MIMO Detection for High-Order QAM Based on a Gaussian Tree Approximation
This paper proposes a new detection algorithm for MIMO communication systems employing high order QAM constellations. The factor graph that corresponds to this problem is very loopy; in fact, it is a complete graph. Hence, a straightforward application of the Belief Propagation (BP) algorithm yields very poor results. Our algorithm is based on an optimal tree approximation of the Gaussian density of the unconstrained linear system. The finite-set constraint is then applied to obtain a loop-free discrete distribution. It is shown that even though the approximation is not directly applied to the exact discrete distribution, applying the BP algorithm to the loop-free factor graph outperforms current methods in terms of both performance and complexity. The improved performance of the proposed algorithm is demonstrated on the problem of MIMO detection.
Finding a linear least squares fit to data is a well-known problem, with applications in almost every field of science. When there are no restrictions on the variables, the problem has a closed form solution. In many cases, a-priori knowledge on the values of the variables is available. One example is the existence of priors, which leads to Bayesian estimators. Another example of great interest in a variety of areas is when the variables are constrained to a discrete finite set. This problem has many diverse applications such as the decoding of multi-input-multi-output (MIMO) digital communication systems. In contrast to the continuous linear least squares problem, this problem is known to be NP hard .
We consider a MIMO communication system with transmit antennas and receive antennas. The tap gain from transmit antenna to receive antenna is denoted by . In each use of the MIMO channel a vector is independently selected from a finite set of points according to the data to be transmitted, so that . The received vector is given by:
The vector is an additive noise in which the noise components are assumed to be zero mean, statistically independent Gaussians with a known variance . The matrix is comprises i.i.d. elements drawn from a complex normal distribution of unit variance. The MIMO detection problem consists of finding the unknown transmitted vector given and . The task, therefore, boils down to solving a linear system in which the unknowns are constrained to a discrete finite set. It is convenient to reformulate the complex-valued model into a real valued one. It can be translated into an equivalent double-size real-valued representation that is obtained by considering the real and imaginary parts separately:
Hence we assume hereafter that H has real values without any loss of generality. The maximum likelihood (ML) solution is:
However, going over all the vectors is unfeasible when either or are large.
A simple sub-optimal solution, known as the Zero-Forcing algorithm, is based on a linear decision that ignores the finite set constraint:
and then, neglecting the correlation between the symbols, finding the closest point in for each symbol independently:
This scheme performs poorly due to its inability to handle ill-conditioned realizations of the matrix . Somewhat better performance can be obtained by using a minimum mean square error (MMSE) Bayesian estimation on the continuous linear system. Let be the mean symbol energy. We can partially incorporate the information that by using the prior Gaussian distribution . The MMSE estimation becomes:
and then the finite-set solution is obtained by finding the closest lattice point in each component independently. A vast improvement over the linear approaches described above can be achieved by the V-BLAST algorithm that is based on sequential decoding with optimal ordering .
These linear type algorithms can also easily provide probabilistic (soft-decision) estimates for each symbol. However, there is still a significant gap between the detection performance of the V-BLAST algorithm and the performance of the ML detector.
Many alternative methods have been proposed to approach the ML detection performance. The sphere decoding (SD) algorithm finds the exact ML solution by searching the nearest lattice point. [8, 25, 5, 28]. Although the SD reduces computational complexity compared to the exhaustive search of ML solution, sphere decoding is not feasible for high-order QAM constellations. While sphere decoding has been empirically found to be computationally very fast for small to moderate problem sizes (say, for for 16-QAM), the sphere decoding complexity would be prohibitive for large , higher order QAM and/or low SNRs . Another family of MIMO decoding algorithms is based on semidefinite relaxation (e.g. [31, 27, 18]). Although the theoretical computational complexity of semidefinite relaxation is a low degree polynomial, in practice the running time is very high. Thus, there is still a need for low complexity detection algorithms that perform well.
This study attempts to solve the MIMO decoding problem using the Belief Propagation (BP) paradigm. It is well-known (see e.g. ) that a straightforward implementation of the BP algorithm to the MIMO detection problem yields very poor results since there are a large number of short cycles in the underlying factor graph. In this study we introduce a novel approach to utilize the BP paradigm for MIMO detection. The proposed variant of the BP algorithm is both computationally efficient and achieves near optimal results. A preliminary version of this paper appears in . The paper proceeds as follows. In Section II we discuss previous attempts to apply variants of the BP algorithm to the MIMO decoding problem. The proposed algorithm which we dub ‘The Gaussian-Tree-Approximation (GTA) Algorithm’ is described in Section III. Experimental results are presented in Section IV.
Ii The Loopy Belief Propagation Approach
Given the constrained linear system , and a uniform prior distribution on , the posterior probability function of the discrete random vector given is:
The notation stands for equality up to a normalization constant. Observing that is a quadratic expression, it can be easily verified that is factorized into a product of two- and single-variable potentials:
where is the -th column of the matrix . Since the obtained factors are simply a function of pairs, we obtain a Markov Random Field (MRF) representation . In the MIMO application the (known) matrix is randomly selected and therefore, the MRF graph is usually a completely connected graph (see an MRF graph illustration in Fig. 1).
The Belief Propagation (BP) algorithm aims to solve inference problems by propagating information throughout this MRF via a series of messages sent between neighboring nodes (see  for an excellent tutorial on BP). In the sum-product variant of the BP algorithm applied to the MRF (7), the message from to is:
In each iteration messages are passed along all the graph edges in both edge directions. In every iteration, an estimate of the posterior marginal distribution (‘belief’) for each variable can be computed by multiplying together all of the incoming messages from all the other nodes:
A variant of the sum-product algorithm is the max-product algorithm in which the summation in Eq. (9) is replaced by a maximization over all the symbols in . In a loop-free MRF graph the sum-product algorithm always converges to the exact marginal probabilities (which corresponds in the case of MIMO detection to a soft decision probability of each symbol ). In a loop-free MRF graph the max-product variant of the BP algorithm always converges to the most likely configuration  (which corresponds to ML decoding in our case). For loop-free graphs, BP is essentially a distributed variant of dynamic programming. The BP message update equations only involve passing messages between neighboring nodes. Computationally, it is thus straightforward to apply the same local message updates in graphs with cycles. In most such models, however, this loopy BP algorithm will not compute exact marginal distributions; hence, there is almost no theoretical justification for applying the BP algorithm (one exception is that, for Gaussian graphs, if BP converges, then the means are correct ). However, the BP algorithm applied to loopy graphs has been found to have outstanding empirical success in many applications, e.g., in decoding LDPC codes . The performance of BP in this application may be attributed to the sparsity of the graphs. The cycles in the graph are long, hence the graph has tree-like properties, so that messages are approximately independent and inference may be performed as though the graph was loop-free. The BP algorithm has also been used successfully in image processing and computer vision (e.g. ) where the image is represented by a grid-structured MRF that is based on local connections between neighboring nodes.
However, when the graph is not sparse, and is not based on local grid connections, loopy BP almost always fails to converge. Unlike the sparse graphs of LDPC codes, or grid graphs in computer vision applications, the MRF graphs of MIMO channels are completely connected graphs and therefore the associated detection performance is poor. This has prevented the BP from being an asset for the MIMO problem. Fig. 2 shows an example of a BPSK MIMO system based on an matrix and (see Section IV for a detailed description of the simulation set-up). As can be seen in Fig. 2, the BP decoder based on the MRF representation (7) has very poor results. Standard techniques to stabilize the BP iterations such as damping the message updates  do not help here. Even applying more advanced versions of BP (e.g. Generalized BP and Expectation Propagation) to inference problems on complete MRF graphs yields poor results . The problem here is not in the optimization method but in the cost function that needs to be modified to yield a good approximate solution.
There have been several recent attempts to apply BP to the MIMO detection problem with good results (e.g. [15, 22, 12, 16]). However, in these methods the factorization of the probability function is done in such a way that each factor corresponds to a single linear equation. This leads to a partition of the probability function into factors each of which is a function of all the unknown variables. This results in an exponential computational complexity when computing the BP messages. Shental et. al  analyzed the case where the matrix is relatively sparse (and has a grid structure) (see Fig. 3). They showed that even under this restricted assumption the BP still does not perform well. As an alternative method they proposed the generalized belief propagation (GBP) algorithm that does work well on the sparse matrix if the algorithm regions are carefully chosen. There are situations where the sparsity assumption makes sense (e.g. 2D intersymbol interference (ISI) channels). However, in the MIMO channel model we assume that the channel matrix elements are i.i.d. and Gaussian; hence we cannot assume that the channel matrix is sparse.
Iii The Gaussian Tree Approximation Algorithm
Our approach is based on an approximation of the exact probability function:
that enables a successful implementation of the Belief Propagation paradigm. Since the BP algorithm is optimal on loop-free factor graphs (trees) a reasonable approach is finding an optimal tree approximation of the exact distribution (11). Chow and Liu  proposed a method to find a tree approximation of a given distribution that has the minimal Kullback-Leibler (KL) divergence to the true distribution. They showed that the optimal tree can be learned efficiently via a maximum spanning tree whose edge weights correspond to the mutual information between the two variables corresponding to the edges endpoints. The problem is that the Chow-Liu algorithm is based on the two-dimensional marginal distributions. However, finding the marginal distribution of the probability function (11) is, unfortunately, NP hard and it is (equivalent to) our final target.
To overcome this obstacle, our approach is based on applying the Chow-Liu algorithm on the distribution corresponding to the unconstrained linear system. This distribution is Gaussian and therefore it is straightforward in this case to compute the two-dimensional marginal distributions. Given the Gaussian tree approximation, the next step of our approach is to apply the finite-set constraint and utilize the Gaussian tree distribution to form a discrete loop free approximation of which can be efficiently globally maximized using the BP algorithm. To motivate this approach we first apply a simplified version to derive the zero-forcing decoding algorithm (4) described in Section I.
where is a Gaussian density with mean and covariance matrix . can be viewed as a posterior distribution of assuming a non-informative prior. Now, instead of marginalizing the true distribution , which is an NP hard problem, we approximate it by the product of the marginals of the Gaussian density :
At this stage we apply the finite-set constraint. From the Gaussian approximation (13) we can extract a discrete approximation:
Since this joint probability function is obtained as a product of marginal probabilities, we can address each variable separately:
Taking the most likely symbol we obtain the sub-optimal Zero-Forcing solution (4).
Motivated by the simple product-of-marginals approximation described above, we suggest approximating the discrete distribution via a tree-based approximation of the Gaussian distribution . Although the Chow-Liu algorithm was originally stated for discrete distributions, one can verify that it also applies for the Gaussian case. For the sake of completeness we give a detailed derivation below.
Iii-a Finding the optimal Gaussian tree approximation
We represent an -node tree graph by the loop-free parent relations such that is the parent node of . To simplify notation we do not separately describe the root node. The parent of the root is implicitly assumed to be the empty set. A distribution is described by a tree if it can be written as . We start with a formula for the KL divergence between a Gaussian distribution and a distribution defined on the same space that is represented by a tree graphical model.
Theorem 1: Let be a multivariate Gaussian distribution and let be another distribution that is represented by a loop-free graphical model (tree). The KL divergence between and is:
such that is the mutual information and is the differential entropy, based on the distribution .
Proof: The definition of the KL divergence implies:
From Eq. (III-A) it can be easily seen that if we fix a tree graph , the tree distribution whose KL divergence to is minimal is . In other words, the best tree approximation of is constructed from the conditional distributions of . For that tree approximation we have:
Moreover, since in Eq. (17) and do not depend on the tree structure, the tree topology that best approximates is the one that maximizes the sum:
A spanning tree of a graph is a subgraph that contains all the vertices and is a tree. Now suppose the edges of the graph have weights. The weight of a spanning tree is simply the sum of weights of its edges. Eq. (18) reveals that the problem of finding the best tree approximation of the Gaussian distribution can be reduced to the well-known problem of finding the maximum spanning tree of the weighted -node graph where the weight of the - edge is the mutual information between and . It can be easily verified that the mutual information between two r.v. and that are jointly Gaussian is:
where is the correlation coefficient between and .
There are several algorithms to find a minimum spanning tree. They all utilize a greedy approach. In this work we use the Prim algorithm  which is efficient and very simple to implement. The Prim algorithm begins with some vertex in a given graph, defining the initial set of vertices . Then, in each iteration, we choose a minimum-weight edge , connecting a vertex in the set to the vertex outside of set . Then vertex is brought in to . This process is repeated until a spanning tree is formed. We can use a heap to remember, for each vertex, the smallest edge connecting the current sub-tree with that vertex. The complexity of the Prim’s algorithm, that finds the minimum spanning tree of an -vertex graph is .
We note in passing that since the Prim algorithm is based on a greedy approach, it only relies on the order of the weights and not on their exact values. Hence, applying a monotonically increasing function on the graph weights does not change the topology of the optimal tree. To find the optimal Gaussian tree approximation we can, therefore, use the weights instead of . The optimal Gaussian tree is, therefore, the one that maximizes the sum of the square correlation coefficients between adjacent nodes. To summarize, the algorithm that finds the best Gaussian tree approximation is as follows. Define as the root. Then find the edge connecting a vertex in the tree to the vertex outside the tree, such that the corresponding square correlation coefficient is maximal and add the edge to the tree. Continue this procedure until a spanning tree is obtained.
Iii-B Applying BP on the tree approximation
Let be the optimal Chow-Liu tree approximation of (12). We can assume, without loss of generality, that is rooted at . is a loop-free Gaussian distribution on , i.e.
where is the ‘parent’ of the -th node in the optimal tree. The Chow-Liu algorithm guarantees that is the optimal Gaussian tree approximation of in the sense that the KL divergence is minimal.
Given the Gaussian tree approximation, the next step of our approach is to apply the finite-set constraint to form a discrete loop free approximation of which can be efficiently globally maximized using the BP algorithm. Our approximation approach is, therefore, based on replacing the true distribution with the following approximation:
The probability function is a loop free factor graph. Hence the BP algorithm can be applied to find its most likely configuration. We next derive the messages of the BP algorithm that is applied on . An optimal BP schedule, when applied to a tree, requires passing a message once in each direction of each edge . The BP messages are first sent from leaf variables ‘downward’ to the root. The computation begins at the leaves of the graph. Each leaf variable node sends a message to its parent. Each vertex waits for messages from all of its children before computing the message to be sent to its parent. The ‘downward’ BP message from a variable to its parent variable is computed based on all the messages received from its children:
If is a leaf node in the tree then the message is simply:
The ‘downward’ computation terminates at the root node.
Next, BP messages are sent ‘upward’ back to the leaves. The computation begins at the root of the graph. Each vertex waits for a message from its parent before computing the messages to be sent to each of its children. The ‘upward’ BP message from a parent variable to its child variable is computed based on the ‘upward’ message received from its parent and from ‘downward’ messages that received from all the siblings of :
If is the root of the tree then the message is simply:
After the downward-upward message passing procedure is completed we can compute the ‘belief’ at each variable which is the product of all the messages sent to the variable from its parent and from its children (if there are any).
In the case is the root node, the ‘belief’ computed is as follows:
Since the approximated distribution (21) is loop free, the general Belief Propagation theory guarantees that (the normalized) belief vector is exactly the marginal distribution of the approximated distribution (21). To obtain a hard-decision decoding we choose the symbol whose posterior probability is maximal:
Above we described the sum-product version of the BP algorithm that computes the marginal probabilities . A variant of the sum-product algorithm is the max-product algorithm in which the summation in Eq. (22)-(25) is replaced by a maximization over all the symbols in . The max-product algorithm finds the most likely pattern of the approximation . We did not observe any significant performance difference using either the sum-product or the max-product variants for decoding MIMO systems presented in the experiment section. The max-product is more computationally efficient since the BP messages can be entirely computed in the log-domain.
Iii-C An MMSE version of a tree approximation
The MMSE Bayesian approach (5) is known to be better than the zero-forcing solution (4). In MMSE we partially incorporate the information that by using the prior Gaussian distribution . In a similar way we can consider a Bayesian version of the proposed Gaussian tree approximation. We can partially incorporate the information that by using the prior Gaussian distribution such that . This yields the posterior Gaussian distribution:
such that and . The MMSE method is obtained by approximating the unconstrained posterior distribution by a product of marginals. In our approach we use, instead, the best loop-free Gaussian approximation. We can apply the Chow-Liu tree approximation on the Gaussian distribution (29) to obtain a ‘Bayesian’ Gaussian tree approximation for . This way we partially use the finite set constraint when we search for the best tree approximation of the true discrete distribution . This is likely to yield a better approximation of the discrete distribution than the tree distribution which is based on the unconstrained distribution .
To summarize, our solution to the MIMO decoding problem is based on applying BP on a discrete version of the Gaussian tree approximation of the Bayesian version of the continuous least-square solution. We dub this method “The Gaussian-Tree-Approximation (GTA) Algorithm”. The GTA algorithm is summarized in Fig. 4. We next compute the complexity of the GTA algorithm. The complexity of computing the covariance matrix is , the complexity of the Chow-Liu algorithm (based on Prim’s algorithm for finding the minimum spanning tree) is and the complexity of the BP algorithm is .
Iv Experimental Results
In this section we provide simulation results for the GTA algorithm over various MIMO systems. The channel matrix comprised i.i.d. elements drawn from a zero-mean complex normal distribution of unit variance. We used realizations of the channel matrix and each matrix was used once for sending a message. The performance of the proposed algorithm is shown as a function of the variance of the additive noise . The signal-to-noise ratio (SNR) is defined as where ( is the number of variables, is the variance of the Gaussian additive noise, and is the mean symbol energy).
We compared the performance of the GTA method to the V-BLAST (MMSE-SIC) algorithm with optimal ordering for the successive interference cancelation , and to the Schnorr-Euchner variant of sphere decoding (SE-SD) with infinite radius [1, 25]. We used sorting of the channel matrix using the SQRD algorithm  and regularization , which substantially reduces the computational complexity.
We also implemented the SDR detector suggested by Sidiropoulos and Luo . Recently it was shown  that for the cases of 16-QAM and 64-QAM this SDR based method is equivalent to the SDR based detection method suggested by Weisel, Eldar and Shamai . The SDRs were solved using the CSDP package . In the SDR Gaussian randomization step, 100 independent randomizations were implemented. All the MIMO detection algorithms were implemented in C for efficiency.
Fig. 5 shows MIMO detection performance for a MIMO system using 16-QAM. The methods that are shown are V-BLAST, GTA, SDR and sphere-decoding. In this case the SDR outperforms both the V-BLAST and the GTA methods. However, in this case it is still feasible to compute the optimal maximum-likelihood algorithm using the sphere decoding algorithm.
The SD-SE is the favorite detection method when the problem size is small or moderate. In this case the SD-SE can always yield the exact ML solution at acceptable computational cost. In large size problems, however, there is still a need for good approximation methods. Fig. 6 shows the SER versus SNR and worst case execution time versus SNR for the system using 64-QAM. Fig. 7 shows the same experimental results for the system using 64-QAM. To assess the computational complexity we used a measure of the worst case rather than the average case since in online applications we have to decode within a specified time. The choice between execution time or number of floating point operations is debatable. The differences in running time between the methods we implemented was in orders of magnitude and running time is easier to appreciate.
As can be seen from Figs. 6 and 7, the performance of the GTA algorithm in high SNR is significantly better than the V-BLAST. The computational complexity of GTA is comparable to V-BLAST and it is two orders of magnitude better than SDR. We note in passing that the performance of the SDR method  in these 64-QAM cases is worse than that of MMSE-SIC and the computational complexity is much higher. From the derivation of the SDR method it can be seen that the relaxation becomes more crude as at higher constellations.
We next show the performance of several variants of the GTA algorithm. The GTA algorithm differs from the ZF, MMSE and MMSE-SIC algorithms in several ways. The first is a Markovian approximation of instead of an approximation based on a product of independent densities. The second aspect is the use of an optimal tree. To clarify the contribution of each component we modified the GTA algorithm by replacing the Chow-Liu optimal tree by the tree . We call this method the ‘Line-Tree’. As can be seen from Fig. 8, using the optimal tree is crucial to obtain improved results. Fig. 8 also shows the results of the non-Bayesian variant of the GTA algorithm. As can be seen, the Bayesian version yields better results. Fig. 8 shows the symbol error rate (SER) versus SNR for a , , real MIMO system. The performance of the GTA method and its variants was compared to the MMSE and the MMSE-SIC algorithms.
We proposed a novel MIMO detection technique based on the principle of a tree approximation of the Gaussian distribution that corresponds to the continuous linear problem. The proposed method outperforms previously suggested MIMO decoding algorithms in high order QAM constellation, as demonstrated in simulations. Although the proposed method yields improved results, the tree approximation we applied may not be the best one (finding the best tree for the integer constrained linear problem is NP hard). It is left for future research to search for a better discrete tree approximation for the constrained linear least squares problem. This paper dealt with a tree approximation approach, more complicated approximations such as multi-parent trees could improve performance and can potentially provide a smooth performance-complexity trade-off.
While the method provides excellent performance, it is worthwhile mentioning that the method provides a-posteriori probabilities for each variable that can be used to improve performance. This is done by applying a Schnorr-Euchner sphere decoding algorithm , where we order the symbols according to their a-posteriori probabilities. This can lead to very close to optimal performance in a significantly reduced complexity. It is because that by computing the a-posteriori probabilities, we have a much higher probability of finding the true solution during the first search, therefore significantly reducing the search radius.
There are several important applications of the proposed technique. We comment here on combining it into communication systems with coding and interleaving. It is useful both for single carrier and OFDM systems. It can serve as a MIMO decoder for wireless communication systems. Using the a-posteriori probability distribution of the symbols we can easily compute the a-posteriori probability and the likelihood ratio for the bits. The technique can be combined with MIMO-OFDM system with bit interleaved coded modulation or with trellis coded modulation by joint coding of over all frequency tones of the OFDM system, and running the decoder for each tone independently.
In this paper we focused on the MIMO detection problem. The proposed method, however, can be applied to solve constrained linear least squares problems which is an important issue in many fields. A main concept in the GTA model is the interplay between discrete and Gaussian models. Such hybrid ideas can be considered also for discrete inference problems other than least-squares.
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