Power Optimal Signaling for Fading Multi-access Channel in Presence of Coding Gap

Transcription

1 Power Optial Signaling for Fading Multi-access Channel in Presence of Coding Gap Ankit Sethi, Prasanna Chaporkar, and Abhay Karandikar Abstract In a ulti-access fading channel, dynaic allocation of bandwidth, transission power and rates is an iportant aspect to counter the detriental effect of tie-varying nature of the channel. Most of the existing work on dynaic resource allocation assues capacity achieving codes for various signaling schees like TDMA, FDMA, CDMA and successive decoding. For the capacity achieving codes, the rate achievable by the user is log( + SNR), where SNR denotes the signal to noise ratio of the user at the receiver side. However, codes that are used in practice have a finite gap to capacity, i.e., the achievable rate is log( + SNR ) for Γ >. The exact value of Γ depends on the Γ coding strategy and the desired bit error rate. Many existing resource allocation techniques that are optial for capacity achieving codes perfor sub-optially in presence of the coding gap. For exaple, successive decoding does not always iniize the su power required for providing the desired rate to each of the users for Γ >. The proble of iniizing the su power while guaranteeing the required rate to each of the users is iportant for both real-tie and non real-tie applications, and is addressed here. We obtain the resource allocation that is optial for the above proble in presence of the coding gap. I. INTRODUCTION We consider a Gaussian ulti-access fading channel with perfect channel side inforation (CSI) at the transitters and the receiver. This odels any iportant practical systes including the uplink of wireless LANs and the cellular systes. In a ulti-access fading channel, dynaic allocation of bandwidth, transission power and rates is an iportant aspect to counter the detriental effect of tie-varying nature of the channel [], [], []. Most of the existing work on dynaic resource allocation assues capacity achieving codes for various signaling schees, such as code-division ultiple access (CDMA), tie-division ultiple access (TDMA) and frequency-division ultiple access (FDMA) [], [], [6]. For capacity achieving codes, the rate achievable by the user is given by log( + SNR), where SNR denotes the signal to noise ratio of the user at the receiver side. However, codes used in practical scenario have a finite gap to capacity. For a variety of uncoded and coded odulations, this gap to capacity can be approxiated by scaling SNR with a factor (/Γ) for Γ > [7], i.e., the achievable rate is approxiately log( + SNR Γ ). Moreover, this gap to capacity is constant with SNR for a nuber of coding techniques and depends only on the probability of error (P e ). For exaple, in case of PAM/QAM, Γ = 9. db at P e = 7. Strictly speaking, this coding gap to capacity is a function of SNR, but it can be This work was done while A. Sethi was a graduate student at Indian Institute of Technology Bobay (IIT Bobay), Mubai, India. P. Chaporkar and A. Karandikar are with IIT Bobay, Mubai, India. (eails: ankits@iitb.ac.in, chaporkar@ee.iitb.ac.in and karandi@ee.iitb.ac.in) This research was supported by TTSL-IIT BOMBAY, Center for Excellence in Teleco (TICET), Mubai. approxiated to be constant over a large range of SNR. The odified SN R can now be used in any optiization setting in the sae way as that for the capacity achieving codes (Γ = ). We consider the syste with M users. (We use the ters user and transitter interchangeably.) A user k requires rate R k in each slot, where slot duration is equal to the channel coherence tie. Thus, the channel gain is assued to be constant in a slot, but it can vary fro slot to slot. Let h(t) = [h (t) h M (t)] denote the channel gains in slot t, i.e., if user k transits at power P in slot t, then the received power is (t)p. The coding strategy, and hence the coding gap Γ is specified. Our ai is to deterine a signaling strategy and the resource allocation for the given signaling strategy so as to iniize the su transit power while providing the desired rate to each of the users for any given h(t). We note that for any given signaling strategy, the resource allocation has to be dynaic depending on h(t). Clearly, this proble is of interest for real tie applications as they require strict delay guarantees. Next, we illustrate why this proble is of interest even for non-real tie applications. For non-real tie applications, let each user k desire a long ter rate r k. Note that unlike real-tie applications, this rate need not be provided in every slot. We can view this syste as follows: higher layer of the protocol stack feeds r k bits to the ultiple access control (MAC) layer buffer of the k th user in every slot. In slot t, MAC layer serves R k (t) bits fro the buffer, where R k (t) Q k (t). Here, Q k (t) is the nuber of back-logged bits in the buffer (queue length) of user k in slot t. Then, to provide the required rate to each user, it suffices to ensure that the expected queue length for each user is bounded (queue is stable), atheatically sup t E[Q k (t)] < for eac. Thus, providing the required long-ter rate to each user is equivalent to ensuring the queue stability for each user. Recently, the proble of iniizing the average su power required for ensuring stability of all the queues has been studied extensively [8]. It has been shown that, in each slot, transitting R k bits fro each user k such that a function V P(R, h(t)) M Q k(t)r k is iniized achieves the required goal, where V is a sufficiently large constant. Here, P(R, h) is the su power required to transit R = [R R M ] bits in the ulti-access channel experiencing channel gains h(t). Note that the function P(R, h(t)) depends on R, h(t) and coding and signaling strategy eployed. The previous work assues that the function P(, ) is given, i.e., it assues that the coding and signaling strategy is specified. The optiality of the above schee is shown for any non-negative P(, ). Thus, for truly iniizing the average su power while guaranteeing stability for a given coding schee, one needs to deterine a signaling schee

2 that achieves the iniu su power while guaranteeing R when channel gains are h. This shows how our proble is of interest for non-real tie applications. Next we review the related work. In the absence of coding gap, the rate region of a ulti-access fading channel is a polyatroid structure as derived in [9]. In [], the proble of iniizing the su power of the users under the constraints of providing the with iniu defined rates is considered. Non-orthogonal signaling like superposition coding along with successive decoding (SCSD) at the receiver is shown to be the optial strategy with appropriate power allocation. However, the analysis here considers only the capacity achieving codes. In presence of coding gap, the results in [9], [] do not hold. Indeed, [7] shows that the SCSD is not an optial signaling in presence of the coding gap. Specifically, [7] considers additive white Gaussian noise (AWGN) ulti-access channel with two users, and shows the existence of rates R, R for which the power optial signaling is FDMA and not SCSD, i.e., the iniu su power under FDMA is lesser than that under SCSD. The su power under a given signaling schee is iniized over all possible resource allocations for that signaling schee. For FDMA, resources are bandwidth and power alloted to each user, while for SCSD, resource is only the power at which each user transits. Authors in [7] also show that FDMA is not power optial signaling for all the values of the required rates, i.e., there exists R and R such that the power optial signaling is SCSD. Thus, the choice of optial signaling depends on the rate requireents. We note that [7] does not consider ulti-path fading, i.e., the channel gains were assued to be tie invariant. In fading channel, the optial signaling depends not only on the rate requireents, but also on the fading state which changes in every slot. Unlike [7], which only provides an existence result, one of our key contribution is to explicitly copute power optial signaling schee for any given rate requireents R and the channel fading state h. Our contributions are explicitly entioned below: As discussed above, for any given R and h either SCSD or FDMA is power optial. Thus, our approach is to obtain the power optial resource allocation for FDMA and SCSD for any given R and h. The optial signaling is then obtained as the one that requires the lesser su power between the two signaling schees. As a first step, we obtain the optial bandwidth and power allocation for FDMA. We note that the concept of changing the bandwidth allotted to users depending on h has not received uch attention in the literature. This is because the rate region achieved by FDMA is a strict subset of the rate region achieved by SCSD when capacity achieving codes are used for any given average power constraint. Thus, priarily, the power optial resource allocation for SCSD is widely explored in the literature [9], []. But, the optiality of SCSD is no longer true in the presence of coding gap [7]. Thus, unlike previous work, we need to deterine optial resource allocation for FDMA signaling. Next, we obtain the optial resource allocation for SCSD signaling. Even though the optial resource allocation for SCSD is well known for the capacity achieving codes, the case with coding gap does not follow directly fro the known results. This is because the optial resource allocation for SCSD is obtained fro the key property that the rate region is a polyatroid structure [9], []. This key property does not hold in the presence of coding gap, and hence the optial resource allocation for SCSD has to be obtained afresh. After the power optial resource allocation is obtained for FDMA and SCSD for the given R and h, the optial signaling is obtained as the one that requires a lower su transit power. Here, this strategy is referred as adaptive strategy. To obtain insights into when a certain signaling schee would perfor better than the other, we investigate how the optial resource allocation depends on Γ and h. The paper is organized as follows. In Section II, we present the syste odel. In Sections III and IV, we obtain the optial resource allocation for FDMA and SCSD, respectively. In Section V, we quantify the dependence of the optial resource allocation on Γ and h. In Section VI, we conclude. II. SYSTEM MODEL We consider a ulti-access channel fading with M users. Tie is slotted. The channel is tie-varying with (t) being the fading state of k th user in slot t. The fading is assued to be flat. We assue AWGN with spectral density σ. All the users use the sae codes and hence have the sae coding gap Γ to capacity. We consider a discrete tie channel M Y (t) = hk (t)x k (t) + Z(t), where Y (t) is the received signal in t th tie slot, X k (t) is the transitted signal of k th user in t th tie slot, and Z(t) is the noise. Let R = [R R M ] denote the rate requireents. The objective is to iniize the average su power constrained to providing a iniu rate R k to each user k in every channel state. Next, we obtain the optial resource allocation for FDMA and SCSD for any given h. III. OPTIMAL RESOURCE ALLOCATION FOR FDMA Here, we deterine the optial bandwidth allocation and power allocation schee that iniizes the su power of the users constrained to providing the iniu defined rate to eac for FDMA signaling. The proble can be atheatically forulated as follows. Let the current channel state be denoted by h. Let a power allocation policy be P(h) = [P (h) P M (h)], and bandwidth allocation policy α(h) = [α (h) α M (h)]. Here, P k (h) is the power allotted to k th user in channel state h, and α k (h) is the fraction of bandwidth allotted to k th user in channel state h. Thus, for every k we have ( R k α k (h)log + P ) k(h) Γσ. () α k (h) Without loss of generality, we assue that the total bandwidth is. Clearly, the su power is iniized when () is satisfied with equality for eac. Thus, fro () we have P k (h) = (e Rk α k (h) )Γσ α k (h).

3 Using the above relation, we get the following optiization in α M Subjected to: (e R k α k )Γσ α k M α k =, and α [, ] M. () We first note that the function (er k /α k )Γσ α k is strictly convex for α k [, ]. This is because the second derivative of the function (= Γσ e R/α k ) is positive for α α k >. Thus, the k objective function is the su of convex functions, and hence it is also a convex function. Clearly, the set of feasible solutions is convex. Thus, the above proble is an instance of the convex optiization proble []. For convex optiization, polynoial coplexity algoriths using interior point ethod have been proposed []. These algoriths can be used to obtain the optial resource allocation for FDMA signaling. IV. OPTIMAL RESOURCE ALLOCATION FOR SUCCESSIVE DECODING For SCSD, we need to specify the decoding order. Let π = [π() π(m)] denote a perutation on the set of users. We say that π is the decoding order if π(m) is decoded first, then π(m ) and so on until π(). Thus, for the π(k) th user, signals fro the users π() to π(k ) act as interference. Note that for a given π, the power allocation P has to satisfy the following relations so as to provide the desired rates to each of the users. For every k, ( ) P π(k) R π(k) log + Γ( k i= h. () π(i)p π(i) + σ ) The objective is in P,π M i= P i, where P and π satisfy (). Clearly, for a given π, the iniization over P happens when () is satisfied with equality for every k. Thus, the proble boils down to finding the optial decoding order. A. Optial Decoding Order and Miniu Su Power In the absence of coding gap (Γ = ), the decoding order that iniizes the su power while guaranteeing the desired rate to each of the users is obtained in []. In [], the authors have shown that the optial decoding order π satisfies h π () h π (M) irrespective of the rate requireents R. Thus, the optial power allocation can be obtained using a greedy procedure. The key property utilized to prove the result is that for any given π and S {,..., M}, ( P k S R k S k = log + P k(π) σ ), where P k (π) is the transit power for user π(k) under decoding order π. This property yields polyatroid structure for the rate region under SCSD (for coplete details, see []). Now, the polyatroid structure is used to derive π. We note that the aforeentioned property does not hold when Γ >, and hence the rate region for SCSD ay not be a polyatroid. Thus, the optial decoding orders for Γ = and Γ > need not be the sae. But, as we show in the next result, the optial decoding order for Γ > is the sae as that for Γ =. First, we note that when () is satisfied with equality, ( ) e R π(k) Γ (σ + ) k i= P i(π)h π(i) P k (π) =. () Theore : Let π (Γ) denote the optial decoding order for a given Γ >. Then, π (Γ) = π for every Γ >. Proof: Suppose, for soe Γ >, π (Γ) π. For brevity, let π = π (Γ). Then, there exists < M such that h π() > h π(+). Let us construct another decoding order π such that π (k) = π(k) for k {, + }, and π () = π( + ) and π ( + ) = π(). In other words, we obtain π by swapping th and ( + ) th user in the decoding order of π. Thus, clearly fro (), P k (π) = P k (π ) for every k <. Now, let us consider the following: + Now, we note that + P k (π) P k (π )h π (k) = P (π)h π() + P + (π)h π(+) P (π )h π () P + (π )h π (+). P (π)h π() P + (π )h π (+) = Γ ( e R π() ) P (π )h π () = Γ ( e R π() )( e R π(+) )( ) σ + P i (π)h π(i).() Siilarly, P + (π)h π(+) P (π )h π () i= = Γ ( e R π(+) ) P (π)h π() = Γ ( e R π() )( e R π(+) )( ) σ + P i (π)h π(i). (6) Fro () and (6), we conclude that + P k (π) = + i= P k (π )h π (k). (7) Fro () and (7), it can be seen that P k (π) = P k (π ) for every k > +. Now, since π is the optial decoding order, we know that M M P k (π) P k (π ) P (π) + P + (π) P (π ) P + (π ) P + (π) P (π ) P + (π ) P (π). (8) h π(+) h π() The relation (8) follows fro () and (6) as h π() = h π (+) and h π(+) = h π () by the construction of π. But, note that (8) provides a contradiction as we have chosen such that h π() > h π(+). This proves the required. Next, using exaples, we deonstrate that indeed for the given R and Γ, there exist channel states h such that FDMA

4 Paraeters User User User Su h α in FDMA P in FDMA P in SCSD TABLE I FDMA ACHIEVES BETTER PERFORMANCE THAN THAT OF SCSD Paraeters User User User Su h α in FDMA P in FDMA P in SCSD TABLE II SCSD ACHIEVES BETTER PERFORMANCE THAN THAT OF FDMA Power (W) 7 x 8 6 Dotted: User Dashed: User Solid: User Coding Gap (a) FDMA Power (W) 6 x 6 Dotted: User Dashed: User Solid: User Coding Gap (b) SCSD gives lesser su power than that of SCSD and vice versa. The exaples are presented in Tables I and II. Here, we assue that the syste has three users, Γ = 7 and R = [...]. V. EFFECT OF Γ AND h ON OPTIMAL RESOURCE ALLOCATION Here, we investigate how resource allocation varies with () coding gap and () channel states. A. Dependence on Coding Gap Γ In this section, we analyze how the optial power allocation in case of FDMA and SCSD depends on Γ under the condition that all other syste variables reain unchange. We already know that when there is no coding gap (Γ = ), the iniu su power for all h is achieved by SCSD. But as the coding gap increases (Γ > ) this is no longer true, i.e., for Γ > there exists h such that the iniu su power is achieved by FDMA. Here, we attept to find a reason behind this. ) FDMA: First, we explore the dependence of optial power allocation under FDMA on the coding gap Γ. Let us fix R and h. Now, let the optial power allocation and bandwidth allocation for FDMA be given by P FDMA and α for soe coding gap Γ. Now, P FDMA and α are the solutions to (). Fro (), it is clear that for the coding gap γγ, the optial power and the bandwidth allocations are γp FDMA and α, respectively. Thus, the power requireent under FDMA increases in proportion to the coding gap. This can be seen in Figure (a). ) Successive Decoding: Now, we explore the dependence of optial power allocation under SCSD on the coding gap Γ. As before, let us fix R and h. Let the optial decoding order and the power allocation be π and P SCSD, respectively. Then, ) (e R π(k) Γ(σ + k i= P π(i)h π(i) ) P π(k) = (fro ()) ) (e R π(k) Γ( k i= P π(i)h π(i) ) ( ) ( ( ) Γ k e R π(k) σ Πi= k e R π(i) )h π(i) Thus, the power of the user π(k) is lower bounded by a quantity that is proportional to Γ k. This has two iplications. Fig.. Variation of power of users with coding gap when (a) FDMA strategy (b) SCSD strategy is used. Here M =, [R, R, R ] = [.,.,.9], [h, h, h ] = [.,.6,.], σ = Avg. su power requireent (nano Watts) SCSD strategy FDMA strategy Adaptive strategy No. of users (a) Γ =. Avg. su power requireent (nano Watts) SCSD Strategy FDMA Strategy Adaptive Strategy 6 8 No. of users (b) Γ =. Fig.. Variation of the iniu average su power of users for FDMA, SCSD and Adaptive Strategy in a Rayleigh Fading channel with nuber of users in the syste. Here MR =, σ =. In (b), the plots for FDMA strategy and Adaptive strategy are overlapping. Firstly, as Γ increases, the transit power of the users increases exponentially, where the exponent depends on π. This can be seen in Figure (b). It follows that as Γ increases the power consuption of a user under SCSD increases at a uch higher rate than that of the respective user under FDMA strategy (except for the user that is decoded last). This explain why FDMA can achieve better perforance than SCSD for Γ >. Secondly, the power consuption of the user to be decoded first is lower bounded by a quantity proportional to Γ M where M is the nuber of users in the syste. Thus, as the nuber of users in the syste will increase, FDMA strategy will start outperforing SCSD and the power consuption in FDMA strategy will eventually converge to that of the Adaptive Strategy. This can be seen in Figure (a) and Figure (b). These show the variation of iniu average su power required by the users in a Rayleigh Fading channel when FDMA strategy, SCSD strategy and Adaptive strategy are used with the nuber of users in the syste. Here, for a given nuber of users in the syste, the iniu rate required for every user is sae i.e R = R.. = R M = R where MR =. B. Dependence on the Channel State h First, we note that changing h to γh is equivalent to changing Γ to Γ/γ while keeping the sae channel state. Hence, the observations in the previous section apply for scaler

5 Bandwidth (W) Power requireent of user (W) x 7 (a) BW allocation Dotted: User Dashed: User Solid: User (c) P under FDMA and SCSD Power requireent of user (W) Su Power requireent (W) 6 x (b) P under FDMA and SCSD 7 x (d) Su Power Fig.. Here M =, Γ = 7, [R, R, R ] = [.,.,.], [h, h, h ] = [.,.6,.], σ =. We plot various perforance easures as channel for the user degrades, i.e., it changes fro h to h /. shift in h. Now, we explore the dependence of the syste perforance on the channel state of an individual user. ) FDMA: Let us consider two channel state vectors h and h such that h k = for all k and h > h. Thus, h corresponds to the channel state vector in which user has worse channel gain than that in h, while the channel gain of all the other users reain unchanged. Let α and α denote the optial bandwidth allocation for h and h, respectively. Then, we show the following. Lea : The optial bandwidth allocation α and α satisfy that α < α and α k α k for every k. The proof for the lea is oitted because of space constraints. Fro the above lea it follows that change in the channel state of one user results in a change in the power allocation for all the users. Specifically, it can be shown that the power for all the users increases when the channel for any of the users degrades. ) Successive Decoding: In this case, when the channel state for a user changes, the optial decoding order also changes. Thus, exact ipact of the change in the channel state of a user on the power allocation policy depends on the placeent of the user in the decoding order before and after the change in the channel state. A special case in which the channel gain of the worst user, i.e., the user with the sallest channel gain, becoes saller, it can be seen fro () that the power requireent of the worst user alone increases and the power requireents of others reain the sae. Note that in this case, the optial decoding order reains the sae. ) Nuerical Evaluation: The results for nuerical study are presented in Fig.. Fig. (a) verifies Lea. Fig. (b) shows the rate at which power for the user, whose channel quality worsens, increases under FDMA and SCSD. The rate is higher under FDMA than that under SCSD. Fig. (c) shows that the power for users whose channel reains sae does not change under SCSD, while under FDMA the power requireent increases. Finally, Fig. (d) shows that the rate of increase of su power under FDMA is higher than that under SCSD. Hence, even when initially FDMA was an optial signaling, SCSD becoes the optial signaling as the channel quality for user k = becoes worse. VI. CONCLUSIONS We addressed the proble of iniizing the su power subject to providing the desired rate to each user in ultiaccess fading channel in the presence of coding gap. We showed that in the presence of coding gap, SCSD is no longer an optial strategy in all the channel states and also that there are certain channel states where FDMA outperfors SCSD. For these channel states, we deterined a power optial bandwidth allocation policy as a function of the channel state vector. This shows the benefit of the dynaic bandwidth allocation in the presence of coding gap (dynaic FDMA). Further, for the channel states where SCSD is optial, we showed that the optial decoding order is to decode the users in the decreasing order of their channel gains independent of their rate requireents. Finally, we developed soe insights on how the iniu su powers for SCSD and FDMA depend on the channel state vector and the coding gap. REFERENCES [] R. Knopp and P. Hublet, Inforation capacity and power control in single-cell ultiuser counications, IEEE International Conference on Counications, Gateway to Globalization, vol., 99. [] A. Lozano, A. Tulino, and S. 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