OFDM Pilot Optimization for the Communication and Localization Trade Off

SPCOMNAV Communications and Navigation OFDM Pilot Optimization for the Communication and Localization Trade Off A. Lee Swindlehurst Dept. of Electrical Engineering and Computer Science The Henry Samueli School of Engineering University of California Irvine http://newport.eecs.uci.edu/~swindle Gonzalo Seco Granados Communications and Navigation Dept. of Telecommunications and Systems Engineering Universitat Autònoma de Barcelona http://spcomnav.uab.es 1

Outline SPCOMNAV Communications and Navigation Motivation and Objectives Pilot Design for Joint Time Delay and Channel Estimation Pilot Reduction Algorithm Averaging over the Channel Distribution Pilot Design for Capacity Maximization Bayesian Approaches Conclusions and Open Problems 2

Motivation and Objectives SPCOMNAV Most wireless Communications systems and Navigation are designed for either high data rate communications or precise positioning. 3

Motivation and Objectives SPCOMNAV Most wireless Communications systems and Navigation are designed for either high data rate communications or precise positioning. The use of one type of system for the other application results in poor performance. GPS / Galileo / GNSS can provide sub meter positioning accuracy, but the data rate is 1kbps or even less. Cellular networks and WLANs (based on IEEE 802.11x) are widespread, but they are seldom used for positioning because of poor accuracy, need for infrastructure investments or generation of databases, etc. A single combined system is of interest for users, operators, funding institutions and for system manufacturers At present, traffic increases exponentially with time but revenues increase only linearly. Operators need a solution for this time bomb. Reduced development and maintenance costs, and reduced terminal costs. Increased capability to exploit the synergy between both applications; more services and more intensive usage. 5

Motivation and Objectives SPCOMNAV Trends in the evolution of communications standards and future positioning Communications and Navigation systems suggest possible convergence. GNSS Next generation may include broadband communications. The integration of Galileo with GMES and ISICOM is being discussed. 3GPP standardization is paying more attention to localization functionality. Multicarrier signals (MC) seem to be a good choice for a combined communications and positioning system. Besides the advantages that have made MC signals the preferred choice for all types of terrestrial communications (DVB T/H, LTE, WiMax, IEEE 802.11a/g/n, xdsl, military systems: Joint Tactical Radio Systems JTRS) : Reduction in equalization complexity Inherent simplicity in their application to multiple access schemes Robustness against narrow band interference and frequency selective fading Adaptation of the data rate of each subcarrier according to its SNR; high scalability of the data rates Possibility of deploying single frequency networks, which is especially attractive for broadcasting applications SATCOM is also starting to use MC formats (DVB SH). 8

SPCOMNAV Motivation and Objectives Communications and Navigation MC signals are also attractive for positioning: Flexibility in shaping the signal spectrum: RMS bandwidth can be easily controlled through the subcarrier power allocation, which can be optimized for different working scenarios Simpler combination of different components (pilots / data) and services Easier channel estimation, potentially robustness against multipath (TBC) In some cases, better spectral containment 9

SPCOMNAV Motivation and Objectives Communications and Navigation Spectrum 4 3.5 3 2.5 2 1.5 1 0.5 POWER ALLOCATION FOR CAPACITY OPTIMIZATION Noise Signal Use only the pilots needed for channel estimation Waterfilling type allocations Low RMS bandwidth reduces timing accuracy Data subcarriers contribute to timing accuracy to a lesser extent than pilot subcarriers. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency -0.5-0.25 0 0.25 0.5 (frequency) 11

SPCOMNAV Motivation and Objectives Communications and Navigation POWER ALLOCATION FOR TIMING ESTIMATION OPTIMIZATION 20 18 Noise Signal Spectrum 16 14 12 10 8 Use mainly pilots located at the band edges; avoid data subcarriers. Data transmission and multiple access capability are impaired. 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5-0.25 Frequency 0 0.25 0.5 (frequency) 12

SPCOMNAV Motivation and Objectives Communications and Navigation POWER ALLOCATION FOR TIMING ESTIMATION OPTIMIZATION 20 18 Noise Signal Spectrum 16 14 12 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5-0.25 Frequency 0 0.25 0.5 (frequency) Use mainly pilots located at the band edges; avoid data subcarriers. Data transmission and multiple access capability are impaired. Data rate and timing accuracy seem to be competing objectives. How to produce a design with reasonable performance for both objectives? 13

The Pilot Allocation Problem SPCOMNAV The objective Communications is to find and a Navigation pilot distribution that addresses the trade off between capacity and timing accuracy. 14

The Pilot Allocation Problem SPCOMNAV The objective Communications is to find and a Navigation pilot distribution that addresses the trade off between capacity and timing accuracy. Improvement in communication capacity beyond a certain threshold requires spectral signal allocations that tend to degrade ranging accuracy, and vice versa. Therefore, there is a fundamental trade off to explore, and designs can be obtained for different working points in the feasible region. The formulation of the capacity under realistic assumptions (i.e. channel and timing estimation errors) is extremely difficult. The capacity (data rate) objective is replaced by the optimization of channel estimation accuracy and the minimization of the # of pilots. Capacity improves with the quality of the channel state information, and smaller # s of pilots allow more data subcarriers. 18

Different Pilot Structure Design Approaches SPCOMNAV Communications and Navigation There are different approaches to optimizing the estimation capabilities of pilot structures. We present three different lines of work: 1. Joint time delay and channel estimation CRB Minimize the CRB of the joint time delay and channel impulse response estimates Expressions may depend on the parameters to be estimated. 2. Joint time delay and channel estimation Bayesian CRB Similar to the previous case but based on a Bayesian bound. Expressions depend on the parameters statistics rather than their actual value. 3. Time Delay Estimation CRB for Random Channels Only interested in minimizing the CRB of the time delay for a given channel distribution 20

Pilot Design for Joint Time Delay and Channel Estimation SPCOMNAV The design of optimal pilot structures for joint time delay and channel estimation is studied Communications and Navigation through the use of the Cramér Rao Bound (CRB). A frequency selective channel with length equal to L samples is assumed. The received signal during one vector is: The vector of parameters to estimate in this formulation is 22

Pilot Design for Joint Time Delay and Channel Estimation The SPCOMNAV joint CRB for the channel and time delay estimate is given by: Communications and Navigation 23

Pilot Design for Joint Time Delay and Channel Estimation: Problem Formulation SPCOMNAV Communications and Navigation To find good pilot structures, we minimize the weighted trace of the CRB for the timing ( ) and the CRB of the channel (CRB 22 ) is proposed: The problem of finding the optimal pilots can be expressed as This problem is convex and can be easily solved numerically. The remarkable fact is that solutions are naturally sparse, without having imposed any related constraints. That is, the optimal pilot distributions naturally have a low number of active components (K). Low means on the order of the L+1 (L is the length of the channel impulse response) which is typically much smaller than N. Plenty of carriers free to allocate data! Bad news: optimal pilot allocation is channel (but not time delay) dependent 26

Pilot Design for Joint Time Delay and Channel Estimation: Example SPCOMNAV Communications and Navigation Cluster Normally L+1 pilots or L+1 clusters of pilots, but there are exceptions. When channel estimation performance is emphasized (α=0) more power is concentrated in the central subcarriers, while for time delay estimation (α=1) power is concentrated at the edges of the bandwidth. 27

Pilot Design for Joint Time Delay and Channel Estimation: Trade off SPCOMNAV Communications and Navigation Performance trade off boundary obtained by varying α from 0 to 1: 28

Statistics on the Optimal Number of Pilots Distributions of Signal pilot Processing order for SPCOMNAV Communications and Navigation 5 x Random Channels of Length 4 when = 0 104 5 x Random Channels of Length 4 when = 1 104 4.5 4 (α=0) 4.5 4 (α=1) 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 4 5 6 7 8 9 10 11 12 13 Number of Pilot Carriers Number of pilots increases as we focus more on channel estimation than time delay estimation. The number of pilots in the optimal distribution is low for any α. 4 5 6 7 8 9 10 11 12 13 Number of Pilot Carriers For α=1, we can show that no more than 2L+1 pilots are needed. For α<1 we have no formal result, but we have never observed more than 2L+4 in the simulations. 0 29

Pilot Reduction Algorithm SPCOMNAV Communications and Navigation Obtaining a single optimal allocation is relatively easy from a computational point of view because the problem is convex. However, there are in general many (infinite) optimal allocations and, by simply executing an optimization routine, we do not have control over the particular solution that we get. The resulting allocation, although optimal, may have more pilots than strictly needed. We have developed a pilot reduction algorithm by formulating the original problem as an SDP and observing the KKT optimality conditions. Starting from an arbitrary optimal pilot distribution, it produces another optimal allocation with a smaller number of active pilots (what implies that more subcarriers are available for data transmission). The algorithm is applied iteratively until a stopping condition is fulfilled. It is computationally simple; each iteration only involves solving a system of linear equations. 32

Example of the Pilot Reduction Algorithm SPCOMNAV As an example Communications and for and the Navigation sake of the clarity of the results, we consider the case of pilot allocations that minimize the channel estimation error in the absence of unknown timing. This problem has been addressed several times in the literature. It is known that equi powered and equi spaced allocations are optimal. But this type of solutions present some problems: For certain values of N and L, this implies that all subcarriers should be occupied by pilots. It is not easy to see how the solution can be modified to comply with additional constraints (e.g. forcing a certain frequency region to be free of pilots). Uniform allocation Periodic allocation 33

Example of the Pilot Reduction Algorithm SPCOMNAV Let us consider Communications that we and want Navigation to avoid pilots in the region of subcarriers 5 and 3. These problems can be avoided because the equipowered and equispaced allocations are not the only optimal solutions. The proposed algorithm allow us to find allocations that are optimal have a small number of pilots fulfill additional constraints 34

SPCOMNAV Example of the Pilot Reduction Algorithm Communications and Navigation x initial allocation o new allocation 35

Fixing the Number of Pilots SPCOMNAV Communications and Navigation In some cases, it may be of interest to allow for an even smaller number of pilots than those provided by the previous algorithm while sacrificing some channel or timing estimation accuracy. As a first step, we propose to solve the following minimization problem The solution to this problem is a distribution with the lowest possible number of pilots that achieves a value of the cost function below some target performance t. In order to find the optimal pilot distribution with a fixed number of pilots, one iteratively solves the problem for increasing values of t until a distribution with the desired number of pilots is found 37

Fixing the Number of Pilots Drawback: SPCOMNAV The 0 norm is non continuous and non differentiable, and thus numerical methods Communications and Navigation such us interior point or active set algorithms cannot be used. We can use an approximation of the behavior of the 0 norm for small ² : Simulation results show that the approximation does not significantly affect performance. Here we set K=5 pilots. In general, the least powerful pilot in each cluster disappears: 39

Fixing the Number of Pilots SPCOMNAV Degradation of the CRB when the number of pilots is fixed to the smallest number Communications and Navigation that makes the problem identifiable (i.e. K=L+1): The worst case degradation is bad, but in the majority of cases the deterioration is almost negligible, which is corroborated by a very low (< 2%) average increase of the CRB. 40

Averaging over the Channel Distribution SPCOMNAV Communications and Navigation To eliminate dependence of the results on the channel, we reformulate the problem as a function of the expected value of the CRB matrix with respect to the channel. The new cost function is: where, using the Laplace approximation, we can obtain that: Thus, this formulation does not depend on the channel realization but on the channel statistics. 42

Averaging over the Channel Distribution SPCOMNAV Results show Communications the shape and of Navigation optimum pilot distributions is similar to the original problem, but The distributions are symmetrical (for α=1) or almost symmetrical (for α 1). The maximum number of pilots is slightly smaller. h optimal (α=0) τ optimal (α=1) 43

Averaging over the Channel Distribution SPCOMNAV Distributions of pilot order Communications and Navigation Channel estimation (α=0) Kmax = 2L+1 Compromise (α=0.5) Kmax = 2L+1 44

SPCOMNAV Averaging over the Channel Distribution Communications and Navigation Time-delay estimation (α=1) Kmax = 2L For α = 1 all the distributions are symmetric, so a distribution with 7 pilot carriers is not to be possible for an even number of carriers. Note that the maximum number of pilots is noticeably lower with this problem formulation. 45

Pilot Design for Capacity Maximization with a Timing/Channel Estimation Accuracy Target SPCOMNAV Communications and Navigation The goal is to introduce link capacity into the design problem (work in progress). We have proposed an algorithm to find the best combination of pilots and pilot power: 1. Fix the desired value of the weighted trace of the CRB. 2. For a pilot structure with the lowest possible number of pilots, obtain the minimum power P P needed to achieve the desired value of the weighted trace. 3. Allocate the remaining power P D =P T P P to the free subcarriers using waterfilling. Compute the capacity. If the obtained capacity is larger than the previous capacity, continue. Else, finish. 4. Find the optimum pilot structure with one additional pilot and obtain the minimum power needed to achieve the desired CRB. Underlying idea: check whether or not increasing pilot order is beneficial. Increasing # of pilots reduces required pilot power => more power but fewer subcarriers available for data. Algorithm evaluates whether the increase in available power compensates for the decrease in # of subcarriers. Capacity is computed using the AWGN formula for each data subcarrier, which is an approximation. 47

Pilot Design for Capacity Maximization with a Timing/Channel Estimation Accuracy Target: Results SPCOMNAV Communications and Navigation We have found that typically the distribution with the lowest amount of pilots is preferable (in terms of capacity) to the distribution that minimizes the CRB. 48

Bayesian CRB for Time Delay and Channel Estimation Bayesian CRB exploits knowledge of an a priori distribution of the parameters. Unlike the SPCOMNAV Communications and Navigation CRB, the Bayesian CRB does not depend on any random parameters, only their distribution. This is different from the expected value of the CRB with respect to the channel distribution, usually referred to as the Expected CRB. It is known that ECRB BCRB. With the BCRB framework, we introduce a priori knowledge about the channel impulse response. This knowledge can take the form of an a priori estimate of the channel plus some uncertainty, which is represented by a covariance matrix: The a priori estimate can be (for instance) a previous channel estimate, or, in a TDD system, the channel estimate obtained in the opposite communication link. 53

Bayesian CRB for Time Delay and Channel Estimation SPCOMNAV Communications and Navigation 1) Estimates the channel 2) Transmits with a pilot distribution adapted to that channel, but taking into account that the channel will have changed to some extent. 55

0.45 Example solutions: SPCOMNAV 0.5 Bayesian CRB for Time Delay and Channel Estimation Signal = 0 Processing for Communications and Navigation Pilot Structure Channel Frequency Response 0.4 0.35 = 0 Channel/Normalized Pilot Magnitude 0.4 0.35 0.3 0.25 0.2 0.15 0.1 Fixed Channel Channel/Normalized Pilot Magnitude 0.3 0.25 0.2 0.15 0.1 Pilot Structure Channel Frequency Response Small Variances 0.05 0.05 0-15 -10-5 0 5 10 15 Discrete Frequency Index 0.4 = 0 0-15 -10-5 0 5 10 15 Discrete Frequency Index 0.35 Channel/Normalized Pilot Magnitude 0.3 0.25 0.2 0.15 0.1 Pilot Structure Channel Frequency Response Large Variances 0.05 0-15 -10-5 0 5 10 15 Discrete Frequency Index 56

Communication and navigation performance goals are often at odds. SPCOMNAV Receivers must Communications estimate and Navigation timing in the presence of an unknown channel (e.g. multipath), and receivers have to estimate the channel in the presence of unknown timing. Both estimation problems have typically been addressed individually, assuming the other parameter is known. This is not representative of real scenarios. We have addressed both problems jointly. A remarkable result: optimal distributions for the joint problem require very few pilots. Best possible performance can be achieved while leaving many free carriers for data transmission. When time delay estimation performance is emphasized, optimal distributions have fewer pilots with energy concentrated at the band edge. The trade off (Pareto frontier) for channel and timing accuracy has been obtained. This allows us to obtain solutions with a pre determined balance of both performance metrics. We derived an algorithm to reduce the number of pilots starting from an arbitrary optimal allocation. Reduction is important to increase available bandwidth for data without degrading the estimation accuracy. The problem admits several formulations: Explicit use of the number of pilots in the optimization goal Expected CRB Bayesian CRB Pilot Allocation: Conclusions 58

Pilot Allocation: Open Problems SPCOMNAV Communications and Navigation Study of the implications of the assumptions made in each of the approaches to the problem. Which one is more suited to specific scenarios? Study the convexity of the different approaches and derive bounds on the length of the optimal allocations New formulations of the problem for the design of both pilot and data subcarriers, taking into account: Power distribution between pilot and data carriers. Contribution of data carriers to channel and time delay estimation. Effects of time delay estimation on capacity. Extend the study to more complex configurations: Joint estimation including the frequency offset Multiple antennas at the Tx and Rx. Pilot allocations in 3D, i.e. different pilots are transmitted at each symbol period, subcarrier and antenna. 59