The Alamouti Scheme with CDMA-OFDM/OQAM
© Chrislin Lélé et al. 2010
Received: 23 June 2009
Accepted: 29 December 2009
Published: 18 January 2010
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 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.
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 . 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 ( ) 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, , 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) , also named oversampled OFDM in , 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  is a pioneering work that illustrates this possibility. Later on an efficient Discrete Fourier Transform (DFT) implementation of the Saltzberg system , named Orthogonally Quadrature Amplitude Modulation (O-QAM), has been proposed by Hirosaki . To the best of our knowledge, the acronym OFDM/OQAM, where OQAM now corresponds to Offset QAM, appeared for the first time in . In  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) , is established in .
It is now recognized in a large number of applications, with cognitive radio being the most recent and important one , 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 , 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  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)  or estimating it at the receiver side (RX) , 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  complicates the RX and introduces a processing delay. The pseudo-Alamouti scheme recently introduced in  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  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 , 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
with the set of integers, an even number of subcarriers, the subcarrier spacing, the prototype function assumed here to be a real-valued and even function of time, and an additional phase term such that where can be chosen arbitrarily. The transmitted data symbols are real-valued. They are obtained from a -QAM constellation, taking the real and imaginary parts of these complex-valued symbols of duration , where denotes the time offset between the two parts [2, 6, 7, 9].
It is well-known  that to satisfy the orthogonality condition (2), the prototype filter should be chosen such that if and .
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 .
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 , but keeping in mind that this modulator corresponds to an FBMC modulator as shown in Figure 1.
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
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 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 .
where 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 .
2.3. OFDM/OQAM with Alamouti Scheme
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 distortion-free 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  significantly increases the RX complexity, while the second one  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
In this section we summarize the results obtained, assuming a distortion-free channel, in  and  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
3.1.2. Interference Cancellation
It is shown in  that if spreading codes are properly selected then the interference is cancelled. The W-H 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.
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  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.
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 , , . 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
3.2.3. Performance Evaluation
flat fading channel (one single Rayleigh coefficient for all 128 subcarriers);
the IOTA prototype filter with length 512,
zero forcing one tap equalization for both transmission schemes,
no channel coding.
4. Alamouti and CDMA-OFDM/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 real-valued symmetric function and also that the W-H codes are selected using the procedure recalled in Section 3.1.2.
4.1. CDMA-OFDM/OQAM with Spreading in the Time Domain
We now propose to consider the transmission of complex data, denoted , using well chosen W-H codes. In order to establish the theoretical features of this complex CDMA-OFDM/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 W-H codes are supposed to be a power of 2, that is, with an integer.
W-H codes satisfy the set of mathematical properties that are proved in .
Using these results, (47) can be proved straighforwardly.
4.2. Alamouti with CDMA-OFDM/OQAM with Spreading in Time
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 :
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
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.
power profile (in dB): 0, -6, -9, -12,
delay profile (in samples): 0, 1, 2, 3,
GI for CP-OFDM: 5 samples,
and the 7-path by
power profile (in dB): 0, -6, -9, -12, -16, -20, -22,
delay profile (in samples): 0, 1, 2, 3, 5, 7, 8,
GI for CP-OFDM: 9 samples;
time invariant channel (no Doppler),
the IOTA prototype filter of length 512,
spreading codes of length 32, corresponding to the frame duration (32 complex OQAM symbols),
zero forcing, one tap equalization,
no channel coding.
The two strategies perform the same until a BER of or for the 4 and 7-path 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 CDMA-OFDM/OQAM strategy 2 with the Alamouti CP-OFDM, 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 7-path channels, respectively.
In this paper, we showed that the well-known 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 CDMA-OFDM/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 CDMA-OFDM/OQAM, we elaborate two strategies for implementing the MIMO space-time 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 CP-OFDM. It was found that, under some channel hypothesis, the combination of Alamouti with complex CDMA-OFDM/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.
The authors would like to thank the reviewers and Professor Farhang-Boroujeny for their careful reading of our manuscript and for their helpful suggestions. This work was partially supported by the European ICT-2008-211887 project PHYDYAS.
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