Feedback can substantially increase the achievable rate for multi-carrier transmission through a frequency-selective fading channel. In [1], the achievable rate ...
Minimum Feedback Rates for Multi-Carrier Transmission With Correlated Frequency–Selective Fading Yakun Sun and Michael L. Honig Department of ECE Northwestern University Evanston, IL 60208
Abstract— We consider multi-carrier transmission through a frequency-selective fading channel with limited feedback. An on-off power allocation activates the set of sub-channels with gains above a threshold. We model the sequence of sub-channel gains as a Markov process, and give a lower bound on the feedback rate in bits per sub-channel needed to specify the sequence of activated sub-channels as a function of the activation threshold. Optimizing the threshold gives the same asymptotic growth in capacity as optimal water-filling as the number of subchannels N goes to infinity. If the ratio of coherence bandwidth to the total available bandwidth is fixed, then the ratio between minimum feedback rates with correlated and i.i.d. sub-channels, respectively, converges to zero with N . For a sequence of Rayleigh fading sub-channels, which are modeledas a first-order log N with autoregressive process, the ratio goes to zero as O N the optimized threshold. We also consider finite-precision rate control on each sub-channel, and show that the feedback rate required to specify the sequence of assigned rate levels across sub-channels gives the same asymptotic increase in achievable rate with N as the (infinite-precision) on-off power allocation.
I. I NTRODUCTION Feedback can substantially increase the achievable rate for multi-carrier transmission through a frequency-selective fading channel. In [1], the achievable rate for multi-carrier transmission over a frequency-selective fading channel is analyzed with limited feedback. In that work, the sub-channels are i.i.d. and time-invariant, and the asymptotic growth in capacity as a function of the number of sub-channels N is specified for different power allocation schemes. For a general class of fading distributions, it is shown that the asymptotic capacity with optimal water-filling grows at the same rate as the capacity corresponding to a uniform power distribution over an optimized set of sub-channels. We refer to the latter scheme as the optimal on-off power allocation. For example, with Rayleigh fading sub-channels, the capacity grows as O(log N ). In practice, the sequence of sub-channels is likely to be correlated. Since the channel capacity depends only on the first-order distribution of channel gains, this correlation does 1 This
work was supported by ARO under grant DAAD19-99-1-0288.
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not affect the growth in capacity with N . However, correlations among sub-channels can be exploited to reduce the minimum amount of feedback, which is needed to achieve this growth. An on-off power allocation can be specified by a binary sequence, corresponding to active and inactive subchannels. The set of active sub-channels have gains, which exceed some chosen threshold. Here we model the sequence of sub-channel gains as a Markov process, and give a lower bound on the feedback rate, in bits per sub-channel, as a function of the threshold. By choosing the optimal threshold, the minimum feedback required to achieve the asymptotic capacity can be obtained. For an on-off power allocation the minimum feedback rate is the entropy rate of the binary on-off sequence. We compute the entropy rate in terms of the channel parameters, and show how it converges to zero as N → ∞. The ratio between the entropy rates for correlated and i.i.d. sub-channels also converges to zero, and we specify the corresponding convergence rate with N . As an example, we consider a first-order autoregressive sequence of complex Gaussian sub-channels. In order to model a wireless system, which occupies a fixed total bandwidth, we constrain the number of coherence bands, which are contained within the N sub-channels. In that case, when the activation threshold grows as O(log N ), which allows the achievable rate to increase as O(log N ), the savings in feedback relative to log N . Numerical examples are i.i.d. sub-channels is O N provided, which show that the asymptotic analysis accurately predicts the average amount of feedback needed in finite-size systems for on-off power control. With finite-precision rate control, additional bits per active sub-channel are needed to choose from a discrete set of rates. We show that adding these bits does not increase the order of the feedback rate with N . Furthermore, even when the rate for each active sub-channel is fixed, and optimized to maximize the total rate, we show that the growth in achievable rate with N matches the optimal growth in capacity. However, as shown in [1], the absolute difference between the achievable rate and capacity converges to a Gaussian random variable, where the mean decreases with the number of possible rate levels.
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In what follows, we say that two sequences {xn } and {yn } are asymptotically equivalent if limn→∞ xn /yn = 1, and write xn yn . The following two theorems are restated from [1]. Theorem 1: If E[µ|µ > x] − x is finite for all x, then
Related work on the performance of different transmission schemes with limited feedback is presented in [2], [3], [4], [5], [6], [7], [8]. Most of that work is on characterizing the effect of imperfect channel estimates on achievable rates. A finite number of feedback bits per dimension is also assumed in [2], [5], [7], although the specific feedback schemes and analysis differ from that presented here.
(wf)
CN where
th
We assume that the received vector corresponding to the i transmitted multi-carrier symbol is given by
CN =
log(1 + Pi µi )
(wf)
(1)
(2)
i=1
1 + ) µi
(9)
512 Rayleigh sub−channels
where the water level λ is determined by (λ −
P log N
where mn and σn are constants. Figure 1 shows plots of mean data rate vs. SNR for multicarrier transmission with 512 Rayleigh sub-channels with unity variance. Achievable rates are shown with water-filling (C (wf) ), the optimal on-off power allocation with infinite precision rate control (C (on-off) ), and on-off power allocation with finite-precision rate control with n = 1, 2 and 4. The figure shows that C (on-off) is very close to C (wf) . The gap between the achievable finite-precision data rate and C (wf) increases with power, and decreases as n increases.
N where Pi is the power on the ith sub-channel, and i=1 Pi ≤ P. With the optimal water-filling power allocation the channel capacity is N 1 + (wf) CN = log 1 + (λ − ) µi (3) µi i=1 N
(on-off)
CN
(8)
Suppose now that the rate Ri for the ith sub-channel is ¯ n }. That is, ¯1, · · · , R chosen from the discrete set R = {0, R the range of channel gains is divided into intervals, each of which is associated with a particular rate. In what follows, we assume that these intervals are selected to maximize the achievable rate [1]. The corresponding total finite-precision N rate is R(fp) = i=1 Ri . Theorem 2: The loss in the total achievable data rate, relative to the on-off capacity, due to finite-precision rate control is a random variable, which satisfies D (10) C (on-off) − R(fp) −→ N mn , σn2
i=1
P=
(7)
is the optimal threshold and satisfies
CN
where s(i) is the vector of transmitted symbols across subchannels at time i, and H is the diagonal channel matrix, which contains the channel coefficients, hi , i = 1, · · · , N . The sub-channel gains are µi = |hi |2 , i = 1, · · · , N , and are identically distributed random variables. Let fµ (·) and Fµ (·) denote the pdf and cdf of µi , respectively, and F¯µ (x) = 1 − Fµ (x). The noise, n(i), is white Gaussian with unity covariance matrix. We assume a total power constraint, i.e., trace{E{ss† }} ≤ P. Conditioned on the sub-channel gains, the channel capacity is given by [9] N
P µ∗0
P E 2 [µ|µ > µ∗0 ] = 2 (E[µ|µ > µ∗0 ] − µ∗0 ) N F¯µ (µ∗0 ) For Rayleigh fading sub-channels,
II. A SYMPTOTIC C HANNEL C APACITY
r(i) = Hs(i) + n(i)
µ∗0
(on-off)
CN
40 Optimal On−Off Equal rate, On−Off 2−level rate−control 4−level rate−control Water−Filling
35
(4)
30
i=1
(on-off)
= max µ0
N
log 1 + P¯ µi 1µi ≥µ0
15
5
0
(6)
0
1
2
3
4
5 SNR (dB)
6
7
8
9
10
i=1
where P¯ depends on µ0 . The optimal threshold that maximizes (on-off) CN is denoted as µ∗0 .
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20
10
Optimizing the threshold gives the corresponding on-off capacity for finite N , CN
25 Data Rate (bit)
To reduce the amount of feedback for power and rate optimization, we consider on-off feedback, in which the transmitter allocates equal power P¯ across a subset of sub-channels with gains that exceed a threshold µ0 . The power constraint then becomes N P¯ 1µi ≥µ0 ≤ P (5)
Fig. 1. Mean channel capacity vs. SNR for water-filling and on-off power allocations with infinite- and finite-precision rate control.
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Theorem 3: If the threshold µ0 → ∞ with N , then
III. F EEDBACK R ATE WITH C ORRELATED FADING For a large class of fading distributions, an achievable rate, which is asymptotically equivalent to the capacity, can be obtained by feeding back an optimized subset of active subchannels. Given a threshold µ0 , let 1 if µi ≥ µ0 χi = 0 if µi < µ0 where 1 ≤ i ≤ N , and χN = (χ1 , χ2 , · · · , χN ) be the N bit on-off sequence. Define the minimum feedback rate as BN , measured in bits per sub-channel. If the sequence of subchannels is a stationary process, then we have lim BN = H(X )
(11)
N →∞
where H(X ) is the entropy rate of the sequence χN . If the sub-channels are i.i.d., then the corresponding feedback rate (iid) BN → −γ log γ − (1 − γ) log(1 − γ), where γ = F¯µ (µ0 ). If µ0 → ∞ with N , then −γ log γ −(1 − γ) log(1 − γ) as γ → 0, and we have 1 (iid) (12) BN (µ0 ) γ log γ (iid)
which goes to zero with N . Therefore BN (µ∗0 ) is a lower bound on the feedback needed to achieve the asymptotic capacity. Now suppose that the sequence of sub-channel gains {µ1 , · · · , µN } is a Markov process with joint second-order pdf g(µi , µi−1 ). Clearly, {χ1 , χ2 , · · · , χN } is a two-state Markov chain, as shown in Fig. 2, with transition probabilities 1 ∞ ∞ g(x, y) dx dy q = P r{χi = 1|χi−1 = 1} = γ µ0 µ0 (13) 1 − (2 − q)γ (14) p = P r{χi = 0|χi−1 = 0} = 1−γ The
corresponding
entropy
rate
is
H(X )
=
(mc)
lim
N →∞
BN (µ0 ) (iid)
BN (µ0 )
=0
Furthermore, if θ(µ0 ) → 0 with N , then 1 γ Given the conditions in(mc) Theorem 3, we therefore have BN (µ0 ) 1−q (iid) BN (µ0 ) (mc)
BN (µ0 ) (1 − q)γ log
(16) (17)
where q, defined in (13), is a function of µ0 . The condition θ(µ0 ) → 0 is satisfied in many situations of interest. In that case (17) implies that for large N , the minimum feedback with correlated sub-channels is much less than that with i.i.d sub-channels. Now suppose that the data rate on each sub-channel is chosen from one of n possible rates. According to Theorem 2, the maximum achievable rate is asymptotically equivalent to the capacity, although there is an absolute (random) loss in rate, which on average increases with the transmitted power. Given a set of n thresholds, the gain of each sub-channel is quantized as χi , where χi ∈ {0, 1, · · · , n} indicates the rate ¯ χ for 0 ≤ χi ≤ n. assigned to sub-channel i. That is, Ri = R i In this case, µ0 is the smallest (activation) threshold, i.e., if ¯ 0 = 0. µi < µ0 , then χi = 0 and Ri = R N The sequence χ = (χ1 , · · · , χN ) is an (n + 1)-state (fp,n) Markov chain, as shown in Fig 3. Let BN (µ0 ) denote the p0n p01 p00 1
0
.........
n
p10 pn0
Fig. 3. (n + 1)-state Markov chain corresponding to feedback with finiteprecision rate control.
1-p
minimum feedback rate with n-level finite-precision rate conn (fp,n) πi pij log p1ij , trol. As N → ∞, BN (µ0 ) converges to
q
p
1
0
i,j=0
where pij is the transition probability from state i to state j, and {πi } is the steady-state distribution. Let
1-q Fig. 2.
Two-state Markov chain corresponding to the sequence χN .
qn
(1−p)H(q)+(1−q)H(p) 2−p−q
= γH(q) + (1 − γ)H(p) [9]. In what follows, we will assume that the ratio of coherence bandwidth to the total available bandwidth is fixed. In that case as N increases, the correlation between neighboring (mc) denote the sub-channels increases, i.e., lim q = 1. Let BN N →∞ asymptotic on-off feedback rate for the preceding Markov chain model, and let θ(µ0 ) =
γ P r {µi ≥ µ0 } = P r {µi < µ0 |µi−1 ≥ µ0 } 1−q
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= P r{χi = 0|χi−1 = 0} = P r{µi ≥ µ0 |µi−1 ≥ µ0 } 1 ∞ ∞ = g(x, y)dxdy. γ µ0 µ0
That is, qn generalizes q in (13) to the (n + 1)-state Markov chain. Theorem 4: As N → ∞, if θ(µ0 ) → 0, then for a fixed number of data rates n,
(15)
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BN
(µ0 ) (1 − qn )γ log
1 γ
(18)
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Comparing the preceding feedback rate with that for i.i.d. subchannels gives (fp,n) BN (µ0 ) (19) 1 − qn (iid) BN (µ0 ) We therefore conclude that given the same activation threshold, finite-precision rate control with a finite number of rate levels n requires the same order of feedback as onoff feedback, which does not depend on n and the set of thresholds. That is, for large N the additional overhead needed to specify one of n data rate levels is negligible compared with the feedback needed to specify the binary on-off sequence. Furthermore, by choosing the optimal threshold set, the corresponding achievable rate is asymptotically equivalent to the capacity. IV. R AYLEIGH FADING We now assume that each sub-channel coefficient, hi , is a complex Gaussian random variable, so that the sub-channel gains have a Rayleigh distribution with unity variance, i.e., fµ (x) = e−x . To model the correlation among sub-channels, we assume that the sequence {hi } is generated from a firstorder autoregressive model, hi = αhi−1 + ξi
(20)
where 0 ≤ α ≤ 1, and ξi is a complex Gaussian random variable, which is independent of hi , and has variance 1 − α2 , so that E[|hi |2 ] = 1. The parameter α determines the correlations between the sub-channels. In what follows, we assume that the total bandwidth is fixed so that the width of the sub-channels tends to zero as N → ∞. Hence the correlation between sub-channels increases, so that α → 1. It is straightforward to show that (20) implies that {µ1 , · · · , µN } is a Markov process, and √ x+y 2α xy 1 − 1−α 2 e I0 (21) g(x, y) = 2 1−α 1 − α2 where I0 (·) is the modified Bessel function of the first kind and zero-order. Assuming the threshold µ0 → ∞ as N → ∞, the transition probability q can be approximated as
1−α 2µ0 (22) q ≈ (1 + α)Q 1+α
∞ x2 where Q(x) = x √12π e− 2 dx. Substituting (22) into (16) and (18) gives the respective asymptotic on-off and finiteprecision feedback rates. For example, if limN →∞ α = 1, then 1 gives the relative feedback choosing the threshold µ0 = 1−α (mc) (iid) gain BN /BN → 0.68. If (1 − α)µ0 → 0, then (22) can be further approximated as 2(1 − α)µ0 (23) q ≈1− π and the ratio of minimum feedback rates for correlatedand i.i.d. sub-channels converges to zero as O (1 − α)µ0 .
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The reduction in feedback obtained by exploiting the subchannel correlations clearly depends on the rate at which α → 1 with N . Here we consider a special case. Let W be the total bandwidth of the channel and Wc be the coherence bandwidth. Specifically, we define the coherence bandwidth Wc = K∆f , where ∆f is the width of the sub-channels, and K = max {k : cov(µi , µi+k ) ≥ ρ}
(24)
where ρ is the correlation between sub-channels separated by Wc , and 0 < ρ < 1. The number of coherence bands spanned c by the channel is assumed to be fixed, i.e., W W = K/N = β where β is a constant. Since cov(µi , µi+k ) = α2k , we can write 1
log 1 ρ
α = ρ 2βN = e− 2βN
(25)
which increases to 1 exponentially with N . µ0 ≥ 1, then the on-off Corollary 1: If limN →∞ 1 log N 2 feedback rate with i.i.d. sub-channels is (iid)
BN
e−µ0 µ0 .
(26)
Furthermore, the on-off and finite-precision feedback rates for the autoregressive sequence of sub-channels satisfy (mc)
BN where β1 =
1 log ρ πβ
(fp,n)
BN
3 β1 √ e−µ0 µ02 N
(27)
.
Comparing the feedback rates for the autoregressive and i.i.d cases gives (mc) (fp,n) BN BN µ0 (28) β 1 (iid) (iid) N BN BN Specifically, if µ0 b log N , where b ≥ 1/2 is a constant, then (mc) (fp,n) √ BN BN log N (29) (iid) bβ1 (iid) N BN BN The condition b ≥ 1/2 is needed so that (23) is valid. Given the total bandwidth W and coherence bandwidth Wc , the minimum feedback rate for on-off and finite-precision rate control can be estimated from (27). For example, choosing the optimal threshold µ∗0 for Rayleigh fading (i.e., µ∗0 log N 3 [1]) gives a minimum feedback rate of O logN N for i.i.d. sub-channels. For the autoregressive model, the minimum feedback relative to the i.i.d. case, is reduced by the factor rate, log N . In both cases, the maximum achievable rate is O N asymptotically equivalent to the capacity, which is O(log N ). Note that the data in rate rate normalized by the feedback √ N creases from O log12 N for i.i.d. subchannels to O log5/2 N for correlated subchannels, where the units are transmitted bits per feedback bit per channel use.
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V. N UMERICAL R ESULTS Here we give a numerical example motivated by a cellular system. The channel bandwidth is 5 MHz, and the coherence bandwidth with ρ = 0.5 is 146 kHz [10]. We take the threshold µ0 = 12 log N . (Taking µ0 > 12 log N complicates the generation of simulated results, since for finite N , active sub-channels occur relatively infrequently.) Figure 4 shows the feedback rate vs. N computed from Corollary 1, and by generating sample sequences of sub-channel gains according to (20), and encoding the on-off sequence with an arithmetic code. Results are shown for both i.i.d. and correlated subchannels with the same threshold µ0 . We observe that the asymptotic bounds for both i.i.d. and correlated sub-channels (mc) decrease logarithmically with N , i.e., log BN decreases as (iid) − log N and log BN decreases as − 12 log N . As N increases, correlated sub-channels greatly reduce the required feedback, relative to i.i.d. sub-channels, as indicated in (28). The variance of the simulated results with correlated channels increases with N , but the asymptotic results predict the mean behavior. 0
10
Feedback Rate (bits/sub−channel)
iid, theoretical bound iid, arithmetic encoding correlated, theoretical bound correlated, arithmetic encoding
−1
10
−2
10
model, the relative reduction in feedback rates can be explicitly computed. Namely, the minimum feedback rate for correlated sub-channels, relative to i.i.d. sub-channels, tends to zero as O
log N N
. In both cases, the achievable rate with feedback
is asymptotically equivalent to the capacity. This is also true with finite-precision rate control, where in that case, the absolute loss in achievable data rate, relative to capacity, is a Gaussian random variable with finite variance. R EFERENCES [1] Y. Sun and M. L. Honig, “Asymptotic Capacity of Multi-Carrier Transmission Over a Fading Channel with Feedback”, IEEE International Symposium on Information Theory, Japan, 2003. [2] A. Narula, M. J. Lopez, M. D. Trott and G. W. Wornell, “Efficient Use of Side Information in Multiple-Antenna Data Transmission over Fading Channels”, IEEE Transactions on Communications, 16(8):14231436, Oct., 1998. [3] G. J¨oongren, M. Skoglund, and B. Ottersten, “Combining Beamforming and Orthogonal Space-Time Block Coding”, IEEE Transactions on Information Theory, 48(3):611-627, March, 2002. [4] S. Bhashyam, A. Sabharwal and B. Aazhang, “Feedback Gain in Multiple Antenna Systems” IEEE Transactions on Communications, 50(5):785800, May, 2002. [5] K. Mukkavilli, A Sabharwal, E. Erkip, and B. Aazhang, “On Beamforming With Finite Rate Feedback In Multiple Antenna Systems”, submitted to IEEE Transactions on Information Theory, 2002. [6] E. Visotsky and U. Madhow, “Space-Time Precoding with Imperfect Feedback”, IEEE Transactions on Information Theory, 47(6):2632-2639, Sep, 2001. [7] W. Santipach and M. L. Honig, “Signature Optimization for DS-CDMA with Limited Feedback”, Int. Symposium on Spread Spectrum Systems and Applications, Prague, Czech Republic, September 2002 [8] V.K.V. Lau, Y. Liu, and T.-A. Chen, “The Role of Transmit Diversity on Wireless Communications–Reverse Link Analysis With Partial Feedback”, IEEE Transactions on Communications, 50(12):2082-2090, Dec., 2002. [9] T. M. Cover and J. A. Thomas, Elements of Information Theory, 1991, John Wiley & Sons, Inc. [10] T. S. Rappaport, Wireless Communications, 1999, Prentice-Hall, PTR.
−3
10
2
3
10
4
10
10
5
10
Number of Sub−Channels
Fig. 4.
Feedback rate vs. number of sub-channels N .
VI. C ONCLUSIONS We have studied the minimum feedback rate required to specify the sequence of active sub-channels for an on-off power allocation. Optimizing the activation threshold gives an achievable rate, which is asymptotically equivalent to the channel capacity. For a sequence of sub-channels generated according to a Markov process, the minimum feedback rate has been computed in terms the transition probability, and the sub-channel activation threshold µ0 . If θ(µ0 ) → 0 with N , where θ(µ0 ) is defined by (15), then the feedback rate, measured in bits per sub-channel, converges to zero. When the number of coherence bands is fixed, the ratio of feedback rates with correlated and i.i.d. sub-channels converges to zero. Given a complex Gaussian sequence of subchannels generated according to a first-order autoregressive
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