GLRT-Optimal Noncoherent Lattice Decoding

IEEE TRANSACTIONS ON SI GNAL PR OCESSING, A CCEPTED TO AP PEAR (A CCEPTED NO V . 2006) 1 GLR T -Optimal Noncoherent Lattice Decoding Daniel J. Ryan †‡∗ , Iain B. Collings ‡ and I. V aughan L. Clarkson ⋆ † School of Elect rical and Information Engineering, Uni versi ty of Sydney , A U S T R A L I A , Phone: +612 9372 4465, Fa x: +612 9372 4490, Email: dan@ee.usyd.edu.au ‡ W ireless T echnologie s Laboratory , CSIR O ICT Centre, A U S T R A L I A , Phone: +612 9372 4120, Fa x: +612 9372 4490, Email: iain.collings@csiro. au ⋆ School of Informat ion T echnology and Electrica l Engineering, Unive rsity of Queensla nd, A U S T R A L I A , Phone: +617 3365 8834, Fax: +617 3365 4999, Email: v.clarkson@itee.uq .edu.au Abstract This paper presen t s ne w low-comple xity lattice-decoding algorithms for noncoher ent block detection of QAM and P AM signals ov er complex -v al ued fading channels. The algorithms are optimal in terms of the generalized likelihood rati o test (GLR T). The compu t ational comple xity is polynomial in the block length; making GL R T -optimal noncohe rent detection feasible for implementation. W e also provide even lower complexity suboptimal algorithms. Simulations sho w t hat the suboptimal algorithms hav e performance indistinguishable from the optimal algorithms. Finally , we consider block based transmission, and propose to use noncoherent detection as an alternativ e to pilot assisted transmission (P A T). The new technique i s shown to outperform P A T . Index T erms Noncoheren t detection, lattice decoding, wireless communications. I . I N T R O D U C T I O N Noncoh erent tr ansmission of d igital signals over unkn own fading chan nels has recently received significant attention especially for the case of the block -fading cha nnel m odel. Application s in clude recovery fr om deep fades in pilot-symbo l assisted modu lation based sche mes, eavesdropping, and no n-data-aided channel estimation . Noncoh erent transmission is particularly applicable to systems exhibiting small coherence intervals wh ere the use of training signals would result in a significan t loss in thr oughpu t. Recently , some elegant inform ation-theoretic results have been deri ved for noncoh erent single and multiple- antenna systems under the assump tion of Rayleigh fading , for I. V . L . Clarkson was on s tudy leav e throughout 2005 at the Dept. of Electric al & Computer Engineeri ng, Univ ersity of British Columbia, Canada . T his paper has appeared in part at VTC-Spring 2006 and ICASSP 2006. IEEE TRANSACTIONS ON SI GNAL PR OCESSING, ACCEPTED TO AP PE AR (A CCE PTED NO V . 2006) 2 example [ 1, 2]. Inform ation theore tic aspects of no ncoherent tran smission wer e co nsidered in [3] wh ich conclude d that at low SNR an d for small co herence inter v a ls there is a significan t capacity pen alty by using tra ining. Under this no ncoherent detection r egime, it has be en shown by n umerical simulation tha t standard mo dulation techniqu es such as quad rature amplitu de modula tion (QAM) can achieve near-capacity in th e single-antenn a noncoh erent block Rayleigh-fading channe l [4]. This paper f ocuses on n oncohere nt receiver design for the block fading channel. A great d eal o f work h as b een perfor med on partially coher ent receivers such as p ilot-symbol a ssisted mo dulation ( PSAM) [5, 6], per-surviv or technique s [7], an d coup led estimators [8]. Ho wev er the cha llenge remain s to develop h igh-perf ormance, low- complexity , fully noncoher ent receivers. V arious subo ptimal algor ithms have been propo sed for block- based nonco herent detection . For slowly fading channels, a b lind phase recovery appro ach was prop osed in [ 9] for nonco herent de tection of differentially encoded QAM [10] whe re the atten uation was assum ed to be kn own exactly at the receiver . In [1 1], a suboptimal techniqu e for PSK was propo sed which in volved for ming a nu mber of e qually spaced channel p hase estimates. An extension to multi-amp litude constellations was also presented , where e very sequence of symbol a mplitudes is co nsidered, and then th e PSK techn ique is applied to determine the phase of the symb ols. Unfortu nately , the com plexity o f this suboptimal appr oach is still exponen tial in th e sequence leng th, a lbeit with a smaller ba se. Recently , lattice deco ding algorithms have been a pplied to nonc oherent a nd differential d etection. For PSK over tempor ally-correlated Rayleigh fading channels, a fo rm of lattice decoding ( namely sphere -decoding ) c an b e applied since it tu rns out that the d etection metric is Euc lidean [12]. Lattice decodin g tech niques ha ve also been used for d if f erential detection o f diag onal space-time b lock code s over Rayleigh fading chan nels, by a pproximatin g the decision metric with a Eu clidean me tric [13 , 14]. I n [1 5], we presented simu lation results for anoth er lattice projection approa ch for suboptimal P AM and QAM detection. Unfortun ately , each of these algor ithms requir e complexity exponential with the b lock length to guaran tee that the optimal estimate is f ound [16 , 17]. Practical implementatio n consideratio ns deman d that low comp lexity alg orithms be developed. For the case of the co nstant en velop e PSK co nstellation, a detectio n algorithm with complexity O ( T log T ) was developed in [18 , 19] (where T is the block length ), which can provide the o ptimal data estimate over an unk nown nonco herent fadin g chann el, in term s of the Generalized Likeliho od Ratio T est (GLR T). The GLR T is equiv alent to joint ML estimation of a co ntinuous valued ch annel pa rameter and discrete-valued d ata param eters. This app roach was g eneralized in [20], wh ere th ey outlin ed a gener al graph -based appro ach which in volved fo rming a spann ing tree. Spe cific details were presented for the cases of QAM over a phase noncohe rent channe l ( i.e. kn o wn channel amplitude) [20 , 21], and f or PSK over a fading chann el with coding [22 ]. The ch allenge r emains to develop efficient IEEE TRANSACTIONS ON SI GNAL PR OCESSING, ACCEPTED TO AP PE AR (A CCE PTED NO V . 2006) 3 algorithm s for op timal n oncoheren t sequen ce d etection o f m ulti-amplitude constellations over fadin g c hannels. In this pap er we propose a new GLR T -optimal nonco herent l attice decod ing approach for QAM and P AM symb ols which has co mplexity po lynomial in the blo ck length. W e start by consid ering detection o f M -ary P AM over r eal- valued fading channe ls (which we shall term r eal-P AM). W e show that the GLR T -optimal codeword estimate is the closest co dew or d (or lattice poin t) in a ngle to the line described by the received vector . W e p ropose an algorithm that search es along the line, and chooses the best codew ord estimate fr om this sear ch. W e provide a th eorem that bound s the search to a segment of the line, lim iting the number o f codewords that n eed to be consider ed. W e show how the search can be do ne in an iterative man ner , and th at the resulting complexity o f th e alg orithm is O ( T log T ) . W e then co nsider the more practical case of M -ary QAM detection over comp lex-v a lued fading chan nels, an d show th at in this case the GLR T -o ptimal codeword estimate is the clo sest c odew o rd in angle to a plan e de scribed by the received vector . W e p ropose an algorithm that searches across the plane, and chooses the best cod e word estimate fro m th is search. W e provide a theorem that bounds the search to a segment of the plane. W e show how the extent of the search can b e furthe r reduced by exploiting th e rotatio nal symmetry of the constellation. Th e resulting plan e sear ch a lgorithm can be per formed with complexity of order O ( T 3 ) . W e also present n e w subop timal nonc oherent QAM detection alg orithms with even lower complexity; by com- bining a cha nnel phase estimator with ou r fast real-P AM algo rithm. W e propo se using O ( T ) instances o f the real-P AM algo rithm. This appro ach the refore has co mplexity o f ord er O ( T 2 log T ) . Simulations in dicate that there is a negligib le perfo rmance loss compared to GLR T , when using this suboptimal techniqu e. Finally , we also p ropose a p ilot-assisted version of our new redu ced-search noncoher ent lattice- decoding algo- rithms. Th e pilo t symbol is used to r emove th e amb iguities inheren t with no ncoherent detection . Our ap proach obtains improved perfor mance compared with standard pilot assisted transmission [6], while main taining the same data rate. I I . S Y S T E M M O D E L A. Sig nal Mod el W e defin e a cod ebook C ( X , T ) as the set o f all po ssible seq uences o f T tran smitted symb ols, x = [ x 1 , . . . , x T ] ′ , such that each x t is in some constellation X . For an M -ary P AM con stellation, X = {± 1 , ± 3 , . . . , ± ( M − 1) } . For QAM, X is a subset of the Gaussian (complex) integers with odd real and imaginary components. For example, for an M 2 -ary square QAM co nstellation, X = { x | Re { x } ∈ X ′ , Im { x } ∈ X ′ } where X ′ , {± 1 , ± 3 , . . . , ± ( M − 1) } . Thus each co debook C ( X , T ) is a set of lattice poin ts drawn fr om a subset of the unit lattice of R T or C T . W e consid er block fading cha nnels and assume that the chann el h is con stant for at least T symbols as in [1–4 , IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 4 11]. W e will co nsider n arrowband fading channels where h is eithe r real-valued and complex-valued channels. Thus we can wr ite the received codeword y = [ y 1 , . . . , y T ] ′ as f ollows , y = h x + n (1) where n = [ n 1 , . . . , n T ] ′ is a vector o f a dditiv e white Gaussian n oise. B. Detection The n oncoheren t detectio n problem is to estimate x based on y without knowledge o f the channel and in the absence of train ing data. The log -likelihood f unction of the m aximum likelihood (ML) detector (of b oth channel and data) is gi ven b y L ( y ; x , h ) = − k y − h x k 2 (2) where constant factors have been discarde d and k·k repr esents the Eu clidean norm . For a gi ven codeword hypothesis ˆ x , the likelihood fu nction is maximized by choosing ˆ h = ˆ x † y k ˆ x k 2 (3) where ( · ) † denotes Herm itian tr anspose. Hence, the ML estimate of x conditioned on th e co rrespondin g chan nel estimate, is given by ˆ x opt = arg max ˆ x ∈C ( X ,T ) L y ; ˆ x , ˆ x † y k ˆ x k 2 ! = arg max ˆ x ∈C ( X ,T )   ˆ x † y   2 k ˆ x k 2 (4) This is th e Generalized Likelihoo d Ratio T est (GLR T) [2 3] con sidered in [11, 20]. Note that ( 4) is equivalently given by ˆ x opt = arg max ˆ x ∈C ( X ,T )   ˆ x † y   2 k ˆ x k 2 k y k 2 (5) = arg max ˆ x ∈C ( X ,T ) cos 2 θ ( ˆ x , y ) (6) where θ ( x , y ) is th e principa l angle b etween x and y [2 4]. Thus ˆ x opt can be f ound by searching th e points of C ( X , T ) to find the one closest in angle to y . For QAM, we can also obtain a ge ometric interpr etation of (4) by expressing the complex vectors in R 2 T . W e will use th e under score notation x to den ote the mapped version of x as follows, x = [ Re { x 1 } Im { x 1 } . . . Re { x T } Im { x T } ] ′ (7) IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 5 and deno te the rea l-valued codebo ok as C R ( X , T ) = { x | x ∈ C } . For M 2 -ary squar e QAM, we therefore hav e C R ( X , T ) = C ( X ′ , 2 T ) where X ′ is an M -ary P AM constellation. W e also define Y ∈ R 2 T × 2 as a basis for the subspace y C mapp ed into th e re al space R 2 T ; that is Y ,    Re { y 1 } Im { y 1 } . . . Re { y T } Im { y T } − Im { y 1 } Re { y 1 } . . . − Im { y T } Re { y T }    ′ . (8) Note that th e column s of Y are ortho gonal. The projection matrix P ( y ) ∈ R 2 T × 2 T is de fined as P ( y ) , Y Y ′ k y k 2 (9) such that P ( y ) x = arg min v ∈ Y R 2 k x − v k . That is, the vector P ( y ) x is the p rojection of x onto the subsp ace Y R 2 . Now , it can be ea sily shown that ˆ x opt = arg max ˆ x : ˆ x ∈C ( X ,T ) cos 2 θ ( ˆ x , P ( y ) ˆ x ) Thus the GLR T -op timal data estimate ˆ x opt , cor responds to the ˆ x ∈ C R ( X , T ) closest in angle to the p lane Y R 2 . It is important to no te th at two forms of a mbiguity exist for this no ncoheren t detection prob lem. Th e first is the well-known p hase ambiguity which occu rs fo r any con stellation that is inv ariant to a par ticular ph ase ro tation. For example, for square QAM co nstellations the following fo ur optimal channel estimate and codeword pairs have the same likelihood: ( ˆ h opt , ˆ x opt ) , ( − ˆ h opt , − ˆ x opt ) , ( − i ˆ h opt , i ˆ x opt ) and ( i ˆ h opt , − i ˆ x opt ) ; correspon ding to th e fo ur π / 2 rotations of the constellation. W e will assume that this typ e of ambig uity can be r esolved, for example, by using the phase of the last symb ol fr om the previous cod e word [4], or by using differential encoding [10]. The second type of amb iguity we call a divisor ambig uity a nd a rises when there are multiple po ints in C ( X , T ) that lie on the same 1 -dimensional (real or complex) sub space e.g. [1 , 1 , 1 ] an d [3 , 3 , 3 ] fo r 4-ary real-P AM with T = 3 . T his produ ces a lower boun d on the noncoher ent blo ck detection error r ate as discussed and analyzed in [25]. I I I . R E D U C E D S E A R C H S PAC E In this section we show that the GLR T -o ptimal data estimate ˆ x opt , can b e foun d without testing all the elements of C ( X , T ) . In the previous section we established that ˆ x opt is th e codeword clo sest in angle to a par ticular subspace, so it naturally makes sense to define a ‘nearest neighb or set’ of the subspace and search within th at set. The subspace of interest has basis vecto r y an d passes throu gh th e orig in. W e show that the nearest n eighbor set fo r th is subspace con tains ˆ x opt . This implies tha t low co mplexity dec oding algorithm s can b e developed, ba sed o n finding this particu lar near est neig hbor set, and sear ching it. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 6 Definition 1: W e define N N ( v ) to b e the point, o r set of points, in C ( X , T ) closest to the arbitrary po int v ( i.e. , the nea rest ne ighbor to v ). T hat is, d is an element of N N ( v ) if k v − d k 6 k v − z k f or all z ∈ C ( X , T ) . Of cou rse, usually N N ( v ) will h a ve a single element, and in this case we can write N N ( v ) = d . Definition 2: W e define N ( C ( X , T ) , y ) to be the nearest neigh bor s ubset of the codebook C ( X , T ) , correspon ding to the sub space with b asis vector y , passing thr ough th e origin. That is, u ∈ N ( C ( X , T ) , y ) if and only if there exists some λ suc h th at N N ( λ y ) = u . Note that f rom a geome trical perspec ti ve, it is useful to think o f λ as being equiv alent to the in verse o f a chann el estimate; implyin g that a p oint u is in the nearest neighb or set if the re is a cha nnel estimate ˆ h such that the d istance | y − ˆ h u | is smaller than for any other point. Consequently we define ˆ λ opt , ( ˆ h opt ) − 1 as the reciprocal of th e o ptimal channel estimate. The following p roperty of the GLR T -optima l codeword estimate ˆ x opt , allo ws us to red uce the set of codewords which need to be tested, to a sm all subset o f the |X | T possible c odew o rds (where |·| deno tes set cardina lity). No te that an equivalent result was pre sented in [20], howev er the geome trical in terpretation of our formulation is more apparen t; and is im portant when developing ou r n e w sear ch a lgorithms later . Pr operty 1 : ˆ x opt ∈ N ( C ( X , T ) , y ) . Pr oof: Consider the case wh ere ˆ x opt / ∈ N ( C ( X , T ) , y ) . From Definition 2 this implies th at the re exists som e ˆ x ∈ N ( C ( X , T ) , y ) such that || ˆ λ opt y − ˆ x || < || ˆ λ opt y − ˆ x opt || howe ver this would imply L ( y ; ˆ x , ˆ h opt ) > L ( y ; ˆ x opt , ˆ h opt ) from (2) an d h ence we have a proof by co ntradiction. I V . P A M D E T E C T I O N F O R R E A L - V A L U E D F A D I N G C H A N N E L S This section presents a low complexity algorithm for GL R T -optim al noncoh erent P AM de tection over rea l-v alued channels. Practically , such channels arise in ba seband transmission (eg. multi-level PCM), o r in cer tain bandpass systems where p hase and freq uency are separately estimated b y a phase- locked loop. W e first present a the orem th at we will u se to red uce the number of co dew or ds that need to be examined , even beyond the limitations imp osed by Pro perty 1. Note th at in this real-valued chann el ca se, the subspace o f interest (defined b y y ) actually reduces to a lin e, y R . Th e theorem implies that only a limited extent of the line needs to be searched ; and that th e exten t depen ds on the largest value of y . W e th en pr opose a fast low-comp lexity iterative algorithm to perfor m the search. In the sequel, we will extend the alg orithm to com plex-valued chann els. Later , we will dir ectly incorp orate the algo rithm from this section into an extremely low comp lexity subop timal alg orithm for nonco herent detection over the mo re comm only enco untered co mplex-valued chan nels. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 7 A. Limiting the Sear ch Spa ce Theor em 1 : For nonc oherent detection o f M -ary P AM codewords of len gth T over a real-valued fading cha nnel   ˆ λ opt y t   6 M + T − 2 for all t = 1 , 2 , . . . , T . Pr oof: Define n , arg max t {| ˆ λ opt y t |} . Note that if | ˆ λ opt y n | 6 M then the theorem is satisfied. Now , consider the alternative case when | ˆ λ opt y n | > M . Rearranging the GLR T -optimal channel estimate in (3) giv es ( ˆ x opt ) ′ ( y − ˆ h opt ˆ x opt ) = 0 and h ence ( ˆ x opt ) ′ ( ˆ λ opt y − ˆ x opt ) = 0 . (10) W e will use this property to bou nd ˆ λ opt . Using (2) and the fact that C ( X , T ) con tains all possible sequence s { x | x t ∈ X ∀ t } , the elements of ˆ x opt can be determined o n a n element-w ise basis a s ˆ x opt t = arg min x ∈X   ˆ λ opt y t − x   . (11) For the case we are c onsidering where | ˆ λ opt y n | > M , it follows that since the largest P AM con stellation values are ± ( M − 1) , th at ˆ x opt n = sgn { y n } ( M − 1) whe re s gn is the signum function. W e n o w substitute (11) into (10) to bo und ˆ λ opt , which gi ves ˆ x opt n  ˆ λ opt y n − ˆ x opt n  = − X t 6 = m ˆ x opt t  ˆ λ opt y t − ˆ x opt t  . (12) Now (11) an d the symmetry of the P AM con stellation implies th at sgn  ˆ x opt t  = sgn { ˆ λ opt y t } . Moreover , since ˆ x opt t ∈ {± 1 , ± 3 , . . . , ± ( M − 1) } it follows form the definitio n of ˆ λ opt that − 1 < ˆ λ opt y t − ˆ x opt t < 1 for all ˆ x opt t except ± ( M − 1 ) . More generally , for all ˆ x opt t , sgn  ˆ x opt t  ( ˆ λ opt y t − ˆ x opt t ) > − 1 an d hence ˆ x opt t  ˆ λ opt y t − ˆ x opt t  > −| ˆ x opt t | . Substituting this into (12) gives ˆ x opt n  ˆ λ opt y n − ˆ x opt n  6 X t 6 = m   ˆ x opt t   and hen ce   ˆ λ opt y n   =      ˆ x opt n + 1 ˆ x opt n X t 6 = n   ˆ x opt t        6 M − 1 + ( T − 1)( M − 1) M − 1 = M + T − 2 . Therefo re, since | ˆ λ opt y t | 6 | ˆ λ opt y n | for a ll t the theorem is proved. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 8 B. P AM Alg orithm In this Section we use Pr operty 1 and Theor em 1 to develop a low-complexity algorithm fo r real-P AM d etection. This algorithm reduces the n umber of codewords f or which the dec ision metric is e valuated to o rder M T , which is mu ch smaller than the set of all possible M T codewords that would be co nsidered by an exhaustiv e search. Furthermo re, we demonstrate how the algorithm can be im plemented in an itera ti ve man ner so that the c omplexity is O ( T log T ) . Property 1 implies that ˆ x opt can be fou nd b y calculating the metr ic in (4) for only tho se ˆ x ∈ N ( C ( X , T ) , y ) , i.e. for only tho se ˆ x ∈ C for wh ich the lin e y R passes throug h its Euclidean nearest neigh bor region . Furthe rmore, Theorem 1 implies that only a finite segment of the line need be con sidered. W e have dem onstrated such a sear ch in Figure 1, w hich shows the po siti ve axes f or 8- ary P AM with T = 2 , w here the shaded region s ind icate the nearest neighbo r regions o f the po ints which need to b e search ed. Th e spe cifics of the a lgorithm are as follows. First, for ease of notation we mo dify th e received c odew o rd y by chan ging the sign s of all negative elements in y . Th is will mean that the correspon ding (mo dified) ˆ x opt will now hav e all po siti ve elements. The true (orig inal) GLR T estimate of x can be obtained b y apply ing the r e verse sign chang es to ˆ x opt . Ob serve that we can do this without loss o f g enerality since the P AM c onstellation is symmetric arou nd zero . Definition 3: W e define P ( ˆ x ) to be the range o f λ such that ˆ x is the nearest n eighbor to λ y , within the limits 0 < λ < λ max , ( M + T − 2) / max t { y t } (where the limits are du e to Th eorem 1 and the fact that all x t are greater than zero for the m odified r ecei ved codeword). Formally , P ( ˆ x ) , { λ | ˆ x ∈ N N ( λ y ) , λ ∈ (0 , λ max ) } . Note that each n on-empty P ( ˆ x ) cor responds to a distinct interval of the line y R . The propo sed algorithm proceeds by enume rating these non-emp ty P ( ˆ x ) ’ s, by first enu merating their boun dary points along the line y R . W e then sort the bo undary po ints so that the decision metrics f or the co rrespondin g ˆ x can be ca lculated in an iterati ve man ner . For real-P AM, the b oundary values of λ can be shown to be giv en by ν t,b = b y t for all t = 1 , . . . , T and b = 2 , 4 , . . . , M − 2 such that 0 < ν t,b < λ max (where the values of b co me from the regular b oundaries in the positive half of th e P AM constellation X ). W e use V 0 to denote th e set of all ( ν t,b , t ) pairs. W e then sort the elem ents of V 0 in ascending o rder of their ν t,b value, and a ppend th e value ( λ max , 0) to the end of the order ed set (since this is the outer boun dary of the segment of the line y R wh ich needs to b e searched , accordin g to T heorem 1; whe re the second elemen t of the pair is arbitrarily set to 0 since it is not needed in the algorithm ). W e den ote the n e wly ord ered set by V , a nd ind ex it by k , ( i.e. we d enote its k th elem ent by ( ν k , t k ) ). These ordered values are shown o n the exam ple case of Figure 1, wh ere th e values of ν k denote the distance alo ng the line y R whe re th e line crosses fro m one n earest neighbo r region in to the next. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 9 W e now show how V can be used to enumer ate the cod e word s wh ich need to be searched, and show how to calculate the correspo nding decision metrics in an iterative m anner . The algorith m c an be v isualized geo metrically as search ing along a segment of the lin e y R , by iterating in k . Whenever the line crosses fr om the n earest neighb or region of one lattice poin t (codeword) to the nearest neighbo r r egion of a nother, we calculate th e metric for the new lattice point. Note that in this context, the value of t k indicates the dimension of the b oundary th at is g oing to be cr ossed (in the T -d imensional space) wh en leaving the k th se g ment of the line . The iterative search starts with the codeword ˆ x (1) = 1 , [ 1 , 1 , . . . , 1] ′ ; wh ich has a correspond ing decision metric L ( 1 ) = ( 1 ′ y ) 2 / k 1 k 2 , wh ere L ( ˆ x ) , ( ˆ x ′ y ) 2 / k ˆ x k 2 is the likeliho od function in (4). W e will use the symbol ˆ λ as a marker fo r the m ost likely co dew or d, an d we initially set it to ˆ λ = ν 1 / 2 ( i.e. during the iteration process, ˆ λ will be u pdated wh enev er a codeword is f ound to have a high er likelihoo d than any previously sear ched c odew o rd, and the v a lue of ˆ λ will be chosen such tha t N N ( ˆ λ ) gives the n e w codew o rd). The iteration proceed s b y noting that each time a nearest n eighbor bound ary is cr ossed, only o ne element o f the T -dimensional n earest neig hbor codeword vector changes (since for real-P AM, the b oundaries are straigh t lines, orthog onal to one of the d imensions, and parallel to all th e other s). T herefore the k th cod e word which ne eds to be considered , is calcu lated f rom th e ( k − 1 ) th codeword, on an elem ent-wise basis as f ollows : ˆ x ( k ) p =      ˆ x ( k − 1) p , for p 6 = t k − 1 ˆ x ( k ) p + 1 , for p = t k − 1 . (13) W e define α k , ( ˆ x ( k ) ) ′ y and β k ,   ˆ x ( k )   2 , and hence L ( ˆ x ( k ) ) = α 2 k /β k is the decision metric for th e k th codeword consid ered. The values α k and β k are calcu lated iteratively as follows, α k = α k − 1 + 2 y t k − 1 (14) β k = β k − 1 + 4 ˆ x t k − 1 + 4 . (15) If L ( ˆ x ( k ) ) improves on the previous best codeword estimate then we upd ate ˆ λ in the interior of P ( ˆ x ( k ) ) , by setting ˆ λ = ( ν k + ν k − 1 ) / 2 . Once all segments of th e line have be en sear ched, we have ˆ x opt = N N ( ˆ λ y ) . Pseudo -code for the algor ithm is giv en in T able III. The complexity o f th e algo rithm is a fu nction of the numb er of intersection points ν t,b , i.e. , N I , |V 0 | , where |·| denotes set cardinality . N I is u pper bou nded by ( M / 2 − 1) T , however in genera l it will be mu ch less than th is due to the restricted line search implied b y Theo rem 1, as shown by simulation in Section VII. The sorting of V 0 can be perform ed u sing standard sortin g techniques in O ( N I log N I ) [2 6]. The u pdates (13), (1 4) and (15) hav e complexity O (1 ) , and the final calculation of ˆ x opt is of order T . Thus the overall complexity is dominated by the IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 10 sorting ope ration, an d hence the complexity of the algorith m is o f order O ( T log T ) ; a significa nt improvement compare d with an exhausti ve search over all M T possible co dew or ds in the co debook C ( X , T ) . V . G L RT - O P T I M A L Q A M D E T E C T I O N F O R C O M P L E X - V A L U E D F A D I N G C H A N N E L S This section pr esents a low complexity algor ithm fo r GLR T -optimal no ncoheren t QAM detec tion over complex- valued fadin g chann els. Similar ly to the the real-P AM case, we first present a theorem th at we will use to reduce the n umber of codewords that n eed to be examine d, beyond the limitations imp osed by Property 1. In the com plex- valued chan nel case, the subspace of interest is the plane Y R 2 , where Y was d efined in (8). Th e theore m implies that only a limited extent of the plane n eeds to be searched; and that the extent de pends on the largest element in y . W e then p ropose a fast low-complexity algo rithm to p erform the searc h for QAM. W e also show h o w P AM detection over complex-valued ch annels can be viewed as a sp ecial case of the QAM algorith m. A. Limiting the Sear ch Spa ce Theor em 2 : For nonco herent detection of M 2 -ary QAM cod e word s of leng th T over a com plex-valued fadin g channel   Re  ˆ λ opt y t    6 M + 2 T − 2 , and   Im  ˆ λ opt y t    6 M + 2 T − 2 , for all t = 1 , 2 , . . . , T . Pr oof: Defin e th e po int v , ˆ λ opt y , alo ng with its correspond ing r eal-valued r epresentation v , as in (7). Also define n , arg max t {| v t |} . Note that if | v n | 6 M then | Re { ˆ λ opt y t }| 6 M and | Im { ˆ λ opt y t }| 6 M for all t and the theo rem is satisfied. Now , con sider the alternative case when | v n | > M . Similarly to the r eal-P AM case, rearr anging the GLR T -optimal channel estimate in (3) gives ( ˆ x opt ) † ( y − ˆ h opt ˆ x opt ) = 0 and h ence ( ˆ x opt ) † ( ˆ λ opt y − ˆ x opt ) = 0 . (16) It follows that ˆ λ opt = k ˆ x opt k 2 | y † ˆ x opt | 2 y † ˆ x opt . Combining this with the the fact that for any vector u ∈ C T , the real-valued rep resentation of the complex scalar y † u is Y ′ u , we ob tain th e real-valued rep resentation of ˆ λ opt as ˆ λ opt = k x k 2   Y ′ x   2 Y ′ x IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 11 and theref ore v = Y ˆ λ opt = k x k 2   Y ′ x   2 Y Y ′ x . It follows that v ′ x = k x k 2 , i.e. ( ˆ x opt ) ′ ( v − ˆ x opt ) = 0 . (17) Using (2) and the fact th at C ( X , T ) co ntains all p ossible seque nces { x | x t ∈ X ∀ t } , the elements of ˆ x opt can be determined on an ele ment-wise basis as ˆ x opt t = arg min x ∈X ′ | v t − x | (18) for all t = 1 , . . . , 2 T where we recall from Section II-A that X ′ = {± 1 , ± 3 , . . . , ± ( M − 1) } . W e now substitute (18) into (17) w hich gives ˆ x opt n  v n − ˆ x opt n  = − X t 6 = n ˆ x opt t  v t − ˆ x opt t  . (19) This is similar to ( 12) in the proof of The orem 1. By following through the subseque nt steps in the proof of The orem 1, and keeping in min d that the d imensions of the vectors are now of dimen sion 2 T , we obtain | v n | 6 M + 2 T − 2 which implies that | Re { ˆ λ opt y t }| < M + 2 T − 2 and | Im { ˆ λ opt y t }| < M + 2 T − 2 for all t = 1 , . . . , T . B. QAM Algorithm In this Section we use Property 1 and Th eorem 2 to develop a low-complexity algorithm for QAM d etection. Property 1 implies that ˆ x opt can b e fou nd by calculating th e m etric in (4) for only th ose ˆ x ∈ N ( C ( X , T ) , Y ) , i.e. for only those ˆ x ∈ C for which the plan e Y R 2 passes throu gh its Euclidean nearest neigh bor region . Furtherm ore, Theorem 2 imp lies that only a finite region of the plane need be con sidered. Conceptu ally , this is a direct extension of the real-P AM case shown in Figu re 1 (con sidered previously). The difference b eing that Figure 1 shows the line y R , but we now have a p lane Y R 2 . Also the num ber of orthogo nal dimen sions do ubles wh en considering complex-valued channels. W e demonstrate d this complex-valued channel QAM case in Figure 2 which is a two dimensiona l plot in the plan e Y R 2 . The parallel lin es (at various angles) are the bou ndaries arising fr om th e QAM constellation, and the shaded r egion indicates the n earest n eighbor r egions of cod e word s which need to be searc hed. The QAM search algorithm we present here, f ollows the same p rinciples as the real-P AM algor ithm o f Section IV -B, where instead of working with boun dary poin ts of line segments, we need to work with boun dary edges o f planar region s. The specifics of the algorithm are as follows. First, for ease o f notation we modif y the received cod e word y by multiplyin g it by the complex scalar y ∗ m / | y m | , where m = arg max t | y t | . This will me an that the m th element of y will be r eal-valued and positiv e. The true IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 12 (origina l) GLR T -optimal estimate of the channel ca n be b e obtain ed by app lying the reverse phase ro tation to ˆ h opt , while the o ptimality of the new GLR T -optimal codeword estimate is unaffected. Hence Theorem 2 implies that the search over the plane Y λ is reduced to th e segment of th e plan e fo r which | λ 1 | , | λ 2 | < λ max where λ max , ( M + 2 T − 2) / | y m | . Fur thermore, as d iscussed in Sectio n II-B b ecause of the π / 2 phase amb iguity in square QAM constellation s, there are four GLR T -optimal inverse channel estimates ± ˆ λ opt , ± i ˆ λ opt (with corresp onding phase ambig uous GLR T -optimal codew o rd estimates). Henc e, we only need to consider the square region of th e plan e S = { λ | λ 1 ∈ (0 , λ max ) , λ 2 ∈ [0 , λ max ) } (20) since exactly o ne of ± ˆ λ opt , ± i ˆ λ opt will exist in th is region of the plane. No te that S is the shaded region in Figure 2 (m entioned pr e v iously). Similarly to the re al-P AM case we m ake the f ollowing definition. Definition 4: W e de fine P ( ˆ x ) to b e th e range of λ ∈ S such that ˆ x is the nearest neig hbor to Y λ . Formally , P ( ˆ x ) , { λ | ˆ x ∈ N N ( Y λ ) , λ ∈ S } . Note that each non-em pty P ( ˆ x ) correspon ds to a d istinct region o f the plane Y R 2 . T he prop osed algorithm proceed s by en umerating these no n-empty P ( ˆ x ) ’ s, b y first enu merating th eir b oundary vertices in the plane. These vertices are found by calculatin g the intersectio n of all the co nstellation-poin t b oundary lines in the plane (e.g. as shown in Figure 2). The vertices are then u sed to calculate an interior-poin t in side each of the neare st neighb or regions in the shade d square S . The respe cti ve n earest n eighbor codeword is calculate d for each inter ior-point, and then it is only these points f or which the likelihood metr ics are calculated. Clearly , this is a significantly reduced search space compared w ith the space of all possible co de words. For QAM the vertices of the n earest-neighbo r regions in the p lane Y R 2 can b e f ound by first notin g that, sin ce ˆ x opt t can be gi ven in o n an element-wise b asis as in (18), P ( ˆ x ) can be w ritten as P ( ˆ x ) , 2 T \ t =1 { λ | x t = arg min x ∈X ′ | ( Y λ ) t − x | , λ ∈ S } where ( Y λ ) t is the t th elemen t of Y λ and we re call tha t X ′ = {± 1 , ± 3 , . . . , ± ( M − 1) } . T his can be wr itten a s the feasible region for the set of linear in equalities correspon ding to the n earest n eighbor region b oundaries in X ′ for each elem ent of ˆ x t , as P ( ˆ x ) = 2 T \ t =1 { λ | l ( ˆ x t ) 6 ( Y λ ) t 6 u ( ˆ x t ) , λ ∈ S } where l ( ˆ x t ) and u ( ˆ x t ) are th e upper an d lower nea rest n eighbor b oundaries in the constellation X ′ . For t / ∈ { 2 m, 2 m − 1 } they take o n values in th e set { 0 , ± 2 , . . . , ± ( M − 2) , ±∞} . For t ∈ { 2 m, 2 m − 1 } we must IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 13 consider intersection s with the bound ary o f S , and therefore in this case l ( ˆ x t ) and u ( ˆ x t ) take on values in the set { 0 , ± 2 , . . . , ± ( M − 2) , ± ( M + 2 T − 2) } . By including the square bou ndary of the region S , all no n-empty P ( ˆ x ) are closed simply conn ected sets on the plane R 2 . Therefore, sinc e P ( ˆ x ) is f ormed from linea r inequa lities it is a co n vex polygo n in R 2 . F or each P ( ˆ x ) , denote B ( ˆ x ) as its p olygonal bound ary and V ( ˆ x ) as the vertices of the po lygon. W e n o w p ropose a method that en umerates all the vertices V ( ˆ x ) for all non-em pty P ( ˆ x ) , an d then uses these vertices to generate a po int in the interio r of all P ( ˆ x ) , which is then u sed to obtain a uniq ue codeword via findin g the nearest neig hbor cod e word to that p oint. Con sider the set of p oints { ν ± ǫ µ | ν ∈ V ( ˆ x ) } . If µ is so me vector that is no t parallel to any side of the p olygon P ( ˆ x ) , an d if ǫ is chosen sufficiently sma ll, then at least on e p oint in this set will be in the interio r of P ( x ) . Sin ce th e received symbol is subject to A WGN, and is therefore irration al with prob ability one, it follows that the arb itrary cho ice of µ , [ 1 1 ] ′ will almost surely gu arantee this, given that ǫ > 0 is cho sen suffi ciently small. In pr actice, simp ly setting ǫ to some small positi ve co nstant will be sufficient to ensure that a point in the interior of P ( ˆ x ) is enum erated. However , in App endix A we present a techn ique to perfor m this in a strictly op timal fashion with complexity per vertex of O ( T ) . Since the vertices are shared b y adjacen t P ( ˆ x ) , ea ch vertex is only requ ired to be enum erated o nce. W e define the set of all vertices within o r on the bo undary of S as V = { ν | ν ∈ V ( ˆ x ) , P ( ˆ x ) 6 = ∅ } . The set V can be enumera ted as the the in tersections of the lines Y t, 1 ν 1 + Y t, 2 ν 2 = b and Y t ′ , 1 ν 1 + Y t ′ , 2 ν 2 = b ′ , for all p airs of t, t ′ and for all nearest neigh bor b oundaries b, b ′ in X ′ . That is    ν 1 ν 2    =    Y t, 1 Y t, 1 Y t, 2 Y t, 2    − 1    b b ′    (21) for all t = 1 , 2 , . . . , 2 T − 1 , t ′ = t + 1 , t + 2 , . . . , 2 T , and b, b ′ ∈ B ( t ) , where B ( t ) , { 0 , ± 2 , . . . , ± ( M − 2 ) } if t / ∈ { 2 m, 2 m − 1 } and for symbol indices t ∈ { 2 m, 2 m − 1 } where we co nsider the squ are bounda ry B ( t ) , { 0 , ± 2 , . . . , ± ( M − 2) , ± ( M + 2 T − 2) } . T o enum erate a p oint in each P ( ˆ x ) , for eac h vertex ν enumerated we calculate th e points o n the plane λ + , ν + ǫ µ , and λ − , ν + ǫ µ . Th en for e ach of these two po ints, if it is in the square S , we calculate the correspo nding codewords N N ( λ + Y ) and/o r N N ( λ − Y ) and the d ecision metrics in ( 4). Pseudo-co de is pr ovided in T able IV. The complexity of the algorithm is a functio n of th e number of cod e words examine d, N C , wh ich is in turn a fun ction of the n umber o f vertices c alculated. The n umber of vertices calculated in (21) corr esponding to th e intersections betwee n lines in where b, b ′ is a bo undary of X ′ and both b an d b ′ are non- zero is T (2 T − 1)[( M − 1) 2 − 1] ; for which at most two co dew or ds a re g enerated for a qu arter of these intersections. For the intersection s IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 14 of the b oundaries of X ′ and the square S the re are 2(2 T − 2 )( M − 1 ) 2 intersections, which for o ne q uarter of these intersections one codeword is gen erated. For the vertices at (0 , 0) and ( λ max , λ max ) one codeword is gene rated. Hence the total n umber of codewords examined is a t most N C 6 T (2 T − 1) 2 [( M − 1) 2 − 1] + (2 T − 2)( M − 1) 2 + 2 . (22) Since th e com plexity of each co de word and decision m etric calcu lation is of orde r T then the overall complexity is of o rder M 2 T 3 (which is linear in the constellation size M 2 ) a significant imp rovement over an exhaustive search over all M 2 T possible codewords in the codebo ok C ( X , T ) . A furth er reduction in comp utational expense, without any loss in o ptimality , c an be ac hie ved by en umerating only one out of each set of four phase am biguous vertices. The tech nique is not pr esented here due to space constraints, however the number of non-zer o vertices examin ed is reduced b y a factor of 4 and 1 / 3 of the matrix in verse calculation s in (21) are av oid ed. C. P AM Over Complex Channe ls P AM d etection over complex fading channels can be viewed as a special case of complex-ch annel QAM, where there is zero im aginary compon ent in th e constellation. In this case, the search over the plane Y R 2 can be restricted by extend ing the pr oof of Theorem 2. T o d o this, we note that the co ndition in (16) holds, which implies that ( ˆ x opt ) ′ (Re { ˆ λ opt y } − ˆ x opt ) = 0 since ˆ x opt is alw ay s real-valued. Th e rest of the proof follows to give the result tha t | Re { ˆ λ opt y t }| 6 M + T − 2 . This fact combined with Prop erty 1 and the π phase ambiguity of P AM constellations, implies that we only con sider codewords ˆ x = N N ( Y λ ) for λ in the region S = { λ | 0 < λ 1 < λ max = ( M + T − 2 ) / | y m | } . The spe cifics of the M -ary P AM algorith m are the same as for the M 2 -ary QAM case, with the exception that the calculatio n of (2 1) to obtain the vertices in the inter ior of the (21) is on ly p erformed fo r all t = 1 , 3 , . . . , 2 T − 1 , t ′ = t + 2 , t + 4 , . . . , 2 T , and b, b ′ ∈ B ( t ) , where B ( t ) , { 0 , ± 2 , . . . , ± ( M − 2 ) } if t 6 = 2 m − 1 and for B ( t ) , { 0 , ± 2 , . . . , ± ( M − 2) , M + T − 2 } if t = 2 m − 1 . The total n umber of cod e word s searched can be shown to be at most N C 6 T ( T − 1) 2 [( M − 1) 2 − 1] + ( T − 1 )( M − 1) + 1 . (23) Since th e com plexity of each co de word and decision m etric calcu lation is of orde r T then the overall complexity is O ( T 3 ) . In the following section, we will see that a simple subop timal approa ch c an ach ie ve even lower comp lexity with near-optima l pe rformance . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 15 V I . S U B O P T I M A L A L G O R I T H M S F O R C O M P L E X - V A L U E D F A D I N G C H A N N E L S In th is sectio n, we propo se even lower comp lexity suboptim al algorith ms fo r detection of QAM and P AM over complex-valued fading cha nnels. W e directly use the GLR T -optimal algo rithm for real-P AM from Section I V a s the basis for the algorith ms. A. Su boptimal P AM algorithm Since fo r P AM constellations, all constellation poin ts lie along the real line in the co mplex plan e, a suboptimal phase estimation tech nique combined with o ur GLR T -optimal algorithm for real-valued fading chann els should b e sufficient to provide n ear - optimal perfo rmance. Th is effectiv ely reduces the search over the whole plane Y R 2 for the GLR T -optimal case, to a search over a single lin e at the g i ven estimated ph ase angle. W e u se the power -law estimator [27 ] which, f or constellations exhibiting a rotation al sym metry o f π radians, is simply ˆ φ PL , 1 2 ∠ T X t =1 y 2 t (24) where ∠ re fers to the complex argument. Detection is perfor med by first rotating th e received codeword y accordin g to this estimate, and then detecting Re { e − j ˆ φ PL y } u sing th e GLR T -optim al algo rithm of P AM over a re al-valued fading channel. B. Su boptimal QAM algorithm Here we propose a subo ptimal algor ithm, which redu ces the overall algorithmic complexity to O ( T 2 log T ) by using O ( T ) instances of the P AM detection algorithm presented in Sec tion IV. Instead of en umerating the intersections o f lines on the ( λ 1 , λ 2 ) -plane, a s we did in Sectio n V -B, her e we p ropose to use a mo dified version of the nea rest-neighbo r real-P AM line-sear ch a lgorithm for L lines of the type pr esented in Section IV. W e generate these lines em anating f rom the origin into S (the shad ed region in Figure 2), evenly spac ed in angle. Of co urse, this does no t guarantee tha t we fully enum erate N ( C ( X ) , T ) sin ce a finite nu mber of rad iating lines can not co mpletely cover a p lane, howe ver, we will see by simu lation in Section V -B that th e p erformanc e is close to the optimal. As in the optim al case, we multiply y by y ∗ m / | y m | so that y m will be real-valued a nd positiv e. In this suboptima l QAM case, this implies that we only exam ine p oints on the p lane Y λ f or λ = [ λ 1 λ 2 ] ′ satisfying 0 < | λ 1 | , | λ 2 | < λ max 0 , wher e λ max 0 , ( M + 2 T − 2) / | y m | . The L d irections of the lines with re spect to the direction of po siti ve λ 1 have an gles Φ , { φ 1 , . . . , φ L } where φ ℓ = ( ℓ − 1) π / (2 L ) . For each an gle φ ℓ , we p erform a ne arest neig hbor line search for the line with basis vector y ℓ [ co s φ ℓ sin φ ℓ ] , as proposed in th e suboptim al P AM algorithm in Section VI -A. The search is per formed for IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 16 the segmen t of the line λ y ℓ where λ ∈ R a nd 0 < λ < λ max ℓ where λ max l , λ max 0 / max { cos φ ℓ , sin φ ℓ } . I n this case the lines search es are perf ormed fo r b locks of length 2 T . There is of cou rse a mod ification required to up date the cod e word metrics in terms of complex num bers. T he first line search perf ormed is for φ 1 = 0 , and hence the line searc h is over λ y 1 = λ y . In this case th e inte rvals of the line P ( ˆ x ) are defin ed a s, P ( ˆ x ) , n λ | ˆ x ∈ N N ( λ y l ) , λ ∈ (0 , λ max l ) o . Hence the alg orithm works by enu merating and calculating the me tric fo r all ˆ x ∈ C ( X ′ , 2 T ) for which P ( ˆ x ) is non-em pty . In this case the set V 0 of bo undary poin ts of the regions P ( ˆ x ) is enum erated by c alculating ν t,b = b/ | y t | f or all t = 1 , . . . , 2 T and b = 2 , 4 , . . . , M − 2 , (which are the nea rest neighb or bound aries in the positive h alf of the constellation X ′ ), and storing only tho se values of ( ν t,b , t ) such that ν t,b < λ max ℓ . The set of or dered bound ary points V is again ob tained b y so rting, and ( λ max ℓ , 0) is ap pended to V as the exten t of th e search . Recall that ( ν k , t k ) are the k th elements of V . The search through the codewords is initialized to the first codeword for the which the line segment passes through, which is given by ˆ x (1) = s where s , sgn  y  . The likelihood upda te v ar iables are initialized to α = ( ˆ x (1) ) † y and β = k ˆ x (1) k 2 . T o regen erate th e optim al codeword, the values of λ and φ are in itialized to λ = ν 1 / 2 an d φ = φ 1 = 0 . The ( k + 1) th codeword con sidered, ˆ x ( k +1) , is calculated from the k th codeword as ˆ x ( k +1) t k = ˆ x ( k ) t k + 2 s t k . (25) T o update the decision me tric we defin e α k , ( ˆ x ( k ) ) † y and β k , k ˆ x ( k ) k 2 , and h ence L ( ˆ x ( k ) ) = | α k | 2 /β k is the decision metric fo r the k th co dew or d consid ered. The values α k are updated as follows, I f t k is od d, th en α k is updated as α k =        α k − 1 + 2 s t k − 1 y ( t k − 1 +1) / 2 , t k − 1 odd α k − 1 − 2 is t k − 1 y t k − 1 / 2 , t k − 1 ev en . (26) The values o f β k are up dated ac cording to β k = β k − 1 + 4 s t k − 1 ˆ x t k − 1 + 4 . (27) If L ( ˆ x ( k ) ) = | α k | 2 /β k improves on th e b est codeword estimate then w e store λ = ( ν k + ν k − 1 ) / 2 an d φ = φ ℓ . T o start the next line search, y is multiplied by e jπ 2 L and the line search is then perfor med again f or th e new value of y . When all line search es have b een perfor med, we calculate ˆ x opt = N N ( λe j φ y ) for the o riginal y . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 17 Pseudo-co de is pr ovided in T able V. The significantly redu ced algor ithmic complexity com pared to th e GLR T -op timal alg orithm is governed by the number of line searches and the complexity of each line search. Sin ce there are L phases, each p erforming a version of th e real- P AM line-search algor ithm of Section IV for the case M -ary P AM detection of 2 T symb ols. Thus N C 6 L (2 T ( M / 2 − 1) + 1) . From Sectio n V -B we have noted that th e number of codewords in N ( C ( X , T ) , y ) is of order M 2 T 2 and thus L must be O ( T ) for it to be possible tha t the majority o f N ( C ( X , T ) , y ) is enumer ated. Hence, if L is incr eased propor tionally to T , th e overall complexity of the algo rithm is O ( T 2 log T ) . No te that howe ver, th e im proved computation al perform ance of the alg orithm is largely d ue to being ab le to choose L small, which correspo nds to av o iding exam ining a sig nificant numb er of the ˆ x with associated P ( ˆ x ) being so small as to imply that ˆ x is not relatively close in angle to the plane Y R 2 . W e will see v ia simu lation in Section VII that small L ( e.g. L = 4 fo r T = 7 16-QAM detectio n) can achie ve ne ar - optimal perf ormance. V I I . S I M U L AT I O N R E S U LT S W e now present simulation results to de monstrate the p erformanc e of the new P AM and QA M nonco herent reduced search lattice-dec oding algo rithms. Simulation s are p erformed to ob tain the co dew or d erro r rate (CER) as a fu nction of SNR for nonco herent d etection of 8- ary P AM an d 1 6-ary squar e QAM. For both case, the simulatio ns are perfor med for cod e word lengths o f T = 3 and 7 over a b lock Ray leigh fading channel where h is i.i.d . circular ly symmetric com plex Ga ussian with unit variance. W e have assumed that the phase ambig uities have been r emoved within each co de word , (fo r example, by the use of d if f erential encod ing [1 0]). Figure 3 presen ts results for 8-ar y P AM for th e GLR T -optimal plane searc h algorithm fro m Section V -C an d the suboptimal p hase-estimator plus line-search algorith m from Section VI-A. W e also c ompare with th e suboptim al grid-search algo rithm propo sed in [20 ] and the quantiza tion b ased receiver proposed in [11]. For the grid- search algorithm we use unif ormly spaced ch annel pha se estimates an d the ch annel attenuation e stimates are chosen unifor mly from the CDF of the Rayleigh fading chan nel distribution. For fairness the numbe r of channel attenuation estimates is adjusted so that th e total number of c hannel estimates was kept equal to the maximum num ber of codeword estimates that potentially cou ld be p roduced b y our GLR T -o ptimal alg orithm. Best results are obtained for choo sing th e channel phase estimates a s 0 and π / 2 , and hence the k th channel amplitude estimate is giv e n by | ˆ h ( k ) | 2 = − lo g(1 − k / (1 + ⌈ N C /L ⌉ )) . For the quantization- based receiver (QBR) considered in [11] , all possible seq uences of (positive) am plitude levels ar e p roduced, and the sig n of each symbol is then determin ed by symbol-wise cohere nt d etection using uniformly sp aced ch annel p hase estimates (a ch annel amplitude estima te is not required since the signal amp litude is assumed known). For QBR, we again use the chann el phase estimates 0 IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 18 and π / 2 . Figure 4 presents th e CER as a fun ction of SNR, for 16-QAM transmission. Results are shown for the GLR T - optimal QAM a lgorithm given in Section V -B and th e suboptimal algor ithm gi ven in Section VI-B. W e also compar e with the grid-based algo rithm, where best p erforman ce for a fixed numb er of codeword es tim ates was obtain ed using L = 4 chann el phase estimates, which we also use f or QBR. For both the P AM an d QAM cases we see tha t the suboptima l line-searc h algorithms provide n egligible p er - forman ce loss compare d to the GL R T -optim al algorithm. For the case o f T = 3 , wher e QBR is computation ally possible, ther e is a no ticeable p erforman ce loss. As discussed in Section I I-B, divisor a mbiguities result in a lower bound on the CER. Expressions for these lower bou nds we re provided in [25] and are also shown in the figure. Clearly , for high SNR, both of our GLR T -optimal alg orithms and bo th suboptimal algorithm s detection achiev e these bound s for both P AM an d QAM. As noted in [11 ], ther e is an inherent subop timality in troduced by quantizing the unboun ded channel atten uation by employing a gr id-search appr oach, and hence the p erformance is clearly inferior . Also, althou gh QBR ach ie ves ne ar - optimal p erforman ce for T = 3 , since the comp lexity o f QBR in creases exponentially with T is n ot possible to p roduce curves f or T = 7 . In T able I we present the relative compu tational comp lexities of the algorithm s for the simulation s in terms of the av er age numb er of codewords examin ed. Th e num bers in b rackets indicate the number of codewords examin ed by the sear ch if the restriction s on the search region provided by Theo rems 1 and 2 ar e not ap plied (and are theref ore slightly grea ter than the worst ca se values g i ven in (2 2) an d (2 3)). W e see that the su boptimal phase-estimato r plus line-search app roaches examine far fewer codewords yet obtains near-optimal perfor mance, and that the com plexity of QBR q uickly becomes infe asible with in creasing T . GLR T -Optimal Phase Estimator QBR Grid Reduce d Search + Line Search Search 8-P AM T = 3 132.3 (173) 7.3 (10) 1 28 174 8-P AM T = 7 772.6 (1093) 16.4 (22) 32768 1094 16-QAM T = 3 52.6 ( 87) 22.9 (28) 108 88 16-QAM T = 7 311.8 (439) 52.9 (60) 8748 440 T ABLE I N U M B E R O F C O D E W O R D S E X A M I N E D F O R N O N C O H E R E N T P A M A N D Q A M D E T E C T I O N V I I I . R E D U C E D A M B I G U I T Y T R A N S M I S S I O N In this section we extend our n e w noncoh erent d etection algorith m to pilot a ssisted transmission ( P A T) systems [6]. Unlike, standard P A T we pro pose to use the pilo t symbo l for non coherent amb iguity reso lution, rath er tha n IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 19 simply fo r chann el estimation. W e pro pose to r eplace the pilot sym bol of P A T with a symbo l generated in the following way . T wo bits are allocated for resolvin g the π / 2 ph ase ambiguity of square QAM, and the remainin g bits in the symb ol are allocated to parity , remove di v isor ambiguities and improve erro r performa nce. Ther efore, this scheme h as the same d ata r ate as P A T an d can be c ompared d irectly . W ith par ity c heck b its in the c odew o rd, we ca n now even f urther red uce the search space of our red uced search GLR T lattice-d ecoding algorithm by o nly conside ring codew o rds which satisfy a parity check. Th is significantly reduces the ambigu ity problem . W e will denote this par ity-aided transmission scheme as reduce d a mbiguity (RA) transmission. An arb itrarily chosen par ity check scheme might reduce the number of divisor am biguities, howe ver since the metric (4) has a ge ometric interpreta tion it may be po ssible to d esign other parity-ch eck sch emes which both resolve ambig uities and o ptimize per formance by providing a m inimum angu lar separation b etween codewords. The r esolution o f ambig uities can be ach ie ved, at lea st for 1 6-QAM, by usin g the f ollowing parity-c heck scheme. T wo parity bits p 1 , p 2 are calculated f rom th e data bits { d 1 , d 2 , . . . , d 2( T − 1) } as follows , p 1 ≡ 1 + 4( T − 1) X t =1 d t (28) p 2 ≡ 1 + 2( T − 1) X t =1 d 2 t (29) where ≡ denotes eq uality in GF (2) . They ar e th en map ped to th e uppe r righ t-hand quad rant o f the QAM constellation of the first (p ilot) symbol in the cod e word as follows: (00) ֌ 1 + j , (01) ֌ 1 + 3 j , (11) ֌ 3 + 3 j and (10) ֌ 3 + j . Effectiv ely this mean s th e first two bits o f the first symb ol of each codeword is chosen such that x 1 , x 2 > 0 , wh ich removes the π / 2 phase amb iguity , and the other two b its are p arity bits, which in this case can be sho wn to co mpletely remove the divisor ambig uities (see Ap pendix B). Figure 5 presents the bit error rate (BER) as a function of SNR for detection of 16-QAM transmitted over a block ind ependent phase-nonco herent A WGN chann el. Aga in we have assumed th at the phase a mbiguities hav e been removed with in each cod e word . Results are sh o wn for three cod e word leng ths T = 3 , 5 , 7 . The fig ure shows curves fo r our new RA reduced -search GLR T -o ptimal algorith m, and compares th em to standar d P A T . Both schem es use a sing le pilot symbol p er cod e word ; which for th e RA scheme is generated as describ ed above, an d for P A T it is a symbol wh ich has en ergy equal to the average e nergy p er symbol. For P A T , the GLR T estimate o f the ch annel (based on th e pilot sym bol) is u sed to perf orm GLR T -optimal d ata d etection, while for RA lattice decodin g we use our redu ced search GLR T -optimal algor ithm. Note that for P A T , the BER is indepen dent of the cod e word len gth T sinc e it is a symbo l-by-symbo l detection scheme, whereas for RA lattice dec oding the BER decreases as T increases since it is a seq uence d etection sch eme. Clearly o ur schem e ou tperforms P A T increasingly with T . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 20 Figure 6 shows the CER for the scenario of Figure 5. This serves to highligh t e ven fu rther the be nefit from our lattice ( sequence) decoding ap proach co mpared with P A T . For P A T , since bit err ors o ccur ind ependently on a symbol-b y-symbol basis, th e CER in creases with T . Howe ver, for RA lattice deco ding th e CER d ecreases. Also the figure high lights th e advantage of using p ilot symbols, compared with f ully non coherent tran smission, by ob serving that the SNR range is sig nificantly lower than fo r Fig ures 3 and 4. I X . C O N C L U S I O N In this p aper we developed polyno mial-time lattice-d ecoding algorithm s for no ncoherent block de tection of P AM and QAM. Fas te r suboptim al algorithms for QAM were also presen ted which have excellent ag reement with th e optimal algorithms. A re duced amb iguity transmission scheme was introdu ced which was shown to outper form pilot assisted transmission over the ph ase n oncoheren t chan nel. A P P E N D I X A. Strictly Optimal Calculation of I nterior P oints For each non -empty region P ( ˆ x ) , there exists a vertex ν ∈ V ( ˆ x ) an d small scalars ν + , ν − > 0 , such that either ν + , ν + [ ν + 0 ] ′ or ν − , ν + [ ν − 0 ] ′ is in th e interior of P ( ˆ x ) . Suppose the first case is tr ue. Now , the line ν + γ [ 1 0 ] ′ intersects an edge of the b oundary of P ( x ) , and we will call this intersection point µ . W e p ropose to choose ν + = γ > 0 so th at ν + is the midp oint of ν and µ . Defining u t as the t th elemen t of u = Y ν , where Y is defined by the or iginal received vector y , we can calculate ν + as follows, ν + =        min t 2 ⌈ u t 2 ⌉− u t 2 y t y t < 0 , min t − 2 ⌊ u t 2 ⌋− u t 2 y t y t > 0 . Note that alm ost surely y t 6 = 0 . Sim ilarly , u sing th e lin e ν − γ [ 1 0 ] ′ we calculate ν − = γ > 0 as ν − =        min t − 2 ⌊ u t 2 ⌋− u t 2 y t y t < 0 , min t 2 ⌈ u t 2 ⌉− u t 2 y t y t > 0 . This process will in general a lw ays calculate a point in each n on-empty P ( ˆ x ) . Howe ver , to av o id calcu lation problem s we first rotate ν b y y m / | y m | , so that th e vectors [ ν + 0 ] ′ and [ ν − 0 ] ′ are no t parallel to any of the edges of P ( ˆ x ) ( e.g. those that are part of S ). This ro tation is later reversed, so that th e points calculated are in the original coord inates. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 21 B. Removal o f Amb iguities in 16-ary QAM In this section we show th at the p roposed RA pilot symbol appro ach (using parity checks, as discu ssed in Section VIII) to tally removes both the p hase and divisor a mbiguities other wise inheren t in a nonco herent detec tion system (as discussed in Section II-A). W e start b y recalling that the pro posed p arity scheme inv olves c alculating the parity bits fro m th e data bits d t for t = 1 , . . . , 4 T as follows. p 1 ≡ 1 + 4( T − 1) X ℓ =1 d t p 2 ≡ 1 + 2( T − 1) X ℓ =1 d 2 t (30) where ≡ denotes eq uality in GF (2) . The data and parity bits are then mapped to the symbols as shown in T ab le I I, wher e we recall fro m the defin ition in (7) that x 2 t − 1 = Re { x t } and x 2 t = Im { x t } . d 2 t − 1 d 2 t x t 00 − 3 01 − 1 11 1 10 3 p 1 p 2 x 1 (pilot symbol) 00 1 + i 01 1 + 3 i 11 3 + 3 i 10 3 + i T ABLE II M A P P I N G O F D AT A A N D PA R I T Y B I T S . Since x 1 is constrain ed to have positiv e real a nd imaginar y compo nents, the p hase amb iguity h as b een re moved. It rema ins to show that all divisor amb iguities have also been removed. T o d o th is, we first defin e the associates of a Gaussian integer g to be the elements o f the set A ( g ) = { g , g i, − g , − g i } . W e also den ote A ( g ) T to be a co de word of len gth T com posed of o nly elements of A ( g ) . For 16- QAM, it can be easily shown that a necessary con dition for a divisor ambigu ity to exist is that there exists codewords x (1) ∈ A ( g 1 ) T , x (2) ∈ A ( g 2 ) T for some g 1 , g 2 ∈ X , { 1 + i, 3 + 3 i, 3 + i, 1 + 3 i } such that g 1 6 = g 2 . For a codeword x and some g ∈ X , we define N 1 , N 2 , N 3 and N 4 as the numbe r of o ccurrences in a c odew o rd of each of the fo ur po ssible rotations of g in the codew o rd, that is g , g i, − g and − g i respectively . No ting that the phase ambig uity has been removed ( since x 1 is constraine d to have positive real and imag inary co mponents), a sufficient condition for two codewords to be unamb iguous is that ther e exists som e t , such th at the t th symbols from the two codewords are in different quadr ants of the comp lex plane. It follows then , that a su f ficien t cond ition for two cod e word s to be un ambiguou s is that they do not have the same v alu es of N 1 to N 4 . W e now use th is property on N 1 to N 4 to show that f or arbitrary T , it is not po ssible for two ambiguou s codewords x (1) ∈ A ( g 1 ) T , x (2) ∈ A ( g 2 ) T , to satisfy the parity check (30) f or any g 1 , g 2 ∈ X such that g 1 6 = g 2 . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 22 W e c onsider e ach g ∈ X (wher e we have previously defined X = { 1 + i, 3 + 3 i, 3 + i, 1 + 3 i } ) in turn, showing that all c odew o rds x ∈ A ( g ) T that satisfy the p arity check , are distinguishable in p hase fro m all parity-satisfy ing codewords considere d u p to th at po int. For 16-QAM this process in volves considerin g th e four Gau ssian integers 1 + i , 3 + 3 i , 3 + i and 1 + 3 i in turn, as detailed in the following fo ur cases. Define x D to be th e data codeword co mponent o f x , i.e. x = [ x 2 . . . x T ] ′ . • Case x D ∈ A (1 + i ) T − 1 : In this case, we show that there d oes not exist any x ∈ A (1 + i ) T that satisfies th e parity check. Using T able I I, the bits ( d 4 ℓ − 3 . . . d 4 ℓ ) are mapp ed to th e symbol x ℓ = x 2 ℓ − 1 + ix 2 ℓ ∈ x D in the fo llo win g way: (1111) ֌ 1 + i, (0111) ֌ − 1 + i, (0101 ) ֌ − 1 − i and (1101) ֌ 1 − i . Clearly fro m (2 9), p 2 ≡ 1 , and therefor e the pilot sym bol x 1 will be either 1 + 3 i or 3 + 3 i . I t follows that x / ∈ A (1 + i ) T . • Case x D ∈ A (3 + 3 i ) T − 1 : In this case, we show th e co nditions u nder which a co dew or d x ∈ A (3 + 3 i ) T satisfies the parity check. The associated b it mapp ings are (1010) ֌ 3 + 3 i, (0 010) ֌ − 3 + 3 i, (0000 ) ֌ − 3 − 3 i and (100 0) ֌ 3 − 3 i . Clearly , p 1 ≡ 1 + N 2 + N 4 and p 2 ≡ 1 . Theref ore, x 1 = ( 1 + 3 i , if ( p 1 p 2 ) = (01) i.e. if N 2 6≡ N 4 , 3 + 3 i , if ( p 1 p 2 ) = (11) i.e. if N 2 ≡ N 4 . Furthermo re it follows that x ∈ A (3 + 3 i ) T only if N 2 ≡ N 4 . • Case x D ∈ A (3 + i ) T − 1 : In this case, we show th e conditions under wh ich a codeword x ∈ A (3 + i ) T satisfies the parity check, and show that und er these condition s there does not exist any ambiguo us codeword from A (3 + 3 i ) T , i.e. from the previous case. The bit mappings are (101 1 ) ֌ 3 + i, (0110 ) ֌ − 1 + 3 i, (0001 ) ֌ − 3 − i and (1100) ֌ 1 − 3 i . In this case, p 1 ≡ 1 + N 1 + N 3 and p 2 ≡ 1 + N 1 + N 2 + N 3 + N 4 ≡ 1 + T − 1 ≡ T . If T is odd, then p 2 ≡ 1 and therefore x 1 ∈ { 1 + 3 i , 3 + 3 i } and therefo re x / ∈ A (3 + i ) T . If T is even then p 2 ≡ 0 and p 1 ≡ 1 + N 1 + N 3 ≡ N 2 + N 4 . Ther efore x 1 = ( 3 + i if ( p 1 p 2 ) = (10) i.e. if N 2 6≡ N 4 , 1 + i, if ( p 1 p 2 ) = (00) i.e. if N 2 ≡ N 4 . It f ollows that x ∈ A (3 + i ) T only if N 2 6≡ N 4 and T is e ven. Recall that in the pr e v ious case, valid parity satisifying codewords only occurr ed if N 2 ≡ N 4 . Therefo re a n ambiguity will not occur between two codewords x ∈ A (3 + i ) T and x (1) ∈ A (3 + 3 i ) T since they will be distinguishable in phase. • Case x D ∈ A (1 + 3 i ) T − 1 : In this case, we sho w the conditions und er which a code word x ∈ A (1+ 3 i ) T satisfies the parity check, and s h ow IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 23 that und er these condition s there d oes not exist any ambiguo us codeword fr om either A (3 + 3 i ) T or A (3 + i ) T , i.e. f rom the previous two cases. T he bit mapp ings are (11 10) ֌ 1 + 3 i, (0 011) ֌ − 3 + i , (0100) ֌ − 1 − 3 i and (1 0 01) ֌ 3 − i . Here, p 1 ≡ 1 + N 1 + N 3 and p 2 ≡ T . If T is even, then p 2 ≡ 0 and ther efore x 1 ∈ { 1 + i, 3 + i } a nd no ambig uity oc curs. 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Collings, “Blind detectio n of P AM and QAM in fading channel s, ” IEEE T rans. Inform. Theory , vol. 52, pp. 1197–1206, Mar . 2006. [26] T . H. Cormen, C. E. Leiserson, R. L. Ri vest, and C. Stein, Intro duction to Algorithms , 2nd ed. Cambridge , MA, USA: McGraw-Hill Higher Education , 2001. [27] M. Moeneclae y and G. de Jonghe, “ML-orient ed NDA carrier synchroni zatio n for general rotational ly symmetric signal constell ations, ” IEEE T rans. Commun. , pp. 2531– 2533, Aug. 1994. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 25 1 begin 2 s := sgn y ; // Store sign of each y t 3 y := s ◦ y ; // M ake each y t positiv e 4 B max := M + T − 2 ; 5 m := arg max t { y t } ; 6 λ max := ( M + 2 T − 2) / | y m | ; // Search region: 0 < λ < λ max 7 V 0 := ∅ ; // Calculate and store P ( x ) boundary points 8 for t := 1 to T do 9 for all b ∈ { 2 , 4 , . . . , M − 2 } do 10 ν := b/y t ; 11 if ν < λ max ; 12 V 0 := {V , ( ν, t ) } ; 13 else break ; 14 end for all ; 15 end for; 16 V := sort ( V 0 ) ; // Sort V 0 in ascending order of ν 17 V := {V , ( λ max , 0) } ; 18 ˆ x := [ 1 1 . . . 1 ] ′ ; // Initialize data estimate 19 α := ˆ x ′ y ; // Initialize likelihood terms 20 β := k ˆ x k 2 ; 21 L := α 2 /β ; 22 λ := V (1 , 1) / 2 ; 23 f or k := 1 to |V | − 1 do // Iterativ e ly examine likelihoods 24 t := V ( k, 2) ; 25 α := α + 2 y t ; // Update likelihood terms 26 β := β + 4 ˆ x t + 4 ; 27 ˆ x t := ˆ x t + 2 ; // Update x 28 if α 2 /β > L // If better x found 29 L := α 2 /β ; // Upda te likelihood 30 λ : = ( V ( k , 1) + V ( k + 1 , 1)) / 2 ; // Store point in P ( ˆ x ) 31 end if ; 32 end for ; 33 retur n ˆ x opt := s ◦ N N ( λ y ) ; T ABLE III M - A RY R E A L - PA M N O N C O H E R E N T L AT T I C E D E C O D I N G A L G O R I T H M IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 26 1 begin 2 m := arg max t {| y t |} ; 3 y := ( y ∗ m / | y m | ) y ; // R otate y so that y m is purely real 4 B max := M + 2 T − 2 ; 5 λ max := B max / | y m | ; // Search boundary (Thm. 2). 6 ˆ x best := N N (( ǫ + iǫ ) y ) ; // Codeword near origin 7 L max := L ( ˆ x best ) ; // L ’hood L ( x ) , ˛ ˛ ˛ x † y ˛ ˛ ˛ 2 / k x k 2 8 B := { 2 , 4 , . . . , M − 2 } ; // Positive NN boundaries // C alculate only intersection points ( i.e. vertices) in // first quadrant us ing (21) by reducing number of // NN boundaries B 1 , B 2 and then rotating. 9 for t := 1 to 2 T − 1 do 10 B 1 := B ; 11 if t ∈ { 2 m − 1 , 2 m } then B 1 := {B 1 , B max } ; 12 fo r t ′ := t + 1 to 2 T do 13 B 2 := B ; 14 if t ′ ∈ { 2 m − 1 , 2 m } then B 2 := {B 2 , B max } ; 15 S := » Y t, 1 Y t, 2 Y t ′ , 1 Y t ′ , 2 – − 1 ; // Matrix in (21) 16 for all b 1 ∈ B 1 17 for all b 2 ∈ B 2 // C alculate intersec tion point; 18 ν := Real-T o-Complex ( S [ b 1 b 2 ] ′ ) ; 19 ν := Rotate-T o-First-Quadrant ( ν ) ; 20 for all s ∈ {− 1 , 1 } 21 λ : = ν + s ( ǫ + iǫ ) ; // Point in s ome partition // Check that λ is in reduce d sea rch region 22 if 0 < Re { λ } < λ max and 0 6 Im { λ } < λ max then 23 ˆ x := N N ( λ y ) ; // Calculate NN 24 i f L ( ˆ x ) > L max // If better x found 25 ˆ x best := ˆ x ; // Upda te codeword estimate 26 L max := L ( ˆ x ) ; // Upda te likelihood 27 en d if ; 28 end if ; 29 end for all ; 30 end for all ; 31 end for all ; 32 e nd for ; 33 end for ; 34 return ˆ x opt := ˆ x best ; T ABLE IV M 2 - A R Y S Q U A R E Q A M N O N C O H E R E N T L AT T I C E D E C O D I N G A L G O R I T H M IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 27 1 begin 2 L := 0 ; // Initialize likelihood 3 λ max 0 := ( M + 2 T − 2) / max t | y t | ; 4 for ℓ = 1 to L then 5 // Search region: 0 < λ < λ max (Theorem 2) 6 λ max := λ max 0 / min ˘ ˛ ˛ cos ℓπ 2 L ˛ ˛ , ˛ ˛ sin ℓπ 2 L ˛ ˛ ¯ ; 7 V 0 = ∅ ; // Calculate and store P ( x ) boundary points 8 for t = 1 to 2 T then 9 for all b ∈ { 2 , 4 , . . . , M − 2 } then 10 ν := b/ | y t | ; 11 if ν < λ max ; 12 V 0 := {V 0 , ( ν, t ) } ; 13 else break ; 14 end for all ; 15 end for; 16 V := sort ( V 0 ) ; // Sort V 0 in ascending order of ν 17 V := {V , ( λ max , 0) } ; 18 s := sgn { y } ; 19 ˆ x := s ; // Initialize data estimate 20 α := ˆ x † y ; // Initialize likelihood terms 21 β := k ˆ x k 2 ; 22 if α 2 /β > L // If better x found 23 L := α 2 /β ; // Update likelihood 24 λ := V (1) / 2 ; 25 φ := ℓ π/ (2 L ) ; 26 end if ; 27 for k := 1 to |V | − 1 do // Iterativ e ly examine likelihoods 28 t := V ( k, 2) ; 29 if t ′ is odd then 30 α := α + 2 s t y ( t +1) / 2 ; 31 else 32 α := α − 2 is t y t/ 2 ; 33 end if 34 β := β + 4 s t ˆ x t + 4 ; 35 ˆ x t := ˆ x t + 2 s t ; 36 if | α | 2 /β > L // If better x found 37 L := | α | 2 /β ; // Update likelihood 38 λ := ( V ( k, 1) + V ( k + 1 , 1)) / 2 ; // Store point in P ( ˆ x ) 39 φ := ℓ π/ (2 L ) ; 40 end if ; 41 end for ; 42 y := y e jπ 2 L ; // Rotate y for next line se arch 43 end for ; 44 retur n ˆ x opt := N N ( λe jφ y ) ; T ABLE V S U B O P T I M A L M 2 - A R Y S Q U A R E Q A M M U LT I P L E L I N E - S E A R C H N O N C O H E R E N T D E T E C T I O N A L G O R I T H M IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 28 0 2 4 6 8 10 12 0 2 4 6 8 10 12 ν 1 ν 2 ν 3 ν 4 ν 5 λ max x 1 x 2 ν 1 ν 2 ν 3 ν 4 ν 5 λ max Fig. 1. Illu s trat ion of noncohere nt detection of 8-ary P AM for T = 2 . The dots are all the (two dimensiona l) P AM code words in the positi ve quarter -plane, and the angle d line is y R , for an exa m ple recei ved codew ord y . The shaded regions indicate the nearest neighbor regions of points which need to be searched. That is, they are in N ( C ( X , T ) , y ) (from Property 1), and they correspond to va lues of λ less than λ max = ( M + T − 2) / max t | y t | = M / max t | y t | (from Theorem 1). −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 λ 1 = Real{ λ } λ 2 = Imag{ λ } Fig. 2. Plot of partitions P ( ˆ x ) on the R 2 plane for 16 -ary QAM detec tion of a sequence of length T = 3 for the recei ved vector y = [ − 0 . 1076 − 0 . 4728 i, − 0 . 7002 − 0 . 0968 i, − 1 . 1228 + 0 . 4955 i ] . The bold square corre s ponds to the search boundary S . IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 29 20 30 40 50 60 70 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) CER T=3 T=7 New GLRT−Optimal Reduced−Search Phase−Estimator + Line Search Grid Search [20] QBR (exponential complexity) Ambiguity lower bound Fig. 3. Plot of Codewo r d Error Rate (C ER) as a function of SNR for an 8-ary P AM system. 20 30 40 50 60 70 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) CER T=3 T=7 New GLRT−Optimal Reduced−Search Phase−Estimator + Line Search (Section VI.B) Grid Search [20] QBR (exponential complexity) Ambiguity Lower Bound Fig. 4. Plot of Codewo r d Error Rate (C ER) as a function of SNR for a 16-ary square QAM system. IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED TO APPEAR (ACCEPTED N O V . 2006) 30 10 12 14 16 18 20 22 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) BER PAT RA T=3 RA T=5 RA T=7 Fig. 5. Comparison of Bi t Error Rate (B ER) as a function of SNR for 16-QAM for P A T versus RA tr ansmission. 10 12 14 16 18 20 22 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) CER PAT T=7 PAT T=5 PAT T=3 RA T=3 RA T=5 RA T=7 Fig. 6. Comparison of Codewo rd E rror Rate (CER) as a function of SNR f or 16-QAM for P A T versus RA transmission.