EURASIP Journal on Advances in Signal Processing
Volume 2010 (2010), Article ID 703513, 13 pages
doi:10.1155/2010/703513
Research Article

The Alamouti Scheme with CDMA-OFDM/OQAM

Chrislin Lélé,1 Pierre Siohan,2 and Rodolphe Legouable2

1CNAM, Laetitia group, 292, rue Saint Martin, 75141 Paris, France
2Orange Labs, 4, rue du Clos Courtel, BP 91226, 35512 Cesson Sévigné Cedex, France

Received 23 June 2009; Revised 4 October 2009; Accepted 29 December 2009

Academic Editor: Behrouz Farhang-Boroujeny

Copyright © 2010 Chrislin Lélé et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 well-known 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 CDMA-OFDM/OQAM combination has to be built taking into account particular features of the transmission channel. Our simulation results illustrate the 2 × 1 Alamouti coding scheme for which CDMA-OFDM/OQAM and CP-OFDM 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, CP-OFDM 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 CP-OFDM 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 ( s i n c ( 𝑥 ) ) behavior in the frequency domain. Thus CP-OFDM 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 Balian-Low theorem, see, for example, [2], it is not possible to get at the same time (i) Complex orthogonality; (ii) Maximum spectral efficiency; (iii) A well-localized pulse shape in time and frequency. With CP-OFDM 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 bank-based 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 CP-OFDM 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 (O-QAM), 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/OQAM-IOTA modem. A formal link between these continuous-time 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 CP-OFDM. However, most of these publications are related to a single user case and to Single-Input-Single-Output (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 Alamouti-like scheme for OFDM/OQAM proposed in [15] complicates the RX and introduces a processing delay. The pseudo-Alamouti 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 CDMA-OFDM/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 CDMA-OFDM/OQAM in order to solve the problem. Indeed, in [11], we have shown that the combination of CDMA and OFDM/OQAM (CDMA-OFDM/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 CDMA-OFDM/OQAM technique achieves similar performance to the Alamouti CP-OFDM system.

2. OFDM/OQAM and Alamouti

2.1. The OFDM/OQAM Transmultiplexer

The baseband equivalent of a continuous-time multicarrier OFDM/OQAM signal can be expressed as follows [7]: 𝑠 ( 𝑡 ) = 𝑀 − 1 𝑚 = 0 𝑛 ∈ 𝐙 𝑎 𝑚 , 𝑛 𝑔 ( 𝑡 − 𝑛 𝜏 0 ) 𝑒 𝑗 2 𝜋 𝑚 𝐹 0 𝑡 𝜈 𝑚 , 𝑛 𝑔 𝑚 , 𝑛 ( 𝑡 ) ( 1 ) with 𝐙 the set of integers, 𝑀 = 2 𝑁 an even number of subcarriers, 𝐹 0 = 1 / 𝑇 0 = 1 / 2 𝜏 0 the subcarrier spacing, 𝑔 the prototype function assumed here to be a real-valued and even function of time, and 𝜈 𝑚 , 𝑛 an additional phase term such that 𝜈 𝑚 , 𝑛 = 𝑗 𝑚 + 𝑛 𝑒 𝑗 𝜙 0 , where 𝜙 0 can be chosen arbitrarily. The transmitted data symbols 𝑎 𝑚 , 𝑛 are real-valued. They are obtained from a 2 2 𝐾 -QAM constellation, taking the real and imaginary parts of these complex-valued symbols of duration 𝑇 0 = 2 𝜏 0 , where 𝜏 0 denotes the time offset between the two parts [2, 6, 7, 9].

Assuming a distortion-free channel, the Perfect Reconstruction (PR) of the real data symbols is obtained owing to the following real orthogonality condition: ℜ 𝑔 𝑚 , 𝑛 , 𝑔 𝑝 , 𝑞 = ℜ 𝑔 𝑚 , 𝑛 ( 𝑡 ) 𝑔 ∗ 𝑝 , 𝑞 ( 𝑡 ) 𝑑 𝑡 = 𝛿 𝑚 , 𝑝 𝛿 𝑛 , 𝑞 , ( 2 ) where ∗ denotes conjugation, ⟨ ⋅ , ⋅ ⟩ denotes the inner product, and 𝛿 𝑚 , 𝑝 = 1 if 𝑚 = 𝑝 and 𝛿 𝑚 , 𝑝 = 0 if 𝑚 ≠ 𝑝 . Otherwise said, for ( 𝑚 , 𝑛 ) ≠ ( 𝑝 , 𝑞 ) , ⟨ 𝑔 𝑚 , 𝑛 , 𝑔 𝑝 , 𝑞 ⟩ is a pure imaginary number. For the sake of brevity, we set ⟨ 𝑔 ⟩ 𝑝 , 𝑞 𝑚 , 𝑛 = − 𝑗 ⟨ 𝑔 𝑚 , 𝑛 , 𝑔 𝑝 , 𝑞 ⟩ . The orthogonality condition for the prototype filter can also be conveniently expressed using its ambiguity function 𝐴 𝑔 ( 𝑛 , 𝑚 ) = ∞ − ∞ 𝑔 𝑢 − 𝑛 𝜏 0 𝑔 ( 𝑢 ) 𝑒 2 𝑗 𝜋 𝑚 𝐹 0 𝑢 𝑑 𝑢 . ( 3 ) It is well-known [7] that to satisfy the orthogonality condition (2), the prototype filter should be chosen such that 𝐴 𝑔 ( 2 𝑛 , 2 𝑚 ) = 0 if ( 𝑛 , 𝑚 ) ≠ ( 0 , 0 ) and 𝐴 𝑔 ( 0 , 0 ) = 1 .

In practical implementations, the baseband signal is directly generated in discrete time, using the continuous-time signal samples at the critical frequency, that is, with 𝐹 𝑒 = 𝑀 𝐹 0 = 2 𝑁 𝐹 0 . Then, based on [9], the discrete-time baseband signal taking the causality constraint into account, is expressed as 𝑠 [ 𝑘 ] = 𝑀 − 1 𝑚 = 0 𝑛 ∈ 𝐙 𝑎 𝑚 , 𝑛 𝑔 ( 𝑘 − 𝑛 𝑁 ) 𝑒 𝑗 2 𝜋 𝑚 ( 𝑘 − ( 𝐿 𝑔 − 1 ) / 2 ) 𝜈 𝑚 , 𝑛 𝑔 𝑚 , 𝑛 [ 𝑘 ] . ( 4 )

The parallel between (1) and (4) shows that the overlapping of duration 𝜏 0 corresponds to 𝑁 discrete-time samples. For the sake of simplicity, we will assume that the prototype filter length, denoted 𝐿 𝑔 , is such that 𝐿 𝑔 = 𝑏 𝑀 = 2 𝑏 𝑁 , with 𝑏 being a positive integer. With the discrete time formulation, the real orthogonality condition can also be expressed as: ℜ 𝑔 𝑚 , 𝑛 , 𝑔 𝑝 , 𝑞 = ℜ ⎧ ⎨ ⎩ 𝑘 ∈ 𝒵 𝑔 𝑚 , 𝑛 [ 𝑘 ] 𝑔 ∗ 𝑝 , 𝑞 [ 𝑘 ] ⎫ ⎬ ⎭ = 𝛿 𝑚 , 𝑝 𝛿 𝑛 , 𝑞 . ( 5 )

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 𝜈 𝑚 , 𝑛 and on the prototype length. Note also that in this scheme, to transmit QAM symbols of a given duration, denoted 𝑇 0 , the IFFT block has to be run twice faster than for CP-OFDM. The polyphase block contains the polyphase components of the prototype filter 𝑔 . At the RX side, the dual operations are carried out.

703513.fig.001
Figure 1: Transmultiplexer scheme for the OFDM/OQAM modulation.

The prototype filter has to be PR, or nearly PR. In this paper, we use a nearly PR prototype filter, with length 𝐿 𝑔 = 4 𝑀 , 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 continuous-time. Thus, in the rest of this paper, we present an OFDM/OQAM modulator that delivers a signal denoted 𝑠 ( 𝑡 ) , 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.

703513.fig.002
Figure 2: The transmission scheme based on OFDM/OQAM.

Let us consider a time-varying channel, with maximum delay spread equal to Δ . We denote it by ℎ ( 𝑡 , 𝜏 ) in time, and it can also be represented by a complex-valued number 𝐻 ( 𝑐 ) 𝑚 , 𝑛 for subcarrier 𝑚 at symbol time 𝑛 . At the receiver side, the received signal is the summation of the 𝑠 ( 𝑡 ) signal convolved with the channel impulse response and a noise component 𝜂 ( 𝑡 ) . For a locally invariant channel, we can define a neighborhood, denoted Ω Δ 𝑚 , Δ 𝑛 , around the ( 𝑚 0 , 𝑛 0 ) position, with Ω Δ 𝑚 , Δ 𝑛 = ( 𝑝 , 𝑞 ) , | | 𝑝 | | ≤ Δ 𝑚 , | | 𝑞 | | ≤ Δ 𝑛 ∣ 𝐻 ( 𝑐 ) 𝑚 0 + 𝑝 , 𝑛 0 + 𝑞 ≈ 𝐻 ( 𝑐 ) 𝑚 0 , 𝑛 0 , ( 6 )

and we also define Ω ∗ Δ 𝑚 , Δ 𝑛 = Ω Δ 𝑚 , Δ 𝑛 − { ( 0 , 0 ) } .

Note also that Δ 𝑛 and Δ 𝑚 are chosen according to the time and bandwidth coherence of the channel, respectively. Then, assuming 𝑔 ( 𝑡 − 𝜏 − 𝑛 𝜏 0 ) ≈ 𝑔 ( 𝑡 − 𝑛 𝜏 0 ) , for all 𝜏 ∈ [ 0 , Δ ] , the demodulated signal can be expressed as [13, 14, 17] 𝑦 ( 𝑐 ) 𝑚 0 , 𝑛 0 = 𝐻 ( 𝑐 ) 𝑚 0 , 𝑛 0 𝑎 𝑚 0 , 𝑛 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 0 , 𝑛 0 + 𝐽 𝑚 0 , 𝑛 0 + 𝜂 𝑚 0 , 𝑛 0 ( 7 ) with 𝜂 𝑚 0 , 𝑛 0 = ⟨ 𝜂 , 𝑔 𝑚 0 , 𝑛 0 ⟩ the noise component, 𝑎 ( 𝑖 ) 𝑚 0 , 𝑛 0 , the interference created by the neighbor symbols, given by 𝑎 ( 𝑖 ) 𝑚 0 , 𝑛 0 = ( 𝑝 , 𝑞 ) ∈ Ω ∗ Δ 𝑚 , Δ 𝑛 𝑎 𝑚 0 + 𝑝 , 𝑛 0 + 𝑞 ⟨ 𝑔 ⟩ 𝑚 0 , 𝑛 0 𝑚 0 + 𝑝 , 𝑛 0 + 𝑞 , ( 8 ) and 𝐽 𝑚 0 , 𝑛 0 the interference created by the data symbols outside Ω Δ 𝑚 , Δ 𝑛 .

It can be shown that, even for small size neighborhoods, if the prototype function 𝑔 is well localized in time and frequency, 𝐽 𝑚 0 , 𝑛 0 becomes negligible when compared to the noise term 𝜂 𝑚 0 , 𝑛 0 . Indeed a good time-frequency localization [7] means that the ambiguity function of 𝑔 , which is directly related to the ⟨ 𝑔 ⟩ 𝑚 0 , 𝑛 0 𝑚 0 + 𝑝 , 𝑛 0 + 𝑞 terms, is concentrated around its origin in the time-frequency plane, that is, only takes small values outside the Ω Δ 𝑚 , Δ 𝑛 region. Thus, the received signal can be approximated by 𝑦 ( 𝑐 ) 𝑚 0 , 𝑛 0 ≈ 𝐻 ( 𝑐 ) 𝑚 0 , 𝑛 0 𝑎 𝑚 0 , 𝑛 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 0 , 𝑛 0 + 𝜂 𝑚 0 , 𝑛 0 . ( 9 )

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 one-tap channel model described as 𝑦 𝑘 = ℎ 𝑘 , 𝑢 𝑠 𝑘 , 𝑢 + 𝑛 𝑘 , ( 1 0 ) where, at time instant 𝑘 , ℎ 𝑘 , 𝑢 is the channel gain between the transmit antenna 𝑢 and the receive antenna and 𝑛 𝑘 is an additive noise. We assume that ℎ 𝑘 , 𝑢 is a complex-valued 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 ℎ 𝑘 , 𝑢 .

The Alamouti scheme is implemented with 2 transmit and one receive antennas. Let us consider 𝑠 2 𝑘 and 𝑠 2 𝑘 + 1 to be the two symbols to transmit at time (time and frequency axis can be permuted in multicarrier modulation.) instants 2 𝑘 and 2 𝑘 + 1 , respectively. At time instant 2 𝑘 , the antenna 0 transmits 𝑠 2 𝑘 / √ 2 whereas the antenna 1 transmits 𝑠 2 𝑘 + 1 / √ 2 . At time instant 2 𝑘 + 1 , the antenna 0 transmits − ( 𝑠 2 𝑘 + 1 ) ∗ / √ 2 whereas the antenna 1 transmits 𝑠 ∗ 2 𝑘 / √ 2 . The 1 / √ 2 factor is added to normalize the total transmitted power. The received signal samples at time instants 2 𝑘 and 2 𝑘 + 1 are given by 𝑦 2 𝑘 = 1 √ 2 ℎ 2 𝑘 , 0 𝑠 2 𝑘 + ℎ 2 𝑘 , 1 𝑠 2 𝑘 + 1 + 𝑛 2 𝑘 , 𝑦 2 𝑘 + 1 = 1 √ 2 − ℎ 2 𝑘 + 1 , 0 𝑠 2 𝑘 + 1 ∗ + ℎ 2 𝑘 + 1 , 1 𝑠 2 𝑘 ∗ + 𝑛 2 𝑘 + 1 . ( 1 1 )

Assuming the channel to be constant between the time instants 2 𝑘 and 2 𝑘 + 1 , we get 𝑦 2 𝑘 𝑦 ∗ 2 𝑘 + 1 = 1 √ 2 ℎ 2 𝑘 , 0 ℎ 2 𝑘 , 1 ℎ 2 𝑘 , 1 ∗ − ℎ 2 𝑘 , 0 ∗ 𝐻 2 𝑘 𝑠 2 𝑘 𝑠 2 𝑘 + 1 + 𝑛 2 𝑘 𝑛 ∗ 2 𝑘 + 1 . ( 1 2 ) Note that 𝐻 2 𝑘 is an orthogonal matrix with 𝐻 2 𝑘 𝐻 𝐻 2 𝑘 = ( 1 / 2 ) ( | ℎ 2 𝑘 , 0 | 2 + | ℎ 2 𝑘 , 1 | 2 ) 𝐼 2 , where 𝐼 2 is the identity matrix of size ( 2 , 2 ) and 𝐻 stands for the transpose conjugate operation. Thus, using the Maximum Ratio Combining (MRC) equalization, the estimates ̂ 𝑠 2 𝑘 and ̂ 𝑠 2 𝑘 + 1 are obtained as ̂ 𝑠 2 𝑘 ̂ 𝑠 2 𝑘 + 1 = √ 2 | | ℎ 2 𝑘 , 0 | | 2 + | | ℎ 2 𝑘 , 1 | | 2 ℎ ∗ 2 𝑘 , 0 ℎ 2 𝑘 , 1 ℎ 2 𝑘 , 1 ∗ − ℎ 2 𝑘 , 0 𝑦 2 𝑘 𝑦 2 𝑘 + 1 = 𝑠 2 𝑘 𝑠 2 𝑘 + 1 + 𝜇 2 𝑘 𝜇 2 𝑘 + 1 , ( 1 3 )

where, 𝜇 2 𝑘 𝜇 2 𝑘 + 1 = √ 2 | | ℎ 2 𝑘 , 0 | | 2 + | | ℎ 2 𝑘 , 1 | | 2 ℎ ∗ 2 𝑘 , 0 ℎ 2 𝑘 , 1 ℎ 2 𝑘 , 1 ∗ − ℎ 2 𝑘 , 0 𝑛 2 𝑘 𝑛 ∗ 2 𝑘 + 1 . ( 1 4 )

Since the noise components 𝑛 2 𝑘 and 𝑛 2 𝑘 + 1 are uncorrelated, 𝐸 ( | 𝜇 2 𝑘 | 2 ) = 𝐸 ( | 𝜇 2 𝑘 + 1 | 2 ) = 2 𝑁 0 / ( | ℎ 2 𝑘 , 0 | 2 + | ℎ 2 𝑘 , 1 | 2 ) , where 𝑁 0 denotes the monolateral noise density. Thus, assuming a QPSK modulation, based on [18], the bit error probability, denoted 𝑝 𝑏 , is given by 𝑝 𝑏 = 𝑄 ⎛ ⎜ ⎜ ⎝ ⎷ ⎛ ⎜ ⎜ ⎝ | | ℎ 2 𝑘 , 0 | | 2 + | | ℎ 2 𝑘 , 1 | | 2 2 ⎞ ⎟ ⎟ ⎠ S N R 𝑡 ⎞ ⎟ ⎟ ⎠ , ( 1 5 ) where S N R 𝑡 denotes the Signal-to-Noise 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, 𝑑 ( 𝑐 ) 𝑚 , 𝑛 , is divided into two real symbols, ℜ { 𝑑 ( 𝑐 ) 𝑚 , 𝑛 } and ℑ { 𝑑 ( 𝑐 ) 𝑚 , 𝑛 } , transmitted at successive time instants, transmission of a pair of data symbols, according to Alamouti scheme, is organized as follows: 𝑎 𝑚 , 2 𝑛 , 0 = ℜ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 , 𝑎 𝑚 , 2 𝑛 , 1 = ℜ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 , 𝑎 𝑚 , 2 𝑛 + 1 , 0 = ℑ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 , 𝑎 𝑚 , 2 𝑛 + 1 , 1 = ℑ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 , 𝑎 𝑚 , 2 𝑛 + 2 , 0 = − ℜ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 ∗ = − ℜ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 = − 𝑎 𝑚 , 2 𝑛 , 1 , 𝑎 𝑚 , 2 𝑛 + 2 , 1 = ℜ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 ∗ = ℜ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 = 𝑎 𝑚 , 2 𝑛 , 0 , 𝑎 𝑚 , 2 𝑛 + 3 , 0 = − ℑ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 ∗ = ℑ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 = 𝑎 𝑚 , 2 𝑛 + 1 , 1 , 𝑎 𝑚 , 2 𝑛 + 3 , 1 = ℑ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 ∗ = − ℑ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 = − 𝑎 𝑚 , 2 𝑛 + 1 , 0 . ( 1 6 )

We also assume that in OFDM/OQAM the channel gain is a constant between the time instants 2 𝑛 and 2 𝑛 + 3 . Let us denote the channel gain between the transmit antenna 𝑖 and the receive antenna at subcarrier 𝑚 and time instant 𝑛 by ℎ 𝑚 , 𝑛 , 𝑖 . Therefore, at the single receive antenna we have 𝑦 𝑚 , 2 𝑛 = ℎ 𝑚 , 2 𝑛 , 0 𝑎 𝑚 , 2 𝑛 , 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 , 0 + ℎ 𝑚 , 2 𝑛 , 1 𝑎 𝑚 , 2 𝑛 , 1 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 , 1 + 𝑛 𝑚 , 2 𝑛 , 0 , 𝑦 𝑚 , 2 𝑛 + 1 = ℎ 𝑚 , 2 𝑛 , 0 𝑎 𝑚 , 2 𝑛 + 1 , 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 1 , 0 + ℎ 𝑚 , 2 𝑛 , 1 𝑎 𝑚 , 2 𝑛 + 1 , 1 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 1 , 1 + 𝑛 𝑚 , 2 𝑛 + 1 , 1 , 𝑦 𝑚 , 2 𝑛 + 2 = ℎ 𝑚 , 2 𝑛 , 0 𝑎 𝑚 , 2 𝑛 + 2 , 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 2 , 0 + ℎ 𝑚 , 2 𝑛 , 1 𝑎 𝑚 , 2 𝑛 + 2 , 1 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 2 , 1 + 𝑛 𝑚 , 2 𝑛 + 2 , 0 , 𝑦 𝑚 , 2 𝑛 + 3 = ℎ 𝑚 , 2 𝑛 , 0 𝑎 𝑚 , 2 𝑛 + 3 , 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 3 , 0 + ℎ 𝑚 , 2 𝑛 , 1 𝑎 𝑚 , 2 𝑛 + 3 , 1 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 3 , 1 + 𝑛 𝑚 , 2 𝑛 + 3 , 1 . ( 1 7 )

Setting 𝑧 𝑚 , 2 𝑛 = 𝑦 𝑚 , 2 𝑛 + 𝑗 𝑦 𝑚 , 2 𝑛 + 1 , 𝑧 𝑚 , 2 𝑛 + 1 = 𝑦 𝑚 , 2 𝑛 + 2 + 𝑗 𝑦 𝑚 , 2 𝑛 + 3 , ( 1 8 ) and using (16), we obtain 𝑧 𝑚 , 2 𝑛 = ℎ 𝑚 , 2 𝑛 , 0 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + ℎ 𝑚 , 2 𝑛 , 1 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 + ℎ 𝑚 , 2 𝑛 , 0 𝑥 𝑚 , 2 𝑛 , 0 + ℎ 𝑚 , 2 𝑛 , 1 𝑥 𝑚 , 2 𝑛 , 1 + 𝜅 𝑚 , 2 𝑛 , 0 , 𝑧 𝑚 , 2 𝑛 + 1 = − ℎ 𝑚 , 2 𝑛 , 0 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 ∗ + ℎ 𝑚 , 2 𝑛 , 1 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 ∗ − ℎ 𝑚 , 2 𝑛 , 0 𝑥 𝑚 , 2 𝑛 + 2 , 0 ∗ + ℎ 𝑚 , 2 𝑛 , 1 𝑥 𝑚 , 2 𝑛 + 2 , 1 ∗ + 𝜅 𝑚 , 2 𝑛 + 2 , 0 , ( 1 9 )

where, 𝑥 𝑚 , 2 𝑛 , 0 = − 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 1 , 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 , 0 , 𝑥 𝑚 , 2 𝑛 , 1 = − 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 1 , 1 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 , 1 , 𝜅 𝑚 , 2 𝑛 , 0 = 𝑛 𝑚 , 2 𝑛 , 0 + 𝑗 𝑛 𝑚 , 2 𝑛 + 1 , 0 , 𝜅 𝑚 , 2 𝑛 , 0 = 𝑛 𝑚 , 2 𝑛 + 2 , 0 + 𝑗 𝑛 𝑚 , 2 𝑛 + 3 , 0 , 𝑥 𝑚 , 2 𝑛 + 2 , 0 = 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 3 , 0 + 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 2 , 0 , 𝑥 𝑚 , 2 𝑛 + 2 , 1 = − 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 3 , 1 − 𝑗 𝑎 ( 𝑖 ) 𝑚 , 2 𝑛 + 2 , 1 . ( 2 0 )

This results in 𝑧 𝑚 , 2 𝑛 𝑧 𝑚 , 2 𝑛 + 1 ∗ 𝑧 2 𝑛 = ℎ 𝑚 , 2 𝑛 , 0 ℎ 𝑚 , 2 𝑛 , 1 ℎ 𝑚 , 2 𝑛 , 1 ∗ − ℎ 𝑚 , 2 𝑛 , 0 ∗ 𝑄 2 𝑛 ⎡ ⎢ ⎣ 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 𝑑 ( 𝑐 ) 𝑚 , 2 𝑛 + 1 ⎤ ⎥ ⎦ 𝑑 2 𝑛 + ℎ 𝑚 , 2 𝑛 , 0 ℎ 𝑚 , 2 𝑛 , 1 0 0 0 0 ℎ 𝑚 , 2 𝑛 , 1 ∗ − ℎ 𝑚 , 2 𝑛 , 0 ∗ 𝐾 2 𝑛 × ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑥 𝑚 , 2 𝑛 , 0 𝑥 𝑚 , 2 𝑛 , 1 𝑥 𝑚 , 2 𝑛 + 2 , 0 𝑥 𝑚 , 2 𝑛 + 2 , 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 𝑥 2 𝑛 + 𝜅 𝑚 , 2 𝑛 𝜅 𝑚 , 2 𝑛 + 1 ∗ 𝜅 2 𝑛 . ( 2 1 )

We note that 𝑄 2 𝑛 is an orthogonal matrix which is similar to the one found in (12) for the conventional 2 × 1 Alamouti scheme. However, the 𝐾 2 𝑛 𝑥 2 𝑛 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 distortion-free channel, we cannot achieve a good error probability since 𝐾 2 𝑛 𝑥 2 𝑛 is an inherent “noise interference” component that, differently from the one expressed in (9), cannot be easily removed. (in a particular case, where ℎ 𝑚 , 2 𝑛 , 0 = ℎ 𝑚 , 2 𝑛 , 1 , 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. CDMA-OFDM/OQAM and Alamouti

3.1. CDMA-OFDM/OQAM

In this section we summarize the results obtained, assuming a distortion-free channel, in [19] and [11] for CDMA-OFDM/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 𝑁 𝑐 the length of the CDMA code used and assume that 𝑁 𝑆 = 𝑀 / 𝑁 𝑐 is an integer number. Let us denote by 𝑐 𝑢 = [ 𝑐 0 , 𝑢 ⋯ 𝑐 𝑁 𝑐 − 1 , 𝑢 ] 𝑇 , where ( ⋅ ) 𝑇 stands for the transpose operation, the code used by the 𝑢 th user. When applying spreading in the frequency domain such as in pure MC-CDMA (Multi-Carrier-CDMA) [20], for a user 𝑢 0 at a given time 𝑛 0 , 𝑁 𝑆 different data are transmitted denoted by: 𝑑 𝑢 0 , 𝑛 0 , 0 , 𝑑 𝑢 0 , 𝑛 0 , 1 , … , 𝑑 𝑢 0 , 𝑛 0 , 𝑁 𝑆 − 1 . Then by spreading with the 𝑐 𝑢 codes, we get the real symbol 𝑎 𝑚 0 , 𝑛 0 transmitted at frequency 𝑚 0 and time 𝑛 0 by 𝑎 𝑚 0 , 𝑛 0 = 𝑈 − 1 𝑢 = 0 𝑐 𝑚 0 / 𝑁 𝑐 , 𝑢 𝑑 𝑢 , 𝑛 0 , ⌊ 𝑚 0 / 𝑁 𝑐 ⌋ , ( 2 2 )

where 𝑈 is the number of users, / the modulo operator, and ⌊ ⌋ the floor operator. From the 𝑎 𝑚 0 , 𝑛 0 term, the reconstruction of 𝑑 𝑢 , 𝑛 0 , 𝑝 (for 𝑝 ∈ [ 0 , 𝑁 𝑆 − 1 ] ) is insured thanks to the orthogonality of the code, that is, 𝑐 𝑇 𝑢 1 𝑐 𝑢 2 = 𝛿 𝑢 1 , 𝑢 2 ; see [21] for more details. Therefore, noise taken apart, the despreading operator leads to ̂ ‌ 𝑑 𝑢 , 𝑛 0 , 𝑝 = 𝑁 𝑐 − 1 𝑚 = 0 𝑐 𝑚 , 𝑢 𝑎 𝑝 𝑁 𝐶 + 𝑚 , 𝑛 0 . ( 2 3 )

In [19], it is shown that, since no CP is inserted, the transmission of these spread real data ( 𝑑 𝑢 , 𝑛 0 , 𝑝 ) can be insured at a symbol rate which is more than twice the one used for transmitting complex MC-CDMA data. Figure 3 depicts the real CDMA-OFDM/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 𝑖 𝑢 , 𝑛 is not detected. This scheme satisfies a real orthogonality condition and can work for a number of users up to 𝑀 .

703513.fig.003
Figure 3: Transmission scheme for the CDMA-OFDM/OQAM system with spreading in frequency of real data.
3.1.2. Interference Cancellation

A closer examination of the interference term is proposed in [11] assuming that the CDMA codes are Walsh-Hadamard (W-H) codes of length 𝑀 = 2 𝑁 = 2 𝑛 , with 𝑛 an integer. The prototype filter being of length 𝐿 𝑔 = 𝑏 𝑀 , its duration is also given by the indicating function 𝐼 | 𝑛 − 𝑛 0 | < 2 𝑏 , equal to 1 if | 𝑛 − 𝑛 0 | < 2 𝑏 and 0 elsewhere. Then, the scalar product of the base functions can be expressed as 𝑔 𝑚 , 𝑛 , 𝑔 𝑝 , 𝑛 0 = 𝛿 𝑚 − 𝑝 , 𝑛 − 𝑛 0 + 𝑗 𝛾 ( 𝑝 , 𝑛 0 ) 𝑚 , 𝑛 𝐼 | 𝑛 − 𝑛 0 | < 2 𝑏 , ( 2 4 ) where 𝛾 ( 𝑝 , 𝑛 0 ) 𝑚 , 𝑛 is given by 𝛾 ( 𝑝 , 𝑛 0 ) 𝑚 , 𝑛 = ℑ ( − 1 ) 𝑚 ( 𝑛 + 𝑛 0 ) 𝑗 𝑚 + 𝑛 − 𝑝 − 𝑛 0 𝐴 𝑔 𝑛 − 𝑛 0 , 𝑚 − 𝑝 . ( 2 5 )

For a maximum spreading length, that is, 𝑀 = 2 𝑁 = 𝑁 𝑐 , based on [11, Equation (18)], the interference term when transmitting real data can be expressed as 𝑖 𝑢 , 𝑛 = 𝑈 − 1 𝑢 = 0 2 𝑏 − 1 𝑛 = − 2 𝑏 + 1 , 𝑛 ≠ 0 𝑑 𝑛 + 𝑛 0 , 𝑢 ⎛ ⎜ ⎝ 2 𝑁 − 1 𝑝 = 0 2 𝑁 − 1 𝑚 = 0 𝑐 𝑝 , 𝑢 0 𝑐 𝑚 , 𝑢 𝛾 ( 𝑝 , 𝑛 0 ) 𝑚 , 𝑛 + 𝑛 0 ⎞ ⎟ ⎠ . ( 2 6 )

It is shown in [11] that if 𝑈 ≤ 𝑀 / 2 spreading codes are properly selected then the 𝑖 𝑢 , 𝑛 interference is cancelled. The W-H matrix being of size 𝑀 = 2 𝑁 = 2 𝑛 can be divided into two subsets of column indices, 𝑆 𝑛 1 and 𝑆 𝑛 2 , with cardinal equal to 𝑀 / 2 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 𝑛 0 = 1 , each subset is initialized by setting: 𝑆 1 1 = { 0 } and 𝑆 1 2 = { 1 } .

Let us now assume that, for a given integer 𝑛 = 𝑛 0 , the two subsets contain the following list of indices: 𝑆 𝑛 0 1 = 𝑖 1 , 1 , 𝑖 1 , 2 , 𝑖 1 , 3 , … , 𝑖 1 , 2 𝑛 0 − 1 , 𝑆 𝑛 0 2 = 𝑖 2 , 1 , 𝑖 2 , 2 , 𝑖 2 , 3 , … , 𝑖 2 , 2 𝑛 0 − 1 . ( 2 7 ) These subsets are used to build two new subsets of identical size such that 𝑆 𝑛 0 1 = 𝑖 2 , 1 + 2 𝑛 0 , 𝑖 2 , 2 + 2 𝑛 0 , 𝑖 2 , 3 + 2 𝑛 0 , … , 𝑖 2 , 2 𝑛 0 − 1 + 2 𝑛 0 , 𝑆 𝑛 0 2 = 𝑖 1 , 1 + 2 𝑛 0 , 𝑖 1 , 2 + 2 𝑛 0 , 𝑖 1 , 3 + 2 𝑛 0 , … , 𝑖 1 , 2 𝑛 0 − 1 + 2 𝑛 0 . ( 2 8 )

Then, we get the subsets of higher size, 𝑛 = 𝑛 0 + 1 , as follows: 𝑆 𝑛 0 + 1 1 = 𝑆 𝑛 0 1 ∪ 𝑆 𝑛 0 1 , 𝑆 𝑛 0 + 1 2 = 𝑆 𝑛 0 2 ∪ 𝑆 𝑛 0 2 . ( 2 9 ) Applying this rule one can check that for 𝑛 = 5 , as an example, we get 𝑆 5 1 = { 1 , 4 , 6 , 7 , 1 0 , 1 1 , 1 3 , 1 6 , 1 8 , 1 9 , 2 1 , 2 4 , 2 5 , 2 8 , 3 0 , 3 1 } , 𝑆 5 2 = { 2 , 3 , 5 , 8 , 9 , 1 2 , 1 4 , 1 5 , 1 7 , 2 0 , 2 2 , 2 3 , 2 6 , 2 7 , 2 9 , 3 2 } . ( 3 0 )

Hence, for a given user and at a given time, we get ̂ ‌ 𝑑 𝑢 , 𝑛 = 𝑑 𝑢 , 𝑛 and 𝑖 𝑢 , 𝑛 = 0 and these equalities hold for a number of 𝑈 users up to 𝑀 / 2 . The complete proof given in [11] takes advantage of three properties of W-H codes.

3.1.3. Transmission of Complex Data Symbols

As the imaginary component can be cancelled when transmitting real data through a distortion-free channel when using CDMA-OFDM/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 𝑑 ( 𝑐 ) 𝑛 , 𝑢 the complex data to transmit, the OFDM/OQAM symbols transmitted at time 𝑛 𝜏 0 over the carrier 𝑚 and for the code 𝑢 are complex numbers, that is, 𝑎 ( 𝑐 ) 𝑚 , 𝑛 , 𝑢 = 𝑐 𝑚 , 𝑢 𝑑 ( 𝑐 ) 𝑛 , 𝑢 are complex symbols. The corresponding complex CDMA-OQAM transmission scheme is depicted in Figure 4. The baseband equivalent of the transmitted signal, with a spreading in frequency, can be written as 𝑠 𝐹 ( 𝑡 ) = 𝑛 ∈ 𝐙 2 𝑁 − 1 𝑚 = 0 𝑥 𝑚 , 𝑛 𝑔 𝑚 , 𝑛 ( 𝑡 ) w i t h 𝑥 𝑚 , 𝑛 = 𝑈 − 1 𝑢 = 0 𝑎 ( 𝑐 ) 𝑚 , 𝑛 , 𝑢 . ( 3 1 ) In this expression, as in [11], we assume that the phase term is 𝜈 𝑛 , 𝑚 = 𝑗 𝑛 + 𝑚 ( − 1 ) 𝑛 𝑚 , that is, 𝜙 0 = 𝜋 𝑛 𝑚 . Then, if the 𝑈 codes are all in 𝑆 𝑛 1 , or 𝑆 𝑛 2 , the interference terms are cancelled and we get ∀ 𝑛 , 𝑢 , 𝑧 ( 𝑐 ) 𝑛 , 𝑢 = 𝑑 ( 𝑐 ) 𝑛 , 𝑢 . ( 3 2 ) Otherwise said, this CDMA-OFDM/OQAM scheme satisfies a complex orthogonality condition, that is, the back-to-back 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 𝑀 / 2 instead of 𝑀 . In both cases the overall data rate is therefore the same.

703513.fig.004
Figure 4: Transmission scheme for the CDMA-OFDM/OQAM system with spreading in frequency of complex data.

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 distortion-free 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 CDMA-OFDM/OQAM?’’. Let us analyze this problem assuming a one-tap equalization.

3.2. Alamouti with CDMA-OFDM/OQAM with Spreading in the Frequency Domain

In a realistic transmission scheme the channel is no longer distortion-free. So, we assume now that we are in the case of a wireless Down-Link (DL) transmission and perfectly synchronized.

3.2.1. Problem Statement

Before trying to apply Alamouti scheme to CDMA-OFDM/OQAM, one must notice that the channel equalization process is replaced by the Alamouti decoding. When adapting Alamouti scheme to CDMA-OFDM/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 MC-CDMA 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 CDMA-OFDM/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 𝑡 𝑖 , 𝛽 𝑖 , 𝜆 𝑖 . Does it sound possible to obtain ∑ 𝑀 − 1 𝑖 = 0 𝛽 𝑖 ( 𝑡 𝑖 / 𝜆 𝑖 ) (equalization  + despreading) from ∑ 𝑀 − 1 𝑖 = 0 𝛽 𝑖 𝑡 𝑖 (despreading)? Here, equalization is materialized by 𝑒 𝑖 = 𝑡 𝑖 / 𝜆 𝑖 and the despreading operation by ∑ 𝑀 − 1 𝑖 = 0 𝛽 𝑖 𝑒 𝑖 . The answer is in general (obviously) NO, except if all the 𝜆 𝑖 are the same, that is, 𝜆 𝑖 = 𝜆 𝑗 = 𝜆 . 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 ℎ 𝑛 , 𝑖 the single channel coefficient between the transmit antenna 𝑖 and the single receive antenna at time instant 𝑛 , the despreaded signal is given by: 𝑧 ( 𝑐 ) 𝑛 0 , 𝑢 0 = ℎ 𝑛 0 , 𝑖 𝑑 ( 𝑐 ) 𝑛 0 , 𝑢 0 , 𝑖 , ( 3 3 ) where 𝑑 ( 𝑐 ) 𝑛 0 , 𝑢 0 , 𝑖 is the complex data of user 𝑢 0 being transmitted at time instant 𝑛 0 by antenna 𝑖 . 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 𝑢 data, denoting by 𝑠 𝑘 , 𝑢 the main stream of complex data for user 𝑢 , we have a t t i m e 2 𝑘 , 𝑑 ( 𝑐 ) 2 𝑘 , 𝑢 , 0 = 𝑠 2 𝑘 , 𝑢 √ 2 𝑑 ( 𝑐 ) 2 𝑘 , 𝑢 , 1 = 𝑠 2 𝑘 + 1 , 𝑢 √ 2 , a t t i m e 2 𝑘 + 1 , 𝑑 ( 𝑐 ) 2 𝑘 + 1 , 𝑢 , 0 = − 𝑠 2 𝑘 + 1 , 𝑢 ∗ √ 2 𝑑 ( 𝑐 ) 2 𝑘 + 1 , 𝑢 , 1 = 𝑠 ∗