 Research
 Open Access
Channelaware adaptive receivers for linearly precoded MIMOOFDM systems with imperfect CSIT
 Felip RieraPalou^{1}Email author and
 Guillem Femenias^{1}
https://doi.org/10.1186/168714992013240
© RieraPalou and Femenias; licensee Springer. 2013
 Received: 14 May 2013
 Accepted: 24 September 2013
 Published: 7 October 2013
Abstract
Within the context of linearly precoded MIMOOFDM (combination of multiple antenna techniques with multicarrier transmission schemes such as orthogonal frequency division multiplexing) systems with multiplestream transmission, maximum likelihood detection (MLD) has been shown to offer large performance gains when compared to an alllinear setup (i.e., linear transmitter/receiver) when either perfect or imperfect channel state information at the transmitter (CSIT) is available. Unfortunately, these gains come at the cost of a higher complexity. In particular, the increase in computational cost is more significant when the receiver is designed to operate with soft information and even more dramatic when, in order to optimise error rate performance, iterative decoding is allowed. In order to exploit the best features of each detection technique, this paper proposes a method to selectively choose the detection strategy (ML or linear) for each individual subcarrier as a function of the instantaneous channel conditions and CSIT accuracy. Numerical results show that a cautious and selective use of ML detection substantially reduces complexity while still reaping most of the performance advantage.
Keywords
 Orthogonal Frequency Division Multiplex
 Power Allocation
 Minimum Mean Square Error
 Orthogonal Frequency Division Multiplex Symbol
 Soft Information
1 Introduction
The combination of multiple antenna techniques (socalled MIMO) with multicarrier transmission schemes such as orthogonal frequency division multiplexing (OFDM), the socalled MIMOOFDM architecture, is now at the heart of most stateoftheart wireless systems and future standards [1–3]. In this context, techniques that exploit the availability of channel state information at the transmitter (CSIT) have been intensively researched (see [4, 5] for a review). It is well known that the capacityachieving strategy when perfect CSIT is available is to precancel interference among simultaneously transmitted streams, a scheme usually referred to as dirty paper coding (DPC) [6]; however, its high computational cost motivates the need for simpler strategies. Palomar et al., in their landmark paper [7], introduced a framework for the optimisation of MIMOOFDM systems with CSIT based on linear processing at the transmitter and receiver. The proposed scheme defines transmit and receiver filters that are based on the singular value decomposition (SVD) of the whitened channel matrix and performs a distribution of the available power among the different transmit modes using waterfilling in accordance with various performance metrics. Further insight on this architecture was provided in [8, 9], where the diversity order performance was analysed for single and multiplestream configurations (i.e. spatial division multiplexing beamforming). These studies showed that such schemes lose diversity when increasing the number of transmitted streams as performance becomes dominated by the worst employed spatial transmission mode. Very recently, it has been shown in [10] that full diversity can be restored by incorporating a linear transformation at transmission spreading the symbols to be transmitted over the available spatial modes. Unfortunately, this diversity advantage comes at the cost of having to rely on joint maximum likelihood detection (MLD) at the receiver.
Most of these results assumed perfect CSIT, which is a rather optimistic hypothesis in practical deployments. Channel feedback delay and quantization noise are typical impairments affecting the quality of CSIT, whose effects should be accounted for. To this end, [11] incorporated channel knowledge imperfections in the design of a linear transmitter/receiver architecture that, as the CSIT approaches perfection, converges towards the solution of [7]. A related work by Sengul et al. [12] proposes a codebook construction methodology based on a Lloyd quantizer design that aims at the improvement of the robustness against imperfect CSIT in linearly precoded bitinterleaved coded modulation (BICM) systems while still relying on linear filters at the receiver side. In [13], precoding strategies combined with forward error correction were considered but again limiting the context to that of linear detectors. Remarkably, it should be noted that, under imperfect CSIT, the optimisation of error rate metrics requires MLDbased reception.
The use of MLD in combination with linear precoding has been extensively studied in [14] and [15] under perfect and imperfect CSIT, showing that large reductions in the bit/packet error rate (BER/PER) are possible at the cost of an increased receiver complexity as even smart implementations (i.e. sphere decoding [16]) are computationally demanding at low signaltonoise ratios (SNRs), where practical systems usually operate [17]. To address this downside, this paper proposes the selective and careful application of MLD only under very specific conditions, which depend on the specific channel realisation and CSIT accuracy, while otherwise relying on linear detection. The introduced technique is shown to be effective with various architectures, namely hard, soft and iteratively decoded receivers. This scheme is specially appropriate for scenarios where the channel and/or CSIT accuracy may vary widely from packet to packet. As an illustrative example of this type of scenario, this paper considers wireless local area networks (WLANs) based on the IEEE 802.11n standard [18], whose multipleaccess policy based on carrier sense multiple access with collision avoidance (CSMA/CA) causes substantial variations in the accuracy of the available channel information at the transmitter. It is worth mentioning that a related idea, but in the context of MIMO systems without CSIT and restricted to the 2×2 MIMO setup, was introduced in [19], where the detection strategy selection was based on the condition number of the channel correlation matrix resulting in the utilisation of linear and ML detection for wellconditioned and illconditioned channels, respectively.
The rest of the paper is organised as follows: Section 2 introduces the system model under consideration including a description of the assumptions regarding the channel model and CSIT accuracy. Section 3 begins by reviewing the two classic detectors at hand, minimum mean square error (MMSE) and MLD, within the context of the considered scenario and subsequently introduces the channelaware adaptive detector. In Section 4, the adaptive detector concept is revisited within the framework of soft and iterative detection strategies. Numerical results are presented in Section 5 illustrating the benefits of the channelaware adaptive detector. Finally, the main outcomes of this work are recapped in Section 6.
This introduction concludes with a brief notational remark. Vectors and matrices are denoted by bold lower case and bold upper case letters, respectively. The superscript (·)^{ H } denotes the complex transpose (Hermitian) of the corresponding variable. The symbol I_{ k } denotes the kdimensional identity matrix, whereas $\mathcal{D}(\mathit{x})$ is used to represent a (block) diagonal matrix having x at its main (block) diagonal and [A]_{i,j} serves to indicate the (i,j)element of matrix A.
2 System model
A MIMOOFDM architecture is considered where the transmitter and receiver are equipped with N_{ T } and N_{ R } antennas, respectively, which are capable of simultaneously transmitting N_{ s }≤ min(N_{ T },N_{ R }) data streams. The available system bandwidth is exploited by means of N_{ c } subcarriers out of which N_{ d } are used to carry data and N_{ p } are destined to pilot signals and guard bands.
2.1 Transmitter processing
where W[ q,n], with dimensions N_{ T }×N_{ s }, represents the precoding matrix and $\mathbf{s}[\phantom{\rule{0.3em}{0ex}}q,n]\phantom{\rule{2.77626pt}{0ex}}={\left({s}_{1}[\phantom{\rule{0.3em}{0ex}}q,n]\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}{s}_{{N}_{s}}[\phantom{\rule{0.3em}{0ex}}q,n]\right)}^{T}$, with s_{ i }[ q,n] denoting the symbol corresponding to the i th stream to be transmitted on the q th subcarrier at time instant n. Finally, the precoded symbols are supplied to an OFDM modulator consisting of an IFFT plus the addition of a cyclic prefix (CP).
2.2 Channel modelling
where an arbitrary entry h_{i,j}[ q,n] (1≤i≤N_{ R }, 1≤j≤N_{ T }) corresponds to the frequency response on subcarrier q of the channel linking Tx antenna j and Rx antenna i. It is assumed that the entries of H[ q,n] are uncorrelated^{a} (i.e. Tx/Rx antennas are sufficiently spaced).
where $\stackrel{\u0304}{\mathbf{H}}[\phantom{\rule{0.3em}{0ex}}q,n]$ represents the channel mean known at the transmitter (estimated channel), Δ[ q,n] denotes the channel estimation noise whose entries are $\mathcal{C}\mathcal{N}(0,1)$, and ρ[ n]∈[0,1] can be a packetdependent random variable effectively modelling the CSIT accuracy for the current packet, which is also known at the receiver side.
2.3 Reception equation
where A[ q] =H[ q]W[ q], and the N_{ R }×1 vector υ[ q] corresponds to the noise samples affecting the q th subcarrier, which are assumed to be i.i.d. and drawn from a zeromean complex Gaussian distribution with variance ${\sigma}_{v}^{2}$. It is assumed that, on average, each subcarrier has unit energy available to transmit N_{ s } symbols and that the channel frequency response is normalised so that the average signaltonoise ratio per subcarrier can be defined as E_{ s }/N_{0}=1/(N_{ s }σ^{2}).
3 Channelaware robust detection
where U[ q] has as columns the eigenvectors of ${\stackrel{\u0304}{\mathbf{R}}}_{q}={\rho}^{2}\stackrel{\u0304}{\mathbf{H}}{[\phantom{\rule{0.3em}{0ex}}q]}^{H}\stackrel{\u0304}{\mathbf{H}}[\phantom{\rule{0.3em}{0ex}}q]\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{N}_{s}\frac{1}{{\sigma}_{v}^{2}}$ corresponding to its N_{ s } largest eigenvalues, and $\mathbf{\Omega}[\phantom{\rule{0.3em}{0ex}}q]=\mathcal{D}\left({\omega}_{q}^{1}\cdots {\omega}_{q}^{{N}_{s}}\right)$ is the power allocation matrix whose coefficients can be found optimising a prescribed objective metric [7]. The matrix C is a (subcarrierindependent) unitary transform that spreads the incoming symbols among the different spatial modes. It has been recently shown in [10] that choosing C to be the product of a unitary transform (e.g. Fourier, Hadamard) and a constellation rotation, in the form of a diagonal matrix with different phase factors, maximises diversity and leads to optimum performance in terms of BER.
3.1 Linear and nonlinear detectors
 1.
The CSIT is perfect.
 2.
The rotation matrix C in (5) is diagonal.
Results in [14, 15] show that, in fully loaded configurations, (8) is very advantageous over (7) in terms of BER, although this comes at the cost of an increased receiver complexity, even when employing efficient implementations such as sphere decoding. Regardless of the detection method, either MMSE or ML, estimated symbols are then demodulated, and the corresponding bits, subsequently deinterleaved, (spatially) deparsed and finally supplied to a Viterbi decoder to obtain an estimate of the transmitted packet.
3.2 Adaptive detector

When conditions 1 and 2 hold, it is obvious that β_{i,j}[ q]=0 ∀i,j and MMSE detection is optimum.

When condition 1 holds and condition 2 does not hold, the magnitude of the interfering terms β_{i,j}[ q] depends on the conditioning of H[ q]. If the matrix is well conditioned, MMSE will perform well, but if it is not, MLD will result in a significant advantage.

When condition 1 does not hold and condition 2 holds, it will depend on the actual realisation of parameter ρ. If ρ≃1, the overall processing matrix will be virtually diagonalized, making MMSE detection optimal. In contrast, the further away ρ is from 1, the more significant interfering terms β_{i,j}[ q] will become, thus requiring MLD for acceptable performance.

When conditions 1 and 2 do not hold, the detection strategy selection will depend on both the channel matrix conditioning and the specific CSIT accuracy.

The subcarrierbased nature of the algorithm is to be emphasised. Most likely, for a given packet, some of the subcarriers will be linearly detected while others will require the use of MLD.

Sphere detectionbased MLD usually starts the search procedure using the zero forcing (ZF) solution as the centre of the sphere. However, if an estimate of the noise power s^{2} is available, centering the search around the MMSE solution is computationally advantageous [20]; thus, the computation of (6), even in the case of eventually relying on MLD, still plays a role.
4 Iterative soft detection
Despite the importance of hard decoding in its own right, most practical deployments are based on the use of softbased decoding principles. Consequently, it is important to consider the performance of the proposed adaptive detection scheme when the component detectors extract soft information, typically in the form of loglikelihood ratios (LLRs), from the received samples. Furthermore, soft detectors are often able to operate iteratively following turbo receiver design principles. In this case, the MIMO detector and channel decoder exchange (soft) information back and forth with the corresponding LLRs becoming more reliable at each iteration [21]. The next subsections describe two popular softbased detection schemes, one based on MLD and another one based on MMSE, and the iterative extension within the context of the considered setup.
4.1 MLDbased soft detection
where the characters ${\mathcal{\mathcal{B}}}_{p,+1}$ and ${\mathcal{\mathcal{B}}}_{p,1}$ represent the sets of ${2}^{{N}_{b}1}$ bit vectors whose p^{th} position is a ‘ +1’ or ‘ −1’, respectively. Moderate values of M and/or N_{ s } make the sets ${\mathcal{\mathcal{B}}}_{p,+1}$ and ${\mathcal{\mathcal{B}}}_{p,1}$ extremely large, making the search in (12) computationally challenging. To address this issue, the LSD limits the search to the sets ${\widehat{\mathcal{\mathcal{B}}}}_{p,+1}={\mathcal{\mathcal{B}}}_{p,+1}\cap \mathcal{C}$ and ${\widehat{\mathcal{\mathcal{B}}}}_{p,1}={\mathcal{\mathcal{B}}}_{p,1}\cap \mathcal{C}$ where is the set containing the bit vectors corresponding to the N_{cand} candidates closer, in a Euclidean sense, to the received samples, i.e. $\mathcal{C}=\left\{{\mathbf{b}}^{1},\phantom{\rule{1em}{0ex}}\dots ,\phantom{\rule{1em}{0ex}}{\mathbf{b}}^{{N}_{\text{cand}}}\right\}$ where ${\mathbf{b}}^{n}={\mathcal{\mathcal{M}}}^{1}({\stackrel{~}{\mathbf{s}}}^{[c]}[\phantom{\rule{0.3em}{0ex}}q])$ with $\left\{{\stackrel{~}{\mathbf{s}}}^{[1]}[\phantom{\rule{0.3em}{0ex}}q],\phantom{\rule{1em}{0ex}}\dots ,\phantom{\rule{1em}{0ex}}{\stackrel{~}{\mathbf{s}}}^{[{N}_{\text{cand}}]}[\phantom{\rule{0.3em}{0ex}}q]\right\}$ being the N_{cand} group candidates for which the Euclidean distance ∥r[ q]−A[ q]s[ q]∥^{2} is smallest.
4.2 MMSEbased soft detection
where D_{I,p} is given by the mappings defined in [22] (14 to 18). LLRs for the bits in quadrature are computed using an analogous procedure.
4.3 Channelaware iterative soft detection
Further performance improvements in the form of lower error rates can be achieved if the detector and channel decoder are allowed to exchange information, specially when the detector is based on ML detection principles [21]. In fact, it has been observed that when detection relies on linear processing techniques such as MMSE, the benefits of iterative reception become rather marginal [23]. Consequently, in this work, the application of iterative processing is limited to those cases where MLD has been selected as the preferred detection strategy.
As shown in Figure 3, each (subcarrierbased) detector operates in accordance with Algorithm 1 to decide which detection strategy, MLD or MMSE, should be used and computes the corresponding LLR for each bit using (12) or (14), respectively. The desegmentation process is then in charge of collecting the LLR values computed for the successive OFDM symbols forming a packet/stream resulting in the LLR streams $\left\{{\mathcal{\mathcal{L}}}_{1}^{D1},\cdots \phantom{\rule{0.3em}{0ex}},{\mathcal{\mathcal{L}}}_{{N}_{s}}^{D1}\right\}$. These LLRs, after subtracting any a priori knowledge available from previous iterations, give rise to the extrinsic information $\left\{{\mathcal{\mathcal{L}}}_{1}^{E1},\cdots \phantom{\rule{0.3em}{0ex}},{\mathcal{\mathcal{L}}}_{{N}_{s}}^{E1}\right\}$, which after suitable deinterleaving and spatial deparsing, results in the input stream to the maximum a posteriori (MAP) decoder (${\mathcal{\mathcal{L}}}_{1}^{A2}$). The MAP decoder has a double output: on one hand, an estimate of the information symbols, and on the other hand, a refined version of the input LLRs. This latter output, ${\mathcal{\mathcal{L}}}_{1}^{D2}$, after subtracting already known information (${\mathcal{\mathcal{L}}}_{1}^{A2}$), results in the extrinsic information ${\mathcal{\mathcal{L}}}_{1}^{E2}$ to be fed back to the detection stage. To this end, signal ${\mathcal{\mathcal{L}}}_{1}^{E2}$ is suitably parsed and interleaved resulting in the sequences $\left\{{\mathcal{\mathcal{L}}}_{1}^{A1},\cdots \phantom{\rule{0.3em}{0ex}},{\mathcal{\mathcal{L}}}_{{N}_{s}}^{A1}\right\}$ forming the a priori information for the next turbo iteration. Note that only the LLRs corresponding to those subcarriers that have been detected using MLD are fed back to the detector (denoted in Figure 3 by ${\left\{{\mathcal{\mathcal{L}}}_{i}^{A1}\right\}}_{\text{ML}}$) while no information is fed back to the MMSEdetected subcarriers.
5 Numerical results
5.1 Simulation setup
The simulation environment has been defined in accordance with specifications from the IEEE 802.11n architectures [18], considering a setup with N_{ T }=N_{ R }=4 antennas transmitting N_{ s }=4 streams. The system operates on a bandwidth of B=20 MHz using N_{ c }=64 subcarriers out of which N_{ d }=52 are used for data transmission and the rest are devoted to pilot signalling and guard bands. For all simulations, transmission modes with either quadrature phase shift keying (QPSK) or 16QAM modulation and a 1/2rate convolutional channel coder with generator polynomials g= [ 133, 171]_{8} have been employed. Full cyclic prefix is used in order to guarantee the avoidance of interference among successively transmitted OFDM symbols. Power allocation matrices are computed according to the ARITHMSE criterion in [7] for hard decoding and uniform power allocation for soft/iterative decoding^{d}. Two different spatial spreading matrices have been considered, $\mathbf{C}={\mathit{I}}_{{N}_{s}}$ (no spreading) and $\mathbf{C}={\mathit{\Psi}}_{{N}_{s}}$ (full spreading), with ${\mathit{I}}_{{N}_{s}}$ and ${\mathit{\Psi}}_{{N}_{s}}$ denoting the identity and rotated WalshHadamard matrices of dimension N_{ s }, respectively.
Interestingly, IEEE 802.11based systems are a representative scenario where the CSIT accuracy may (widely) vary over a short time frame. This is due to the channel contention mechanism that, based on CSMA/CA, causes the time span between the reception of channelrelated feedback at Tx and its utilisation to fluctuate on a packet basis and, moreover, to make it heavily dependent on the number of active users in the system. Note that when users enter or exit the system, the average delay in using the acquired CSIT for the rest of the active users is likely to vary, thus effectively implying a degradation or improvement in the CSIT accuracy. For the results shown here, the channel, generated following the specifications in [24], is assumed to remain static over the duration of a packet and vary independently from packet to packet (block fading).
5.2 Harddecoded results
5.3 Softdecoded results
5.4 Iteratively decoded results
When the statistical CSIT quality is considerably degraded (μ=1.31), results are significantly different. As it can be observed in the middle plot of Figure 8, and in line with results in the previous subsection, noniterative ML detection provides a gain of nearly 3 dB with respect to MMSE. Furthermore, when the ML receiver is allowed to iterate, a further 1 to 1.5dB gain is achieved. When using channelaware detection, it is observed that Algorithm 1, either with or without iterations, rightly chooses to rely on ML detection owing to the rather large interfering terms the receiver observes when evaluating (10) and (11).
5.5 An important remark
It is worth emphasising that the merits of the adaptive detector should be valued by globally appreciating the results on each of the considered scenarios (hard, soft and iterative decoding): a single configuration for each type of receiver ($\alpha =0.75,\mathcal{F}(\mathbf{x})=min(\mathbf{x})$ for hard decoding, $\alpha \phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}0.95,\mathcal{F}(\mathbf{x})\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}min(\mathbf{x})$ for soft decoding and $\alpha =0.975,\mathcal{F}(\mathbf{x})=min(\mathbf{x})$ for iterative decoding) leads to a strategy able to attain virtually optimum PER performance while potentially offering a very significant complexity reduction with respect to the full use of MLD. In other words, Algorithm 1 provides the receiver with the capability of distinguishing when MLD will be effective and when MMSE will suffice. This scheme can therefore be very attractive in those scenarios where the quality of CSIT may vary over time such as it occurs in today’s WLAN environments depending on the number of users in the system or changes in the environment. Furthermore, note that the parameter α acts as a performance/complexity trading knob, thus enabling the reconfiguration of the system as a function of, for instance, the available battery power or required processing latency.
6 Conclusions
This paper has proposed an adaptive detection technique that allows the receiver of a MIMOOFDM linearly precoded system to toggle between the use of MMSE and MLD depending on the CSIT accuracy and/or channel conditioning. The introduced technique works on a persubcarrier basis and is compatible with different receiver architectures, namely hard, soft and iterative decoding schemes. Numerical results have shown that regardless of the receiver setup, the adaptive detector is able to distinguish the system conditions that allow MLD to boost performance from those where the much simpler MMSE detector would perform (near) optimally.
Endnotes
^{a} The incorporation of antenna correlation effects to the current system model is trivial; however, it unnecessarily complicates notation without providing any further insight or significantly altering numerical results.
^{b} This is a realistic assumption since the receiver should be aware of the last CSI information sent to the transmitter. Alternatively, if the transmitter sends pilot symbols through the precoder (and channel), the receiver can also deduct the precoding filter used in transmission.
^{c} To simplify the notation, the subcarrier and stream indices are skipped when referring to the bits.
^{d} The (possibly iterative) utilisation of soft information at the receiver suggests using capacitybased measures to optimise the power allocation. Unfortunately, under imperfect CSIT, no closedform solutions are available and power allocation solutions require convex numerical optimisation procedures (see [5] for a detailed discussion on this issue).
^{e} For the results shown here, only two iterations between detector and decoder were allowed as it was observed that further iterations did not bring along any significant performance benefit.
Declarations
Acknowledgements
Work funded by MINECO and FEDER under project AM3DIO (TEC201125446), Spain.
Authors’ Affiliations
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