# On Maximum Likelihood Detection and the Search for the Closest Lattice Point

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2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 49, NO X, 003 Let U denote a squared QAM signal set with Q signal points [1] Assume that the transmitter maps blocks u U L with independent and uniformly distributed entries (information symbols) onto M T transmit arrays of the form S(u) = L (F l u l + G l u l ) (6) l=1 where {F l, G l C M T : l = 1,, L} are the linear dispersion code generators [11], [1], [13] Then, the array S(u) = [s 1 (u),, s T (u)] is transmitted column by column in T channel uses Due to the linearity of the channel (5) and of the encoder (6), one can easily see that there exists a matrix B R NT L such that the ML detection can be written in the form (4) In general, matrix B depends on the physical channel matrix H and on the code generators {F l, G l } (see [11] for an explicit expression of the matrix B in terms of H and the code generators) A particularly simple case corresponds to the transmission of uncoded QAM symbols In this case, L = M, T = 1, S(u) = u and B is given explicitly by [ ] 1 SNR Re{H} Im{H} B = M(Q (7) 1) Im{H} Re{H} Block Transmission over Time-selective or Frequencyselective Fading: In [14], linearly block-coded transmission for time-selective fading was considered The complex baseband channel is given by r = HMu + ν, where now H = diag(h 1,, h N ), {h i } is the sequence of scalar fading coefficients, and M is a suitable rotation matrix Similarly, for frequency-selective slow fading we can work in the frequency domain to obtain linearly precoded OFDM schemes (see for example [15] and references therein) in the same form as above, where H = diag(h 1,, H N ) and H i denotes the fading channel frequency response at the i-th OFDM subcarrier 3 Multiple-access Gaussian Waveform Channel (CDMA): The canonical baseband complex model for a K-user synchronous CDMA system is given by [16] y t = HWu t + ν t (8) where the columns of H C N K represent the (discrete-time) users signature waveforms, W is a diagonal K K matrix of amplitudes and u t is a K 1 vector containing the modulation symbols transmitted by the users in the t-th symbol interval In most CDMA systems the user symbols take on values in the 4- QAM signal set Again, after suitable translation and scaling, the ML joint detection of the K user symbols can be put in the form (4) [17], [18] It is well-known that the minimization (4) for arbitrary B and y is NP-hard (see [3], [6], [19], [0]) Nevertheless, it has been shown recently that in many relevant cases, for a certain range of system parameters such as SNR, m, n and Q, the average complexity of some algorithms implementing or approximating ML detection is polynomial in m (roughly, O(m 3 )) The reason of this behavior is that, in (1), the received point y is not arbitrary, but it is obtained by perturbing the transmitted point Bx by the additive noise z Therefore, it can be expected that as the SNR increases the average complexity decreases, for fixed m, n and Q This fact has been shown by theoretical analysis in [1] in the case of uncoded QAM over a N M frequency flat MIMO channel with N M, H having iid entries N C (0, 1), and the basic sphere decoder, known as Pohst enumeration (see details later) However, the same fact has been observed in more general cases by computer simulations in several recent works (eg, [], [3], [4]) While the exact average complexity analysis of the basic sphere decoder for general linear-dispersion codes and, a fortiori, for improved sphere decoding algorithms as those proposed in this work, appear to be intractable, finding improvements on the existing algorithms and illuminating the trade-offs and relationships between the different approaches is relevant on its own Developing efficient sphere decoders to solve or approximate (4) has recently gained renewed attention mainly because of their applications to multiple-antenna systems [8] This interest is due to the significant performance gain achieved by sphere decoders compared to other sub-optimal detection schemes [4], [5], [6], and to their average polynomial complexity (experimentally demonstrated) at medium to high SNR Moreover, the ML detector can be easily augmented to provide symbolby-symbol soft output, in the form of approximated posterior marginal probabilities p(x i y) The resulting soft-output detector forms the core of some iterative decoders based on Belief- Propagation (see for example [7], [8], [9], [30], [31]) The main contribution of this paper is a fresh look at the class of sphere decoding algorithms We start by reviewing the basic Pohst and Schnorr-Euchner enumeration strategies for infinite lattices in Section II In Section III we propose two variants of the basic algorithms in order to take into account finite PAM signal sets We also show that the proposed algorithms can be interpreted as a chain of sequential decoding stages, where each stage is based on a special case of the stack algorithm [3] This observation makes precise the intuition given in [7] that sphere decoding and sequential decoding are similar algorithms, and provides a path for cross-fertilization between the rich bodies of work on sequential and lattice decoding In Section IV, we discuss some heuristics for pre-processing aimed at transforming the problem so that the resulting algorithm has lower average complexity while still being ML (or near ML) In Section V we discuss some extensions of the proposed algorithms to handle general lattice codes and non-invertible channel matrices Finally, the performance of the proposed algorithms and the impact of various pre-processing methods are investigated through computer simulations in a number of relevant examples in Section VI This simulation study, along with some intuitive arguments, lead to the conclusion that one of the proposed algorithms uniformly outperforms all known sphere decoders in terms of receiver complexity, while offering near-ml detection performance (ie, the performance is within a very small fraction of a db from ML in all considered scenarios)

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 49, NO X, II THE POHST AND SCHNORR-EUCHNER ENUMERATIONS For B R n m and y R n, consider the minimization x = arg min x Z y Bx (9) m This is analogous to (4) but Z Q is replaced by Z, the (infinite) ring of integer numbers The set Λ = {Bx : x Z m } is an m- dimensional lattice in R n [33] The search in (9) for the closest lattice point to a given point y has been widely investigated in lattice theory In general, the optimal search algorithm should exploit the structure of the lattice For general lattices, that do not exhibit any particular structure, the problem was shown to be NP-hard In [], however, Pohst proposed an efficient strategy for enumerating all the lattice points within a sphere with a certain radius Although its worst-case complexity is exponential in m, this strategy has been widely used ever since in closest lattice point search problems due to its efficiency in many useful scenarios (see [6] for a comprehensive review of related works) The Pohst enumeration strategy was first introduced in digital communications by Viterbo and Biglieri [4] In [5], Viterbo and Boutros applied it to the ML detection of multi-dimensional constellations transmitted over single antenna fading channels, and gave a flowchart of a specific implementation More recently, Agrell et al [6] proposed the use of the Schnorr-Euchner refinement [7] of the Pohst enumeration in the closest lattice point search They further concluded, based on numerical results, that the Schnorr-Euchner enumeration is more efficient than the Viterbo-Boutros (VB) implementation The Pohst enumeration is briefly outlined as follows Assume that n m and rank(b) = m Let C 0 be the squared radius of an n-dimensional sphere S(y, C 0 ) centered at y We wish to produce a list of all points of Λ S(y, C 0 ) By performing the Gram-Schmidt orthonormalization of the columns of B (equivalently, by applying QR decomposition on B), one writes [ ] B = [Q, Q R ] (10) 0 where R is an m m upper triangular matrix with positive diagonal elements, 0 is an (n m) m zero matrix and Q (resp, Q ) is an n m (resp, n (n m)) unitary matrix The condition Bx S(y, C 0 ) can be written as y Bx C 0 [ ] [Q, Q ] T R y x 0 C 0 Q T y Rx C 0 (Q ) T y y Rx C 0 (11) where y = Q T y and C 0 = C 0 (Q ) T y Due to the upper triangular form of R, the last inequality implies the set of conditions y j r j,l x l C 0, i = 1,, m (1) j=i l=j By considering the above conditions in the order from m to 1 (akin to back-substitution in the solution of a linear upper triangular system), we obtain the set of admissible values of each symbol x i for given values of symbols x i+1,, x m More explicitly, let x m l = (x l, x l+1,, x m ) T denote the last m l+1 components of the vector x For a fixed x m i+1, the component x i can take on values in the range of integers I i (x m i+1 ) = [A i (x m i+1 ), B i(x m i+1 )] where A i (x m i+1) = 1 y i r i,j x j C 0 r i,i 1 r i,i B i (x m i+1) = y i m r i,j x j + C 0 m m y j r j,l x l, l=j y j r j,l x l If y j r j,l x l > C 0 l=j or if A i(x m i+1 ) > B i(x m i+1 ), then I i (x m i+1 ) = (the empty set) In this case, there is no value of x i satisfying the inequalities (1) and the points corresponding to this choice of x m i+1 do not belong to the sphere S(y, C 0 ) The Pohst enumeration consists of spanning at each level i the admissible interval I i (x m i+1 ), starting from level i = m and climbing up to level i = m 1, m,, 1 At each level, the interval I i (x m i+1 ) is determined by the current values of the variables at lower levels (corresponding to higher indices) If I 1 (x m ) is non-empty, the vectors x = (x 1, (x m )T ) T, for all x 1 I 1 (x m ), yield lattice points Bx S(y, C 0 ) The squared Euclidean distances between such points and y are given by d (y, Bx) = j=1 y j r j,l x l The algorithm l=j outputs the point x for which this distance is minimum If, after spanning the interval I m corresponding to x m (ground level), no point in the sphere is found, the sphere is declared empty and the search fails In this case, the search squared radius C 0 must be increased and the search is restarted with the new squared radius The Pohst enumeration is based on the so-called natural spanning of the intervals I i (x m i+1 ) at each level i, ie, x i takes on values in the order A i (x m i+1 ), A i(x m i+1 ) + 1,, B i(x m i+1 ) Schnorr-Euchner enumeration is a variation of the Pohst strategy where the intervals are spanned in a zig-zag order, starting from the mid-point S i (x m i+1 ) = 1 y i r i,i m l=j r i,j x j (14) (13)

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 49, NO X, (where denotes rounding to the closest integer) Hence, the Schnorr-Euchner enumeration will produce at each level i the (ordered) sequence of values if y i x i {S i (x m i+1), S i (x m i+1) + 1, S i (x m i+1) 1, S i (x m i+1) +, S i (x m i+1), } I i (x m i+1) r i,j x j r i,i S i (x m i+1) 0, or the (ordered) sequence of values if y i x i {S i (x m i+1 ), S i(x m i+1 ) 1, S i(x m i+1 ) + 1, S i (x m i+1), S i (x m i+1) +, } I i (x m i+1) m r i,j x j r i,i S i (x m i+1 ) < 0 Similar to the Pohst enumeration, when a given value of x i results in a point segment x m i outside the sphere, the next value of x i+1 (at level i + 1) is produced Note that with the Schnorr-Euchner enumeration one can set C 0 = Obviously, in this way the event of declaring an empty sphere never occurs It is also easy to see that the first point found with C 0 = corresponds to the Babai point [6] In the communication theory parlance this point is referred to as the nulling and canceling [5], or zero-forcing decision-feedback equalization (ZF-DFE) point Explicitly, this is given by the back-substitution with integer quantization (slicing) x zf dfe i = S i (x zf dfe i+1,, xm zf dfe ) = 1 m y i r r i,j xj zf dfe i,i (15) for i = m, m 1,, 1 As we shall see in Section VI, the drawback of setting C 0 = is that the distance d (y, Bx zf dfe ) might be quite large, therefore the algorithm will eventually span a large number of points before finding the ML solution III ML DETECTION OF FINITE ALPHABET CONSTELLATIONS Probably the most immediate application of the Pohst enumeration to solve (4) consists of the following steps: 1 Fix C 0 according to some criterion (see the discussion in Section VI) Apply the Pohst enumeration with the interval boundaries modified as A i (x m i+1 ) = max {0, 1 y i r i,j x j C 0 r i,i B i (x m i+1) = min {Q 1, 1 m y i r i,j x j + C 0 r i,i m m y j r j,l x l l=j y j r j,l x l l=j and obtain the list of all vectors x Z m Q such that Bx S(y, C 0 ) 3 If the list is non-empty, output the point achieving minimum distance (ie, the ML decision) Otherwise, increase C 0 and search again The average complexity of this simple algorithm has been given in closed form in [1] for the special case where B is a random matrix with iid entries N (0, 1) As anticipated in the introduction, this corresponds to the transmission of uncoded QAM over a MIMO frequency-flat independent Rayleigh fading channel The above sphere decoder can be improved in several ways The VB implementation is essentially the same algorithm given above, but C 0 is changed adaptively along the search: as soon as a vector x Z m Q is found such that Bx S(y, C 0 ), then C 0 is updated as C 0 d (y, Bx) and the search is restarted in the new sphere with the smaller radius The drawback of this approach is that the VB algorithm may re-span values of x i for some levels i, 1 < i m, that have already been spanned in the previous sphere Next, we give the details of two new sphere decoding algorithms The first, namely Algorithm I, is similar to the VB algorithm, but avoids re-spanning already spanned point segments The second, namely Algorithm II, can be seen as a modification of the Schnorr-Euchner enumeration in order to take into account the finite signal set boundary Interestingly, both algorithms are functionally equivalent to a chain of stack sequential decoders [3], where the stack content and path metric of each decoder depend on the outcome of the previous decoder Algorithm I (Input C 0, y, R Output x): Step 1 (Initialization) Set i := m, T m := 0, ξ m := 0 d c = C 0 (current sphere squared radius) Step (Bounds on x i ) If d c < T i go to 4 Else, y i ξ i d c T i r i,i }, B i (x m i+1 ) := A i (x m i+1 {0, { ) := max y min Q 1, i ξ } i+ d c T i r i,i, and set x i := A i (x m i+1 ) 1 Step 3 (Natural spanning of the interval I i (x m i+1 )) x i := x i +1 If x i B i (x m i+1 ) go to 5, else go to 4 Step 4 (Increase i: move one level down) If i = m terminate, else set i := i + 1 and go to 3 (16)

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 49, NO X, { Step 5 (Decrease i: move one level up) If i > 1, then let ξ i 1 := r i 1,j x j, T i 1 := T i + y i ξ i r i,i x i, let i := j=i i 1 and go to } Step 6 (A valid point is found) Compute d := T 1 + y 1 ξ 1 r 1,1 x 1 If d < dc { let d c := d, save x := x, and update the upper boundaries B l (x { m l+1 ) := y min Q 1, l ξ l + } } d c T l r l,l, for all l = 1,, m Go to 3 As in the VB implementation, if no valid point is found, C 0 is increased and the algorithm is restarted Note that the variable ξ i, i = m,, 1, is the decision feedback of a ZF-DFE when the decisions on the symbols from i + 1 to m are the current values of (x i+1,, x m ) As it will become clear in the following, it is useful to visualize sphere decoders for finite PAM signals as a bounded search in a tree In fact, thanks to the upper triangular form of R, the symbol vectors x Z m Q can be represented as paths in a tree of depth m, where the possible values of symbol x i at level i correspond to branches at depth m i + 1 For example, Fig 1 shows the tree for Q = and m = 4, corresponding to the case of uncoded 4-QAM transmission over a multiple antenna channel with M = and N Each branch at depth m i + 1 is labeled by the branch metric defined by w i (x m i ) = y i r i,j x j j=i (17) where x m i = (x i, x i+1,, x m ) T are the symbols labeling the path connecting the branch with the root The path metric for path x m i is defined as M(x m i ) = w j (x m j ) (18) and coincides with the term T i 1 in Algorithm I Generally speaking, a sphere decoding algorithm explores the tree of all possible symbol sequences and uses the path metric in order to discard paths corresponding to points outside the search sphere The main advantage of Algorithm I over the VB algorithm is that once we find a lattice point, we just update all the upper bounds of the intervals without restarting In other words, partial paths in the tree that have already been examined will not be reconsidered after reducing the sphere radius We can prove the following Proposition 1: For a given R, y and C 0, the number of tree nodes visited by Algorithm I is upper bounded by the number of tree nodes visited by the original Pohst enumeration and by the VB algorithm j=i Proof: We give the proof for the VB implementation; the proof is clear for the original Pohst algorithm To this end, it suffices to show that the lower bounds A i (x m i+1 ) when updating d c by d and restarting from Step 1 are smaller than the current values of the components of x (Step 6 in Algorithm I) But this is clear since the current point is inside the sphere of squared radius d Sequential decoders comprise a set of efficient and powerful decoding techniques able to perform close to ML decoding, without suffering the complexity of exact ML decoding, for coding rates not too close to the channel capacity [3], [34] Next, we interpret Algorithm I as a chain of sequential decoders To facilitate this interpretation, we make use of the stack sequential decoding algorithm, briefly summarized as follows Consider a tree of depth m, where each branch at level m i + 1 is labeled by x i Z Q and is associated with a weight w i (x m i ) which depends, in general, on both i and the path x m i connecting the branch with the root The path metric M(x m i ) associated with the path x m i is given by (18) The stack algorithm is defined by a sorting rule used in conjunction with the above path metric At the beginning, the stack contains only the root of the tree with an associated metric equal to zero At each step, the algorithm sorts the stack according to the sorting rule and expands the path at the top of the stack, say x m i, by generating the Q extensions {x m i 1 : x i 1 Z Q } Then, for each extension, the algorithm computes the associated path metric as M(x m i 1 ) = M(xm i ) + w i 1(x m i 1 ), and substitutes the original path with all its extensions The algorithm stops when one path in the stack is fully extended (ie, it reaches depth m) Now, consider the branch metric (17) and the following stack sorting rule (denoted by the ordering relation ): Sorting rule I If I i 1 (x m i ) =, the path xm i cannot be extended and should be eliminated from the stack Consider two paths x m i and u m j such that both I i 1 (x m i ) and I j 1(u m j ) are non-empty Then, x m i u m j if either i < j or i = j and x i u j In words, rule I says that path x m i has priority higher than path u m j if it can be extended without violating the sphere (condition I i 1 (x m i ) ) and either it has depth larger than um j or, if both paths have the same depth, it has a lexicographic priority (the condition x i u j for i = j) Lexicographic ordering corresponds to the natural spanning of the Pohst enumeration Suppose that, by applying the stack algorithm for the first time with the above metric and sorting rule, we obtain one point in the sphere, say x 1 Then, we can apply the stack algorithm again, by letting C 0 = d (y, B x 1 ) and keeping in the stack all the paths generated in the first round This will eventually produce a second point x or declare an empty stack If a second point is found, we repeat the stack algorithm a third time, by letting C 0 = d (y, B x ) and keeping in the stack all the partial paths after the second round We continue in this way until the stack is empty, ie, no more paths in the stack can be extended It is immediate to see that this sequence of concatenated stack sequential decoding steps is functionally equivalent to Algorithm I, in the sense that both algorithms produce the same set of can-

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 49, NO X, didate ML vectors in the same order However, we hasten to say that Algorithm I, as given above, is much more efficient (complexity-wise) than the equivalent sequential decoding formulation, since thanks to the lattice structure of the signal set and the choice of this particular sorting rule, the path at the top of the stack can be predicted at each step without explicitly maintaining and sorting a stack Our second sphere decoding algorithm can now be obtained as a sequence of concatenated stack sequential decoding steps, with the following enhanced sorting rule: Sorting rule II If I i 1 (x m i ) =, the path xm i cannot be extended and should be eliminated from the stack Consider two paths x m i and u m j such that both I i 1 (x m i ) and I j 1(u m j ) are non-empty Then, x m i u m j if either i < j or i = j and M(x m i ) M(um j ) Rule II is identical to rule I, except for the fact that equal length paths are sorted according to their accumulated path metric instead of lexicographic ordering If the stack algorithm based on rule II is applied with C 0 =, the resulting x coincides with the ZF-DFE point given by (14), with the additional constraint that slicing is forced to give a value in Z Q for each component (notice the analogy with the Schnorr-Euchner enumeration) Algorithm II can be concisely formulated as follows Algorithm II (Input C 0, y, R Output x): Step 1 (Initialization) Put the root of the tree in the stack, with the associated metric equal to zero, let d c = C 0 and k = 1 Step (k-th Stack sequential decoding stage): if the stack is empty, terminate, else expand the path at the top of the stack and order the stack with rule II If the top path has depth m go to 3, else repeat step Step 3 (A valid point is found) Let x denote the depth-m path found Then, remove it from the stack, let d c := M(x), save x := x, let k := k + 1 and go to As for Algorithm I, if no valid point is found, C 0 is increased and the algorithm is restarted Again, thanks to the particular structure of the problem and the appropriate choice of the sorting rule, Algorithm II can be implemented in a much more efficient way that does not require maintaining and sorting a stack explicitly This is given as follows Algorithm II, smart implementation (Input C 0, y, R Output x): Step 1 (Initialization) Set i := m, T m := 0, ξ m := 0, and d c := C 0 (current sphere squared radius) Step (DFE on x i ) Set x i := (y i ξ i)/r i,i and i := sign(y i ξ i r i,i x i ) Step 3 (Main step) If d c < T i + y i ξ i r i,i x i, then go to 4 (ie, we are outside the sphere) Else if x i / [0, Q 1] go to 6 (ie, we are inside the sphere but outside the signal set boundaries) Else (ie, we are inside the sphere and signal set boundaries) { if i > 1, then let ξ i 1 := r i 1,j x j, T i 1 := T i + y i j=i } ξ i r i,i x i, i := i 1, and go to Else (i = 1) go to 5 Step 4 If i = m, terminate, else set i := i + 1 and go to 6 Step 5 (A valid point is found) Let d c := T 1 + y 1 ξ 1 r 1,1 x 1, save x := x Then, let i := i + 1 and go to 6 Step 6 (Schnorr-Euchner enumeration of level i) Let x i := x i + i, i := i sign( i ), and go to 3 The main difference with Algorithm I is that given the values of x i+1,, x m, taking the ZF-DFE on x i avoids re-testing other nodes at level i in case we fall outside the sphere Notice also that in the implementation of the algorithm above, the branch metric y i ξ i r i,i x i needs to be computed only once in steps 3 and 5 (even if it appears twice in step 3 and once in step 5) By setting d c =, one ensures that the first point found by the algorithm is the ZF-DFE (or the Babai) point (15) This choice, however, may result in some inefficiency if the distance between the ZF-DFE point and the received signal (referred to as the Babai distance) is very large This inefficiency becomes especially significant at very large dimensions, as the algorithm zig-zags its way in the tree from the ZF-DFE point to the ML point By setting d c to a finite value C 0, one informs the algorithm that the ML solution lies in a sphere of a squared radius C 0, which allows it to retrace its path in the tree if the squared Babai distance is larger than C 0 IV PRE-PROCESSING AND ORDERING The complexity of sphere decoders depends critically on the pre-processing stage, the ordering in which the components of x are considered, and the initial choice of C 0 The standard pre-processing and ordering (implicitly assumed in the formulation of Algorithms I and II) consists of the QR decomposition of the channel matrix B and the natural back-substitution component ordering, given by x m, x m 1,, x 1 However, different pre-processing/ordering approaches may yield a lower expected complexity 1 Columns Ordering According to the Euclidean Norm: In [3], Fincke and Pohst proposed two modifications of the preprocessing stage to further reduce the complexity of the Pohst enumeration for infinite lattices (a) Apply the LLL reduction algorithm [35] on the upper triangular matrix R The LLL algorithm finds a unimodular 1 matrix U such that G = RU and G has reduced lattice vectors and almost orthogonal columns (b) Order the columns of G according to their Euclidean norms in a non-decreasing order Clearly, GΠ where Π is the column ordering permutation matrix, is also a generator matrix for the lattice Λ generated by R 1 A square matrix U is said to be unimodular if it has integer components and its inverse has also integer components or, equivalently, if it has integer components and a determinant equal to ±1

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