Spreading Signals in the Wideband Limit

Spreading Signals in the W ideband Limit Elchanan Zwecher an d Dana Porrat The Hebrew Uni versity of Jerusalem Jerusalem, Israel 9 1904 elchanz w , dporrat@cs.huji.ac.il Abstract — Wideband communications are impossible with sig- nals that are spread o ver a ve ry large band and ar e t ransmitted ov er multipath channels unknown ahead of ti me. Thi s work exploits the I-mmse connection to bound the achieva ble data- rate of spre ading si gnals in wi deband settings, and to conclude that the achieva ble data-rate dimin ishes as the bandwidth in- crease s due t o ch annel uncertainty . Th e result applies to all spreading modulations, i.e. signals that are evenly spread ov er the ban dwidth a v ailable to the communication system, with SNR smaller than log(W/L)/ (W/L) and holds for communications ove r channels where the nu mber of paths L is unbound ed by sub- linear in the bandwid th W . I . I N T R O D U C T I O N This work analy zes the perfo rmance of wideband com muni- cation systems, we character ize the SNR regime that does not allow commun ications in the wideband limit. The under lying reason fo r th e inab ility to com municate is chann el uncerta inty , and our result applies to channels where the number of multipath compon ents is unbound ed and sub -linear in the bandwidth . Our proof is based o n the I-mmse connection [3] and hin ges on a calculation of the m inimum mean squa re erro r (mmse) estimate of the u nknown channel. W e consider signals that spr ead their power over the en tire bandwidth , such as PPM or impulse radio [14], where the pulse shape and duratio n determin e the bandwidth , or d irect sequence sp read spectr um, wher e the chip duratio n determines the freq uency spr ead of the sign al. Our result app lies also to OFDM-type signals, if th e entire av ailable b andwidth is used concur rently . Examples of signals that are not spread over bandwidth are FSK and multi-tone FSK [6], [5], where each symbol concen trates p ower on a small span of fr equencies, although the entire range of symbols ma y span a very large bandwidth . Our result shows that chan nel un certainty is detrimental to spreading systems oper ating over mu ltipath chann els where the numb er of appare nt paths is unbo unded but sub-linear with the system ban dwidth [11], [10], essentially becau se of the signal uses to o many eigen -modes of the chan nel. Our mod el of ch annel variation in time is a simplistic b lock coheren t one, where the channel is fixed for k nown leng ths of time (co herenc e period s) an d realizations over different coheren ce periods are IID. Th is channel mo del offers the advantage of eigen-modes that are p articularly simple, as harmon ic signals ar e eigen-modes of any lin ear time in v ariant channel. Th e channel u ncertainty a communication system faces when o perating over the block– coheren t chan nel is thus limited to the eigen-values of th e cha nnel, or in o ther words to the co mplex chann el gain ov er the frequency b and the system uses. The essen tial featur e of spreading signals that rend ers them ineffecti ve over wide b ands, is that they use the entire ran ge of ch annel eigen-modes concu rrently , an d ar e thu s expo sed to uncertainty of a large num ber of parameters (c hannel eige n- values). Mo dulation schemes of the FSK type, that exp loit a small number of channel eigen-modes per symbol, are exposed to uncer tainty in only a small number of p arameters. Our result can be extended to more complex channels, where the variation in time is d escribed u sing the Dop pler spectrum rather than by block–c oheren ce [9], [12], [2]. T he eigen-mo des of such chann els are approx imately g i ven by orthog onal W eyl-Heisenberg b ases [4] in the under-spread case, i.e. when the channel’ s response is highly concentrated in the delay– Doppler plain. The essential f eature that determines whether co mmun ications are possible in the wideban d limit is the spreading of symbol power over the unknown ch annel eigen-mo des. In c hannels whe re the eigen-mo des are not known in advance, it is very difficult to overcome ch annel uncertainty . Related work has sh own that pulse-po sition modulatio n (PPM) systems are unable to handle uncertainty in the delays of m ultipath comp onents [7], [ 8]. W e showed that the r eceiv er is unable to de tect the chan nel p aths if th e bandwidth is large enoug h, whether it uses a thr eshold detector [7] o r a max imum likelihood d etector [8]. This work extends the scope o f past results to a wider f amily of signals, and makes a stateme nt on the achievable r ate. The com parison of our resu lt to that of T elatar & Tse [ 13] is also interesting. Th e T&T results discussed con tinuous signals, i.e. ef fectiv ely u sing a signal to noise ratio (SNR) that in versely depend s on the system band width SNR = θ  1 W  , whereas our result is more general in the sense that it spec ifies the range o f SNR dependen cies on bandwidth that does not allo w commun ications in the wideb and regime. In o ther words, our result ap plies to impulsive systems and imp licitly ind icates the minimal lev el of imp ulsiv eness that allows com municatio ns in the limit of large bandw idth. Another point of compar ison of our result to that of [13] is the type of signals to wh ich they ap ply . T&T consider two types of spread ing signals: I ID comp lex signals and signals with a very low cro ss-correlation : P i X i X i − n ∼ const that does n ot d epend on the len gth of the vector . Our result a pplies to a wider family of signals, wh ere the cross correlatio n may as hig h as θ  √ W  . Th is is significant becau se spreadin g signals generate d b y IID or p seudo-r andom sequenc es have an empir ical corre lation that varies as θ  √ W  . The results in [ 8], that connect the nu mber of ch annel paths to the level of im pulsiveness are also rele vant here. The impulsiveness pa rameter there is easily tr anslated to the SNR per active b urst o f transmission, and th e results in [8] b asically determine a lower bound on t he SNR, above which the channel uncertainty penalty is insignificant in th e wideband limit. Our new r esult inv olves an upper bou nd on the SNR, below which the ch annel unce rtainty penalty pr ev ents commun ications. I I . M O D E L W e consider communication systems with a (single sided) bandwidth W , operatin g over blo ck-coh erent multipath chan- nels. W e use a real discrete m odel of the system, after sampling at the recei ver at rate W . T he mod el over a single coherence period is giv en by Y = √ SNR X ⋆ ˜ H + Z (1) where Y is the r eceived s ignal over an en tire co herence per iod of leng th T c , this is a vector with K c = T c W entries. The vector X of len gth K c and average energy K c represents th e transmitted signal, the multipath channel is rep resented by th e vector ˜ H o f len gth K c and ⋆ mark s a con volution. Z is white standard Gaussian n oise (IID with zer o mean and variance one) and SNR is the sign al to no ise ratio. The SNR can b e understoo d as the signal to no ise ratio per fr equency resolution bin or per degree of freedom. W e neglect in (1) the edge effects at the beginning of the coherence period . W e impo se a pro babilistic energy constrain t: P  k X k 2 > (1 + o (1)) K c  − − − − → W →∞ 0 (2) Note that IID signaling satisfies this assump tion. The tra nsmitted signal may be impulsi ve, we co nsider the signal to noise ratio during active transmission so th ere is no need to explicitly address the impulsi veness used by the system (i.e. the duty cycle ratio). W e assume that the tra nsmitter does n ot use informatio n on the channel realization , and the transmitted sign al does n ot depe nd on it. The transmitted signa l X is wideban d: its empirical auto- correlation is upper boun ded by    D X i , X j E    ≤ B 4 p K c i 6 = j i, j = 1 , . . . , K c (3) with B 4 is a constant th at does not d epend o n the bandwidth. The notation < , > is u sed for th e inner p roduc t of vectors, and the notation X i is u sed for a vector X that is cyclicly shifted by i position s, i.e. X i =      X (1 − i ) X (2 − i ) . . . X ( K c − i )      (4) where ( − ) indicates a mo d K c difference. The ch annel is compo sed of L path s, each with a delay in the range [0 , T d W ] where T d is the delay spread . The channel is block-co nstant with co herence time T c , i. e. it has IID realizations over different co herence periods. W e assume T d ≪ T c and thus justify to an extent our loo se treatment of edge effects at the beginning of each co herenc e p eriod. W e approx imate (1) with a circularly-shif ted matrix: Y = √ SNR x ˜ H + Z (5) where x =      X 1 X K c . . . X 2 X 2 X 1 X K c . . . . . . . . . . . . X K c . . . X 2 X 1      =  X 0 X 1 . . . X K c − 1  (6) The ch annel mo del is real, L ch annel ga ins are IID and zero mean, with variance 1 /L , so the en ergy in the channel’ s impulse respo nse equals on e on a verage. W e assume an upper bound on path g ains | H i | > B 1 √ L , with a constant B 1 that do es not depen d on the bandwidth . The choice of the L non zero taps is uniform over the  K c L  possibilities. The num ber of paths L di verges as th e bandwid th inc reases in a sub -linear m anner [1 1], [ 10], i.e. L − − − − → W →∞ ∞ and L/W − − − − → W →∞ 0 . W e make a probab ilistic assumptio n o n th e cha nnel’ s re- sponse P         K c X j =1 , j 6 = i ˜ H j D X i , X j E       > B 3 p K c   − − − − → W →∞ 0 i = 1 , 2 , . . . , K c (7) with a con stant B 3 that does not depend on the b andwidth. The typical v alue of the correlatio n in (7) is √ K c , so this assupmtion is a n atural one. By takin g a large co nstant B 3 we ensure th at our r esult hold s for alm ost all v alues of i . I I I . R E S U LT Theor em 1: Com municatio n systems modeled by (5), that use spread ing signals and op erate over mu ltipath channels as described in Section II, with SNR ≪ log W/L W/L have a diminishing ra te in the limit of large bandwidth: I ( Y ; x ) 1 2 K c SNR − − − − → W →∞ 0 with pro bability 1. W e prove the theo rem in Section V by showing th at lim W →∞ I  Y ; ˜ H | x  1 2 K c SNR = 1 and ap plying I ( Y ; x ) = I  Y ; ˜ H , x  − I  Y ; ˜ H | x  (8) ≤ 1 2 K c SNR − I  Y ; ˜ H | x  (9) ~ 1 2 c K SNR ~ I(Y;H,x) upper bound SNR I(Y;x) upper bound I(Y;H|x) log(W/L)/(W/L) Fig. 1. A ske tch of the I-SNR relat ionship for spread signals for a very larg e bandwidth. The coherent datarat e upper bound (top graph) is linear in the lo w SNR regime, and con vex. The channel uncertai nty penalty in the bottom graph is linear for lo w SNR value s, and satur ates at the channel’ s entropy . The incoherent datarat e (middle graph) is not con v ex. The term I  Y ; ˜ H | x  in (9) is the datarate pen alty du e to channel un certainty . I V . D I S C U S S I O N The proof of Theorem 1 is based o n calculating the mmse estimate of the chan nel response ˜ H , given the tr ansmitted and the r eceived signals. W e sh ow that this mmse estimate is a vector with an o (1) norm in low SNR cond itions, essentially because th e n oise Z overwhelms the information ca rrying signal. Theorem 1 shows that in the wideband limit, the low SNR regime can b e divided in to parts: SNR ≪ log K c /L K c /L (10) where spr eading signals ar e no t effecti ve, and SNR > log K c /L K c /L (11) where although the SNR d iminishes in the limit, it ena bles a positive datarate. The chan nel un certainty penalty is upper b ounde d by the channel en tropy , and the b ottom grap h of Figure 1 thus saturates at SNR = log K c /L K c /L + o ( L ) K c , where the first pa rt correspo nds to he entr opy of the paths’ delays and th e second to th eir gains. V . P R O O F O F T H E O R E M 1 The proo f is based o n the I-mmse conn ection, in particular Theorem 2 of [3] . This theorem gives a simple for mula to the achievable ra te of co mmunica tions over a known vector channel in term s o f the err or o f the mm se estima te of the transmitted sign al. W e reverse the roles o f H and x in our usage of Theo rem 2 o f [3], i.e. co nsider x a s known an d H as the estimated party . Using our notation, the I-mmse connection says that as long as the vector H satisfies E k H k 2 < ∞ we have I  H ; √ SNR xH + Z  = 1 2 Z SNR 0 mmse ( SNR ) d SNR (12) where mm se ( SNR ) is given b y mmse ( SNR ) = E     x ˜ H − x ˆ H ( Y ; SNR )    2  (13) and ˆ H is the m mse estimate of ˜ H given both x and Y . W e will show th at the mmse estimate is a vector with an o (1) n orm in low SNR conditions, thus the minimal m ean square error (13) conv erges to K c , the mutu al info rmation (12) conv erges to 1 2 K c SNR an d (9 ) diminishe s. The mm se estimate is given b y ˆ H = E [ H | Y , x ] (14) W e lower b ound the mmse by calculating the minimum mean square erro r in a system that is given a dditional inf ormation on ˜ H , namely wh ich of its position s satisfies (7). ˆ H ′ = E h H | Y , x, I ( 7 ) i (15) where I ( 7 ) is a list of ind ices { i } where ˜ H i satisfies (7). The additional info rmation can o nly red uce the m mse. ˆ H ′ = Z Z . . . Z H f ( H | Y , x ) dH 1 dH 2 . . . dH L (16) The co nditional p robability d ensity in (16) is man ipulated using the indepen dence of the tran smitted signal from the channel. ˆ H ′ = R R . . . R H f ( Y | x, H ) f ( H ) dH 1 dH 2 . . . dH L R R . . . R f ( Y | x, H ) f ( H ) dH 1 dH 2 . . . dH L (17) The cond itional probability density in (1 7) is Gaussian as Y | x, H ∼ N ( xH, I ) . W e deno te by f s ( ) the probability density of a K c long vector of II D standa rd Gaussian variables f s ( S ) = 1 (2 π ) K c / 2 exp  − 1 2 k S k 2  (18) and proceed to examine the componen ts of th e vector ˆ H ′ . Consider first positions (indices) j where ˜ H j = 0 . At these positions, any non-ze ro v alue of ˆ H ′ j increases the estima tion error and can be disregarded in the calculation of a lower bound o n th e mm se. Let us examine the p ositions w here ˜ H i 6 = 0 and ( 7) holds, an d lo ok at the estimates of each su ch v alue: ˆ H ′ i = R R . . . R H i f s  √ SNR x  ˜ H − H  + Z  f ( H ) dH 1 dH 2 . . . dH L / R R . . . R f s  √ SNR x  ˜ H − H  + Z  f ( H ) dH 1 dH 2 . . . dH L (19) W e p rove that th e mmse estimate (1 9) is very small b y comparin g the integral in the nom inator to th e integral in the denomin ator, tha t effecti vely sums over a bigger group of assignments of H . Assuming (7) we show that the n ominator of (19) is negligible when compared to its denominato r . W e first app roximate both integrals in (19) b y sums over sampled grou ps of values of each positions in the vectors H . The s ampling is done over a tight enough grid that the resulting errors ar e small. An upper b ound to (1 9) is calculated by breaking th e sum in th e de nominato r to a series o f su ms over gr oups of v alues of H , where e ach g roup cor respond s to a single assignment of H in the nominator . T he set B consists of assign ments with a non-ze ro v alue H i 6 = 0 . Rewriting the d iscrete ap prox imation of (19) we get ˆ H ′ i = P H ∈B H i f s  √ SNR x  ˜ H − H  + Z  p ( H ) P G ∈B + B c f s  √ SNR x  ˜ H − G  + Z  p ( G ) (20) the notatio n G was introdu ced to improve clarity we define p ( H ) = f ( H )∆ H L and ∆ H is the sampling step of each dimension of the vector H . W e proceed to divide the entire r ange of vector s G into non- overlapping subgr oups, such that for each H ∈ B we have a correspo nding subgroup A ( H ) , su ch th at J ( H ) = H i f s  x  ˜ H − H  + Z  p ( H ) P G ∈A ( H ) f s  x  ˜ H − G  + Z  p ( G ) (21) diminishes in the limit of large ban dwidth. The con vergence of (20) to zero f ollows directly , as the n ominator of ( 20) is a sum of the nomin ators of J ( H ) f or all H ∈ B . The subgrou ps A ( H ) ar e created ra ndomly . For each as- signment of G that has a n on-zer o value in the i th position, it is put in the su bgrou p A ( G ) . For a vector G with G i = 0 we (uniform ly) cho ose one of its n on-zer o tap s and replace it to the i th position. T o clarify the process, let us say that the j th position o f the vector G was ch osen. W e calcu late a ne w vector H by H i = G j ; H j = 0 ; H k = G k for k 6 = i, j (22) and assign the vector G to the sub grou p A ( H ) . Each gr oup A ( H ) contains H and about ( K c − L ) /L o ther members, each different fr om H in exactly two p ositions. W e ensure th at gr oups’ sizes do not d eviate significantly fr om ( K c − L ) /L by reloc ating mem ebers from large grou ps in to suitable smaller ones. W e denote b y H i → k a member of A ( H ) that differs from H by e xchang ing the values in its i th and k th positions, an d define H i → i = H . The set K ( H ) holds the values of k such that H i → k ∈ A ( H ) . The term s p ( H ) in the nominator of (21) and p ( G ) in th e denomin ator are identical fo r all members of the group A ( H ) because of our assum ptions on IID ga ins and a u niform spr ead of the path delay s. J ( H ) = H i exp  − 1 2    Y − √ SNR xH i → i    2  P k ∈K ( H ) exp  − 1 2    Y − √ SNR xH i → k    2  (23) The d enomin ator of (2 3) c ontains a sum ov er about K c /L exponents with different v alues o f k , inclu ding k = i and the nom inator hold s a sign al such factor with k = i . W e take a close look at their exponent and in troduce the notation I ( H i , k ) for a K c -long vector with the v alue H i at the k th positions an d zeros e lse where. − 1 2    Y − √ SNR xH i → k    2 = − 1 2 k Y k 2 (24) − 1 2    √ SNR x  H i → k − I ( H i , k )     2 (25) − 1 2    √ SNR xI ( H i , k )    2 (26) − D √ SNR x  H i → k − I ( H i , k )  , √ SNR xI ( H i , k ) E (27) + D √ SNR x ˜ H , √ SNR x  H i → k − I ( H i , k )  E (28) + D √ SNR x ˜ H , √ SNR xI ( H i , k ) E (29) + D Z, √ SNR x  H i → k − I ( H i , k )  E (30) + D Z, √ SNR xI ( H i , k ) E (31) we now deal with each line (24)-(31) separately , to show that the nom inator of ( 23) is much smaller than the den ominator . (24): The term − 1 2 k Y k 2 does not depend on k . (25), (28), (30): Th ese term s do no t dep end on k because the vectors H i → k − I ( H i , k ) are iden tical over k ∈ K ( H ) . (26): The term − 1 2    √ SNR xI ( H i , k )    2 depend s on k , but the norm is co nstant over k ∈ K ( H ) . The vector I ( H i , k ) essentially extracts a single colu mn of the matrix x and multiplies it by H i . The matrix x is circularly symmetric and thu s (26) is fixed. (27): The term a k = − D √ SNR x  H i → k − I ( H i , k )  , √ SNR xI ( H i , k ) E = − SNR H i K c X j =1 j 6 = k H j D X j − 1 , X k − 1 E is sig nificantly smaller than (29) at th e n ominator of (23), or in o ther words an ord er of magn itude smaller than K c SNR L . W e prove th is by calculating a i = − SNR H i K c X j =1 ,j 6 = i H j D X j − 1 , X i − 1 E (32) The typical value is o n the order of B 1 SNR √ K c √ L and a loose upper bound is gi ven by Using condition (7) on ˜ H , that basically ensur es a low corr elation in (32), we have | a i | ≤ B 1 B 3 SNR r K c L (33) (29): The term b k = D √ SNR x ˜ H , √ SNR xI ( H i , k ) E = SNR H i K c X j =1 ˜ H j D X j − 1 , X k − 1 E is the dominant term in the no minator of (23), i.e. for k = i . b i = SNR H i K c X j =1 ˜ H j D X j − 1 , X i − 1 E (34) The dominan t term in the sum is the i th , where D X i − 1 , X i − 1 E ≤ (1 + o (1)) K c with h igh prob ability . | b i | ≤ B 2 1 (1 + o (1)) SNR K c L + B 1 B 3 SNR √ K c √ L = B 2 1 (1 + o (1)) SNR K c L (35) the last app roximate eq uality is tigh t in th e limit of large bandwidth . (31): The term c k = D Z, √ SNR xI ( H i , k ) E is the dominan t term in the d enomin ator o f (2 3). The sum o f exponents of { c k } is lower boun ded by a single expo nent with k ⋆ ∈ K ( H ) . W e use asymptotic or der statistics to show that there is k ⋆ ∈ K ( H ) such that c k ⋆ = q K c SNR L q 2 lo g K c L in the limit. T o prove the existance o f k ⋆ we examine the joint probab ility d ensity o f { c k } . These are mutually Gau ssian zero mean ran dom variables, with variance v ar ( c k ) = H 2 i SNR D X k , X k E ≤ H 2 i SNR (1 + o (1)) K c and covariance cov ( c k , c m ) = H 2 i SNR D X k , X m E ≤ H 2 i SNR B 4 p K c W e collect { c k } into the vector C of length M = |K ( H ) | and mark its corrlation matrix by R c . R c is positive definite, it has a constant and large value on its diago nal, and significantly smaller values off-diago nal. The mean and variance of th e maximal o f M IID ∼ N  0 , σ 2  random v ariables are gi ven b y [1]: the me an equals σ  √ 2 ln M − ln ln M + ln 2 π − 2 C 2 √ 2 ln M + O  1 ln M  (36) and the v ariance is π 2 σ 2 12 ln M + O  1 ln 2 M  (37) where C ≈ 0 . 5772 is Euler’ s con stant. Note that for a large M the v ariance diminishes. T hese results cannot b e directly applied to the maximal { c k } because th ese variables ar e correleated . W e sh ow that the co rrelations am ong { c k } are insignificant in th e limit o f large band width in the sense that there is a c k ⋆ that is very similar to the maxim al of IID Guassians, and coclude that in the limit c k ⋆ → (1 + o (1)) q H 2 i SNR K c √ 2 ln M (38) In the nomin ator of (23) the term c i is in significant, it has zero mea n and a small variance on the order of q K c SNR L . T o sum marize the discussion of (24)-(31), we can upper bound (23) in the limit of large band width using the significant terms in the nominato r and den ominato r: J ( H ) ≤ H i exp  − 1 2    Y − √ SNR xH i → i    2  max k ∈K ( H ) exp  − 1 2    Y − √ SNR xH i → k    2  ≈ H i exp { a i + b i − c k ⋆ } (39) ≤ H i exp ( 3 B 2 1 K c SNR L − r K c SNR L r 2 lo g K c L ) (40) and for SNR = o  log K c L K c L  the expone nt of (4 0) div erges to −∞ a s the bandwidth increases, a nd J ( H ) − − − − → W →∞ 0 . Replacing K c by W T C , the proof of Theorem 1 is complete. R E F E R E N C E S [1] Harald Cram ´ er . Mathemat ical Met hods of Statistics . Princ eton Univ er - sity Press, 1946. [2] Giuseppe Durisi , H elmut Bolcskei , and Shlomo Shamai . Capacit y of undersprea d W SSUS fad ing channels in the wideband regime. In IEEE Internati onal Symposium on Information Theory , pages 1500–150 4, Jul. 2006. [3] Dongning Guo, Shlomo Shamai, and Ser gio V erd ´ u. 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