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in multipath fading channels , using multiple antennas at the transceivers is known to provide large capacity gains . such a capacity gain comes at the cost of high decoding complexity at the receiver . it is known that a high - complexity joint ml decoder can be employed at the receiver to reliably recover the information . on the other hand , the _ linear receivers _ such as the zf and the mmse receiver @xcite reduce the decoding complexity trading - off error performance .    the integer forcing ( if ) linear receiver has been recently proposed  @xcite this new architecture obtains high rates in mimo fading channels . in this approach , the transmitter employs a layered structure with identical lattice codes for each layer . then each receive antenna is allowed to find an integer linear combination of transmitted codewords . the decoded point will be another lattice point because any integer linear combination of lattice points is another lattice point . this idea has been brought to mimo channels from the compute - and - forward protocol for physical layer network coding  @xcite . in the mimo if architecture , a filtering matrix @xmath1 and a non - singular integer matrix @xmath2 are needed such that @xmath3 with minimum quantization error at high signal - to - noise ratio ( @xmath4 ) values . the exhaustive search solution to the problem of finding @xmath5 is addressed in  @xcite . it is prohibitively complex already for @xmath6 real mimo and becomes untractable for @xmath7 complex mimo and beyond . a smart practical method of finding @xmath2 based on hkz and minkowski lattice reduction algorithms has been proposed recently  @xcite . this provides full receive diversity with much lower complexity in comparison to exhaustive search . the major differences between integer - forcing linear receivers and lattice reduction aided mimo detectors  @xcite are also presented in  @xcite .    in this paper , we propose a low - complexity method for choosing the above matrices . in @xcite , a @xmath8-layered scheme is considered with real lattice codebook for each layer . unlike the model there , we work on complex entries and we lift that set - up to complex case . the proposed method is a combination of three low - complexity methods which are based on complex lattice reduction ( clll )  @xcite technique for a lattice , and singular value decomposition ( svd ) of matrices . for the @xmath0 mimo channel , we compare the performance ( in terms of ergodic rate and uncoded probability of error ) of the proposed low - complexity solution with the known linear receivers and show that the proposed solution ( _ i _ ) provides a lower bound on the ergodic rate of the if receiver , ( _ ii _ ) outperforms the zf and mmse receivers in probability of error , and ( _ iii _ ) trades off error performance for computational complexity in comparison with exhaustive search and other lattice reduction methods including hkz and minkowski algorithms . the rest of the paper is organized as follows . in section  [ sec : back ] , we give a brief background on lattices . we present the problem statement along with the signal model in section  [ sec : model ] . in section [ sec : methods ] , we study the solution to the if receiver via two clll algorithms . a complexity comparison for different known approaches is also given in this section . in section  [ sec : simulations ] , we show some simulation results on the performance of if receiver in ergodic mimo setting . finally , we present concluding remarks in section  [ sec : conclusion ] .    _ notations_. row vectors are presented by boldface letters , and matrices are denoted by capital boldface letters . let @xmath9 be a vector , @xmath10 denotes transposition , and @xmath11 denotes the hermitian transposition . we show the @xmath12 identity and zero matrix as @xmath13 and @xmath14 respectively . for a matrix @xmath15 , the element in the @xmath16-th row and @xmath17-th column of @xmath15 will be denoted by @xmath18 . the sets @xmath19 , and @xmath20 $ ] denote the set of all complex numbers , and the gaussian integer ring , respectively , where @xmath21 . if @xmath22 , then @xmath23 is the real part and @xmath24 is the imaginary part of @xmath25 . let @xmath26 denote the modulus of a complex number . the @xmath27 operation denotes the norm square of a vector . for a complex number @xmath25 , the closest gaussian integer to @xmath25 is denoted by @xmath28 , which we refer as the quantization of @xmath25 . the notation @xmath29 is the component - wise quantized version of the vector @xmath9 . the hermitian product of @xmath30 and @xmath31 is defined by @xmath32 . finally , the set of orthogonal vectors generated by the gram - schmidt orthogonalization procedure are denoted by @xmath33 . a lattice @xmath34 with basis @xmath35 , where @xmath36 , is the set of all points @xmath37}^n\}$ ] . a generator matrix for @xmath34 is an @xmath12 complex matrix @xmath38^t$ ] . the gram matrix of @xmath34 is @xmath39 . the @xmath17th successive minima of @xmath34 , denoted by @xmath40 , is the radius of the smallest possible closed ball around origin containing @xmath17 or more linearly independent lattice points .    in complex lattice reduction , we let @xmath41 , where @xmath42 is an unimodular matrix . let us define @xmath43 where @xmath44 . a generator matrix @xmath45 is said to be clll - reduced if the following two conditions are satisfied  @xcite :    1 .   for @xmath46 @xmath47 2 .   for @xmath48 , @xmath49 where @xmath50 $ ] is a factor selected to achieve a good quality - complexity tradeoff . an algorithm is provided in  @xcite to evaluate a clll - reduced basis matrix @xmath51 of a lattice @xmath34 with a generator matrix @xmath52 . the input of this algorithm is the matrix @xmath53 and a factor @xmath54 , and the outputs of the algorithm are the unimodular matrix @xmath42 and the clll - reduced basis matrix @xmath45 such that @xmath41 . [ sec : model ] a flat - fading mimo channel with @xmath55 transmit antennas and @xmath55 receive antennas is considered . the channel matrix @xmath56 is in @xmath57 , where the entries of @xmath56 are i.i.d . as @xmath58 this channel coefficient remains fixed for a given interval ( of at least @xmath59 complex channel uses ) and take an independent realization in the next interval . we use a @xmath55-layer transmission scheme where the information transmitted across different antennas are statistically independent . for @xmath60 , the @xmath17-th layer is equipped with an encoder @xmath61 . this encoder maps a vector message @xmath62 , where @xmath63 is a ring , into a lattice codeword @xmath64 . if @xmath65 denotes the matrix of transmitted vectors , the received signal @xmath66 is given by @xmath67 , where @xmath68 and @xmath69 denotes the average signal - to - noise ratio at each receive antenna . the entries of @xmath70 are i.i.d . and distributed as @xmath58 . we also assume that the channel state information is only available at the receiver . the goal of if linear receiver is to approximate @xmath56 with a non - singular integer matrix @xmath2 . since we suppose the information symbols to be in the ring @xmath63 , we look for an invertible matrix @xmath2 over the ring @xmath63 . thus , we have @xmath71 as @xmath72 is the useful signal component , the effective noise is @xmath73 . in particular , the power of the @xmath17-th row of effective noise is @xmath74 . hence , we define @xmath75 where @xmath76 and @xmath77 denote the @xmath17-th row of @xmath2 and @xmath78 , respectively . note that in order to increase the effective signal to noise ratio for each layer , the term @xmath79 has to be minimized for each @xmath17 by appropriately selecting the matrices @xmath2 and @xmath80 . we formally put forth the problem statement below ( see  @xcite ) :    given @xmath56 and @xmath81 , the problem is to find the matrices @xmath82 and @xmath83^{n \times n}$ ] such that    * the @xmath84 is minimized , and * the matrix @xmath2 is invertible over the ring @xmath63 .    in order to have zf receiver , we put @xmath85 and @xmath86 . if we let @xmath87 where @xmath88 and @xmath86 , then we get the linear mmse receiver .    in @xcite , the authors have proposed a method to obtain @xmath2 and @xmath80 , which reduces @xmath79 for each @xmath17 . we now recall the approach presented in @xcite . first , conditioned on a fixed @xmath89 , the term @xmath90 is minimized over all possible values of @xmath77 . as a result , the optimum value of @xmath77 can be obtained as @xmath91 then , after replacing @xmath77 of in @xmath90 , one has to minimize @xmath92 over all possible values of @xmath30 to obtain @xmath93 as @xmath94 . the last expression can be written as @xmath95 where @xmath96 is the matrix composed of the eigenvectors of @xmath97 and @xmath98 is a diagonal matrix with @xmath17-th entry @xmath99 , where @xmath100 is the @xmath17-th singular value of @xmath56 . with this , we have to obtain @xmath55 vectors @xmath101 which result in the first @xmath55 smaller values of @xmath102 along with the non - singular property on @xmath2 . in order to get @xmath76 , @xmath103 , the authors of  @xcite have suggested an exhaustive search for each component of @xmath76 within a sphere of squared radius @xmath104 where @xmath105 . it has also been pointed out in  @xcite that this search can be accelerated by means of a sphere decoder on the lattice with gram matrix @xmath106 . for a fixed @xmath81 , the complexity of this approach is of order @xmath107 . it is also shown in  @xcite that the exhaustive search approach provides a diversity order of @xmath55 and a multiplexing gain of @xmath55 . the outage rate and probability of this scheme are studied in  @xcite . at this stage , we note that the exhaustive computation of @xmath76 has high computational complexity , especially for large values of @xmath81 and @xmath55 , and hence the approach in  @xcite is not practical even for the @xmath6 complex case . the best possible @xmath76 s for this problem are the set of all successive minimas of @xmath108 , which can be approximately computed by either the minkowski or hkz lattice reduction algorithms  @xcite . in  @xcite , using hkz and minkowski reduction algorithms , we obtain the matrices @xmath2 and @xmath80 and show that these practical algorithms achieve full receive diversity in terms of error performance with reasonable complexity . since the clll algorithm has lower complexity than hkz and minkowski reduction algorithms , it could be employed to get matrix @xmath2 as well . hence , in the rest of the paper we focus on using clll algorithm to find the best possible @xmath2 for our problem . [ sec : methods ] in this section , we propose three low - complexity methods to obtain some candidates for the rows of @xmath2 . the first two are based on clll algorithm and the last one is based on svd decomposition . then , we propose a selective combining technique to choose the rows of @xmath2 from the candidate rows . once we obtain @xmath2 , we obtain @xmath80 as @xmath109 , where @xmath110 is given in . henceforth , we only address systematic methods to obtain @xmath2 . it is pointed out in @xcite that the minimization problem in is the shortest vector problem for a lattice with gram matrix @xmath111 . since @xmath112 is a positive definite matrix , we can write @xmath113 for some @xmath114 by using choelsky decomposition . with this , the rows of @xmath115 generate a lattice , say @xmath116 . based on , a set of possible choices for @xmath117 is the set of complex integer vectors whose corresponding lattice points in @xmath116 have lengths at most equal to the @xmath55-th successive minima of @xmath116 . however , finding these vectors is again computationally complex . * input : * @xmath118 , and @xmath81 . + * output : * a set of @xmath55 candidates for the rows of @xmath2 .    1 .   form @xmath115 of a lattice @xmath116 . 2 .   reduce @xmath119 to @xmath120 by clll algorithm . the @xmath55 rows of @xmath121 provides @xmath55 values of @xmath76 . we now use the clll algorithm  @xcite to obtain the complex integer vectors . in particular , we use the clll algorithm to reduce the basis set in @xmath119 to obtain a new basis set represented by the rows of @xmath120 . for each @xmath60 , it is known that the length of the @xmath17-th row vector in @xmath120 is upper bounded by a scaled version of the @xmath17-th successive minima of @xmath116  @xcite . hence , the rows of @xmath121 can be used to obtain @xmath55 possible choices for the desired matrix @xmath2 . we note that the structure of the above algorithm is exactly the same as the one presented in  @xcite with only one difference : we used hkz and minkowski lattice reduction algorithms instead of clll algorithm . given @xmath56 and @xmath81 , let us define a @xmath123dimensional complex lattice @xmath34 generated by @xmath124   \in \mathbb{c}^{2n \times 2n}.}\ ] ] the gram matrix of this lattice is @xmath39 . the schur s component of @xmath125 denoted by @xmath126 is given by @xmath127 , while the schur s decomposition of @xmath125 can be derived as , @xmath128 \left[\begin{array}{c|c } { \bf s } & { \bf 0}_{n}\\ \hline { \bf 0}_{n } & ( { \bf m}|{\bf s } ) \end{array}\right ] \left[\begin{array}{c|c } { \bf i}_n & -{\bf s}^{-1}{\bf h}\\ \hline { \bf 0}_{n } & { \bf i}_n \end{array}\right]}.\ ] ] replacing @xmath56 by its svd representation , one can easily observe that @xmath129 . consider @xmath130 where @xmath131 and @xmath132^{2n}$ ] and @xmath133 $ ] forming by adjoining row vector @xmath30 after @xmath31 for some @xmath134^n$ ] . for @xmath135 $ ] with @xmath136^n$ ] , we define @xmath137 . further , we expand the term @xmath138 to obtain @xmath139 . note that @xmath140 . therefore , the solutions to the minimization of @xmath138 are also the solutions to the minimization of @xmath141 . towards minimizing @xmath138 , one can immediately recognize that solving minimization of @xmath138 is nothing but finding the shortest vector of @xmath34 . for the matrix @xmath52 in , let @xmath142 denote the @xmath8dimensional clll - reduced generator matrix of @xmath34 . using the short vectors in @xmath143 , we obtain the complex integer matrix @xmath42 with @xmath8 row vectors @xmath144 $ ] such that @xmath145 . with this , we have obtained complex integer vectors @xmath146 which results in smaller values for @xmath147 . hence , the vectors @xmath146 can be readily used for the if architecture as @xmath148 , and @xmath149 . however , we do not use @xmath150 as it is a complex integer vector . in such a case , @xmath151 even for large @xmath4 values which in turn results in an error floor in the probability of error . instead , we only use @xmath93 , and subsequently obtain @xmath150 using . this algorithm can be summarized as :    * input : * @xmath118 , and @xmath81 . + * output : * a set of @xmath8 candidates for the rows of @xmath2 .    1 .   form the matrix @xmath52 as in  ( [ g ] ) . 2 .   reduce @xmath52 to @xmath143 by clll and compute @xmath152 . 3 .   use @xmath153 $ ] , choose @xmath154 for @xmath155 and put @xmath150 as in . it is important to highlight the difference between algorithm @xmath156 and algorithm @xmath122 . in algorithm @xmath156 , a set of candidates for @xmath93 are chosen via the clll reduction technique on a @xmath55-dimensional complex lattice . however , in algorithm @xmath122 , we jointly obtain @xmath157 and @xmath158 on a @xmath8-dimensional complex lattice , and only use @xmath149 as candidates for the rows of @xmath2 . note that vectors @xmath93 delivered from algorithm @xmath122 can be different from that of algorithm @xmath156 , since algorithm @xmath122 attempts to minimize @xmath159 , whereas algorithm @xmath156 attempts to solve minimization of @xmath160 . note that , the clll algorithm guarantees the existence of at least @xmath55 vectors of the lattice @xmath34 such that the vector in the first row of @xmath161 is most likely the shortest vector of the lattice .      for large values of @xmath81 , we can  @xcite write @xmath163 in as @xmath164 , where @xmath165 denotes the @xmath16-th row of @xmath96 . from the svd property , we have @xmath166 and @xmath167 . if @xmath168 for @xmath169 , then the above equation suggests us to select all @xmath76 , @xmath103 , along the direction of @xmath170 , and as short as possible . on the other hand , if @xmath171 is large for each @xmath16 , and are comparable , then a set of good candidates can come from choosing a complex integer vector along each @xmath165 .    * input : * @xmath118 , and @xmath81 . + * output : * a set of @xmath55 candidates for the rows of @xmath2 .    1 . obtain the svd of @xmath56 as @xmath172 . 2 .   choose @xmath173 for @xmath60 . therefore , another @xmath55 possible choices for integer vectors can be obtained from the rows of @xmath96 as @xmath174 for @xmath175 , where @xmath176 denotes the nearest integer of a real number .      till now , we have proposed three different low - complexity algorithms to obtain candidate vectors for the rows of @xmath2 . algorithm @xmath156 gives us @xmath55 choices , algorithm @xmath122 delivers @xmath8 choices , whereas algorithm @xmath162 brings another @xmath55 possible choices . overall , we can use all the @xmath177 candidate vectors for @xmath76 and obtain the corresponding vectors @xmath77 , @xmath178 using the relation in . with this , we have @xmath177 pairs of @xmath179 . now , we proceed to sort these @xmath177 vectors in the increasing order of their @xmath180 values . then , we find the first @xmath55 vectors which form an invertible matrix over the ring @xmath63 . note that all the three algorithms have low computational complexity . as a result , the combining algorithm has lower computational complexity than the exhaustive search and other lattice reduction algorithms like hkz and minkowski , and hence , the proposed technique is amenable to implementation . we refer to the combined solution as  combined clll - svd solution " . it is clear that the combined clll - svd method includes at most a svd algorithm and a matrix inversion with complexity @xmath181 , a @xmath8-dimensional clll algorithm with complexity @xmath182 following by a sorting algorithm of size @xmath177 with complexity @xmath183 . therefore the complexity of this approach is @xmath181 and it is obviously independent of @xmath81 . the complexity of the proposed exhaustive search in  @xcite is known to be @xmath107 . the complexity of other reduction algorithms like hkz and minkowski are given in  @xcite . in table [ table : complexitycompare ] , we compare the complexity of finding matrix @xmath2 for different known methods .    [ cols="^,^,^",options="header " , ] [ sec : simulations ] [ sec_4 ]     ergodic rate of various linear receivers for @xmath0 mimo channel.,title="fig:",width=287 ]     in this section , we present simulation results on the ergodic rate ( see  @xcite or  @xcite for definition of ergodic rate ) and the probability of error for the following receiver architectures on @xmath0 mimo channel : ( _ i _ ) if linear receiver with exhaustive search , ( _ ii _ ) if linear receiver with minkowski lattice reduction solution  @xcite , ( _ iii _ ) if linear receiver with combined clll - svd solution , ( _ iv _ ) the zf and mmse linear receiver , and ( _ v _ ) the joint maximum likelihood ( ml ) decoder . for the if receiver with exhaustive search , the results are presented with the constraint of fixed radius for the exhaustive search . we have not used the radius constraint given in as the corresponding search space increases with @xmath81 . instead , we have used a fixed radius of @xmath184 for all values of @xmath81 . by relaxing this constraint , we have reduced the complexity of brute force search , noticeably .    in fig . [ fig : ergodicratecapacity22 ] , we present the ergodic rate of the above listed receivers , wherein , for the case of ml receiver , the ergodic capacity  @xcite of @xmath0 mimo channel has been presented . the rate for if linear receiver with minkowski lattice reduction solution is cited from  @xcite . we observe that the combined clll - svd solution performs pretty much the same as if receiver with exhaustive search and if based on minkowski lattice reduction solution at low and moderate @xmath4s . also , note that the combined clll - svd and minkowski lattice reduction solutions give lower bounds on the ergodic rate of exhaustive search based if receiver while the latter one is tighter . for the ergodic rate results of the if receiver , we have used @xmath2 matrices which are invertible over @xmath20 $ ] . now , we present the uncoded bit error rate ( ber ) for the above receiver architectures with @xmath185-qam constellation . we use the finite constellation @xmath186 carved out of the infinite lattice @xmath20 $ ] . note that @xmath20 = \mathcal{s } \oplus 2\mathbb{z}[i]$ ] . in this method , the received vector is of the form , @xmath187 where @xmath188 . for the above setting , we use modulo lattice decoding at the receiver . each component of @xmath189 is decoded to the nearest point in @xmath20 $ ] and then modulo @xmath122 " operation is performed independently on its in - phase and quadrature component . with this , we get @xmath190 from both the components of @xmath189 . further , we solve the system of linear equations @xmath191 over the ring @xmath63 to obtain the decoded vector @xmath192 . ber for various linear receivers with @xmath185-qam constellation.,title="fig:",width=287 ]     in fig . [ fig : qpsk22 ] , we present the ber results for all the five receiver architectures . note that both the if receiver with combined clll - svd and minkowski lattice reduction solutions outperform the zf and mmse architectures , but trades - off error performance for complexity in comparison with brute force search . in particular , the combined clll - svd approach fails to provide diversity results as that of the exhaustive search and minkowski lattice reduction approaches . this diversity loss is due to the larger value of @xmath193 delivered by the combined clll - svd algorithm in comparison with the optimum solution . for the probability of error results of the if receiver , we have used @xmath2 matrices which are invertible over @xmath194 $ ] . we have observed similar results for @xmath195 mimo channels as well . [ sec : conclusion]in  @xcite , algorithm @xmath156 along with hkz and minkowski lattice reduction algorithms was employed to find the matrix @xmath2 . algorithm @xmath156 which includes the complex lll method as an alternate technique for the above reduction methods turned out to be not satisfactory in terms of ergodic rate and error performance . hence , we have tried to improve effectiveness of clll algorithm by combining it with other approaches such as algorithms @xmath122 and @xmath162 . we have proposed a low - complexity systematic method called the combined clll - svd algorithm for the mimo if architecture . simulation results on the ergodic rate and the probability of error were also presented to reveal the effectiveness of combined clll - svd solution versus other linear receivers . the proposed combined algorithm trades - off error performance for complexity in comparison with both if receivers based on exhaustive search and minkowski or hkz lattice reduction algorithms . further improvements are required to achieve results which are competitive with if receivers based on exhaustive search and the ones presented in  @xcite . a.  sakzad , e.  viterbo , y.  hong , and and j.  boutros ,  on the ergodic rate for compute - and - forward , " _ proceeding of international symposium on network coding 2012 ( netcod 2012 ) , mit university , boston , ma , usa_.            j.  zhan , b.  nazer , u.  erez , and m.  gastpar , `` integer - forcing linear receivers , '' _ information theory proceedings ( isit ) , 2010 ieee international symposium on _ , pp .  10221026 , 2010 . extended version is available at : http://arxiv.org/abs/1003.5966 . w.  zhang , s.  qiao , and y.  wei , `` hkz and minkowski reduction algorithms for lattice - reduction - aided mimo detection , '' _ to appear in ieee trans . on signal processing_. available at : http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=@xmath196arnumber=6256756 .
integer - forcing ( if ) linear receiver has been recently introduced for multiple - input multiple - output ( mimo ) fading channels . 
 the receiver has to compute an integer linear combination of the symbols as a part of the decoding process . 
 in particular , the integer coefficients have to be chosen based on the channel realizations , and the choice of such coefficients is known to determine the receiver performance . 
 the original known solution of finding these integers was based on exhaustive search . a practical algorithm based on hermite - korkine - zolotareff ( hkz ) and minkowski lattice reduction algorithms 
 was also proposed recently . in this paper 
 , we propose a low - complexity method based on complex lll algorithm to obtain the integer coefficients for the if receiver . for the @xmath0 mimo channel , we study the effectiveness of the proposed method in terms of the ergodic rate . 
 we also compare the bit error rate ( ber ) of our approach with that of other linear receivers , and show that the suggested algorithm outperforms the minimum mean square estimator ( mmse ) and zero - forcing ( zf ) linear receivers , but trades - off error performance for complexity in comparison with the if receiver based on exhaustive search or on hkz and minkowski lattice reduction algorithms . 
 clll algorithm , mimo , linear receivers .