Improved Target Acquisition Rates With Feedback Codes

Abstract

This paper considers the problem of acquiring an unknown target location (among a finite number of locations) via a sequence of measurements, where each measurement consists of simultaneously probing a group of locations. The resulting observation consists of a sum of an indicator of the target's presence in the probed region, and a zero mean Gaussian noise term whose variance is a function of the measurement vector. An equivalence between the target acquisition problem and channel coding over a binary input additive white Gaussian noise (BAWGN) channel with state and feedback is established.Utilizing this information theoretic perspective, a two-stage adaptive target search strategy based on the sorted Posterior Matching channel coding strategy is proposed. Furthermore, using information theoretic converses, the fundamental limits on the target acquisition rate for adaptive and non-adaptive strategies are characterized. As a corollary to the non-asymptotic upper bound of the expected number of measurements under the proposed two-stage strategy, and to non-asymptotic lower bound of the expected number of measurements for optimal non-adaptive search strategy, a lower bound on the adaptivity gain is obtained. The adaptivity gain is further investigated in different asymptotic regimes of interest.Preliminary versions of this work were presented at the 50th Asilomar Conference on Signals, Systems, and Computers, and at 2017 International Symposium on Information Theory.A. Lalitha N. Ronquillo, and T. Javidi are with the Z n (S n ) is equivalent to channel coding over a binary additive white Gaussian noise (BAWGN) channel with state and feedback (in Section 4.6 [1]). This allows us not only to retrofit the known channel coding schemes based on sorted Posterior Matching (sort PM) [2] as adaptive search strategies, but also to obtain information theoretic converses to characterize fundamental limits on the target acquisition rate under both adaptive and non-adaptive strategies. As a corollary to the non-asymptotic analysis of our sorted Posterior-Matching-based adaptive strategy and our converse for non-adaptive strategy, we obtain a lower bound on the adaptivity gain. A. Our ContributionsOur main results are inspired by the analogy between target acquisition under measurement dependent noise and channel coding with state and feedback. This connection was utilized in [3] under a Bernoulli noise model. In this paper, in Proposition 1, we formalize the connection between our target acquisition problem with Gaussian measurement dependent noise and channel coding over a BAWGN channel with state. Here, the channel state denotes the variance of the measurement dependent noise |S n |δσ 2 . Since feedback codes i.e., adapting the codeword to the past channel outputs, are known to increase the capacity of a channel with state and feedback. This motivates us to use adaptivity when searching, i.e., to utilize past observations {Y 1 , Y 2 , . . . , Y n−1 } when selecting the next measurement vector S n . Furthermore, this informati...

Active Learning and CSI Acquisition for mmWave Initial Alignment

“…• We formulate the initial beam alignment for massive MIMO as active learning of the AoA through multiple sequential and adaptive search beams. Our approach draws heavily from our prior work on algorithms for noisy search [2], active learning [16], and channel coding with feedback [10]. • We propose a new adaptive beamforming strategy that utilizes the hierarchical beamforming codebook of [1].…”

“…The proposed adaptive strategy, hierarchical Posterior Matching (hieP M ), accounts for the measurement noise and selects the beamforming vectors from the hierarchical beamforming codebook based on the posterior of the AoA. The design and analysis of hieP M extends our prior work of sorted posterior matching for noisy search [2] and [10] in that it restricts the search and the measurements to the practical and hierarchical beamforming patterns of [1]. • We analyze the proposed hieP M strategy and give an upper bound on the expected acquisition time of a variablelength hieP M search strategy required to reach a fixed (predetermined) target resolution and error probability in the AoA estimate.…”

“…15 and 5G standards, requires a relatively long initial access time that linearly grows with the angle resolution (highest resolution being the number of the antenna elements). On the theoretical front, in contrast, prior work which is based on simple measurement models noted that the problem of CSI acquisition in mmWave is closely related to that of noisy search, which itself has been shown to be closely related to the problem of channel coding over a binary input channel with ([8], [9], [10]) and without ( [11]) feedback. Under various noise models, it is shown that the number of measurements can be kept to grow only logarithmically with the angular resolution and target error probability [9] and [10].…”

“…On the theoretical front, in contrast, prior work which is based on simple measurement models noted that the problem of CSI acquisition in mmWave is closely related to that of noisy search, which itself has been shown to be closely related to the problem of channel coding over a binary input channel with ([8], [9], [10]) and without ( [11]) feedback. Under various noise models, it is shown that the number of measurements can be kept to grow only logarithmically with the angular resolution and target error probability [9] and [10]. While these early studies did not take the practical beam patterns into account, this logarithmic scaling was later also confirmed and reported in more practical systems with realistic arXiv:1812.07722v4 [cs.IT] 4 Sep 2019 Fig.…”

Sequential Learning of CSI for MmWave Initial Alignment

“…More specifically, at any given time t, a beamforming vector w t ∈ W S is selected sequentially as a function of previously observed signals (y t−1 ). We focus on strategies based on posterior matching, where the problem is connected to channel coding over a binary input channel with ([10], [11], [12]) and without ([13]) feedback. We use the hierarchical posterior matching (hieP M ) scheme from our prior work [8], described in detail in Sect.…”