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ISIT 2004, Chicago, USA, June 27 – July 2, 2004 Adaptive Joint Source/Channel Rate Allocation Policies for Delay Sensitive Applications over Fading Channels C. Thejaswi, Vinod Sharma 1 90 22 80 20 70 18 Avg. Queue length→ Abstract — We consider a problem of optimal source-channel coding rate pair allocation over a wireless link based on the channel state and the queue length at the transmitter. We show that by this scheme, one can gain substantially in performance as compared to only source or channel rate optimization. Avg. Queue length→ Department of Electrical Communication Engg. Indian Institute of Science, Bangalore, INDIA. e-mail: [email protected], [email protected] 60 PolicyŦA 50 40 30 16 14 12 10 PolicyŦB 8 20 I. Introduction JSCŦpolicy We consider a slotted system with a single user communicating over a fading wireless channel. The data to be transmitted which arrives from a higher layer, is subjected to a variable rate compression depending on queue length, power and distortion requirements and is placed in a transmission buﬀer of inﬁnite capacity. We assume that the channel and buﬀer state information is available both at the transmitter and the receiver. There are N channel uses in a slot. The channel fading gain remains constant during a slot. The fading process {Gn } is assumed to be a stationary, ergodic Markov process with a compact state space G and the average number of jumps required to move from one state to any other state is ﬁnite. We also assume that the fading process is positively correlated and hence stochastically nondecreasing. The symbols are transmitted over the fading channel with an additive white Gaussian noise (AWGN). The transmitter, with the knowledge of the buﬀer and channel states, obtains optimum source and channel coding rates and transmission power. These actions minimize the mean queue length subject to long run average transmit power and distortion constraints. Let Sn be the amount of data in the buﬀer at the beginning of the nth slot where Sn ∈ {0, 1, 2, .....}. The state of the system in the nth slot is Xn := (Sn , Gn ). At time n the transmitter places An bits in the buﬀer and removes Un bits for transmission. The buﬀer state evolves as Sn+1 = Sn −Un +An where, Un ∈ {0, 1, ..., Sn }, An ∈ A := {0, 1, 2, ..., Ā}, Ā < ∞. For the nth slot, the transmitter incurs a cost of distortion D(An /J) to encode J source symbols into An bits and power P (Un , Gn ) for transmission of Un bits when the fading gain is Gn . D(.) is the distortion-rate function. The objective is to minimize the mean delay with constraints on the mean distortion and power. Since the delay process is more complicated, instead we try to minimize the mean workload process. The above problem can be considered as a constrained Markov Decision Problem which can be converted into an unconstrained one via the Lagrangean approach. The corresponding Lagrangean functional is J π (x) = lim sup M M −1 π 1 Ex M Sn + βP (Un , Gn ) + θD(An /J) , n=0 1 This work was partially supported by the DRDO-IISc program on advanced research in Mathematical Engineering. ,((( 10 6 0 4 8 8.5 9 9.5 10 10.5 11 11.5 12 JSCŦpolicy 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Avg.SNR (in dB)→ Avg. SNR (in dB)→ Figure 1: Comparison of avg.queue length of JSC with policy A and B respectively for diﬀerent avg.SNR and same avg. distortion. where x is the initial state, π is the given policy and β and θ are Lagrange multipliers. The aim is to ﬁnd π that minimizes J π (x). We have shown that this problem has an optimal solution which is also optimal for the constrained problem. The optimal policy is independent of x and can be obtained as the limit of optimal solutions corresponding to α−discount problems as α → 1. Furthermore, for the optimal solution (u, a), u(s) − a(s) is nondecreasing in s. Recently we have extended this work to MIMO and multiuser uplink channels which will be reported elsewhere. II. Results In this section, we obtain the optimal average cost solution for an example and compare the performance of the optimal policy (JSC) with the following two suboptimal policies. Policy A. Source Coder (SC) has only the buﬀer state information and the Channel Coder (CC) has only channel state information. This provides the well known water-ﬁlling power control. Policy B. SC with no state information and CC with complete state information. The SC computes the minimum source coding rate which satisﬁes the distortion constraint. The optimal CC policy for this case is studied in [1] and [2]. The source is iid N (0, 1), the fading process is Jake’s Rayleigh process (with 8 multipath components and Doppler BW of 5Hz) discretized into an 8-state Markov chain and the AWGN is N (0, 1). Figure 1 gives the comparison of JSC with policy A and with policy B. Policy B performs much better than policy A. However JSC provides signiﬁcant improvement over policy B also (at 8.5 dB 40% and at 10dB 29%). References [1] R. A. Berry, R. G. Gallager, ”Communication over Fading Channels with Delay Constraints”, IEEE Transactions on Inf.Theory, Vol 48, no. 5, 1135-1149, 2002. [2] M. Goyal, A. Kumar and V. Sharma, ”Power Constrained and Delay Optimal Policies for Scheduling Transmission over a Fading Channel,” IEEE INFOCOM 2003, San Francisco, USA.