Noncoherent Communication over the Doubly Selective Channel via Successive Decoding and Channel Re-Estimation

For noncoherent communication over fading channels, pilot-aided transmission is a practical scheme whichallows the receiver to compute channel estimates for subseque nt use in coherent decoding. We propose an improved scheme whereby the transmitter interleaves several pilot substreams with independently-coded data substreams to facilitate su ccessive decoding and channel re-estimation at the receiver. In part icular, an initial pilot-aided channel estimate is used to coherent ly decode the first data substream, which is then used to refine the chann el estimate before coherently decoding the next data substrea m, and so on. Assuming knowledge of the channel statistics, the pil ots and the rates of the data substreams can be chosen to ensure reliable decoding. While similar schemes have been propose d for channels that are either time-selective or frequency-sele ctive, ours is focused on doubly selective channels. We derive a lower bo und on the achievable rate of our strategy and further character ize the achievable rate at high SNR. When the channel satisfies a complex-exponential basis expansion model, we show that the pre-log factor of the high-SNR achievable rate expressi on coincides with that of the ergodic capacity expression. Forthe same channel, we propose a pilot/data power allocation stra tegy that maximizes a lower bound on the achievable rate. 1 I. I NTRODUCTION Practical wireless communication is noncoherent in that th e channel state is never known a priori to the transmitter nor receiver. As a result, practical wireless transmissions mu t be designed with a structure that facilitates reliable recept ion in the absence of channel state information (CSI). Pilot-aide d transmission (PAT) [1], [2] is perhaps the most common means of providing this structure. With PAT, “one-shot” schemes are common, whereby the receiver computes a pilot-aided channel estimate and subse quently uses it for coherent data decoding. In this case, cha nnel estimation error acts as additional noise which degrades de coding performance and thus the rate of reliable communication [3]. Though channel estimation can be improved by allocatin g more transmission resources (e.g., rate and power) to pilot s, doing so limits the resources that remain for data transmiss ion. Hence, information-theoretic analysis are useful to under stand the optimal allocation of resources between pilots and data . Several information-theoretic analyses of PAT with one-sh ot estimation/decoding have appeared, e.g., in [4]–[10]. 1This work supported by Motorola Inc., the National Science F oundation under CAREER grant 237037, and the Office of Naval Research un der grant N00014-07-1-0209. As an improvement to one-shot estimation/decoding of PAT, several authors have considered iterative (i.e., “tur bo”) estimation/decoding strategies, whereby soft decoder out puts are employed to refine channel estimates, which can then be used for improved decoding, and so on [11]–[14]. Such systems are generally suboptimal and difficult to analyze. More recently, the use of block interleaving with successiv e decoding has been proposed as a more structured approach to joint estimation/decoding of PAT [15], [16]. There the idea is to split the information stream into independently coded substreams and decode them successively. While a pilot-aid ed channel estimate is used to decode the first substream, relia bly decoded symbols can be employed to refine the channel estimates used by later decoding stages. For long coding blocks and properly chosen substream rates (e.g., assuming known channel statistics), each substream can be reliably decode d, greatly simplifying the design and analysis of such systems . PAT with successive decoding has been used successfully in time-selective and frequency-selective SISO channels [16 ] as well as time-selective MIMO channels [15]. In this paper, we propose a scheme for noncoherent communication over doubly (i.e, timeand frequency-) selecti ve fading channels that uses successive decoding and channel re-estimation at the receiver. Assuming perfect decoding o f each stream, we calculate an achievable-rate lower-bound a nd use it to infer a set of substream rates which are sufficient to ensure perfect decoding. We also characterize the high-SNR spectral efficiency of the proposed communication strategy . To highlight certain design issues, we consider the special ca se of doubly selective channel which satisfies a complex-exponen tial basis expansion model (CE-BEM). For this channel, we design a suitable pilot pattern and through it verify that the pre-l og factor of the high-SNR achievable rate expression coincide s with that of the ergodic capacity [10], [17]. We also propose a pilot/data power allocation strategy that maximizes a low er bound on the achievable rate. The paper is organized as follows: A description of the system model appears in Section II, the reception strategy is described in Section III, and achievable-rate expressio n are derived in Section III-B. The specific case of the doubly selective CE-BEM channel is considered in Section IV, and conclusions are drawn in Section V. Notation: In the manuscript, (·)T denotes transpose, (·)∗ denotes conjugate, and (·)H denotes conjugate-transpose. [B]m,n denotes the element in the m row andn column of B, where row and column indices begin with zero. 0m×n denotes them×n zero matrix,IK denotes theK×K identity matrix, ande K denotes theq th column ofIK . For matricesA andB, A ≥ B means that A−B is positive semi-definite. The trace of a matrix is denoted bytr(·), and the Kronecker product of two matrices is denoted by ⊗. Also, δl denotes the Kronecker delta sequence, 〈·〉 denotes the modulo operator, and C the set of all complex numbers. Expectation is denoted by E(·) and auto-covariance byΣb := E(bb ) − E(b) E(b). II. SYSTEM MODEL A. Transmission Model Consider a scheme in which information is transmitted throughNs substreams, each of which uses the channel Nb times. In particular, say that sk(i) denotes thei sample of the k substream. The first Np substreams (i.e., {sk(i)}b i=0 for k = 0 . . .Np − 1) are dedicated to pilots while the remaining Ns−Np substreams (i.e., {sk(i)}b i=0 for k = Np . . . Ns−1) are dedicated to data. The data substreams are independentl y encoded at rates that ensure reliable decoding, as will be discussed later. For this, we assume that the transmitter kn ows the channel statistics, but not the channel state. The average transmission power is constrained to Etot Joules per-channel-use, Ep of which is allocated to pilots and the remainder of which is divided equally among the data substreams. Thus, each data substream has power σ s = Etot − Ep Ns −Np . (1) For analytical tractability, we assume the use of i.i.d. Gau ssian codebooks. With this assumption, the power constraints can be expressed as Np−1 ∑ k=0 |sk(i)| = Ep ∀i (2) E{sNp(i1)sNp(i2) } = σ sINs−Npδi1−i2 (3)