that is, , see [21] for more details. Therefore, the despreading operator leads to
We now propose to consider the transmission of complex data, denoted , using well chosen WH codes. In order to establish the theoretical features of this complex CDMAOFDM/OQAM scheme, we suppose that the transmission channel is free of any type of distortion. Also, for the sake of simplicity, we now assume a maximum spreading length (in time domain, ). We denote by the complex data and by the complex symbol transmitted at time over the carrier and for the code . As usual, the length of the WH codes are supposed to be a power of 2, that is, with an integer.
The block diagram of the transmitter is depicted in Figure 8. For a frame containing OFDM/OQAM data symbols, the baseband signal spread in time, can be written as
In (39), we assume that the phase term is as in [7]. Let us also recall that the prototype function satisfies the real orthogonality condition (2) and is realvalued and symmetric, that is, . To express the complex inner product of the base functions , using a similar procedure that led to (24), we get
where is given by
As the channel is distortionfree, the received signal is and the demodulated symbols are obtained as follows:
In this configuration, the demodulation operation only takes place when the whole frame is received. Then, the despreading operation gives us the despreaded data for the code as
Replacing and by their expression given in (39) and (40), respectively, we get:
Splitting the summation over in two parts, with equal to or not to , (44) can be rewritten as:
Considering the WH codes, we obtain
In [11], for WH codes of length , we have shown that for ,
where is given by
To prove the result given in (47), we had the following requirements:

(i)
WH codes satisfy the set of mathematical properties that are proved in [11].

(ii)
Since is a realvalued function, is real valued and the ambiguity function of the prototype function also satisfies the identities and .
Using these results, (47) can be proved straighforwardly.
It is worth mentioning that the above requirements are independent of the phase term and thus are satisfied in the case of the CDMAOFDM/OQAM system with spreading in time. It can also be shown that the modification of the phase term leads to the substitutions and , in obtaining (48) from (41). Accordingly the second term on the right hand side of (46) vanishes and we obtain
4.2. Alamouti with CDMAOFDM/OQAM with Spreading in Time
Now, if we consider the CDMAOFDM/OQAM with spreading in time, contrary to the case of a spreading in frequency domain, as long as the channel is constant during the spreading time duration, we can perform despreading before equalization. At the equalizer output we will have a complex orthogonality. Indeed, considering at first a SISO case, if we denote by the channel coefficient between a single transmit antenna and the receive antenna at subcarrier , the despreaded signal is given by
where is the complex data of user being transmitted at subcarrier by antenna . Thus, we can easily apply the Alamouti decoding scheme knowing the channel is constant for each antenna at each frequency. Otherwise said, the method becomes applicable for a frequency selective channel. Actually two strategies can be envisioned.
(1) Strategy 1. Alamouti performed over pairs of frequencies. If we consider a system with 2 transmit antennas, 0 and 1, and if we apply the Alamouti coding scheme to every user data, that is, if we denote by the main stream of complex data for user , then we have the following at subcarrier :
and at subcarrier ,
Then, considering a flat fading channel, the despreaded signal for user is given by
Therefore, we get
That means, when assuming the channel to be flat over two consecutive subcarriers, that is, for all , we have exactly the same decoding equation as the Alamouti scheme presented in Section 3.2, by permuting the frequency and time axis. Then, the decoding is performed in the same way.
(2) Strategy 2. Alamouti performed over pairs of spreading codes. In this second strategy, we apply the Alamouti scheme on pairs of codes, that is, we divide the codes in two groups (assuming to be even). That is, we process the codes by pair . We denote by the main stream of complex data for user pair . At subcarrier , antennas 0 and 1 transmit
At the receiver side we get,
Then, we do not need to consider the channel constant over two consecutive subcarriers. We have exactly the same decoding equation as the Alamouti scheme presented in Section 3.2. Hence, the decoding is performed in the same way.
We have tested two different channels considering each time the same channel profile, but with different realizations, between the 2 transmit antennas and one receive antenna. The Guard Interval (GI) is adjusted to take into account the delay spread profiles corresponding to a 4path and to a 7path channel. The 4path channel is characterized by the following parameters:

(i)
power profile (in dB): 0, 6, 9, 12,

(ii)
delay profile (in samples): 0, 1, 2, 3,

(iii)
GI for CPOFDM: 5 samples,
and the 7path by

(i)
power profile (in dB): 0, 6, 9, 12, 16, 20, 22,

(ii)
delay profile (in samples): 0, 1, 2, 3, 5, 7, 8,

(iii)
GI for CPOFDM: 9 samples;
We also consider the following system parameters:

(i)
QPSK modulation,

(ii)
subcarriers,

(iii)
time invariant channel (no Doppler),

(iv)
the IOTA prototype filter of length 512,

(v)
spreading codes of length 32, corresponding to the frame duration (32 complex OQAM symbols),

(vi)
number of CDMA WH codes equals to 16 in complex OFDM/OQAM, with symbol duration and this corresponds to 32 codes in OFDM, with symbol duration , leading to the same spectral efficiency

(vii)
zero forcing, one tap equalization,

(viii)
no channel coding.
In Figures 9 and 10, the BER results of the Alamouti CDMAOFDM/OQAM technique for the two proposed strategies are presented.
The two strategies perform the same until a BER of or for the 4 and 7path channel, respectively. For lower BER the strategy 2 performs better than the strategy 1. This could be explained by the fact that strategy 1 makes the approximation that the channel is constant over two consecutive subcarriers. This approximation leads to a degradation of the performance whereas the strategy 2 does not consider this approximation. If we compare the performance of Alamouti CDMAOFDM/OQAM strategy 2 with the Alamouti CPOFDM, we see that both system perform approximately the same. It is worth mentioning that however the corresponding throughput is higher for the OFDM/OQAM solutions (no CP). Indeed, it is increased by approximately 4 and 7% for the 4 and 7path channels, respectively.
5. Conclusion
In this paper, we showed that the wellknown Alamouti decoding scheme cannot be directly applied to the OFDM/OQAM modulation. To tackle this problem, we proposed to combine the MIMO Alamouti coding scheme with CDMAOFDM/OQAM. If the CDMA spreading is carried out in the frequency domain, the Alamouti decoding scheme can only be applied if the channel is assumed to be flat. On the other hand, for a frequency selective channel, the CDMA spreading component has to be applied in the time domain. For the Alamouti scheme with time spreading CDMAOFDM/OQAM, we elaborate two strategies for implementing the MIMO spacetime coding scheme. Strategy 1 implements the Alamouti over pairs of adjacent frequency domain samples whereas the strategy 2 processes the Alamouti coding scheme over pairs of spreading codes from two successive time instants. Strategy 2 appears to be more appropriate since it requires less restrictive assumptions on the channel variations across the frequencies. We also made some performance comparisons with Alamouti CPOFDM. It was found that, under some channel hypothesis, the combination of Alamouti with complex CDMAOFDM/OQAM is possible without increasing the complexity of the Alamouti decoding process. Furthermore, in the case of a frequency selective channel, OFDM/OQAM keeps its intrinsic advantage with a SNR gain in direct relation with the CP length. To find a simpler Alamouti scheme, that is, without adding a CDMA component, remains an open problem. Naturally, some other alternative transmit diversity schemes for OFDM/OQAM, as for instance cyclic delay diversity, could also deserve further investigations.
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Acknowledgments
The authors would like to thank the reviewers and Professor FarhangBoroujeny for their careful reading of our manuscript and for their helpful suggestions. This work was partially supported by the European ICT2008211887 project PHYDYAS.
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Lélé, C., Siohan, P. & Legouable, R. The Alamouti Scheme with CDMAOFDM/OQAM. EURASIP J. Adv. Signal Process. 2010, 703513 (2010). https://doi.org/10.1155/2010/703513
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Keywords
 Orthogonal Frequency Division Multiplex
 Code Division Multiple Access
 Multiple Input Multiple Output
 Cyclic Prefix
 Frequency Selective Channel