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` TUNG T. KIM, GIUSEPPE CAIRE, MIKAEL SKOGLUND
Stockholm 2007 IR–EE–KT 2007:024
On the Outage Exponent of Fading Relay Channels with Partial Channel State Information T`ung T. Kim
Giuseppe Caire
Mikael Skoglund
School of Electrical Engineering Royal Institute of Technology 10044 Stockholm, Sweden
[email protected]
Department of Electrical Engineering University of Southern California Los Angeles, CA 90089
[email protected]
School of Electrical Engineering Royal Institute of Technology 10044 Stockholm, Sweden
[email protected]
Abstract—The problem of optimal dimension and power allocation to maximize the asymptotic outage exponent over a decodeand-forward fading relay channel with quantized channel state feedback (CSF) is studied. Three different scenarios are considered: relay-source, destination-relay, and destination-source feedback. It is found that even with just one bit of CSF from the relay to control the source power is sufficient to achieve the multiantenna upperbound in a range of multiplexing gain, with fixed-length codes, i.e., with coding schemes significantly simpler than “dynamic decode-and-forward.” CSF from destination to control the relay power slightly outperforms dynamic decodeand-forward at high multiplexing gains. CSF from destination when the source-relay channel gain is unknown to the destination, achieves a diversity exponent equal to K times the diversity exponent of dynamic decode and forward, for any multiplexing gain. This linear growth of diversity with the number of feedback levels stands in sharp contrast to the exponential growth for multiantenna channels (fully cooperative transmitters).
I. I NTRODUCTION Recently, motivated by the potential of having simple singleantenna nodes to cooperate and approach the promising performance of multiantenna systems, there has been a renewed interest in the classical relay channels, e.g., in [1]–[3]. It is known that resource allocation over relay channels can enhance the performance significantly [4]–[7]. Previous work on resource allocation often assumed perfect channel state information (CSI) at both the source and the relay. In [8], amplify-and-forward relaying with power control and quantized feedback is studied but diversity analysis is not pursued. This work considers a cooperative communication channel subject to very slowly-varying, but random, channel gains (quasi-static fading), under different forms of limited channel state feedback (CSF). We study the regime of asymptotically high signal-to-noise ratio (SNR) in terms of the diversitymultiplexing tradeoff [9]–[13]. A preview of the results is given in the following. For a statistically symmetric Gaussian decode-and-forward (DF) relay channel (described more precisely later) we have: Relay-source CSF: • With dimension allocation (herein the allocation of channel uses to different transmission phases is referred to as dimension allocation), but no power allocation (shortterm power constraint), a considerable performance gain can be achieved even with heavily quantized CSF from
the relay to the source. Non-orthogonal schemes provide additional, but not very significant gains. • With both dimension and power allocation (long-term power constraint) at the source, even one bit of CSF is sufficient to achieve the cooperative bound DMISO (r) = 2 − 2r over a wide range of multiplexing gains. • Relay-source CSF systems quickly approach dynamic decode-and-forward (DDF) [12] performance as the quality of feedback increases. Our results strongly advocate the use of orthogonal DF scheme with as few as one bit of feedback from the relay, in combination with power control at the source. Destination-relay CSF: We show that our proposed nonorthogonal scheme with power control at the relay outperforms DDF for large values of multiplexing gain. Furthermore, the performance of one bit feedback and perfect feedback is virtually indistinguishable. Our results advocate the use of onebit to control relay transmit power for high-rate systems. Destination-source CSF: • By developing novel upperbounds to the diversity exponent for channels with restricted CSF (see Section VI), we show that under the relatively realistic assumption that the source-relay channel gain is unknown to the destination, the diversity exponent only grows linearly in the number of feedback levels (also referred to as feedback resolution). In contrast, the diversity exponent of a MISO channel (fully cooperative transmitters) grows exponentially in the feedback resolution K. • We propose a scheme, where the destination sends quantized CSF to control power at both the source and the relay, that achieves the upperbound for r ≤ 1/2. Interestingly, for a given multiplexing gain, the diversity of the proposed scheme is exactly K times that of DDF. The details of the proofs of all results exposed here are omitted because of space limitation, and can be found in [14]. II. S YSTEM M ODEL Consider the complex baseband model of a frequencynon-selective fading relay channel. The channel is quasistatic fading, i.e., the channel gains are constant during a fading block consisting of T 1 channel uses, but change
independently from one block to the next. We exclusively consider the case when a transmission codeword spans a single fading block to identify the gain of spatial cooperation. The channel is assumed to be statistically symmetric. In particular, the channel (power) gains between source-destination, source-relay and relay-destination, indicated by h, h1 , and h2 , respectively, are independent and identically distributed (i.i.d.) complex Gaussian random variables with mean 0 and variance 1 (independent Rayleigh fading). We also define the channel power gains g = |h|2 , γ1 = |h1 |2 and γ2 = |h2 |2 . In the largeT regime, without loss of generality we can assume perfect CSI at all receivers. We also assume perfect synchronization. We consider a half-duplex channel, where the relay cannot transmit and receive simultaneously. Communication between source and destination takes place in two phases. In the learning phase, the source transmits, the relay and the destination listen. In the relaying phase the relay transmits based on what it has learned and the source may transmit more symbols but no new messages (in non-orthogonal schemes) or remain silence (in orthogonal schemes). The destination attempts to decode based on the entire signals received during both phases. The received signals in any channel use during the learning phase are given by y1 = hs1 + w1 yR = h1 s1 + wR .
(1)
In the relaying phase, the received signal at the destination is y2 = hs2 + h2 sR + w2
(2)
where for an orthogonal scheme s2 = 0 (this will be discussed in more detail later on). The noise w1 , w2 , wR are mutually independent temporally white complex Gaussian with zero mean and unit variance. We assume individual power constraints at the source and the relay, and consider both a peak (or short-term) power constraint, or an average (or long-term) power constraint. In the first case, transmitters must send at constant power in any signaling block. In the second case, transmitter may vary their power according to the CSF message, subject to a constraint on the average transmit power (averaged over the channel statistics). We will also refer to systems under a short-term (resp., long-term) power constraint as ones without power control (resp., with power control). Define the exponent equality [9] . f (SNR) = SNRa ⇔
lim
SNR→∞
log f (SNR) = a. log SNR
At each value of SNR, the system is designed to serve a rate of R = r log SNR nats per channel use, where r ∈ (0, 1) is the multiplexing gain. The system is said to achieve a diversity exponent of d if its average message error probability satisfies . Pe (r log SNR) = SNR−d . (3) √ Consider the Gaussian channel y = gx + w, where g is random but constant over the block length T . For any input
probability measure PX , there exist codes of rate R achieving an average probability of error not larger than Pout (R) = P (Ig (X; Y ) ≤ R) where Ig (X; Y ) denotes the channel mutual information for given value of channel parameter g. The achievable diversity exponents in this paper are obtained by assuming Gaussian i.i.d. inputs. The converse results (upper bounds) are obtained by considering suitably enhanced channels, for which we can apply Fano inequality (similarly to what done in [9] for the single-user MIMO channel). We do not make any achievability statement for finite T . The bottom line of this short discussion is that we can replace Pe (R) in (3) by Pout (R), where the input-output mutual information for given channel gains is suitably defined according to the relaying protocol and CSF case considered. III. D ECODE - AND -F ORWARD WITHOUT CSF Assume that the source and relay have no transmitter CSI. In this case the system can only optimize the SNR exponent using the knowledge about the statistics of the channel. Such an optimization has been partially considered in [15], [16], and the most general multiple-relay case is studied in [17]. For the sake of later comparison, we briefly review the no-CSF results in the following. The learning phase uses T1 = βT channel uses and the relaying phase uses T2 = (1 − β)T channel uses where 0 < β < 1 is the fraction of dimensions that we want to optimize over. Assume that the destination knows if the source-relay link is in outage or not. This requires the transmission of one bit of information from the relay to destination, which is insignificant in the SNR scale of interest. If the relay successfully decodes, it re-encodes the information message and sends the corresponding codeword to the destination in the relaying phase. If the relay fails, the destination decodes based on the signal received from the source only. Then, Proposition 1: [17] The diversity exponent achieved by an optimized orthogonal DF system with no CSF is 2 − 3r if r < 13 , DONF (r) = 2−2r otherwise. 1+r The diversity exponent achieved by an optimized nonorthogonal DF system with no CSF is √ √ 2 − 3+2 5 r if r < 3−2 5 , NF DNO (r) = (1 − r)(2 − r) otherwise. Both the fixed DF schemes are unsurprisingly outperformed by the DDF scheme [12] for any r ∈ (0, 1). However, compared to the DDF scheme, fixed-dimension DF systems offer a much lower complexity. The considered schemes use fixed-rate codes at the source and the relay, and do not require rateless (or incremental redundancy) codes and acknowledgment from the relay. We will further improve the above simple schemes with the help of quantized CSI in the next sections.
IV. R ELAY-S OURCE C HANNEL S TATE F EEDBACK We assume that, given the gain γ1 , the relay sends an index I(γ1 ) ∈ {1, . . . , K} back to the source via an error-free zero-delay feedback link. Herein the positive integer K is the resolution of the CSF link. The source allocates dimension (under a short-term power constraint) and both dimension and power (under a long-term power constraint). Since the two channel gains g and γ2 to the destination are unknown at the source and relay, the diversity exponent of this scheme can be upper-bounded by that of a MISO 2×1 channel with no-CSIT, i.e., DMISO (r) = 2 − 2r. A. Short-Term Power Constraint Let βk be the dimension allocation fraction given feedback index k, with β1 < β2 < · · · < βK . Consider the index mapping ⎧ ⎪ ⎨K if βK log(1 + γ1 SNR) < r I(γ1 ) = min {k ∈ {1, . . . , K subject to ⎪ ⎩ βk log(1 + γ1 SNR) ≥ r log SNR} otherwise. That is, from a finite set of {βk } the source allocates the minimum dimension so that the rate can be supported by the source-relay link. If the rate is not supportable even with the largest dimension βK , then the index I = K is fed back so that the direct link can benefit the most in the orthogonal case (in the non-orthogonal case, whatever index sent back when the relay fails does not matter). We formulate the optimization problem w.r.t. the fractions RF−K (r) of this {βk } that determines the diversity exponent DO-NPC scheme (see [14]). Unfortunately, in general, the problem is not convex, making the evaluation of the exponent difficult. Herein we present tight upperbounds and lowerbounds to the diversity exponent, which are simpler to compute. The nonorthogonal case is omitted due to space limitation. RF−K (r) is lowerProposition 2: The diversity exponent DO-NPC bounded by ⎧ K r ) ⎨ 2(1−r)−2(1−r)( 1−r if r = 1/2 K RF−K r 1−2r ( 1−r ) DLB-O-NPC (r) = ⎩ 2K if r = 1/2. 1+2K
The lowerbounds, which are tight for r ≥ 1/2, are explicitly given. As for the upperbounds, which are only necessary when r < 1/2, we consider all possible cases β1K ∈ [1/2, 1 − r) × · · · × [1/2, 1 − r) ×
l times
× [1 − r, 1) × · · · × [1 − r, 1),
K−l times
for l = 0, . . . , K where A × B denotes the Cartesian product. Due to the constraint β1 < · · · < βK , we only need to consider K + 1 such regions (and not 2K ). Over each of the K + 1 regions we can efficiently solve for the optimum using convex optimization methods. The problem is convex because over either {βk−1 , βk } ∈ [1/2, 1) × [1/2, 1 − r) or {βk−1 , βk } ∈ [1/2, 1) × [1 − r, 1), Dk in (4) is concave, and the pointwise minimum of a family of concave function is concave. Finally, the maximum of the K + 1 solutions gives the desired upperbound. It turns out that a relatively large improvement over the nofeedback case can be achieved (not plotted herein), but this gain diminishes as K increases from 2 to 3. The proposed CSF scheme is outperformed by DDF, but gradually approaches the performance of DDF as the feedback quality improves. We formalize this observation in the following. Corollary 1: We have 2 − 2r if r < 12 RF−K RF−K (r) = lim DNO-NPC (r) = lim DO-NPC r K→∞ K→∞ if r ≥ 12 . 1−r where the last expression is the diversity exponent of the DDF scheme [12]. Our result essentially shows that non-orthogonality is not necessary to achieve the DDF bounds when the source has perfect knowledge of γ1 and the systems fully adapts the available dimension to the channel condition. This effect practically shows up already for very low-rate CSF. For K as low as 2 (i.e., 1 bit of feedback) there is insignificant gain by using non-orthogonal schemes. Our result advocates the use of an orthogonal, low-rate CSF scheme from relay to source in order to achieve a substantial portion of the cooperative gain. B. Long-Term Power Constraint
We now relax the power constraint at the source, so that both dimension and power allocation are available. Let r < β1 < For r ≥ 1/2, this lower bound is tight. RF−K < 1 be defined as before and P1 < P2 < · · · < PK . . . < β K For r < 1/2, DO-NPC (r) is upper-bounded by be the set of power levels used at the source. We consider the RF−K following index mapping DUB-O-NPC (r) = sup min (D1 , . . . , DK , ⎧ 1/2≤β1
