 Research
 Open Access
Unified tensor model for spacefrequency spreadingmultiplexing (SFSM) MIMO communication systems
 André LF de Almeida^{1}Email author and
 Gérard Favier^{2}
https://doi.org/10.1186/16876180201348
© de Almeida and Favier; licensee Springer. 2013
 Received: 20 June 2012
 Accepted: 15 February 2013
 Published: 15 March 2013
Abstract
This paper presents a unified tensor model for space–frequency spreadingmultiplexing (SFSM) multipleinput multipleoutput (MIMO) wireless communication systems that combine space and frequencydomain spreadings, followed by a space–frequency multiplexing. Spreading across space (transmit antennas) and frequency (subcarriers) adds resilience against deep channel fades and provides space and frequency diversities, while orthogonal space–frequency multiplexing enables multistream transmission. We adopt a tensorbased formulation for the proposed SFSM MIMO system that incorporates space, frequency, time, and code dimensions by means of the parallel factor model. The developed SFSM tensor model unifies the tensorial formulation of some existing multipleaccess/multicarrier MIMO signaling schemes as special cases, while revealing interesting tradeoffs due to combined space, frequency, and time diversities which are of practical relevance for joint symbolchannelcode estimation. The performance of the proposed SFSM MIMO system using either a zero forcing receiver or a semiblind tensorbased receiver is illustrated by means of computer simulation results under realistic channel and system parameters.
Keywords
 Blind receiver
 MIMO–OFDM communications
 Parallel factor analysis
 Space–frequency spreadingmultiplexing
 Tensor modeling
1 Introduction
Wireless communication systems employing multiple antennas at both ends of the link, commonly known as multipleinput multipleoutput (MIMO) systems, are being considered as one of the key technologies to be deployed in current and upcoming wireless communication standards [1]. In this context, the integration of multipleantenna systems with codedivision multipleaccess (CDMA) transmission and/or orthogonal frequency division multiplexing (OFDM) has also been the subject of several works over the past few years [2–4].
Different combinations of OFDM and CDMA have been reported in a number of works. Multicarrier (MC)CDMA performs spreading of the information symbols across the different subcarriers [5, 6], but suffers from limited frequency diversity gains like conventional CDMA systems. MC directsequence (MCDS)CDMA differs from MCCDMA by performing the spreading operation in the timedomain at each subcarrier [7]. For combating frequencyselective fading, MCDSCDMA requires forward errorcorrection coding and frequencydomain interleaving. In [8], a hybrid of MCCDMA and OFDM systems with orthogonal transmission in the frequencydomain was proposed, which ensures interferencefree transmission/reception regardless of the multipath channel profile. A related approach, called multicarrier blockspread (MCBS)CDMA, was introduced in [9] by capitalizing on redundant block spreading and frequencydomain linear precoding to preserve orthogonal multipleaccessing and to enable full multipath diversity gains. The receiver is based on a lowcomplexity singleuser equalization.
By introducing the spatial dimension at the transmit processing, jointly with time and/or frequency dimensions, a number of different space–frequency MIMO transceivers were proposed to enable orthogonal multipleaccess in multiuser systems combining OFDM and CDMA techniques. A spread spectrumbased transmission framework was proposed in [10], therein called multicarrier spread space spectrum multiple access (MCSSSMA), with the idea of fully spreading each user symbol over space, time, and frequency. MCSSSMA is a generalization of its singlecarrier counterpart proposed in [11]. Despite the achieved spectral efficiency gains, the design of [10] was restricted to the case where the number of transmit and receive antennas is equal to the spreading gain. In [12], space–time–frequency spreading was proposed for MCCDMA based on the concatenation of a space–time spreading code with a frequencydomain spreading code.
A common characteristic of all these works is the assumption of perfect channel knowledge at the receiver. When the channel is not known, as it is the case in practice, the receiver design is generally based on suboptimum (linear or nonlinear) filtering/equalization/signal separation structures that use training sequences for channel acquisition and tracking, before decoding the transmitted data. However, practical limitations such as the receiver complexity and the training sequence overhead (which implies a reduction of the information rate) may be too restrictive and prohibitive in some cases.
Recently, tensor modeling has successfully been applied to the design of MIMO transceivers based on spatial multiplexing and/or space–time coding [13–19]. Relying on the use of spreading codes, the common feature of these works is the fact that the received signal can be modeled as a thirdorder tensor, the dimensions of which are associated with space, time, and code diversities [20]. Due to the uniqueness properties of tensor models, these tensorbased MIMO–CDMA transceivers afford blind multiuser detection and channel estimation under more relaxed conditions compared with conventional matrixbased receivers. The approach of [13] relies on pure spatial multiplexing by means of a parallel factor (PARAFAC) model [21]. The work of [14] deals with a multipleaccess MIMO antenna system relying on a block tensor model [22]. In [15], a constrained “blockstructured” PARAFAC model is proposed for allowing multiuser space–time spreading in the uplink. The multiuser downlink case is treated in [16]. More general tensorbased space–time spreading and multiplexing structures were also proposed relying on the constrained factor (CONFAC) model [17, 18] and on PARATUCKtype models [19, 23].
In this article, we present a unified tensor model for space–frequency spreadingmultiplexing (SFSM) MIMO wireless communication systems combining both space and frequency spreadings along with a space–frequency multiplexing. On one hand, spreading across space (transmit antennas) and frequency (subcarriers) potentially provides robustness against frequencyselective fading and channel illconditioning while providing transmit diversity gains. On the other hand, an orthogonal space–frequency multiplexing enables interferencefree multistream transmission. For this system, we adopt a tensorial formulation of the transmitted and received signals that jointly incorporates space, frequency, time, and code dimensions by means of a PARAFAC tensor model. From this tensorial formulation, we show how several existing multipleantenna CDMAbased systems can be derived by making appropriate simplifications on the unified tensor model structure.
We also address the problem of joint symbolchannelcode estimation for the proposed system by capitalizing on the uniqueness properties of the PARAFAC model. By exploiting the space, time, frequency, and code diversities inherent to the unified SFSM tensor model, we obtain new results providing useful bounds on the required number of transmit and receive antennas, subcarriers, and spreading length for ensuring a unique recovery of users’ symbols, channels, and codes. A performance evaluation of the SFSM MIMO system is also carried out considering a zero forcing (ZF) receiver and a semiblind alternating least squares (ALS) receiver that only requires a single pilot symbol per transmitted data stream in order to remove the scaling factor introduced by the estimation process.
The remainder of this article is organized as follows. In Section 2, the main building blocks of the SFSM transmitter are detailed and the transmitted signal model is formulated. In Section 3, we present the received signal model and also derive the proposed unifying tensor model and its special cases. A ZF receiver with joint blockdecoding and equalization is formulated in Section 4. Section 5 is dedicated to the problem of joint symbolchannelcode estimation for the unified SFSM MIMO system, where bounds on the required numbers of transmit/receive antennas, subcarriers, spreading length, and the number of symbols per data stream are provided. The semiblind ALS receiver is also presented in this section. In Section 6, the performance of the SFSM MIMO system is evaluated by means of computer simulations under different system parameter settings. The article is concluded in Section 7.
where ∗ denotes the Hadamard (elementwise) matrix product.
2 SFSM: transmitted signal model
2.1 Spacedomain spreading
and represents the n th space spread symbol of the r th data stream transmitted by the M _{ t }th antenna.
As shown in [24], the Vandermonde structure minimizes an upper bound of the pairwise error probability at high signaltonoise ratios (SNRs). Moreover, this structure yields a good coding gain and makes the transmission more robust to illconditioned/rankdeficient MIMO channels [25].
2.2 Frequencydomain spreading
The second operation consists in jointly spreading and coding each component ${\stackrel{\u0304}{s}}_{{m}_{t},n,r}$ in the frequencydomain. This operation is implemented by means of linear precoding, which adds transmit redundancy in the frequencydomain before the multicarrier modulation. Each data symbol is transmitted simultaneously (in parallel) on different subcarriers in a way similar to an MCCDMA system with frequencydomain spreading [26]. In addition to provide frequency diversity gains, frequencydomain spreading adds resilience to symbol detection even in the presence of a deep channel fade over one or more subcarrier channels.
which is the (f,m _{ t },n,r)th element of the fourthorder tensor $\stackrel{~}{S}\in {\u2102}^{F\times {M}_{t}\times N\times R}$ representing the space–frequency spread signal s _{ n,r } associated with the n th symbol period and r th data stream.
The reason for choosing the Vandermonde structure for the frequency spreading matrix follows that of the space spreading matrix. Some designs for Θ have been reported in the literature (we refer the interested reader to [27] for further details).
Note that spreading in the spacedomain consists in multiplying the symbol s _{ n,r } by a complex code that depends on the transmit antenna number M _{ t } while spreading in the frequencydomain results in a multiplication of the same symbol by a complex code that depends on the frequency number f, as shown in (6).
2.3 Spacefrequency multiplexing
The third operation of the SFSM transmitter consists in a multiplexing of the R space–frequency spread symbols. Using conventional direct sequence (DS) spreading, each space–frequency symbol ${\stackrel{~}{s}}_{f,{m}_{t},n,r}$ is spread by a factor P using a specific spreading code. Due to spectrum spreading at the subcarrier level, each subcarrier signal constitutes a DS spread signal. Consequently, the frequency spectrum associated with each subcarrier is allowed to overlap in order to achieve high spectral efficiency.
2.4 Multicarrier modulation
where ξ _{ j,k }=[Ξ]_{ j,k } and ${x}_{j,{m}_{t},n,p}$ is a typical element of the transmitted signal tensor $\mathcal{X}\in {\u2102}^{J\times {M}_{t}\times N\times P}$.
3 SFSM: received signal model
where ${\stackrel{\u0307}{h}}_{j,{m}_{r},{m}_{t}}$ is an element of the tensor $\stackrel{\u0307}{\mathcal{\mathscr{H}}}\in {\u2102}^{J\times {M}_{r}\times {M}_{t}}$, ${\stackrel{\u0307}{\mathbf{H}}}_{\phantom{\rule{0.3em}{0ex}}\xb7\phantom{\rule{0.3em}{0ex}}{m}_{r}{m}_{t}}\in {\u2102}^{J\times 1}$ being the impulse response of the channel linking the M _{ r }th receive antenna to the M _{ t }th transmit antenna.
In the next section, we show how the tensor model (14) satisfied by the received signals can be cast into a PARAFAC model by contracting the first two modes of the transmitted and received signal tensors. Our motivation behind the use of PARAFAC modeling comes from the possibility of studying identifiability by resorting to the wellknown results available in the literature.
3.1 PARAFAC model formulation
where ${a}_{{i}_{n},r}^{\left(n\right)}$ is the entry (i _{ n },r) of the n th mode matrix factor ${\mathbf{A}}^{\left(n\right)}\in {\u2102}^{{I}_{n}\times R}$, n=1,2,3. When R is minimal, it is called the rank of $\mathcal{X}$.
which corresponds to a thirdorder PARAFAC model for the transmitted signal tensor $\stackrel{\u0304}{\mathcal{Z}}\in {\u2102}^{M\times N\times P}$, with matrix factors (U,S,C).
Note that the contracted received signal tensor $\stackrel{\u0304}{\mathcal{Y}}\in {\u2102}^{I\times N\times P}$ given by (19) follows a PARAFAC model with matrix factors ($\stackrel{\u0304}{\mathbf{H}}(\mathit{\Theta}\u25c7\mathit{\Omega})$, S, C). In fact, models (17) and (19) for the transmitted and received signal tensors, respectively, differ only in their firstmode matrix factors, which are related by (20).
where ${\left[{\stackrel{\u0304}{\mathbf{Y}}}_{1}\right]}_{(p1)I+i,n}={\left[{\stackrel{\u0304}{\mathbf{Y}}}_{2}\right]}_{(i1)N+n,p}={\left[{\stackrel{\u0304}{\mathbf{Y}}}_{3}\right]}_{(n1)P+p,i}={\stackrel{\u0304}{y}}_{i,n,p}$.
3.2 Multiuser case
where $\mathbf{S}=[{\mathbf{S}}^{\left(1\right)},\dots ,{\mathbf{S}}^{\left(Q\right)}]\in {\u2102}^{N\times R}$, $\mathbf{C}=[{\mathbf{C}}^{\left(1\right)},\dots ,{\mathbf{C}}^{\left(Q\right)}]\in {\u2102}^{P\times R}$, $\mathbf{G}=[{\mathbf{G}}^{\left(1\right)},\dots ,{\mathbf{G}}^{\left(Q\right)}]\in {\u2102}^{I\times R}$. Therefore, the PARAFAC model (17) is equally valid for the multiuser case by simply interpreting its factor matrices as blockmatrices.
3.3 Special cases
The proposed structured PARAFAC model (19) of the received signal is general in the sense that it incorporates several existing multipleaccess/multipleantenna signaling schemes. By making appropriate assumptions, the proposed model can gradually be simplified, so that we obtain different tensorbased transceiver models as special cases:

Space–time spreading CDMA (STSCDMA): For F=1, which corresponds to a singlecarrier transmission over a flatfading channel, we can abandon the frequencydependent index and eliminate the frequency spreading matrix $\mathit{\Theta}={\mathbf{1}}_{R}^{T}$, so that $\mathbf{G}=\stackrel{\u0304}{\mathbf{H}}\mathit{\Omega}$. Thus, the trilinear model (21) reduces to classical space–time spreading using multiple spreading codes and can be written as:${\stackrel{\u0304}{\mathbf{Y}}}_{1}=(\mathbf{C}\u25c7\stackrel{\u0304}{\mathbf{H}}\mathit{\Omega}){\mathbf{S}}^{T}\in {\u2102}^{P{M}_{r}\times N}.$(26)

This model is valid for modeling the multipleantenna transmission systems proposed in [25, 29].

Spatial multiplexing CDMA (SMCDMA): In SMCDMA systems, the space spreading operation (which is responsible for spreading R data streams across M _{ t } transmit antennas) is eliminated. In other words, each data stream is transmitted by a different transmit antenna. Still considering F=1, in this case we have R=M _{ t }, $\mathit{\Omega}={\mathbf{I}}_{{M}_{t}}$, and $\mathit{\Theta}={\mathbf{1}}_{R}^{T}$, which implies $\mathbf{G}=\stackrel{\u0304}{\mathbf{H}}$, and model (21) becomes:${\stackrel{\u0304}{\mathbf{Y}}}_{1}=(\mathbf{C}\u25c7\stackrel{\u0304}{\mathbf{H}}){\mathbf{S}}^{T}\in {\u2102}^{P{M}_{r}\times N}.$(27)

This model covers a spatial multiplexing/multipleaccess CDMA system using a different spreading code per transmit antenna [2], and is the same as the PARAFACCDMA model proposed in the seminal paper [20]. It also coincides with the KhatriRao space–time (KRST) coding model of [13].

Multicarrier CDMA systems (MCBSCDMA /MCDSCDMA/ MCCDMA): We consider the transmission model of a MCDSCDMA system where frequencydomain spreading and orthogonal multiplexing take place (e.g. see [26, 30]). This is a singleinput singleoutput (SISO) antenna system (M _{ r }=M _{ t }=1), which means that the channel matrix $\stackrel{\u0304}{\mathbf{H}}$ reduces to an F×F diagonal matrix, and we can eliminate the space spreading matrix $\mathit{\Omega}={\mathbf{1}}_{R}^{T}$ so that $\mathbf{G}=\stackrel{\u0304}{\mathbf{H}}\mathit{\Theta}\phantom{\rule{0.3em}{0ex}}\in {\mathcal{C}}^{F\times R}$. Consequently, the general PARAFAC model (21) becomes:${\stackrel{\u0304}{\mathbf{Y}}}_{1}=(\mathbf{C}\u25c7\stackrel{\u0304}{\mathbf{H}}\mathit{\Theta}){\mathbf{S}}^{T}\in {\u2102}^{\mathit{\text{PF}}\times N}.$(28)

It is worth noting that this special model can be interpreted as the tensorial formulation of the MCBSCDMA system proposed in [9]. In particular, if frequencydomain spreading is not used, we have Θ=I _{ R } so that (28) reduces to a PARAFAC model for a MCDSCDMA system with directsequence spectrum spreading at the subcarrier level [7]. In the SISO case, where $\stackrel{\u0304}{\mathbf{H}}\phantom{\rule{0.3em}{0ex}}\in {\u2102}^{F\times F}$ is diagonal, if space–frequency blockcoding is not used (P=1 and $\mathbf{C}={\mathbf{1}}_{R}^{T}$), then (28) reduces to traditional MCCDMA, and we have:${\stackrel{\u0304}{\mathbf{Y}}}_{1}=\stackrel{\u0304}{\mathbf{H}}\mathit{\Theta}{\mathbf{S}}^{T}\in {\u2102}^{F\times N}.$(29)

Conventional spatial multiplexing: This is the wellknown singleuser singlecarrier MIMO system with spatial multiplexing (such as the VBLAST system of [31]). Then, we have F=P=1, R=M _{ t }, and $\mathbf{C}=\mathit{\Theta}={\mathbf{1}}_{R}^{T}$, $\mathit{\Omega}={\mathbf{I}}_{{M}_{t}}$. In this case, the general PARAFAC model (21) simplifies to the conventional matrixbased model:${\stackrel{\u0304}{\mathbf{Y}}}_{1}=\stackrel{\u0304}{\mathbf{H}}{\mathbf{S}}^{T}\in {\u2102}^{{M}_{r}\times N}.$(30)
Equivalent tensorial formulation for different systems
Systems  ( F,M _{ t }, R,M _{ r })  Θ  Ω  Rx signal  Matrix factors 

(1,M _{ t },R,M _{ r })  ${\mathbf{1}}_{R}^{T}$  Full  $\mathcal{Y}\in {\u2102}^{{M}_{r}\times N\times P}$  $(\stackrel{\u0304}{\mathbf{H}}\mathit{\Omega},\mathbf{S},\mathbf{C})$  
(1,M _{ t },M _{ t },M _{ r })  ${\mathbf{1}}_{R}^{T}$  ${\mathbf{I}}_{{M}_{t}}$  $\mathcal{Y}\in {\u2102}^{{M}_{r}\times N\times P}$  $(\stackrel{\u0304}{\mathbf{H}},\mathbf{S},\mathbf{C})$  
MCBSCDMA [9]  (F,1,R,1)  Full  ${\mathbf{1}}_{R}^{T}$  $\mathcal{Y}\in {\u2102}^{F\times N\times P}$  $(\stackrel{\u0304}{\mathbf{H}}\mathit{\Theta},\mathbf{S},\mathbf{C})$ ^{a} 
MCDSCDMA [7]  (F,1,R,1)  I _{ R }  ${\mathbf{1}}_{R}^{T}$  $\mathcal{Y}\in {\u2102}^{F\times N\times P}$  $(\stackrel{\u0304}{\mathbf{H}},\mathbf{S},\mathbf{C})$ ^{a} 
MCCDMA^{b}[5]  (F,1,R,1)  Full  ${\mathbf{1}}_{R}^{T}$  ${\stackrel{\u0304}{\mathbf{Y}}}_{1}\in {\u2102}^{F\times N}$  $(\stackrel{\u0304}{\mathbf{H}}\mathit{\Theta},\mathbf{S},{\mathbf{1}}_{R}^{T})$ ^{a} 
Spatial multiplexing^{b}[31]  (1,M _{ t },M _{ t },M _{ r })  ${\mathbf{1}}_{R}^{T}$  ${\mathbf{I}}_{{M}_{t}}$  ${\stackrel{\u0304}{\mathbf{Y}}}_{1}\in {\u2102}^{{M}_{r}}\times N$  $(\stackrel{\u0304}{\mathbf{H}},\mathbf{S},{\mathbf{1}}_{R}^{T})$ 
Remark 1 (subcarrier grouping)
In order to reduce the complexity of the receiver, we can resort to subcarrier grouping [32, 33]. It consists in dividing the set of F subcarriers into μ nonintersecting subsets of K equispaced subcarriers, where K can be chosen equal to the number of independent multipaths. Since both F and K can be viewed as system design parameters, we choose them so that μ=F/K is an integer. Information recovery can be carried out independently within each subcarrier group at the receiver (after FFT demodulation). This lowcomplexity detection strategy will be considered later in our simulations. We have chosen to not explicitly model subcarrier grouping in order to avoid unnecessary complicated mathematical notation in the formulation of the transmitted and received signal models.
4 ZF receiver
Since $\mathbf{C}\u25c7\mathbf{G}\phantom{\rule{0.3em}{0ex}}\in {\u2102}^{\mathit{\text{PF}}{M}_{r}\times R}$ must be full columnrank to be leftinvertible, the ZF receiver requires that P M _{ r } F≥R.
From the structure of (32), we can observe that the ZF receiver does not require codeorthogonality to jointly estimate the transmitted signals. In Section 5, we propose a PARAFACbased receiver that can blindly operate, i.e. without a priori knowledge of the space–frequency MIMO channel.
4.1 Space–frequency linear combiner
5 Semiblind ALS receiver
The goal of the base station receiver is to separate the cochannel transmissions while recovering the data transmitted by each user. In our proposed SFSM MIMO system, cochannel transmissions are represented by the R data streams accessing simultaneously the space, time, and frequency channel resources. We are interested in a semiblind receiver that neither requires prior knowledge, or estimation, of channel and antenna array responses, nor relies on statistical independence between the transmitted signals. These properties are distinguishing features of the PARAFAC modeling and constitute the main motivation for using the unified tensor model.
Moreover, the proposed receiver is called semiblind in the sense that it relies only on a single pilot symbol inserted at the beginning of each data stream. This pilot symbol is used to remove the scaling factor introduced by the estimation process.
We now study the joint symbolcodechannel recovery by capitalizing on the fundamental uniqueness property of the PARAFAC model (19). This property allows to establish several practical corollaries, which provide lower bounds on the required number of transmit/receive antennas, subcarriers, symbol periods, and the spreading length for ensuring a semiblind symbolcodechannel estimation. They also clearly illustrate the underlying tradeoffs involving space, frequency, and code diversities.
The first inequality comes from the full columnrank requirement of C◇G and G◇S, while the second one comes from the full columnrank requirement of (S◇C)(Θ◇Ω)^{ T }. These necessary conditions are useful when one is interested in eliminating system configurations leading to a nonidentifiable model. We emphasize that conditions (36) do not imply model identifiability since it is not a sufficient condition.
where k _{(·)} denotes the Kruskalrank^{3} of a matrix.
We now use the fact that $\mathbf{G}=\stackrel{\u0304}{\mathbf{H}}\mathbf{U}$, with U given in (16) and consider particular cases leading to simplifications of (38) which are of practical relevance for the unified SFSM MIMO system. Interesting tradeoffs for joint symbolchannelcode estimation can explicitly be obtained.
5.1 Singlecarrier transmission (F=1)
 1.. We have $\mathbf{G}=\stackrel{\u0304}{\mathbf{H}}\mathit{\Omega}$. Assuming that $\stackrel{\u0304}{\mathbf{H}}$ is full columnrank and Ω is full rank due to its Vandermonde structure, it follows that rank(G)=rank(Ω)=min(M _{ t },R), and (38) becomes:$\underline{{M}_{r}\ge {M}_{t}}$$\text{min}({M}_{t},R)+\text{min}(N,R)+\text{min}(P,R)\ge 2R+2.$(39)
 2.. In this case Ω is full rowrank due to its Vandermonde structure. Assuming that $\stackrel{\u0304}{\mathbf{H}}$ is modeled by i.i.d entries (which corresponds to scatteringrich propagation) and thus is full rank, it follows that $\text{rank}\left(\mathbf{G}\right)=\text{rank}\left(\stackrel{\u0304}{\mathbf{H}}\right)=\text{min}({M}_{r},{M}_{t})$, which implies:$\underline{R\ge {M}_{t}}$$\text{min}({M}_{r},{M}_{t})+\text{min}(N,R)+\text{min}(P,R)\ge 2R+2.$(40)
and can be interpreted in the following way.
Corollary 1
For M _{ r }≥M _{ t }, spreading across M _{ t }=2 transmit antennas is sufficient for joint symbolcodechannel recovery, regardless of the number R≥2 of data streams, for large enough number of symbols and code spreading factors.
Corollary 2
For R≥M _{ t }, M _{ r }=2 receive antennas are sufficient for joint symbolcodechannel recovery, regardless of the number M _{ t }≥2 of transmit antennas, for large enough number of symbols and code spreading factors.
5.2 Singleantenna transmission (M _{ t }=1)
and we obtain:
Corollary 3
For M _{ t }=1, spreading across F=2 subcarriers is sufficient for joint symbolcodechannel recovery, regardless of the number R≥2 of data streams, for large enough number of symbols and code spreading factors.
Note that this condition is independent on the number M _{ r } of receive antennas, which means that joint symbolcodechannel recovery is achieved even with one receive antenna. This clearly illustrates the tradeoff between frequency diversity and space diversity at the receiver, which is inherent to this trilinear PARAFAC model.
5.3 Small spreading lengths (P<R)
The simplified condition (45) results in the following corollary:
Corollary 4
For M _{ r }≥M _{ t }≥R, spreading across P=2 chips is sufficient for joint symbolcodechannel recovery, regardless of the number R≥2 of data streams and receive antennas.
This condition establishes a tradeoff between code diversity (spreading length) and space diversity afforded by the proposed trilinear PARAFAC modeling.
Remark 2
When subcarrier grouping is used, receiver processing is parallelized into μ independent detection “layers”, each one associated with K=F/μ subcarriers. For this reason, identifiability can be studied groupwise (i.e., what matters for identifiability is K and not F) since the results obtained for a given subcarrier group are equally valid for all the other groups.
Note that this condition is more relaxed than Kruskal’s condition (37). In connection with [36], it is shown in [37] that this condition is valid if G and C are randomly sampled from an (F M _{ r }+P)Rdimensional continuous distribution. In a recent work [38], a mathematical proof is provided to the case of nonrandom G and C matrices.
5.4 Receiver algorithm
where ${\stackrel{~}{\mathbf{Y}}}_{i}={\stackrel{\u0304}{\mathbf{Y}}}_{i}+{\mathbf{B}}_{i}$, i=1,2,3, is the noisy version of ${\stackrel{\u0304}{\mathbf{Y}}}_{i}$, and B _{ i } is a matrix representing the additive complexvalued white Gaussian noise.^{4} We can rely on the knowledge of the space and frequency spreading matrices Θ and Ω to directly obtain an LS estimate of $\hat{\stackrel{\u0304}{\mathbf{H}}}$, provided that the second inequality of (36) is satisfied, i.e., if R≥F M _{ t }. From (51), and using (20), we have ${\hat{\stackrel{\u0304}{\mathbf{H}}}}^{T}={\left[(\mathbf{S}\u25c7\mathbf{C}){(\mathit{\Theta}\u25c7\mathit{\Omega})}^{T}\right]}^{\u2021}{\stackrel{~}{\mathbf{Y}}}_{3}$. On the other hand, if R<F M _{ t }, a unique estimation of $\stackrel{\u0304}{\mathbf{H}}$ is not guaranteed, although we can still estimate S, C and G from (49), (50), and (51), respectively.
The ALS algorithm always monotonically converges to (at least) a local minimum. Convergence to the global minimum can sometimes be slow if all the matrix factors $\stackrel{\u0304}{\mathbf{H}}$, S, and C are unknown. Several alternative algorithms have been proposed in the literature to alleviate the slow convergence problems caused by a random initialization of the algorithm. For instance, an eigenanalysis solution based on compression of the tensor dimensions can be used [20]. The study of [37] proposes a generalization of the eigenanalysis solution by means of simultaneous matrix diagonalization. The convergence can also be improved by means of enhanced line search [41, 42] or, using a nonlinear optimization algorithm such as the Levenberg–Marquardt algorithm [43]. The ALS algorithm rapidly converges when one of the three matrix factors of the model is known. This is typically the case in the SFSM MIMO system when relying on the knowledge of the code and spreading matrices (C,Ω,Θ).
where ${D}_{1}\left(\hat{\mathbf{S}}\right)$ is the diagonal matrix formed from the first row of $\hat{\mathbf{S}}$.
In principle, the ALS receiver is capable of processing a higher number of users as long as condition (38) is satisfied. Regarding the computation complexity, three matrix inverses are performed at each iteration of the algorithm. The asymptotic complexity is therefore O(R ^{3}) per iteration. Consequently, a joint detection of a very large number of users can be prohibitive. This is generally a common limitation of multiuser detection receivers. Note that the computational complexity can be reduced if users’ codes are mutually orthogonal. In this case, their symbol matrices can be estimated separately using (34).
6 Simulation results
System parameters
Chip rate  4.096 ×10^{6} cps 
Number of subcarriers (F)  64 
Number of subcarriers per group (K)  2 or 4 
Number of subcarriers groups (μ)  32 or 16 
CP length  5 (Chan. A)/20 (Chan. B) 
Number of symbols per data stream (N)  10 
Modulation  QPSK 
where $\mathcal{\mathcal{B}}\in {\u2102}^{F\times {M}_{r}\times N\times P}$ is the additive noise tensor, whose entries are circularly symmetric complex Gaussian random variables. Note that this SNR measure takes all the received signal dimensions into account, i.e., the number F of subcarriers, the number M _{ r } of receive antennas, the number N of symbol periods, and the spreading length P. At each run, the additive noise power is generated according to this SNR measure. The BER curves represent the performance averaged over the R transmitted data streams and 1,000 independent Monte Carlo runs.
Parameters of the ITU pedestrian channel A
Path  Excess delay (ns)  Average relative power (dB) 

1  0  0 
2  110  −9.7 
3  190  −19.2 
4  410  −22.8 
Parameters of the ITU pedestrian channel B
Path  Excess delay (ns)  Average relative power (dB) 

1  0  0 
2  200  −0.9 
3  800  −4.9 
4  1200  −8.0 
5  2300  −7.8 
6  3700  −23.9 
6.1 Semiblind ALS versus ZF receivers
 1.
To compare the performance of the semiblind ALS receiver with that of the perfect ZF receiver;
 2.
To compare the SFSM MIMO system with other CDMA–MIMO systems when ALS estimation is used;
 3.
To evaluate the channel estimation accuracy as a function of the SNR.
All the simulations were performed assuming F=64 subcarriers divided into groups of K=2 or K=4 subcarriers.
6.2 Performance for different system loads
6.3 Comparison with the MCDSCDMA system
6.4 Comparison with the SSSMA system
6.5 Channel estimation performance
7 Conclusion
We have proposed a unified tensor model for MIMO communication systems with SFSM. The proposed model unifies several existing multipleaccess/multipleantenna communication systems. We have shown that the received signal can be formulated as a trilinear PARAFAC model, and capitalizing on its uniqueness property we have put in evidence lower bounds on the design parameters (number of transmit/receive antennas, subcarriers, symbols per data stream, and spreading length) for a joint symbolcodechannel recovery. The obtained conditions help the understanding of the existing tradeoffs involving space, frequency, and code diversities that are inherent to the SFSM MIMO system. The performance of the proposed receiver using a semiblind ALS algorithm has been illustrated by means of computer simulations under realistic channel models and system parameters, and a comparison with other multipleantenna CDMAbased systems has been made. Perspectives of this work include an investigation of the impact of different transmit antenna, spreading code, and subcarrier allocation schemes on the design and performance of the proposed tensorbased receiver. We believe that these features could be integrated into the SFSM system by modeling the received signals using a CONFAC tensor model [18]. In this case, identifiability can be investigated using the recently established results on the partial uniqueness of constrained tensor decompositions [46, 47]. The impact of nonperfect users’ synchronization on the receiver performance is also a subject for a future work.
Endnotes
^{1} For notational convenience, we omit the noise terms in the following developments. They will be added later, when the receiver algorithm is presented.
^{2} This means that any alternative triplet $\{\stackrel{~}{\mathbf{G}},\stackrel{~}{\mathbf{S}},\stackrel{~}{\mathbf{C}}\}$ satisfying model (19) is related to the true triplet {G,S,C} by the following equalities: $\stackrel{~}{\mathbf{G}}=\mathbf{G}\mathit{\Pi}{\mathbf{\Delta}}_{1}$, $\stackrel{~}{\mathbf{S}}=\mathbf{S}\mathit{\Pi}{\mathbf{\Delta}}_{2}$, $\stackrel{~}{\mathbf{C}}=\mathbf{C}\mathit{\Pi}{\mathbf{\Delta}}_{3}$, where Π is a permutation matrix and Δ _{ i }, i=1,2,3, are diagonal (scaling) matrices such that Δ _{1} Δ _{2} Δ _{3}=I _{ R }.
^{3} The Kruskalrank of A is equal to κ if every subset of κ columns of A is linearly independent.
^{4} See [20, 40] for further details about the ALS algorithm.
Declarations
Acknowledgements
This study was supported by the Ericsson Research and Development Centre, Ericsson Telecommunications S.A., Brazil. André L. F. de Almeida is partially supported by the CNPq and FUNCAP funding agencies.
Authors’ Affiliations
References
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