Cooperation Strategies for Two Co-located receivers, with no CSI at the transmitter

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1 Cooperation Strategies for Two Co-located receivers, with no CSI at the transmitter Amichai Sanderovich, Avi Steiner and Shlomo Shamai (Shitz) Technion, Haifa, Israel Astract We study the case of a single transmitter, which communicates to two co-located users, through an independent lock Rayleigh fading channel. The co-location nature of the users allows cooperation, which increases the overall achievale rate, from the transmitter to oth users. The transmitter is ignorant of the fading coefficients, while receivers have access to perfect channel state information (CSI). This gives rise to the roadcast approach used y the transmitter. The roadcast approach facilitates reliale transmission rates adapted to the actual channel conditions, designed to maximize average throughput. With the roadcast approach, users can decode partly the total message, with almost any fading realization. The etter the channel quality, the more layers that can e decoded. Such an approach is useful when considering average rates, rather than outage vs. rate (outage never occurs). The cooperation etween the users is performed over the co-location channel, modeled as separated additive white Gaussian channels (AWGN), with an average power constraint, and limited or unlimited andwidth. New achievale rates when comining cooperation with the roadcasting approach are presented, through simple change of power allocation, sustantial gains are demonstrated. We consider oth amplify-and-forward (AF) and Wyner-Ziv compress-and-forward (CF), as cooperation approaches, and also compare to decode-and-forward (DF). We extend these methods using the roadcast approach, and also to include separated processing of the layers, realized through multi-session cooperation. Further, novel closed form expressions for infinitely many multi-session AF and recursive expressions for the more complex multi-session CF are given. Numerical results for the various cooperation strategies demonstrate the efficiency of multi-session cooperation. Our results can e extended straightforwardly to a setting of a single transmitter sending common information for two users. I. INTRODUCTION In recent years, cooperation schemes constitute a significant research topic. Cooperating receivers is a particular example in this class [],[2],[3] (among many others). Such cooperation can e achieved using compress-and-forward (CF) and amplify-and-forward (AF) techniques, which use lossy source coding techniques, so the cooperative receiver does not need to decode the message, or also decode-and-forward (DF), which requires decoding at the cooperative receiver [4]. Here, we deal with one transmitter that sends the same information to two co-located users, through independent, lock Rayleigh fading channel [5]. We discuss oth cases the information is directed to a single destination and/or oth co-located users are interested in decoding relialy the message. The Shannon capacity of this channel is zero, and usually one turns to rate versus outage proaility [6] in such cases. When considering the average throughput or the delay as figures of merit, it may e eneficial to use the roadcast approach. The roadcast approach for a single-user facilitates reliale transmission rates adapted to the actual channel conditions, without providing any feedack from the receiver to the transmitter [7]. The single-user roadcasting approach hinges on the super-position coding for the roadcast channel, every fading gain is associated with another user. In [8], a similar network setting is considered, with a single source transmitting to two co-located users, a Wyner-Ziv (WZ) CF single session cooperation is studied, in a different setting. In this paper, we consider the case the two receivers can cooperate etween themselves over separated channels, so that they can improve each other s reception via source related techniques (AF and CF). Since these users are co-located, the separated cooperative channels can use different frequency ands, so that unlike the channel from the transmitter, the proaility of a multi-path non-line-of-sight channel is low, and the cooperation takes place over an additive white Gaussian (AWGN) channel, with a power constraint, and either limited or unlimited andwidth. We present, in addition to the single session cooperation (referred to as naive AF and CF) also multi-session cooperation scheme, like was done y [9] for the inary erasure channel. By comining the roadcasting approach with multi-session cooperation, we can enhance the efficiency of each session, y accounting for the information that was already decoded. Thus, the performance of the naive cooperation schemes is surpassed. The rest of the paper is organized as follows. We give the prolem setting, definitions and the used notations in sections II and II-A. Upper and lower ounds are stated for completeness in section III. Section IV deals with cooperation through the simpler amplify-and-forward, and section V improves the achievale rates of the previous section, y using the Wyner- Ziv compression, namely the CF. II. PROBLEM SETTING We study the prolem of a single transmitter, which does not posses any knowledge of channel state information (CSI), communicating to a destination, as the destination and a co-located relay have access to perfect CSI. The perfect CSI is otained through some preamle, and then the destination and the relay exchange their CSI, through the cooperation channels. The power used for this exchange is not included in the average power, since it is independent of the lock

2 X h 2 h y y 2 Coop links Fig.. Schematic diagram of a source transmitting to two co-located users, with multi-session cooperation. length. The wireless network setting is illustrated in Figure. Our results for this prolem setting are also valid for the case a transmitter sends common information to two co-located users. See section VIII for more details. A. Channel Model Consider the following single-input multiple-output (SIMO) channel (we use oldfaced letters for vectors), y i = h i x s + n i, i =, 2 () y i is the received vector y user i, of length L, which is also the transmission lock length. x s is the transmitted vector, which satisfies E x s 2 P s. n s is the additive noise vector, with elements that are circularly symmetric Gaussian i.i.d with zero mean and unit variance, denoted CN (, ), and h i CN (, ) (i =, 2) are two independent (scalar) lock fading coefficients. Note that this results with the same average SNR for the two receivers, which is realistic due to their physical co-location. Without loss of generality we assume here that the desired destination is user i =, unless explicitly stated otherwise. The cooperation channels etween the users are modeled y AWGN channels as follows y (k) 2, = x(k) + w (k) y (k),2 = x(k) 2 + w (k) 2 y (k) 2, is the second user s received cooperation vector (of length L) from the destination (i = ), on the k th cooperation link, and vise-versa for y (k),2. x(k) i is the cooperation signal from user i, on the k th cooperation link, and w i CN (, ) is the additive noise. The power constraint here is E K k= (2) x (k) i 2 P r (i =, 2), for a single session cooperation K =, and for a multi-session cooperation, K >. The andwidth expansion factor is also associated with K, that results from the multi-session cooperation, which is modeled y K parallel cooperation channels. Notice that unlike the common reference to the relay channel, all these channels are separated, so no interference is introduced, unlike [8]. Also note that the users use half duplex communication equipment. E stands for the expectation operator. Naturally, the link capacity of a single session narrow-and cooperation is given y C coop,nb = log( + P r ). (3) In the limit of K with a power constraint for multisession cooperation, the cooperation link capacity is given y C coop,w B = dr(s) = ρ(s)ds = P r, (4) dr(s) is the fractional rate in session associated with parameter s, and dr(s) = log( + ρ(s)ds). The fractional power at the s th session is ρ(s). The multi-session power constraint implies ρ(s)ds = P r, which justifies the last equality in (22). In view of the simplicity assumed y the naive AF strategy, we do not consider any andwidth expansions, and use only the original signal andwidth in a single session. However, all approaches may utilize a cooperation channel andwidth expansion of the form C coop,w B (22) for improving the cooperation efficiency. III. UPPER AND LOWER BOUNDS In order to evaluate the enefit of cooperation among receivers in a fading channel following the model descried in ()-(2), we present two lower and upper ounds (LB and UB respectively) for oth the roadcasting and outage techniques [7] that can e used y the transmitter. The ounds consist of outage and roadcasting average rates that are computed for a single user, assuming there are no availale users for cooperation, for the lower ound and for a single receiver with two antennas and optimal processing for the upper ound, see []. All ounds are calculated assuming independent lock Rayleigh fading. First, we address in the following the standard outage policy and the corresponding ounds read: R outage,lb = max {( F LB (u th )) log( + u th P s )} (5) u th R outage,ub = max {( F UB (u th )) log( + u th P s )} (6) u th R s,lb = e e zl + 2Ei (z l ) 2E i () (7) R s,ub = s e zu e z u 3Ei (z u ) (z u e zu e z u 3Ei (z u )) (8) F LB (x) = e x F UB (x) = ( + x)e x z l = 2( + + 4P s ) z u = I UB (P s) z u = I UB (), I UB (x) = ( + x x 2 )/x 3. See [7] 2. 2 E i (x) = x e t dt t

3 IV. AMPLIFY-AND-FORWARD COOPERATION For AF, we consider three types of cooperation schemes: ) Naive amplify-and-forward (naf) - Each user operates standard AF using single session (K = ), the transmitter uses the optimal power allocation I(x) (see [7]). 2) Separate preprocessing amplify-and-forward (spaf) - The AF policy is used with the modification of removal of separately decoded layers prior to the forwarding. 3) Multi-session amplify-and-forward (msaf) - Multisession AF (K = ) repeatedly uses the separate preprocessing per session, and a total power constraint P r for all the cooperation sessions. Clearly, when each scheme is operating at optimal setting, we have: R NAF R spaf R msaf, (9) R naf, R spaf and R msaf correspond to the average rates of the naf, spaf and msaf strategies, respectively. A. Naive AF Cooperation In the naive AF strategy, the relaying user (i = 2) scales its input to the availale transmit power P r, and forwards the signal to the destination user (i = ) using a single session (K = ). The received signal at the destination after AF is then y a = (y, y (),2 ) () +α y(),2 = βx s + w 2. The normalized noise vector w 2 CN (, ) hence the normalized signal gain after the P scaling of user i = 2 is β = r s 2 +P s s 2 +P r, s i = h i 2. The achievale rate as a function of the channel fading gains is given y the following mutual information L I(x s; y a h, h 2 ) = log( + P s (s + β)) ( )) P r s 2 = log ( + P s s +. () + P s s 2 + P r Therefore the continuous roadcasting equivalent fading parameter is s a = s + β, rather then s, if no cooperation was used. This requires the derivation of the CDF of s a, []. For a Rayleigh fading the CDF of s a is explicitly given y F sa (x) = x Pr Psx Pr u (+Pr )x (+Pr )x e Pr Psx + due u x+ +Psu+Pr due u x+ P r u +Psu+Pr x < Pr P s x P r P s. (2) We get the following proposition: Proposition 4.: Using naive AF, at the co-located users, and roadcasting with the matching power allocation at the transmitter, the following average rate is achievale: R naf = x x [ 2( Fsa (x)) dx + ( F s a (x))f ] s a (x), (3) x f sa (x) x and x are determined from the oundary conditions I r (x ) = P s and I r (x ) =, I r (x) F s a (x) xf sa (x) f sa (x)x 2 (4) and F sa (x), f sa (x), f s a (x) are the CDF of s, its first and its second derivatives, respectively. For the Rayleigh channel, with transmitter power P s and cooperation power P r, F sa (x) is (2) Proof: See [].The proof follows from the optimization of the average rate over the power allocations, through calculus of variations. The resulting power allocation used y the transmitter is denoted y I r (x). B. AF with Separate Preprocessing Let us write the received signal at the i th user y y i = h i (x s,di + x s,i(si)) + n i, (5) x s,di is the part of the signal successfully and separately decoded y user i. The residual interference signal is then denoted x s,i(si ), which includes coded layers which could not get separately decoded. As [], define the power allocation according to I(s i ) E x s,i(si ) 2. Since the users have full CSI, they can agree on the minimum fading j = argmin i {s i } for determining the commonly decoded signal x s,dj (notice that it depends only on the two fading realizations, not on the AWGN). For example, when s s 2, after removing commonly decoded layers up to s 2 (at oth users), the residual interference power is given y I(s 2 ). The signals to e scaled and amplified are then given y: y i,i = h i x s,i(sj) + n i. Following the same lines of the naive AF (), the equivalent fading gain at user i =, after amplifying and forwarding y 2,I(sj ), is s a = s + P r s 2 + s 2 I(s j ) + P r. (6) Proposition 4.2: In an AF with separate preprocessing cooperation strategy, with a single cooperation session K = (power P r ) and power allocation I(x) (power constraint I(x ) = P s ) at the transmitter, the rate xi (x) R spaf = ( F sa (x)) dx (7) + xi(x) x is achievale. The CDF F sa (x) is calculated from (6) and given in []. Proof: See []. Note that the cooperation power can e increased y P s, P s is the proaility that the destination will successfully decode all layers without cooperation, so that power

4 can e saved. Oserve that since (6) includes I(x), so does F sa (x)). This turns the optimization prolem of the power allocation to e a difficult one, unlike the naive AF. C. Multi-Session AF with Separate Preprocessing In this scheme, we repeatedly use the technique of reducing commonly decoded information and then forwarding. The total power allocation availale for all sessions is still P r, unlike previous schemes, here K =. We find the average rate for unlimited numer of sessions, assuming only an overall power constraint for all sessions P r. It should e emphasized that the multi-session is performed over parallel channels (for example, OFDM), in such way that the source transmission is lock-wise continuous. So that we use wideand cooperation channel here. In the case of unlimited sessions, the scalar equivalent fading gain can e derived for a given roadcasting power allocation I(s). Proposition 4.3: In a multi-session AF (K =, cooperation power constraint P r ) with separate preprocessing cooperation strategy, the highest decodale layer is associated with an equivalent fading gain determined y s a, (assuming s > s 2 ). To compute it, start with s, which is the solution of the following equation, s s s 2 (s + s 2 σ) 2 [ + s I(σ)]dσ = P r, (8) and y using s, Z(s) = s Z(s a = s + s ) 2 + Z(s (9) ). s s 2 + s I(σ) s dσ. (2) ( + s 2 I(σ)) (s + s 2 σ) Since the agents are equivalent, the averaged rate received y the destination is R msaf = 2 E[log 2( + s ap s ) + log 2 ( + s P s)]. Proof: See []. From the equivalent fading gain (9) and the statistics of s, s 2, a CDF for s a can e computed, from which the average achievale rate can e otained. In order to improve performance, the aove approach uses power allocation also for the cooperation channels, not only for the transmitter, which is denoted y ρ(s) (that is, E x (k) i 2 = (k) s ρ(s)ds, s (k) s (k ) is equivalent fading after the k th session). The optimal power allocation for the cooperation sessions as a function of the achieved gain s, at the weaker user i = 2, given I(x) and s, s 2 is ρ(s) = ( + s I(s)) (s + s 2 s) 2. (2) Since the cooperation is performed over parallel channels, with infinitesimal power allocated per channel, the capacity of this wide-and cooperation link is C coop = dr(s) = s ρ(s)ds = P r. (22) Notice that the previous AF schemes (sections IV-A and IV- B), can not effectively use such capacity increase (P r > log(+p r )), ecause they do not encode the transmission prior to forwarding. In any case, the aove approach outperforms all spectral extension techniques for a wide-and naive AF, since here each session reduces the effect of the additive noise that is eing forwarded. The exact solution for optimal I(x) is an open prolem. So for the numerical results we use I(s) corresponding to optimal roadcasting in presence of optimal joint decoding. This selection is demonstrated (see Section VII) to e a good one, particularly for high P s and P r, as such conditions allow approximation of optimal performance with multi-session AF cooperation. V. COMPRESS-AND-FORWARD COOPERATION In this section we consider compress-and-forward cooperation. Both users are capale of performing compressing and forwarding (CF) of a quantized signal to one another. The compression here relies on the well known Wyner-Ziv [] compression using side information at the decoder, which in this case, is the received channel output. Similar to the AF, here too, we consider three ways of implementing the asic cooperation. A. Naive CF Cooperation Consider the channel model in ()-(2). The signal to e sent to the destination, which the compressed version of the received signal at the relay ŷ 2, is given y ŷ 2 = y 2 + n c = h 2 x s + n 2 + n c, (23) n c CN (, σ 2 ) is the compression noise, which is independent of y 2. Then the maximal achievale rate to the destination, is given y R CF, (h, h 2 ) = L I(x s; y, ŷ 2 h, h 2 ) s.t. I(y 2 ; ŷ 2 h 2 ) I(y ; ŷ 2 h, h 2 ) LC coop (24) R CF, (h, h 2 ) is maximized when the constraint is met with equality. The constraint C coop represents the cooperation link capacity. According to the channel model there is unlimited andwidth and a P r power limitation. We mention two cases for the naive CF: ) Narrow-and naive CF: K =, and therefore C coop = log( + P r ). 2) Wide-and naive CF: K = and therefore C coop = P r. Only the first one is considered, as it already significantly outperforms the AF schemes. The resulting quantization noise is E n c 2 = σ 2 which for the narrow-and naive CF equals σnb 2 = +s P s +s 2 P s P r (+s P s ). This leads to the next proposition: Proposition 5.: In a Narrow-and Naive compress-andforward strategy, given s, s 2, and P r, P s, as defined in section II-A (with K = ), the highest decodale layer in user i is associated with an equivalent fading gain determined y s ncf,i = s i + s 3 i ( + s i P s )P r ( + P r )( + s i P s ) + s 3 i P s. (25)

5 The distriution of s ncf,i for a Rayleigh fading F sncf (u) can e expressed in a closed expression, as done for the AF, which is given in [], for the sake of revity. B. CF with separate preprocessing We repeat what was done in section IV-B, also for the CF. That is, instead of first performing CF, oth users can separately decode, and then use CF, which is now performed with etter side information, in the form of the commonly decoded information. For consistency, assume that s > s 2, and then replace P s from σ 2 NB y I(s 2), the equivalent signal to noise ratio at i =, after the first iteration is now written y (6) and (25): s () s 2 P r ( + s I(s 2 )) = s + ( + P r )( + s I(s 2 )) + s 2 I(s 2 ). (26) C. Multiple sessions with CF and separate preprocessing As was done for the amplify-and-forward, can e repeated for the compress-and-forward processing. Define the auxiliaries variales (i =, 2) ŷ (k) i = y i + n (k) c,i, n(k) c,i is independent with y i, y 3 i, as the compression of y i, which is decoded at the other user (3 i) during the k th session. Since we use multiple WZ compressions, each with increased side information, we refer the reader to [2], for the successively refinale Wyner-Ziv. Here, we deal with the case the message that is transmitted in each session has etter side information than the previous session, since more layers are decoded. Further, the second session can use the information sent y all the previous sessions, in order to improve performance. Since the power that is used y each session is a control parameter, rather than a fixed parameter, as in the multi-session AF, the use of an auxiliary compression variale that is transmitted during a session, ut decoded only at the next session (due to the etter side information, declared as V in [2]) is inefficient. Next, using [2], the following Markov chains are defined, unlike [2], we are interested in independent averaged distortion, rather than plain averaged quadratic distortion. y 2 x s y ŷ (k) ŷ (k ) ŷ () (27) y x s y 2 ŷ (k) 2 ŷ (k ) 2 ŷ () 2 (28) The resulting equivalent fading gains after every iteration of the multi-session cooperation are stated in the following proposition. Proposition 5.2: The achievale rate in the multi-session with separate preprocessing and successively refinale WZ is given in a recursive form for the k th session, for user i =, for s, s 2, P r, I(x): R (k) CF,i= = E log( + s (k) s (k) P s) (29) s (k) s 3 i i = s i + (3) + (σ (k) 2 )2 and ( σ (k) i ) 2 = ( σ (k ) i { ( + s 3 i I(s (k ) )) ) 2 ( ) + s i I(s (k ) ) + s 3 i I(s (k ) ) [ ( ( ) )] 2 + δ (k) i + σ (k ) i + s i I(s (k ) )( + δ (k) i )} (3) for i =, 2, and E x (k) i 2 = δ (k) i and s (k ) min i=,2 {s (k ) i }. Proof: See []. VI. DECODE FORWARD COOPERATION For a fair comparison of DF cooperation to the other multisession techniques we consider oth wide-and cooperation, C coop = P r (22), and narrow-and cooperation (corresponding to the single session relaying techniques), the cooperation link capacity is only C coop = log( + P r ). The DF strategy for our setting can e descried as follows. The source performs continuous roadcasting, and two copies of the transmitted signal are received at the destination and the relay users, as descried y the channel model ()-(2). Recalling that the destination is denoted y user i =, then for s s 2 the destination user can decode at least as many layers as the relay user. Hence there is place for DF cooperation only when s < s 2, as in this case the relaying user can decode more layers than the destination. The additional layers decoded y the relay (for s (s, s 2 ]) are encoded y the relay and then forwarded, constrained y the capacity of the cooperation channel. Thus for P r >> P s, which is a practically unlimited cooperation channel, all additional information may e sent to destination and the strongest user upper ound is otained. Denote the decodale rate associated with a fading gain s y R(s), R(s) = s du I (u)u +I(u)u [7]. Say that efore cooperation starts user i decodes R(s i ). Hence for the pair (s, s 2 ), the achievale roadcasting rate is given y { min {R(s ) + C R DF (s, s 2 ) = coop, R(s 2 )} s 2 > s R(s ) otherwise. (32) The optimal roadcasting power distriution maximizes the average rate, and the optimization prolem is stated as follows, R DF = max ρ(s), s.t. E s,s 2 R DF (s, s 2 ) dsρ(s) Ps = max ρ(s), s.t. ρ(s)ds P s { s 2 ds 2 ds f(s )f(s 2 ) min {R(s ) + C coop, R(s 2 )} } + ds 2 ds f(s )f(s 2 )R(s ). s 2 (33) Finding the optimal power allocation seems intractale analytically, however R DF could e computed for su-optimal power distriutions, such as the strongest user I sel,opt (s), or for the no cooperation I SU,opt (s), and also for I Joint,opt (s). These are defined and demonstrated in section VII.

6 Rate [Nats per channel use] Broadcasting LB Naive AF Separate Preprocessing AF Multi Session AF NB ncf I LB NB ncf I UB NB ncf I optimal Decode & forward UB Outage UB Broadcasting UB P r /P s [db], P s = 2 db Fig. 2. Broadcast: Average rates of Naive AF, AF with separate preprocessing, multi sessions AF and narrow-and naive CF (with I(x) taken from (7) for I LB,(8) for I UB and the optimal, related to (25), respectively) compared to upper and lower ounds, as function of the channels quality ratio Pr, when P s P s = 2 db. VII. NUMERICAL RESULTS In this section, we compare the various methods, we use narrow and for all the schemes, esides the multi-session. Figure 2 shows a comparison etween naf, spaf,msaf, and narrow-and ncf, as function of the channels quality ratio P r /P s. Where we used the power allocation of naf in spaf and I(s) = s + 3 s 2 s (optimal for joint) in msaf, oth are suoptimal. As may e noticed from the figure, the lower P r /P s, the higher the rate gains of spaf, over the naf. For P s = 2 db and P r /P s, oth approaches outperform the upper ound for the achievale rate when using the outage approach. In moderate to high P s and P r, the multi-session AF approximates the upper ound of the achievale rate when using roadcasting. The naive CF, again, outperforms all other approaches, and approximates the roadcasting upper ound even on a wider range of P r values. The figure also plots an upper ound on the performance of DF, which for P r > P s.5db, are inferior to all the others. The figure also demonstrates the implications of using su-optimal power allocation for roadcasting in the narrow-and naive CF approach. It may e noticed that the difference etween the optimal and UB optimal allocation is rather small, unlike the LB optimal allocation. This assures us that the evaluated msaf, is not far from the optimal msaf. We suspect that separate processing and multi-session CF could close further the gap to the roadcasting UB. VIII. DISCUSSION AND CONCLUSION We have considered several cooperation strategies for transmission to co-located users. The original data is intended to one of the users and, in the network setting examined, a colocated user receives another copy of the original signal and cooperates with the destination user to improve decoding at the destination. As the transmitter has no access to CSI, the roadcast approach is used along with various cooperation strategies. We have presented and examined the naive AF, along with improved versions, denoted y separate preprocessing AF and multi-session AF. In separate preprocessing AF, the users individually decode as many layers as they can, sutract the common information and forward a scaled version of the residual signal. In a multi-session AF approach, this is repeated infinitely. We gave an explicit formulation for all these techniques. Another cooperation approach considered was CF, which was also improved through preprocessing. Using multisession CF rought notions such as successive refinale WZ coding. Explicit expression was derived for the naive CF and numerical results showed that naive CF outperforms all other AF approaches (for which we computed average rates). The multi-session AF with a su-optimal roadcasting power allocation also presented good results, although only over a wide-and cooperation channel. The results here are also valid, through straightforward transformation, for the case when a single source sends common information to two cooperating users. The impressive performance of the CF scheme, and the improvement of multi-session over single session in the AF, indicate the possile performance enefits of using multisession CF (mscf). However, the optimization prolem for mscf seems to e un-solvale analytically. ACKNOWLEDGMENT This research has een supported y the REMON consortium. REFERENCES [] C. T. 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