Spreading in Block-Fading Channels.
2
Muriel Medard
David N.C. Tse2
1
Massachusetts Institute of Technology
[email protected] University of California at Berkeley
1
[email protected]
that similar results to those which apply to continuously varying doubly selective channels apply to doubly block fading channels, which are block fading in time and frequency. We are mainly interested in determining over what range of bandwidths spreading is bene cial. While all of the results mentioned above show that excessive spreading is detrimental in terms of capacity, we suspect that spreading is bene cial as long as it remains above some threshold. In Section 3, we develop a lower bound to capacity. Combining our upper and lower bounds, we show in Section 4 how we can nd upper and lower bounds to the optimal spreading bandwidth. We use a channel model where each block in frequency fades according to the model in [2]. Over each coherence bandwidth of size W , the channel experiences Rayleigh at fading. All the channels over distinct coherence bandwidths are independent, yielding a block-fading model in frequency. We transmit over coherence bandwidths. The energy of the propagation coeÆcient F [i]j over coherence bandwidth i at sampled time j is F2 . For input X [i]j at sample time j (we sample at the Nyquist rate W ), the corresponding output is Y [i]j = F [i]j X [i]j + N [i]j , where the N [i]j s are samples of WGN bandlimited to a bandwidth of W . The time variations are block-fading nature: the propagation coeÆcient of the channel remains constant for T symbols (the coherence interval), then changes to a value independent of previTW ous values. Thus, F [i](jTj +1) W +1 is a constant vector TW and the F [i](jTj +1) W +1 are mutually independent for j = 1; 2; : : :. For the signals over each coherence bandwidth, the second moment is upper bounded by X 2 E and the amplitude is upper bounded by
Abstract We consider wideband fading channels which are block fading in time and in frequency (doubly block fading). We show that, as bandwidth increases to in nity, capacity goes to zero when signaling is constrained in its second moment and peak signal amplitude. This result is consistent with similar results which do not consider doubly block-fading channels. None of these results, however, o er a range for the optimal spreading bandwidth, because they consider only upper bounds to capacity. While we know that spreading over very large bandwidths is detrimental in terms of capacity, we wish to determine over what range of bandwidths spreading is bene cial. We are able to give a range for the optimal spreading bandwidth by combining our upper bound with a suitable lower bound.
1
Introduction.
Recent results in the area of wideband fading channels have shown that, as the bandwidth over which we transmit becomes arbitrarily large, capacity goes to zero if we scale the signal inversely with the bandwidth. Several models have been considered for these channels variations. In [4], a nite number of timevarying paths are considered and, if paths remain unresolvable, capacity is shown to go 0 as as bandwidth becomes arbitrarily large. In [1], a general doubly selective fading channel model is considered, in which paths never become resolvable. References [1, 3] consider DS-CDMA transmissions over channels which are block fading in frequency but continuously fading in time. The above models all exhibit continuous variations in time. In this paper, we rst show, in Section 2, 1
p
2 E X .
2
1 I X [j ](i+1)T W ; Y [j ](i+1)T W
T
Upper bound on wideband ca-
We rst note that we can restrict ourselves, in the upper bound, to signals which satisfy the second moment constraint with equality. Indeed, let us suppose that we obtain an upper bound using a signaling scheme which does not meet the second moment constraint with equality. Then, by multiplying our signals by some > 1, we can achieve the second moment constraint with equality. Moreover, at the receiver we could divide the received signal by , in e ect reducing the AWGN. Thus, by the data processing theorem, the capacity of the channel with the input multiplied by would be greater than that of the channel not multiplied by . The following lemma gives our upper bound.
T
iT W +1
iT W +1
iT W +1
(3)
We may upper bound the rst term of (3) by: 1 h Y [j ](i+1)T W
T
iT W +1
1 T W 2T log (2e) Y [j]iTi W T W because entropy is maximized by a Gaussian distribution for a given autocorrelation matrix ! TY W 1 F2 X2 [i] + 1 2T log (2e)T W i=1 from Hadamard's inequality T W X = W log (2e) + 1 log F2 X2 [i] + 1 2 2T i=1
( +1) +1
Lemma 1
C W; E ; F2 ; T; ; W 2 E 2 log F + 1
iT W +1
TW = T1 h Y [j ](iTi+1) W +1 1 h Y [j ](i+1)T W jX [j ](i+1)T W :
pacity.
!
E 2
2 2 log T W + (1) 1 2T 2 E F
W2 log (2e) + W2 log F E + 1 ; 2
(4)
where the last inequality uses the concavity of the log and our average energy constraint. We First, we express C in terms of mutual informa- nowfunction proceed to minimize the second term of (2). tions. From our model, we have that TW (i+1)T W Note that, conditioned on X [j ](iTi+1) W +1 , Y [j ]iT W +1 TW is Gaussian and is Gaussian, since F [j ](iTi+1) W +1 (i+1)T W C W; E ; F2 ; T; ; = N1iT W +1 is AWGN. Hence, we have that 0 k 1 X X 1 I X [j ](i+1)T W ; Y [j ](i+1)T W (2) A @ lim max iT W +1 iT W +1 1 h Y [j ](i+1)T W jX [j ](i+1)T W k!1 pX k i=1 j =1 T iT W +1 iT W +1 T
where the fourth central moment of X [j ]i is and = 1 EX log (2e)T Y [j ] i T W : (5) 2T iT W its average energy constraint is E . Since we have no sender channel side information and all the band- i T W has kth diagonal term 2 x[k]2 + 1 and F Y [j ]iT W width slices are independent, we may use the fact j ) term equal to x(k)x(j )F2 , condithat mutual information is concave in the input dis- o -diagonal (k;(i+1) TW = x = [x(1); : : : ; x(T W )]. The tribution to determine that selecting all the inputs to tioned on X [j ]iT W +1 eigenvalues of are 1 for j = 1 : : : T W 1 and be IID maximizes the RHS of (2). We rst rewrite j Y 2 2 the mutual information term as: jjxjj F + 1 for j = T W . Proof of Lemma 1.
2
( +1) +1
( +1) +1
2
Hence, we may rewrite (4) as
3
Lower bound on wideband capacity.
1 h Y (i+1)T W jX (i+1)T W
T
iT W +1
Our lower bound on capacity is obtained by choosing the X[j]s to be:
iT W +1
T W 2 2 = 21T EX log x(iTi+1) + 1 F W +1 (6) + W2 log(2e):
8
E0 . Let us assume that + 4 We W 2 p E2 = nE1 , for n integer. Then, we have that E E W E W = 4 lim 1 C W; E ; 2 ; T; 1; !1 W 2 e 0
1 4
E
W
1
C E A W
W
+e
p pE + WE 1 e 4 W 2
q
E We B log @ 1 1 q E + e 1 2
4
3 2
3 2
1 2
1 4
q
1
E We
3 2
2
E
W
+e
E
=
2E
W
up
n
2
F
n W; E1 ; F2 ; T; 1;
Clow
(15)
Let us now assume that mE0 = E1 . Then, we have that
!
W
Cup
2E
= 1
W
m
4
W; E0 ; F2 ; T; 1;
Clow
W; E1 ; F2 ; T; 1; :
(16)
Cup(E) Clow(E)
E0
E2
E1
E
Figure 1: Upper and lower bounds for capacity and the graphical interpretation of the upper and lower bounds to the optimal For = 1, E1 is an upper bound to the optimal
E and E is a lower bound. Ifi we x E = , then h 0
must be in the range
;
E1 E0
.
References
[1] M. Medard, R.G. Gallager \Bandwidth Scaling for Fading Multipath Channels", submitted to IEEE Transactions on Information Theory. [2] T.L. Marzetta, B.M. Hochwald, \Capacity of a Mobile Multiple-Antenna Communication Link in Rayleigh Flat Fading", IEEE Transactions on Information Theory, vol. 45, no. 1, January 1999, pp. 139{ 157. [3] M. Medard, \Bound on Mutual Information for DSCDMA Spreading over Independent Fading Channels", proceedings of Asilomar Conference on Signals, Systems and Computers, November 1997, pp. 187{191. [4] E. Telatar, D. Tse, \Capacity and Mutual Information of Wideband Multipath Fading Channels",IEEE Transactions on Information Theory , Vol. 46, no. 4 , July 2000, pp. 1384 -1400.
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