Repetition Protocols for Block Fading Channels ... - Semantic Scholar

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Repetition protocols for block fading channels that combine transmission requests and state information Jean Perret, [email protected]

Daniela Tuninetti, [email protected]

INPG (Institut National Polytechnique de Grenoble, France)

UIC (University of Illinois at Chicago)

Abstract—In this paper, we study the throughput performance of incremental redundancy repetition protocols over block fading channels. We propose new protocols that use the available feedback bits not only to request a retransmission, but also to inform the transmitter about the channel quality. We give an explicit protocol construction for any number of retransmissions and any number of feedback bits. We show remarkable throughput improvements, especially at low and moderate SNR, where our protocols can perform power control thanks to the partial channel state obtained through feedback. For the case of a single retransmission and a single feedback bit, we show that the repetition is not needed at low SNR and that the throughput improvement is due to power control only. On the other hand, at high SNR the repetition is useful and the performance comes form a combination of power control and ability to resend.

I. I NTRODUCTION In today networks, error correction is achieved by a combination of FEC (forward error correction) and ARQ (automatic repetition request). In classical repetition protocols, a receiver requests a retransmission (sends a negative acknowledgment, or NACK) in case decoding fails. In block-fading Gaussian channels, with transmit power Pt , fading power γt , and transmission rate R, the receiver fails to decode when the instantaneous channel capacity is below R, that is, NACK ⇐⇒ log(1 + γt Pt ) < R ⇐⇒ γt
0 and let the transmit power be R parametrized as Pt = e s−1 , i.e., R = log(1 + P s), where P = E[Pt ] is the average received SNR (signal to noise ratio). A 1 bit feedback used for ACK/NACK at the end of the slot indicates to the transmitter that γt < s. The probability of successful decoding is Pr[γt ≥ s] and the outage capacity is ηACK = Pr[γt ≥ s] log(1 + P s). On the other hand, consider the case where the 1 bit feedback is used at the beginning of the slot to indicate to the transmitter

the event γt < s. The transmitter uses this information to turn off transmission if the channel is bad (i.e., if γt < s) and send R with power Pt = e s−1 otherwise. The average transmit power R of this simple power allocation scheme is P = e s−1 Pr[γt ≥ s]. Hence, the outage capacity as a function of P is thus s . ηCSI = Pr[γt ≥ s] log 1 + P Pr[γt ≥ s] It is immediate to see that, for the same set of parameters and one bit of feedback, the outage capacity with CSI is larger than the outage capacity with ACK/NACK. Motivated by this observation on outage capacity, in [2] we proposed a protocol that combines CSI and NACK in one single bit of feedback for the case where the transmitter has one retransmission attempt. We showed the impressive result that the protocol achieves no less that 67% of the ergodic water filling capacity (that requires the transmitter to know the channel perfectly and retransmit until the receiver is able to decode!) The present work builds on the intuitions developed in [2] and proposes protocols that combine CSI and NACK for any number of retransmissions and any number of feedback bits. Our new protocols improves on the one proposed in [2] for the same set of parameters. For the case of a single retransmission and a single feedback bit, we show that the repetition is not needed at low SNR, and that the improvement over classical protocols that use the bit for ACK/NACK only is entirely due to ability to perform power control. On the other hand, at high SNR, the repetition is useful, and the performance improvement comes form a combination of power control and the ability to resend. To the best of our knowledge, works available in the literature considering quantized and/or noisy channel state information only focused on ergodic capacity or outage probability, but not on repetition protocols. For example, in [3] the authors consider power control policies for minimizing the outage probability with partial channel state information. The derived power policy shows benefits with respect to the case of complete absence of channel state information, even if the channel knowledge is noisy and partial, especially at low SNR. In [4], the authors studied the outage capacity with quantized channel state information and multi-layer communication. Their setting differs from ours in that the transmitter sends the superposition of several codewords. The receiver decodes as may layers as possible given its actual channel condition. It is found that, multi-layer transmission offers little

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benefits when power control at the transmitter is possible. This result reinforces our intuition that in fading channels, channel state information in critical and holds the key to great performance improvement. In [5], the authors considered the outage capacity with the so called “broadcast approach”, that is, a multi-layer approach with infinitely many layers. Also in this work it is found that even one bit of channel state information greatly impact the performance. The rest of paper is organized as follows. In section II we introduce the system model; in section III we described our new protocols and derived their throughput performance; in section IV we give numerical results, and in section V we state our conclusions and future directions for this work. II. S YSTEM M ODEL Notation: 1{x∈A} is the indicator function of the set A that equals one whenever x ∈ A and zero otherwise; FX (x) = Pr[X < x] is the cumulative distribution function of the random variable X for x ∈ R; E is the expected value. This work considers fixed-rate single-layer transmission with Gaussian codebook over a slowly fading Gaussian channels. The receiver has perfect instantaneous knowledge of the channel fading at the beginning of each slot (coherent communication due to perfect channel state estimation). On the other hand, the transmitter does not know the fading, unless specifically informed by the receiver. For this reason, the transmitter cannot adjust the rate of communication in each slot and hence sends at a fixed rate. The possibly partial channel knowledge obtained at the transmitter is used for power allocation only. Each codeword spans a fading block over which the fading stays constant and whose length suffices to guarantee reliable communication if the accumulated mutual information at the receiver is above the communication rate [1]. The fading changes in an iid fashion from slot to slot. A delay-free and error-free feedback channel with capacity log2 (F ) bits per slot is available for communication of lowrate information between the receiver and the transmitter; the receiver can feedback a retransmission request to the transmitter at the end of a slot, or quantized channel state information at the beginning of a slot, or any other information representable on log2 (F ) bits at any point during the slot. Feedback bits cannot be accumulated over successive slots. To make up for decoding errors, the transmitter can retransmit at most M −1 times (each data packet can take up at most M slots). We consider three protocols [6]. ALO: like in slotted Aloha, the transmitter keeps sending the same codeword and the receiver attempts decoding by using only the most recently received codeword. RTD: the transmitter keeps sending the same codeword and the receiver performs maximum ratio combining of all the received packets, thus realizing Repetition Time Diversity. INR: at each retransmission request, the transmitter sends new redundancy bits and the receiver optimally combines them. In all these protocols, it is customary to assume that the 1 bit of reliable feedback per channel packet is used to signal ACK/NACK at the end of the slot only. For ergodic capacity, CSI dramatically improves performance at

low SNR because it allows to save power when the channel is in a deep fade. At high SNR, CSI is not critical, and indeed water-filling and constant power allocation perform the same asymptotically [7]. We expect that combining repetition requests with CSI will offer similar benefits. Our performance measure is long-term throughput vs. longterm power (other common performance measures, such as are ergodic capacity and outage capacity, are spacial case of our settings for M = ∞ and M = 1 respectively). As pointed out in [7], in delay constrained scenarios, the assumptions about the dynamics of the fading process with respect to the the time over which power constraints are enforced is critical. Here, we are going to consider a power constraint over a time horizon comprising many slots (i.e., much bigger than the maximum number of retransmission) referred to as longterm power constraint. Under these assumptions, the received signal in slot t is √ L Y t = γt X t + Z t ∈ C , t ∈ N, where: L is the slot length, the data signal X t has iid components N(0, Pt ); the noise Z t has iid components N(0, N0 ); the channel fading power gain γt , a scalar, is iid form block to block. Without loss of generality we let E[γt ]/N0 = 1. In [6] we gave a general framework to evaluate the longterm throughput of a systems employing Gaussian codebooks and repetition protocols. We showed that system can be completely characterized by the triplet (T, R, P): • T ∈ {1, ..., M } represents the random number of slots needed to transmit a given data packet (referred to as inter-renewal time) with distribution Pr[T ≥ m]; • R ∈ {0, R} is number of bits successfully decoded when transmission s ends (referred to as renewal); • P ∈ { m=1 Pm }s=1,...,M is the total transmit power when transmission ends (referred to as cost); As a direct application of the renewal-reward theorem [2], it can be shown that the long-term average throughput η as a function of the long-term average power P is the solution of R 1 − Pr[T = M, no dec.] (F ) ηM (P ) = max M {R, Pm } m=1 Pr[T ≥ m] M m=1 E[Pm |T ≥ m] Pr[T ≥ m] ≤P s.t. M m=1 Pr[T ≥ m] Depending on the amount of channel knowledge, the powers Ps , s = 1, ..., M , can depend on the channel gain vector (γ1 , ..., γs ) (notice the causal dependency). III. M AIN R ESULTS Let Bm denote the feedback received before the m-th transmission occurs. We propose the following power policy Pm =

F −1 f =1

eR − 1 eR − 1 1{Bm =f } + 1{Bm =0} τm,f sm

for m = 1, ..., M with 0 ≤ sm and ∆

∆

0 = τm,0 ≤ τm,1 ≤ τm,2 . . . ≤ τm,F −1 ≤ τm,F = +∞


Our protocols use the thresholds {τm,f }F f =0 to define a quantizer for γm . The main idea is that the receiver feeds back Bm = f > 0 to indicate that the power (eR − 1)/τm,f suffices to successfully decode the current data packet when the previous m − 1 transmissions are appropriately combined with the current transmission at power (eR − 1)/τm,f . Upon receiving Bm = f > 0, the transmitter is certain that the receiver will decode correctly; hence, after transmission with power power (eR − 1)/τm,f , he prepares to sent a new data packet using power policy P1 . On the other hand, the receiver feeds back Bm = 0 when none of the powers (eR − 1)/τm,f , ∀f > 0, would guarantee successful decoding. In response to Bm = 0, the transmitter sends with power (eR − 1)/sm and prepares to retransmit the same data packet in the next slot using power policy Pm+1 , unless m = M . On the last transmission, i.e., m = M , the transmitter prepares to sent a new data packet with power policy P1 regardless of the value of the feedback since no more transmissions are possible. From this informal description of the protocol, it is clear that: transmission will not end after m received slots if all the feedback values received were zero, unless m = M , i.e., Pr[T ≥ m] = Pr[B1 = 0, ..., Bm−1 = 0], m = 1, ..., M, the average transmit power per data packet is M eR − 1 Pr[B1 = 0, ..., Bm = 0] E[P] = sm m=1

that is, if τ1,F −2 ≤ γ1 < τ1,F −1 . By continuing our reasoning in this manner, the receiver sends B1 = f

if γ1 ∈ [τ1,f , τ1,f +1 ),

f = 0, ..., F − 1.

so that the thresholds {τ1 f }F f =0 define a quantizer for γ1 . In response to B1 = f > 0 the transmitter sends with power θ/τ1,f that guarantees successful decoding for the whole range of fading values in [τ1,f , τ1,f +1 ). This power value can be interpreted as “channel inversion for the worst channel in the f -th quantization range”. After transmission, the transmitter prepares to send a new packet by using power policy P1 . In response to B1 = 0, the transmitter sends with power θ/s1 that suffices for successful decoding only if γ1 ≥ min{s1 , τ1,1 }. However, the transmitter can not know whether the actual γ1 is above or below min{s1 , τ1,1 }, and hence prepares to retransmit the same packet by using power policy P2 . From the second transmission onwards, the mode of operation depends on the protocol used. We will describe the three protocols separately. a) ALO: In the ALO protocol only the most recent received slots is used for decoding. Because of this feature of ALO, one would be tempted to use in every slot the same feedback policy Bm = f

if

γm ∈ [τm,f , τm,f +1 ),

and set sm = ∞ (zero power) for all m < M , since the accumulated mutual information on the previous slots −1 R M F is not used for decoding. However, assume that in the first e −1 Pr[B1 = 0, ..., Bm−1 = 0, Bm = f ], + transmission the fading satisfied γ1 ≥ min{s1 , τ1,1 }. The τm,f m=1 f =1 receiver knows that the transmitter will resend the same data and the average number of successfully decoded bits when packet because it had received B1 = 0. The receiver can transmission ends is “trick” the transmitter into believing that he will be able to decode in the second slot by sending B2 = F − 1. Clearly, E[R] = R 1−Pout , Pout = Pr[B1 = 0, ..., BM = 0, no dec]. this second transmission is waste of power, but the receiver has All the probabilities can be easily expressed as a function of not way to inform the transmitter of his successfully decoding pm,f for m = 1, ..., M and f = 0, ..., F − 2, where at the end of the slot having already used all its feedback bits at the beginning of the slot. Among all possible powers that ∆ pm,f = Pr[B1 = 0, ..., Bm−1 = 0, Bm ≤ f ] the transmitter could have used for the second transmission, θ/τ2,F −1 is the lowest. and with pm,F −1 = pm−1,0 . The throughput becomes For our ALO protocol, the feedback policy is hence   M −1 P (1 + m=1 pm,0 ) 1−Pout γm−1  . log 1+ M F −1 pm,f γ1 , the transmitter checks whether the second lowest available power θ/τ1 F −2 would suffice for decoding. He sends γ1 ≥ log(1 + θ) B1 = F − 2 if log 1 + θ τ1,F −2

t=1

for f = 0, ..., F − 2 and with pm,F −1 = pm−1,0 , and that the probability of outage is Pout =

M t=1

Fγ (min{st , τt,1 }).


It is immediate to see that only sm < τm,1 , or sm = ∞ (zero power), are of interests (if sm ≥ τm,1 , power θ/sm is insufficient for decoding when γm < τm,1 ). b) RTD: Recall that a second transmission is triggered by B1 = 0, which corresponded to having sent with power θ/s1 . On the second transmission, at the beginning of the slot the receiver measures γ2 and checks whether the lowest available power θ/τ2,F −1 suffices for decoding, and sends γ1 γ2 ≥ log(1 + θ) B2 = F − 1 if log 1 + θ + θ s1 τ2,F −1 that is, B2 = F − 1

γ1 γ2 + ≥ 1. s1 τ2,F −1

iif

If γ1 ≥ s1 , then a retransmission was not necessary in the first place and the power θ/τ2,F −1 is wasted, as for the ALO γ2 < 1 (which implies that γs11 < protocol. If instead γs11 + τ2,F −1 1) the receiver checks whether the second lowest available power θ/τ2,F −2 suffices for decoding, and sends γ1 γ2 ≥ log(1 + θ) B2 = F − 2 if log 1 + θ + θ s1 τ2,F −2 that is, B2 = F − 2 if

τ2,F −2 ≤

γ2 < τ2,F −1 . 1 − γs11

By proceeding with this reasoning, we see that the thresholds γ2 γ1 {τ2,f }F f =0 define a quantizer for 1− γ1 when s1 < 1. Notice s1 that the quantized value is larger than the actual fading γ2 in order to account for the SNR already “harvested” at the receiver in the first transmission. When γ1 ≥ s1 the receiver feeds back B2 = F − 1. In general, if the m-th transmission is required, then the receiver has already accumulated an equivalent SNR of m−1 γt t=1 st from the previous m − 1 transmissions, all in response to a zero-value feedback value. The receiver uses the thresholds {τm,f }F f =0 to define to quantizer for γm as follows Bm = f ⇐⇒

1−

Bm = F − 1 ⇐⇒

γm m−1 t=1 m−1 t=1

γt st

∈ [τm,f , τm,f +1 ) and

γt ≥ 1, st

m−1 t=1

Again, if γ1 ≥ s1 then B2 = F −1. If γ1 < s1 , we can rewrite the above condition as 1 + θ γs11 ≥ τ2,F −1 B2 = F − 1 if γ2 1 − γs11 By proceeding with this reasoning, we see that the thresholds γ1 γ1 {τ2,f }F f =0 define a quantizer for γ2 (1 + θ s1 )/(1 − s1 ) when γ1 < s1 . This scaled version of γ2 accounts for the mutual information already accumulated at the receiver in the first transmission. When γ1 ≥ s1 , the receiver feeds back B2 = F − 1. For a general m, we have γm ∈ [τm,f , τm,f +1 ) and ξm > 0 Bm = f ⇐⇒ ξm   1 + θ 1 ∆ − 1 ≤ 0. Bm = F − 1 ⇐⇒ ξm =  m−1 γt θ 1+θ t=1

st

The throughput can be expressed as a function of t−1 γ γt pm,f = Pr < 1 + θ, 1+θ 1+θ s min{st , τt,1 } =1 γm ∀t ≤ m − 1, < τm,f +1 ξm for f = 0, ..., F − 2 and with pm,F −1 = pm−1,0 , and Pout

t−1 γ γt < 1 + θ, ∀t ≤ M . 1+θ 1+θ Pr s min{st , τt,1 } =1

γt < 1 In general, there is not a closed form available for pm,f , not st even for Rayleigh fading. However, for the iid Rayleigh fading case, tight upper and lower bounds for pm,f can be developed based on order statistics [8].

with γ0 /s0 = 0, for m = 1, ..., M and f = 0, ..., F − 1. The throughput can be expressed as a function of t−1 γ γt < 1, ∀t ≤ m − 1, + pm,f = Pr s min{st , τt,1 } =1

γm < τm,f +1 , m−1 1 − =1 γs for f = 0, ..., F − 2 and with pm,F −1 = pm−1,0 , and

t−1 γ γt < 1, ∀t ≤ M Pout = Pr + s min{st , τt,1 } =1

In general, there is not a closed form available for the above probabilities, unless it is possible to evaluate the density of t−1 random variables of the type =1 γs . This is the case for Rayleigh fading [8]. c) INR: Recall that a second transmission is triggered by B1 = 0, which corresponds to having sent with power θ/s1 . At the beginning of the slot, the receiver measures γ2 and sends γ2 γ1 1+θ ≥1+θ 1+θ B2 = F − 1 if s1 τ2,F −1

IV. N UMERICAL R ESULTS In this section, we give a detailed analysis of the performance of these new protocols that combine ACK and CSI for the case of M = 2 (at most one retransmission) and F = 2 (1-bit feedback). The Fig. 1 shows, for a given (γ1 , γ2 )-pair, the bits fed back to the transmitter with the INR protocol. On the first transmission we have B1 = 1 if γ1 ≥ τ1,1 , and zero otherwise, for all protocols. If B1 = 1 decoding is successful. On the second transmission, which occurs when B1 = 0, B2 = 0 in filled region in Fig.1, and B2 = 1 in the complement of the region {B1 = 0, B2 = 0} with respect to the region {B1 = 0}. For the RTD (ALO) protocol the filled region in Fig.1 would be a triangle (rectangle).


γ2 B1 = 0, B2 = 1 OK decoding

B1 = 1 OK decoding

min{s2 , τ21 }

τ21 B1 = 0, B2 = 0 OK decoding

τ11

B1 = 0, B2 = 0 NO decoding

Fig. 1.

s1

γ1

Feedback regions for the INR protocol in the case s1 > τ1,1 .

1 0.9 0.8 0.7 0.6 0.5

(ack) (F=∞) /η 2,ALO ∞ (ack) (F=∞) η /η 2,RTD ∞ (ack) (F=∞) η /η 2,INR ∞ (F=2) (F=∞) η1 /η∞ (F=∞) (F=∞) η /η 1 ∞ (F=2) (F=∞) η /η 2,INR ∞

η

0.4 0.3 0.2 0.1 0 −25

−20

−15

Fig. 2.

−10

−5

0 P [dB]

5

10

15

20

capacity (M = 1) is also reported for comparison (for M = 1 all three protocols have the same performance, thus we do not mention the protocol in the subscript.) The throughput (F =∞) (dotted line in Fig. 2) is the outage capacity with η1 perfect channel state information, for which the optimal power allocation is “truncated channel inversion” as shown in [9]. We see that the protocols that combine repetition request and channel state information (upper solid line in Fig. 2) dramatically outperform the classical repetition protocols that use the 1-bit feedback for CSI only, especially at low SNR. Indeed, at low SNR it critical to be able to save power when the channel is in deep fade. By providing the transmitter with a 1-bit quantization of the current channel gain, we enable the transmitter to do so. The same gains could not be realized if a peak power constraint would be imposed, instead of an average power constraint. At low SNR, the repetition is not needed (ALO with M = 1 is optimal.) While at hight SNR, the repetition helps, and the throughput is about 80% of the water-filling capacity. We did not report the RTD and ALO curve for M = 2 with CSI as they do not differ much from ALO with M = 1. V. C ONCLUSIONS

25

Throughput as a function of the SNR.

The probability that a transmission takes two slots is Pr[T = 2] = Pr[B1 = 0] = Fγ (τ1,1 ), and hence the average interrenewal time is readily computed as E[T] = 1 + Pr[T = 2] = 1 + Fγ (τ1,1 ). Notice that in this case, the probability of decoding failure in the first slot is p(1) = Fγ (min{s1 , τ1,1 }) = Pr[T = 2], i.e., a successful decoding does not imply a renewal. Notice also that E[T] and p(1) are the same for all protocols for the same set of parameters (s1 , τ1,1 ). The outage probability is the probability of red zone in Fig 1, whose expression is given in the previous section. The average transmitted power is 1−Fγ (τ1,1 ) Fγ (τ1,1 ) Fγ (τ1,1 ) − q q E[P] = + + + , R e −1 τ1,1 s1 τ2,1 τ2,0 with q = Pr[B1 = 0, B2 = 0] is the green&red zone in Fig 1 The outage probability and the average power depend on the protocol used. Fig. 2 shows the ratio of the throughput of different protocols and the ergodic water-filling capacity (indicated as (F =∞) as its achievablity requires perfect channel knowledge, η∞ that is, infinite bits of feedback, and infinite number or retransmissions) with M = 2, 1 bit of feedback and Rayleigh fading. The superscript “(ack)” refers to the protocols in which the 1 bit of feedback is used for ACK/NACK only. The outage

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