C up (E) C low (E) E 2 E 1 E 0

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1 Spreading in lock-fading hannel. Muriel Médard David N.. Te medardmit.edu Maachuett Intitute of Technoy dteeec.berkeley.edu Univerity of alifornia at erkeley btract We conider wideband fading channel which are block fading in time and in freuency (doubly block fading). We how that, a bandwidth increae to innity, capacity goe to zero when ignaling i contrained in it econd moment and peak ignal amplitude. Thi reult i conitent with imilar reult which do not conider doubly block-fading channel. None of thee reult, however, offer a range for the optimal preading bandwidth, becaue they conider only upper bound to capacity. While we know that preading over very large bandwidth i detrimental in term of capacity, we wih to determine over what range of bandwidth preading i benecial. We are able to give a range for the optimal preading bandwidth by combining our upper bound with a uitable lower bound. Introduction. Recent reult in the area of wideband fading channel have hown that, a the bandwidth over which we tranmit become arbitrarily large, capacity goe to zero if we cale the ignal inverely with the bandwidth. Several model have been conidered for thee channel variation. In [], a nite number of timevarying path are conidered and, if path remain unreolvable, capacity i hown to go a a bandwidth become arbitrarily large. In [], a general doubly elective fading channel model i conidered, in which path never become reolvable. Reference [, ] conider DS-DM tranmiion over channel which are block fading in freuency but continuouly fading in time. The above model all exhibit continuou variation in time. In thi paper, we rt how, in Section, that imilar reult to thoe which apply to continuouly varying doubly elective channel apply to doubly block fading channel, which are block fading in time and freuency. We are mainly intereted in determining over what range of bandwidth preading i benecial. While all of the reult mentioned above how that exceive preading i detrimental in term of capacity, we upect that preading i benecial a long a it remain above ome threhold. In Section, we develop a lower bound to capacity. ombining our upper and lower bound, we how in Section how we can nd upper and lower bound to the optimal preading bandwidth. We ue a channel model where each block in freuency fade according to the model in []. Over each coherence bandwidth of ize W, the channel experience Rayleigh flat fading. ll the channel over ditinct coherence bandwidth are independent, yielding a block-fading model in freuency. We tranmit over coherence bandwidth. The energy of the propagation coefcient F [i] j over coherence bandwidth i at ampled time j i fff. For input X[i] j at ample time j (we ample at the Nyuit rate W ), the correponding output i Y [i] j = F [i] j X[i] j + N[i] j, where the N[i] j are ample of WGN bandlimited to a bandwidth of W. The time variation are block-fading nature: the propagation coefcient of the channel remain contant for T ymbol (the coherence interval), then change to a value independent of previou value. Thu, F [i] (j+)tw i a contant vector jtw+ and the F [i] (j+)tw jtw+ are mutually independent for j = ; ;:::. For the ignal over each coherence bandwidth, the econd moment i upper bounded by X» and the amplitude i upper bounded by

2 p fl X. Upper bound on wideband capacity. We rt note that we can retrict ourelve, in the upper bound, to ignal which atify the econd moment contraint with euality. Indeed, let u uppoe that we obtain an upper bound uing a ignaling cheme which doe not meet the econd moment contraint with euality. Then, by multiplying our ignal by ome ff >, we can achieve the econd moment contraint with euality. Moreover, at the receiver we could divide the received ignal by ff, in effect reducing the WGN. Thu, by the data proceing theorem, the capacity of the channel with the input multiplied by ff would be greater than that of the channel not multiplied by ff. The following lemma give our upper bound. Lemma W; ;fff ;T;;fl» W ff + F Tfl TW fl ff F + () Proof of Lemma. Firt, we expre in term of mutual information. From our model, we have that W; ;fff ;T;;fl = lim max k px k kx X i= j= T I X[j] (i+)tw it W + ; Y [j](i+)tw it W + () where the fourth central moment ofx[j] i i fl and it average energy contraint i. Since we have no ender channel ide information and all the bandwidth lice are independent, we may ue the fact that mutual information i concave in the input ditribution to determine that electing all the input to be IID maximize the RHS of (). We rt rewrite the mutual information term a: T I X[j] (i+)tw it W + = T h Y [j] (i+)tw it W + T h Y [j] (i+)tw it W + ; Y [j](i+)tw it W + jx[j](i+)tw it W + : () We may upper bound the rt term of () by: T h Y [j] (i+)tw it W +» T (ße) TW Λ (i+)tw Y [j] it W + becaue entropy i maximized by a Gauian ditribution for a given autocorrelation matrix» TW Y (ße) T TW i= from Hadamard' ineuality = W (ße)+ T TW X i=» W (ße)+W ff F ff F ff X[i] + ff F ff + X[i] + ; () where the lat ineuality ue the concavity of the function and our average energy contraint. We now proceed to minimize the econd term of (). Note that, conditioned on X[j] (i+)tw it W + i Gauian, ince F [j] (i+)tw it W + N (i+)tw it W + i WGN. Hence, we have that T h Y [j] (i+)tw it W +» = T X Λ Y [j] (i+)tw it W +, Y [j](i+)tw it W + i Gauian and jx[j](i+)tw it W + (ße) T Λ Y [j] (i+)tw it W + : (5) ha k th diagonal term ff F x[k] + and off-diagonal (k; j) term eual to x(k)x(j)fff, conditioned on X[j] (i+)tw it W + = x =[x();:::;x(tw)]. The eigenvalue j of Λ Y are for j =:::TW and jjxjj fff +for j = TW.

3 Hence, we may rewrite () a T h = T X + W Y (i+)tw (i+)tw it W jx + it W +» x (i+)tw it W + ff F + (ße): (6) We eek to minimize the RHS of (6) ubject to the econd moment contraint holding with euality and the ubject to the peak amplitude contraint. The ditribution for X which minimize the RHS of (6) ubject to our contraint can be found uing the concavity of the function. The ditribution i uch that the only value which jxj can take are and fl p with probabilitie fl and fl, repectively. Thu, we maylower bound (6) by T h Y (i+)tw (i+)tw it W + jx it W + Tfl TW fl ff F + ombining (7), () and () yield (). + W (ße):(7) Q..D. We obtain immediately ;T; from () that lim W; ;fff. The RHS of () increae with T. Intuitively, we expect the real capacity to alo increae with T,ince a longer coherence time entail better meaurement of the channel, and thu channel behavior which i cloe to that of an WGN channel over every coherence time. Moreover, RHS of () increae with W. gain, a longer coherence bandwidth entail better meaurement of the channel. Finally, we may note that, a fl increae, the bound in () converge more lowly. For any, the limit a fl i ff W F. + Lower bound on wideband capacity. Our lower bound on capacity i obtained by chooing the X[j] to be: X[j] = 8 < : with probability with probability : (8) Moreover, we elect the X to be IID. The channel we conider for our lower bound i a S with croover probability ffl =Φ. Thu, the capacity of our original channel i lower bounded by time the capacity of thi S channel W; ;ff N ;T;;fl ( H(ffl)) = ( + ffl (ffl)+( ffl) ( ffl)) : (9) We may bound ffl a follow: ffl = Z p ß e x dx» p ß e» e : () Hence, we may upper bound H(ffl) in the following manner: H(ffl)» + e e = 6 + e e + + e e e e

4 = + + e e e + e () Uing the fact that ln( x) x, we can write the bound: e (e) 8 e e : () Thu, from (9,, ), we obtain Lemma W; ;ffn ;T;;fl e p e e + e () We may examine the behavior of the RHS of () a. We have lim e =. The limit of the econd term of the RHS of () can be examined uing L'Hopital' rule: = lim p W p p e lim e e + e p W + W e e + e + e e + e p p W + W = + e e () Hence, the limit of the RHS of () a i, and Lemma and provide tight bound. Upper and lower bound to the optimal preading bandwidth We maynow ue our upper and lower bound to nd a range for Λ, the optimal value of to which to pread. Firt note that preading from to n i benecial iff W; ;ff F ;T;;fl» W; ;ff F ;T;n;fl. We alo note that, given the convexity of capacity, we have ;T;;fl that W; ;fff = n W; n ;fff ;T;;fl. Let u x =. Our lower and upper bound from Lemma and are denoted, repectively, by the function low and up. Figure how low and up a a function of, with all other parameter kept contant. Thee function have the property that they are convex for mall value of and concave forlargevalue of. They each exhibit a ingle addle point. The line tangent to low emanating from the origin i tangent to low at. It interect up at and, where >. Let u aume that = n,forninteger. Then, we have that n up W; ;fff ;T;;fl = low n W; ;ff F ;T;;fl (5) Let u now aume that m =. Then, we have that up W; ;ff F ;T;;fl = m low W; ;ff F ;T;;fl : (6)

5 up () low () Figure : Upper and lower bound for capacity and the graphical interpretation of the upper and lower bound to the optimal For =, i an upper bound to the optimal and i a lower bound. h If we x = ο, then Λ mutbeintherange ο ο ; i. Reference [] M. Médard, R.G. Gallager andwidth Scaling for Fading Multipath hannel", ubmitted to I Tranaction on Information Theory. [] T.L. Marzetta,.M. Hochwald, apacity of a Mobile Multiple-ntenna ommunication Link in Rayleigh Flat Fading", I Tranaction on Information Theory, vol. 5, no., January 999, pp [] M. Médard, ound on Mutual Information for DS- DM Spreading over Independent Fading hannel", proceeding of ilomar onference on Signal, Sytem and omputer, November 997, pp []. Telatar, D. Te, apacity and Mutual Information of Wideband Multipath Fading hannel",i Tranaction on Information Theory,Vol.6,no., July, pp