Full Diversity Blind Signal Designs for Unique Identification of Frequency Selective Channels

Full Di v ersity Blind Signal Designs for Unique Identificati on of Frequenc y Selecti v e Channels Jian-Kang Zhang Departmen t of E lectrical an d Com puter En gineerin g, McMaster Un iv ersity , Hamilto n, Ontario , Canad a. Email: jkz hang@mail.ec e.mcmaster .c a Chau Y uen Institute f or I nfoco mm Research, Ro om 0 3-07 21 H eng Mui Keng T errace, Sin gapore 11 9613. Email: cyuen@i2r .a -star .edu .sg Abstract — In this paper , we develop two kinds of n ov el closed- fo rm decompositions on p hase shift keying ( PSK) constell ations by exploiting linear congruence equation theory: the one for factorizing a pq -PS K constellation into a product of a p -PSK constellation and a q -PSK constellation, and the other for decomposing a specific complex n umber i nto a diff erence of a p -PSK constellation and a q -PS K constellation. With this, we propose a simple signal design technique to blindly and uniq uely identify frequency selecti ve channels with zero-padded block transmission under noise-free en vironments by only using the first two b lock r eceiv ed signal vecto rs. Furthermore, a closed-form solution to d etermine the transmitted signals and the channel coefficients is o btained. In the Gaussian noise and Ray leigh fading en vironment, we pro ve that the n ewly proposed signaling sch eme enables non-coherent full diversity for the Generalized Likelihood Ratio T est (GLR T) receiv er . I . I N T RO D U C T I O N In this paper, we conside r wir eless commun ication systems with a sin gle transmitting antenn a and a single receiving antenna which tran smit d ata over a fr equency-selective fading channel. The systems which we consider mitigate the inter- symbol inter ference gen erated by the chann el by transmitting the data stream in consecutive eq ual-size blocks, which are subsequen tly processed at the receiver on a block-b y-block basis, see, e.g., [1] –[3]. I n order to remove interblo ck in - terference , some redu ndancy is added to each block b efore transmission. There are several ways to add redun dancy (e.g., [1], [2]), but in this paper we will focu s on block-b y- block communic ation systems with zero-p adding red undan cy; e.g., [1]–[ 3]. When the receiv er possesses p erfect knowledge of the channel an d employs maximum likelihood (M L) detection , it w as shown [3], [4] th at such a system not o nly enables full diversity but also p rovide the maximu m codin g g ain. Unfortu nately , per fect channel state information at the receiver , in practice, is not easily attainab le. If the coherent time is sufficiently long, then, the transmitter can send training signals that allow the receiver to estimate the channel coefficients accurately . For mobile wireless com munication s, h owe ver, the fading coefficients may chan ge so rapidly that the coherent time may be too sh ort to allow reliable estimation of the coefficients, especially in a system with a large num ber of antennas. T herefo re, the time spent on sen ding tra ining sign als cannot b e ig nored bec ause o f th e n ecessity of send ing mo re training signals f or the a ccurate estimation of the chann el [5] , [6]. T o av o id ha ving to transmit training signals, considerable research efforts have been dire cted to de velop techniqu es of “blind chann el estimation” [7], [8] rece ntly . These techniqu es identify and estimate the tr ansmission chan nel u sing only the recei ved (perhaps noisy) signals at the recei ver . The essence of th ese algor ithms is to exploit the structu re o f the c hannel and /or the property of transmitted signals. The subspace me thod is one such meth od exploitin g the chann el structure an d the second order statistics o f inpu t signals [ 8], [9]. In digital communicatio n applicatio ns, the input signals have finite alphabet prope rty su ch as the constant mo dulus for PSK mod ulation or integers for QAM, wh ich can be further exploited to estimate the chan nel with sho rter coh erent time than the subspace metho d b y n umerically solving some opti- mization problems [10]. Thu s far, all currently a vailable blind methods without either training or pilots f or the frequency selection channel estimation incur scale ambiguity and as a consequen ce, cannot uniquely identify the channel coef ficients. In ad dition, wir eless comm unication app lications deman d the accurate estimate o f th e channel a s well a s o f the signal. The resulting scale amb iguity fr om the chann el estimation will result in significant erro r prob ability for the signal estimation ev en in a no ise-free environment. T o resolve this issue, we pro pose a novel sign aling and transmitting techniqu e fo r the freque ncy selecti ve c hannel with zero-padd ing block transmission, in which neither the transmitter or the receiv er knows the channel state information. Our m ain co ntributions in this p aper are as f ollows: 1) A n ovel signal d esign techniq ue u sing a p air of co- prime p -PSK and q -PSK co nstellations for th e first two block transmissions is p roposed to blind ly and uniq uely identify frequency selectiv e channels with zer o-padd ed block transm ission by on ly processing th e first tw o blo ck received sign als. Fu rthermo re, a closed-form solu tion to determ ine th e transmitted signals an d the ch annel coefficients is obtained by utilizin g the linear co ngru ence equation theory . 2) In the Ga ussian noise and Rayleigh fading environment, we prove that the newly pro posed scheme e nables full div ersity for the GLR T receiver . Here, we should point out that the similar hybrid signaling schemes [11]– [15] were used to eliminate the ambiguity of the blind o rthogo nal space-time block co des. Notatio n : Colu mn vectors an d matric es are bold face low- ercase and uppercase letters, respectiv ely; the matrix trans- pose, the co mplex co njugate, the Hermitian are den oted by ( · ) T , ( · ) ∗ , ( · ) H , respectively; I N denotes the N × N identity matrix; g cd( m, n ) deno tes the greatest co mmon divisor of positive integers m and n . Particularly when g cd( m, n ) = 1 , we say that m and n are coprime integers; ϕ ( n ) deno tes th e Euler func tion, i.e., the number of all positi ve integers th at do not exceed n an d are pr ime to n . T he ( i, j ) th eleme nt of matrix A is de noted b y a ij . I I . C H A N N E L M O D E L A N D B L I N D S I G N A L D E S I G N S If the chann el is assumed to be of leng th at most L (i.e., if L is an upper bou nd on the delay spread), th en, the block transmission system with z ero-pad ding operate as follows: First, L − 1 zeros are ap pend to x to form x ′ which is of length P = K + L − 1 . The elem ents of x ′ are serially transmitted throug h th e channel. Th e channel imp ulse response is den oted by h = [ h 0 , h 1 , · · · , h L − 1 ] T . The leng th P received signal vector r can be written as r = H s + ξ , (1) where r is a P × 1 received signa l vector, s is a K × 1 transmitted signal vector , ξ d enotes the P × 1 vector of noise samples at th e receiver , an d H denote s th e P × K T o eplitz matrix [1]–[ 3] H = 0 B B B B B B B B B B B B B B @ h 0 0 . . . 0 h 1 h 0 . . . 0 . . . h 1 . . . . . . h L − 1 . . . . . . h 0 0 . . . . . . h 1 . . . . . . . . . . . . 0 . . . 0 h L − 1 1 C C C C C C C C C C C C C C A P × K . For blin d channel estimation , it is convenient to rewrite the channel m odel (1) as r = T ( s ) h + ξ , (2) where we hav e used the following fact Hs = T ( s ) h with T ( s ) defined by T ( s ) = 0 B B B B B B B B B B B B B B @ s 1 0 . . . 0 s 2 s 1 . . . 0 . . . s 2 . . . . . . s K . . . . . . s 1 0 . . . . . . s 2 . . . . . . . . . . . . 0 . . . 0 s K 1 C C C C C C C C C C C C C C A P × L . Now we assume that during a T block transmission period, the channel coefficients keep constant and after th at, will random ly change. Our novel blind modulation scheme is described as follows: Du ring the first b lock tran smission, each symbol o f a transmitted signal vector s = x is chosen from p -PSK constellation X ; i.e., r 1 = T ( x ) h + ξ 1 , x ∈ X K . During the second b lock tr ansmission, each sy mbol of a tran smitted signal vector s = y is chosen from q -PSK con stellation Y ; i.e., r 2 = T ( y ) h + ξ 2 , y ∈ Y K , wher e p and q are co prime. Then, du ring the i th block transmission fo r 3 ≤ i ≤ T , each symbol o f a transm itted sign al vector s = z i can be ch osen from any con stellation Z ; i.e ., r i = T ( z i ) h + ξ i , z i ∈ Z K . Collecting all the T block rece iv ed signals, we have z = Sh + η , (3a) where η = ( ξ T 1 , ξ T 2 , · · · , ξ T T ) T and S =  T T ( x ) , T T ( y ) , T T ( z 3 ) , · · · , T T ( z T )  T (3b) for x ∈ X K , y ∈ Y K , z i ∈ Z K . Throu ghou t this pap er , we make the following assumptio ns: 1) The chann el coefficients h ℓ for ℓ = 0 , 1 , · · · , L − 1 are samples of ind ependen t circularly symmetric zero- mean complex white Gaussian random v ar iables with unit variances and remain co nstant for the first P T ( T ≥ 2 ) time slots, after which they chang e to new in depend ent values that are fixed for next P T time slots, an d so o n. 2) The elements η is circu larly symm etric zer o-mean com - plex Gaussian samples with covariance matrix σ 2 I P T ; 3) During P T ob servable time slots, T consecutive block s z i are tran smitted with each entry z ik for i = 3 , 4 , · · · , T and k = 1 , 2 , · · · , P being indepen dently and equa lly likely chosen from the constellation Z , while comp onents x k and y k for k = 1 , 2 , · · · , P in th e previous two b locks are independ ently and equ ally likely chosen fro m the respective p - PSK an d q -PSK constellation s X and Y , where p and q are coprime. 4) Channe l state inf ormation is not available at either the transmitter or the rec eiv er . Our goal in this paper is to prove that our de signed signaling scheme (3) 1) enables the uniq ue ide ntification of th e chann el and the transmitted sig nals for any g iv en no nzero recei ved signal vector r in a noise-fr ee case and 2) provide s full di versity f or the GLR T receiver in the Gaussian n oise and Rayleig h fading environment. I I I . B L I N D U N I Q U E I D E N T I FI C A T I O N A N D F U L L D I V E R S I T Y In th is section, we first develop some decomp osition prop - erties o n a pair of PSK constellations and then, prove that our blind m odulation scheme prop osed in the p revious section enables the un ique identification of the channel coefficients and the transmitted signals in a n oise-free case as well as full div ersity for the GLR T receiver in a noise e n vironm ent. A. Decompo sitions of PSK Constellations Pr op osition 1: L et two po siti ve integers p a nd q b e co- prime. Th en, for a ny in teger k with 0 ≤ k < pq , there exists a pair o f x and y such that xy = exp  j 2 π k pq  . (4) Furthermo re, s p and s q can b e un iquely d etermined by x = exp  j 2 π k q ϕ ( p ) − 1 p  (5a) y = exp  j 2 π k p ϕ ( q ) − 1 q  . (5b) Proposition 1 tells us that any pq -PSK sy mbol can be u niquely factored into th e pro duct of a pair of the coprime p -PSK sym - bol an d q - PSK sym bol. This factorization was first discov ered by Zhou, Zhang and W ong [ 11], [13]. N ow , Pro position 1 significantly simp lifies th e or iginal rep resentation. Th e fo llow- ing pr operty gives a necessary and su fficient co ndition fo r a complex number to be able to be decomp osed into a difference of a p air o f the c oprime p -PSK symbol an d q -PSK symbol. Pr op osition 2: L et w 6 = 0 be a gi ven non -zero com plex number and w = | w | e j θ . Then, there exists a pair of x ∈ X and y ∈ Y satisfying eq uation x − y = w (6) if and on ly if th ere exist three integers m, n and k such th at θ = π ( pn + q m ) pq + k π + π 2 (7a) | w | = ( − 1) k +1 sin  π ( pn − q m ) pq  . (7b) Furthermo re, under the co ndition (7), if p and q are co prime, then, equatio n (6) has the unique solution that can be explicitly determined as follows: 1) If w = 0 , then, x = y = 1 . 2) If w 6 = 0 , then, x an d y are giv en by x = exp  j 2 π ℓq ϕ ( p ) − 1 p  (8a) y = exp  j 2 π ℓp ϕ ( q ) − 1 q  (8b) with integer ℓ = pq  θ π − 1 2  . The proo fs of Pro positions 1 and 2 are omitted becau se of space limitatio n. B. Blind uniqu e identifica tion of the cha nnel Our main p urpose in this sub section is to prove that th e blind signaling sche me (3 ) is capab le of uniquely identif ying the channel coefficients and the transmitted sym bols. T o do this, Let u and v be tw o consecutive blo ck received signal vectors in the first two block tran smission f rom the cha nnel model (2 ) in a n oise-free e n vironm ent; i.e., u = T ( x ) h x ∈ X K , (9) v = T ( y ) h y ∈ Y K . (10) Now , we fo rmally state the first result. Theor em 1: (Unique Identification ) Let u = ( u 1 , u 2 , · · · , u P ) T and v = ( v 1 , v 2 , · · · , v P ) T be the first two consecutive block no nzero received s ignal vector s giv en by (9) and (10 ), respectiv ely , an d p and q be co-prime. If r den otes th e maximum integers such that u 1 v 1 = u 2 v 2 = · · · = u r v r = 0 , then, h 0 = h 1 = · · · = h r − 1 = 0 . In ad dition, the other remaining channel coefficients h r , h r +1 , · · · , h L − 1 and all the transmitted symbo ls in x and y can b e u niquely determin ed as f ollows. 1) Let w 1 be defined by w 1 = u r +1 v r +1 = | w 1 | e j θ 1 and ℓ 1 = pqθ 1 2 π . Th en, we h ave x 1 = exp  j 2 π ℓ 1 q ϕ ( p ) − 1 p  (11a) y 1 = exp  − j 2 π ℓ 1 p ϕ ( q ) − 1 q  (11b) h r = x ∗ 1 u r +1 . (11c) 2) For 1 < m ≤ L − r , let w m be d efined by w m = x ∗ 1  u r + m − P m − 2 i =1 h r + i x m − i  h r − y ∗ 1  v ℓ − P m − 2 i =1 h r + i x m − i  h r . (1 2) a) If w m = 0 , then, we h av e x m = x 1 (13a) y m = y 1 (13b) h r + m − 1 = x ∗ 1 ` u r + m − m − 2 X i =1 h r + i x m − i ´ . (13c) b) If w m 6 = 0 , let w m = | w m | e j θ m and ℓ m = pq  θ m π − 1 2  . Then, we h ave x m = x 1 exp ` j 2 π ℓ m q ϕ ( p ) − 1 p ´ (14a) y m = y 1 exp ` j 2 π ℓ m p ϕ ( q ) − 1 q ´ (14b) h r + m − 1 = x ∗ 1 ` u r + m − m − 2 X i =1 h r + i x m − i ´ . (14c) 3) For L − r + 1 ≤ m ≤ K , we have x m = u m + r − P L − r − 1 i =1 h L − i x m − L + r + i h r (15a) y m = v m + r − P L − r − 1 i =1 h L − i y m − L + r + i h r . (1 5b) Pr oo f : Basically , the proof of Th eorem 1 captur es th e follow- ing f our steps. Step 1 . First, we c onsider th e first received signals in each block. I n this case, we have h 0 x 1 = u 1 , ( 16a) h 0 y 1 = v 1 . (16b) Therefo re, if either u 1 or v 1 is zero, then, h 0 is zero. Similarly , we can o btain h 1 = h 2 = · · · , h r − 1 = 0 if u 1 v 1 = u 2 v 2 = · · · = u r v r = 0 . Step 2 . W e continue to proceed the ( r + 1 ) -th receiv ed signals for each block. I n th is case, w e have receiv ed h r x 1 = u r +1 , (17a) h r y 1 = v r +1 . (17b) Since u r +1 v r +1 6 = 0 , elim inating h 1 from (17) results in x 1 y ∗ 1 = u r +1 v r +1 = w 1 . (18) Now , by Prop osition 1, x 1 and y 1 can be uniquely deter mined by ( 11a) and (1 1b), r espectively , and thus, h r is uniquely determined by (11c). Step 3 . Let us consider the ( r + 2) -th received signals for each b lock: h r +1 x 1 + h r x 2 = u r +2 , (19a) h r +1 y 1 + h r y 2 = v r +2 . ( 19b) Eliminating h r +1 from (1 9) yields x ∗ 1 x 2 − y ∗ 1 y 2 = u r +2 x ∗ 1 − v r +2 y ∗ 1 h r = w 2 (20) Since x i ∈ X an d y i ∈ Y for i = 1 , 2 , we have x ∗ 1 x 2 ∈ X and y ∗ 1 y 2 ∈ Y too . Now , by Proposition 2, x 2 and y 2 can be uniq uely determined by (1 3a) or (14a) and (13b) or (14b), respectively , and thus, h r +1 is uniquely determined by (13c) or ( 14c) with m = 2 . In g eneral, w e proceed to con sider determ ining the ( r + m ) - th ch annel coefficient for 2 < m ≤ L − r − 1 . In th is case, we have receiv ed h r + m − 1 x 1 + h r + m − 2 x 2 + · · · + h r x m = u r + m (21a) h r + m − 1 y 1 + h r + m − 2 y 2 + · · · + h r y m = v r + m (21b) This is e quiv alent to h r + m − 1 x 1 + h r x m = u r + m − m − 2 X i =1 h r + i x m − i (22a) h r + m − 1 y 1 + h r y m = v r + m − m − 2 X i =1 h r + i x m − i (22b) Eliminating h r + m − 1 from (2 2) yields x ∗ 1 x m − y ∗ 1 y m = x ∗ 1  u r + m − P m − 2 i =1 h r + i x m − i  h r − y ∗ 1  v r + m − P m − 2 i =1 h r + i x m − i  h r = w m . (23) Now , b y Proposition 2 , x m and y m can be unique ly d etermined by (1 3a) or (1 4a) and (13 b) or (1 4b), respectively , and th us, h r + m − 1 is uniqu ely determined by (13c) o r ( 14c). Step 4 . L − r − 1 < m ≤ K . In this case, since the channel coefficients have been d etermined by the previous three steps, we can dete rmine the oth er remaining tran smitted signals fr om the remaining received signals of each block. In this case, we have h L − 1 x m − L + r + 1 + h L − 2 x m − L + r + 2 + · · · + h r x m = u r + m (24a) h L − 1 y m − L + r + 1 + h L − 2 y m − L + r + 2 + · · · + h r y m = v r + m (24b) From this we can obtain ( 15). T his com pletes th e proo f of Theorem 1.  W e would like to make the following observations on Theorem 1. 1) Theo rem 1 no t only tells us tha t the chann el coefficients can be u niquely id entified by on ly transmitting two block signals with each sym bol selected from two co-prime PSK constellation s, but also provides simple and closed- form solutions to both the ch annel co efficients a nd the transmitted symbols. The tr aditional blind metho d [9] based on the secon d order statistics requ ires a lot of data blocks. Even so, it still cannot uniquely identify the channel co efficients as we ll a s th e transmitted sign als. 2) If we set p = 2 m and q = 2 m + 1 with m bein g a positive integer , then, it is clear that p and q are co prime. Therefo re, Theorem s 1 ho lds fo r such a pair of p and q . In add ition, if we want the original symbol sets X and Y to contain the same in teger bits, we can delete the only one comm on elemen t 1 fro m Y ; i. e., Y = Y − { 1 } . Thu s, there are totally 2 m elements in the remaining set Y and Theorem s 1 still holds for such a pair of constellation s X and Y . 3) By Theorem 1, if we let m 0 denote the maximum positi ve integer such that u r + m − P m − 2 i =1 h r + i x m − i = 0 for m 0 < m ≤ L − r but u r + m 0 − P m 0 − 2 i =1 h r + i x m 0 − i = 0 , then, the length of the channel is actually equal to m 0 − r + 1 . Therefore, ou r blind mo dulation scheme en ables th e receiver to exactly determin e the length o f th e channel by o nly utilizing the fir st two bloc k received signals. C. Full diversity The GLR T requires n either th e k nowledge of the fad- ing an d noise statistics, nor the knowledge of their real- izations [ 16]. The criterion can b e simply stated as ˆ S = arg max S { z H S  S H S  − 1 S H z } In fact, the GL R T pro jects the received signal z on th e different sub spaces spa nned by S and then calculate the energies o f all the projections and choose the projection that maximizes the energy . Now , in order to examine full div ersity , f or any pair of d istinct codew ords S and e S , let  S H e S H   S , e S  = A . Breh ler and V aranasi [ 17] proved the following lemm a. Lemma 1 : If matrices A have full column rank for all pair of distinct codew ords S and e S , then , the co de-bo ok provid es full d iv ersity for the GLR T re ceiv er . Now , we are in position to formally state the secon d our main r esult. Theor em 2: The blind modulation designed in Section II with p and q bein g coprime enables full di versity for the GLR T receiver . Pr oo f : By Lemma 1, we only need to prove that ( S , e S ) has full column rank for any p air o f distinct sig nal matrices S and e S . Otherwise, if th ere existed a pair of distinct codeword matrices S and e S for which the m atrix ( S , e S ) d oes not h ave full column rank, then, the linear equation s with respec t to v ariables h and − e h ,  S e S   h − e h  = 0 , would have a nonzero solution h 0 and e h 0 . L et r 0 = Sh 0 . Then, we would also have r 0 = e S e h 0 . In oth er words, for a gi ven nonzero received signal r 0 , eq uation r 0 = Sh has two distinct pair of solution s. By Theorem 1 , we have th at T ( x ) = T ( e x ) an d T ( y ) = T ( e y ) . Since S 6 = e S , ther e is a p air o f distinc t signal sub-matrices in S and e S , T ( z i ) and T ( e z i ) fo r some 3 ≤ i ≤ L . That being said , z i 6 = e z i . If we let z i = ( z i 1 , z i 2 , · · · , z iK ) T and e z i = ( e z i 1 , e z i 2 , · · · , e z iK ) T , then , there exists a p ositiv e integer k su ch that z ik 6 = e z ik but z iℓ = e z iℓ for ℓ = 1 , 2 , · · · , k − 1 . For n otional simplicity , we u se B [ M : N ] to deno te the sub - matrix of a m atrix B consisting of all the columns and the rows from M to N . Then, we have the result (25), which is shown in the b ottom of th is page, wh ere we h av e u sed the fact that T ( e z i )[ k : K + k − 1] − T ( z i )[ k : K + k − 1] is actually a K × K lower triangu lar matrix with the d iagonal entries being all equ al to z ik − e z ik . Th erefore, the sub-m atrix of ( S , e S ) ,  T ( x i )[1 : K ] T ( e x )[1 : K ] T ( z i )[ k : K + k − 1 ] T ( e z i )[ k : K + k − 1 ]  is a 2 K × 2 K invertible ma trix and hence, ( S , e S ) has full column ran k, wh ich con tradicts with the previous assumption. This completes th e proof of T heorem 2.  So far, we have shown that o ur blind mo dulation scheme not only en ables the unique iden tification of the c hannel coefficients in the noise-free case but also full di versity in the noise environmen t. Ho wever , the tradition al blind method [9] based on the second order statistics can provide neither the unique identification o f the ch annel coefficients nor e stimation of th e transmitted sign als with f ull d iv ersity reliability . Similar to the Com ment 2) on Th eorem 1, ou r T heorem 2 is also true for both a par ticular pair o f p = 2 m and q = 2 m + 1 and the derived pair of constellation s. I V . C O N C L U S I O N In this paper, we pro posed a novel blind modulatio n tech- nique to uniqu ely identif y freq uency selectiv e ch annels with zero-pa dded block transmission under no ise-free environments by only pro cessing the first two bloc k received signal vectors. Furthermo re, a closed-for m solution to determ ine the transmit- ted sign als an d th e channe l coefficients was derived by using linear c ongru ence equ ation theo ry . 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Theory , vol. 47, pp. 2383–2999, Sept. 2001. ˛ ˛ ˛ ˛ T ( x )[1 : K ] T ( e x )[1 : K ] T ( z i )[ k : K + k − 1] T ( e z i )[ k : K + k − 1] ˛ ˛ ˛ ˛ = ˛ ˛ ˛ ˛ T ( x ) [1 : K ] T ( e x )[1 : K ] − T ( x )[1 : K ] T ( z i )[ k : K + k − 1] T ( e z i )[ k : K + k − 1] − T ( z i )[ k : K + k − 1] ˛ ˛ ˛ ˛ = ˛ ˛ ˛ ˛ T ( x )[1 : K ] 0 T ( z i )[ k : K + k − 1] T ( e z i )[ k : K + k − 1] − T ( z i )[ k : K + k − 1] ˛ ˛ ˛ ˛ = x K 1 ( z ik − e z ik ) K 6 = 0 . (25)