Contractual Date of Delivery: T (months) Actual Date of Delivery: Editor(s) name (s) Participant(s): Work package:

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1 MULTIPOS D4.5 Version 1. Technical report: Design of signals supporting both accurate positioning and high data-rate communications for static channels Contractual Date of Delivery: T + 16 (months) Actual Date of Delivery: Editor: Editor(s) name (s) Author(s): Arash Shahmansoori Participant(s): UAB Work package: WP4 Technical report on the design of signal supporting both accurate positioning and high data-rate communications for static channels Version: 1. Total number of pages: 55 Page 1 (55)

2 Abstract 1: A key aspect to design an OFDM system for combined positioning and high-data-rate communications is to find optimal data and pilot power allocations. Previously, A capacity maximizing design by taking into account the effects of channel and time-delay estimation for finite number of subcarriers and channel taps has been investigated. Increasing the number of subcarriers and channel taps make the matrix inversions in the non-asymptotic bounds close to singular or badly conditioned. Furthermore, computational complexity of such a system designed by non-asymptotic bounds grows significantly. In this paper, a method based on the asymptotic expected Cramér-Rao bound of joint time-delay and channel coefficients by increasing the number of subcarriers and channel taps has been proposed. The method reduces the complexity of the design considerably. Specifically, by increasing the number of channel taps the number of operations to compute matrix inversions is significantly reduced by asymptotic bounds. Numerical results show that as the number of subcarriers increases, the asymptotic bounds converge to the non-asymptotic bounds. Moreover, even for a finite number of subcarriers or channel taps the difference between joint data and pilot power allocations is negligible compared to the nonasymptotic expected Cramér-Rao bounds. Abstract 2: The accuracy of the estimation of time-delay and channel coefficients in Orthogonal frequency division multiplexing (OFDM) communication systems can be improved by reducing the variability of channel coefficients, i.e. reducing channel covariance and increasing channel mean for a given power. First, we prove that the effect of channel variability between different OFDM symbols cannot be directly captured by extending expected Cramer Rao bound (ECRB) for one OFDM symbol to M OFDM symbols. Then, the effect of channel variations between different OFDM symbols is modelled as the variations of channel covariance and channel mean for one OFDM symbol and a given channel power. A pilot design approach based on per-symbol signal to noise and interference ratio (SINR) for a given time-delay estimation accuracy is investigated. The results show that reducing channel covariance and improving channel mean for a given channel power leads to more accurate estimation of time-delay and channel coefficients. Furthermore, one can save the total power for a given estimation accuracy and channel capacity. Disclaimer: This document solely relies on the main results of three papers in ICL-GNSS, SPAWC IEEE conference 214, and ISWCS conference 214. Obviously, the report focus is on the main results and does not include all the technical materials. Further, the unpublished researches are not included in this report. Page 2 (55)

3 Document Control Version Details of Change Review Owner Approved Date Page 3 (55)

4 Executive Summary This document describes the design of OFDM sequences for positioning and communications using the joint Asymptotic ECRB of time-delay and channel coefficients. Also, a method for combined signal design for communications and positioning is proposed. First, we find an expression for joint ECRB of time-delay and channel coefficients for the case of limited number of subcarriers and channel taps. Second, we develop the asymptotic expressions of the bounds by letting the number of subcarriers and/or channel taps to be sufficiently large. Third, the concept of ergodic capacity for imperfect channel state information (CSI) at the transmitter is investigated for the problem. Finally, joint design of OFDM sequences for a given time-delay estimation accuracy based on maximization of the ergodic capacity is proposed. Also, the effect of channel variability on the time-delay estimation and data transmission rate has been investigated. The goal is to analyze the effect of different channel uncertainties on the combined signal design for positioning and data rate communications. Page 4 (55)

5 Authors Partner Name Phone / Fax / UAB Arash Phone: Shahmansoori arash.shahmansoori@uab.cat. Page 5 (55)

6 Table of Contents Document Control... 3 List of Acronyms and Abbreviations... 8 Chapter 1: A Background on LTE & General Description of the Research 1. A Background on LTE General Context of the PhD State-of-the-Art Chapter 2: Minimum Number of Non-Zero Pilots in ECRB 1. Minimum Number of Non-Zero Pilots in ECRB Simulation Results Chapter 3: Joint Communication and Positioning by Asymptotic ECRB 1. Introduction System Model and Preliminaries OFDM Signal Model Expected Cramér-Rao Bound Asymptotic Expected Cramér-Rao Bound Asymptotic Behavior for Finite Channels Asymptotic Behavior for Long Channels Channel Capacity Power Allocation Optimization Simulation Results Parameters Asymptotic Behavior of the ECRB Joint Design Conclusions References Page 6 (55)

7 Chapter 4: Effect of Channel Variability on Joint Design of Data and Pilot 1. Introduction System Model and ECRB Signal Model ECRB for One OFDM Symbol ECRB for M OFDM Symbols Pilot Design Modeling the Channel Variability Optimization Problem for Pilot Design Simulation Results Conclusions Appendix B References Chapter 5: Forthcoming Research Chapter 6: List of Publications Page 7 (55)

8 List of Acronyms and Abbreviations Term ECRB CSI OFDM CRB ECRB h ECRB ττdd as ECRB h aaaa EEEEEEEE ττdd EEEEEEEE aaaa h EEEEEEEE aaaa ττdd Description expected Cramér-Rao bound channel state information Orthogonal frequency division multiplexing Cramér-Rao bound expected Cramér-Rao bound of channel coefficients expected Cramér-Rao bound of time-delay asymptotic expected Cramér-Rao bound of channel coefficients asymptotic expected Cramér-Rao bound of time-delay asymptotic expected Cramér-Rao bound of channel coefficients for long channels asymptotic expected Cramér-Rao bound of time-delay for long channels Page 8 (55)

9 Chapter 1: General Description of the Research Page 9 (55)

10 1. A Background on LTE Long Term Evolution (LTE) moves towards the fourth generation (4G) of mobile communications. LTE, as a transition from the 3rd generation (3G) to the 4th generation (4G), has achieved great capacity and high speed of mobile telephone networks. It is combined of the top-of-the-line radio techniques to outperform Code Division Multiple Access (CDMA) approaches. LTE provides scalable carrier bandwidth from 1.4 to 2 MHz and frequency division duplexing (FDD) together with time division duplexing (TDD). LTE has to satisfy the following requirements: Reduced cost per bit, Simple architecture, Flexibility usage of frequency bands, Reasonable terminal power consuming, lower cost and high speed. LTE is composed of many new technologies such as: OFDM (Orthogonal Frequency Division Multiplex), MIMO (Multiple Input Multiple Output). OFDM: It provides resilience to reflections and interference that leads to high data bandwidth. Furthermore, OFDMA (Orthogonal Frequency Division Multiple Access) is used for downlink and SC-FDMA (Single Carrier- Frequency Division Multiple Access), is used for uplink. This has the advantage of smaller peak to average ratio and constant power able to get RF power amplifier efficiency. MIMO: MIMO operations include spatial multiplexing as well as pre-coding and transmit diversity. These operations addressed the problems of multiple signals arising from many reflections, which were encountered by previous telecommunications systems. Moreover, using MIMO also increases the throughput via the additional signal paths after those operations. MIMO requires two or more different antennas with different data streams to distinguish the different paths, such as the schemes using 2 x 2, 4 x 2, or 4 x 4 antenna matrices. Page 1 (55)

11 The uplink and downlink LTE specifications are listed in the following table: Parameters Details Data type All packet switched data (voice and data). No circuit switched Modulation types QPSK, 16QAM, 64QAM (UL and DL) Access schemes OFDMA (DL); SC-FDMA (UL) Duplex schemes FDD and TDD Peak DL speed 64QAM (Mbps) SISO, 2*2 MIMO, 4*4 MIMO Downlink positioning procedure uses different time of arrivals of downlink radio signals to compute user position. The method relies on a network based strategy. In summary, first user requiest the information about the position, the measurements will be sent to location server (E- SMLC (Enhanced Serving Mobile Location Center)) by LTE positioning protocol (LLP), E- SMLC estimates the position using trilateration technique. Positioning Reference Signal (PRS) is a dedicated signal for positioning purposes in release nine of LTE systems that mitigates the near-far effect, due to a higher frequency reuse factor (i.e. of six).the PRS is scattered in time and frequency in the so-called positioning occasion that allocates consecutive positioning sub-frames with a certain periodicity. Page 11 (55)

12 2. General Context of the PhD So far, the wireless transmission systems are designed to fulfil one of the objectives: positioning or communications. For instance, Global Positioning System (GPS) provides information about user position of the order of meter. An example for a communication system is Wireless Local Area Network (WLAN) which provides data rates up to 6 Mbits/s. there is a clear user demand for combined communication and positioning systems, one only needs to observe the large amount of technologies that are embedded in current mobile devices: UMTS, GPS, WiFi, inertial sensors, etc., which often complement each other. The existence of a single system for communications and navigation would bring important advantages to operators, manufacturers and users, such as reduced equipment cost, savings in equipment form factor (yielding smaller devices) and power consumption (yielding improved battery life), enlarged coverage, further possibilities for development of new value-added services, etc. The combination of communication and positioning offers a wide range of advantages like: Reducing user equipment cost. Enhanced resource allocation or improved power control in cellular networks. Applications such as locating emergency calls, tracking, and guiding firefighters or policemen on a mission, or location-based services become feasible. However, designing a system which can be used for both purposes is a challenging problem. In fact, wireless communication systems designed for one purpose act poorly for the other purpose. This is due to the fact that by increasing channel capacity, or equivalently data rate, the accuracy of the time-delay estimation which is used for positioning purposes is reduced. In a similar way, by improving the accuracy of the time-delay estimation the data rate is considerably reduced. LTE-Standard wireless communications systems incorporate components for positioning purposes like: Positioning Reference Signal (PRS) in the downlink. Sounding Reference Signal (SRS) in the uplink. However, improving the reference signals capabilities for positioning purposes without affecting the data-rate transmission is an open problem. Page 12 (55)

13 3. State-of-the-Art A possible solution to design a system that can be used for both purposes is to design a signal to maximize the capacity and at the same time minimize lower bound of the time delay estimation. The properties of multicarrier signals, more specifically Orthogonal Frequency Division Multiplexing (OFDM) signals, make them the best candidate on which to base the design of a combined system. Basically the main technical part of our research includes: We address flexible-multicarrier signals (e.g., OFDM signals) as the best option for multi-purpose signal design (e.g., positioning and communication). We derive timing accuracy bounds that avoid the usual simplifying assumptions about the channel knowledge and the modeling of the data, e.g., Gaussian bounds like: GCRB. We use the powerful tools of convex optimization for multi-purpose signal design. Optimizing the power of signals to maximize the capacity and minimize time delay error results a power allocation optimization problem. Consequently some of the subcarriers are used for data transmission and the other ones act as pilot symbols to estimate channel coefficients and time-delay. However, since each subcarrier can either be used as pilot or data the problem is turned to a non-convex combinatorial problem that can be relaxed and solved as a convex optimization. Page 13 (55)

14 Chapter 2: Minimum Number of Non-zero Pilots in ECRB Page 14 (55)

15 1. Minimum Non-zero Pilot in ECRB: In this section, we briefly discuss the topic of minimum required number of non-zero pilots to minimize the expected CRB (ECRB) of time-delay and channel coefficients separately and for weighted values of time-delay and channel coefficients. The problem can be formalized as the zero norm minimization problem that is NP-Hard and therefore non-convex. In other words, we have min pp pp subject to 11 TT pp = PP tt EEEEEEBB ww εε (2.6) where the optimization is with respect to the number of pilots and pp stands for pilots and. stands for the norm zero. Since the norm zero cost function is non-convex one can approximate it by its logarithmic approximation or other approximations. This is not the subject of our discussion here. The first two constraints assure the pilots to be non-negative and their sum be equal to the total required power for the design PP tt. The last constraint limits the accuracy of weighted ECRB of time-delay and channel coefficients that is weighted as ECRB w = ww ECRB τd + (1 ww) trace(ecrb h ) The ECRB of time-delay ECRB τd and trace(ecrb h ) depend on the pilots and their expressions are proposed in the next chapters of this report. Here instead of solving the direct non-convex optimization problem we design the pilots to minimize the weighted ECRB for different weights and random channel covariance matrices and count the number of nonzero pilots. This way the problem is turned into a convex problem since the weighted ECRB is a convex function with respect to the pilots. Next we present the simulation results for different weights and several channel realizations. Note that adding the problem of data transmission and combined design of data and pilots reduce the number of non-zero pilots to increase the capacity at the same time. Page 15 (55)

16 2. Simulation Results: Here we use the following simulation parameters. Parameters Number of subcarriers Number of Channel Taps Description N=32 L=5 Number of Realizations M=1 Channel Covariance Matrix Diagonal We do the simulations for weighting factor w= (channel coefficients), and w=1 (time-delay). Figure 2-1 Number of channel realizations with respect to number of non-zero pilots EEEEEEEE hh. Figure 2.1 shows the number of channel realizations with respect to number of non-zero pilots. It is clear that the maximum number of non-zero pilots for 1 channel realizations to minize the trace of ECRB of channel coefficients is 18. However, this is least likely to happen. Page 16 (55)

17 Figure 2.2 shows the number of channel realizations with respect to number of non-zero pilots. It is clear that the maximum number of non-zero pilots for 1 channel realizations to minize the ECRB of time-delay is 18. However, this is least likely to happen. Consequently, the number of required non-zero pilots to minimize the trace of ECRB of channel coefficients is much bigger than the number of required non-zero pilots to minimize the ECRB of time-delay. Figure 2-2 Number of channel realizations with respect to number of non-zero pilots EEEEEEEE ττdd. We propose two simple examples of the pilot designs based on minimization of the ECRB of time-delay and channel coefficients to demonstrate how the minimum non-zero required number of pilots to minimize the ECRB is more for the case of channel coefficients comparing to the optimization based on time-delay. The reason behind this is that the more the unknowns are the more one needs pilots to estimate them. For the case of optimization with respect to pilot vector based on ECRB of channel coefficients we need more number of non-zero pilots for estimation. Next we propose two simple examples for the pilots used to minimize ECRB of time-delay and channel coefficients with the number of channel taps LL = 5 and number of subcarriers NN = 32. Note that here the goal is not to formulate the results since these are the results that are already available but we want to provide insight to the concept of number of non-zero pilots and its effect on the data transmission in the future chapters. Page 17 (55)

18 Figure 2-3 shows the pilot design for minimization of ECRB of time-delay. As it is clear from the simulations the number of non-zero pilots is 2222 = This means that 11 pilots are enough to minimize the ECRB of time-delay. Note that ECRB of time-delay is is the result of joint calculation of Fisher Information Matrix for tim-delay and channel coefficients and this is the reason that we still need 1111 pilots to minimize it otherwise if the channel is known much less amount of pilots are required. Figure 2-3 Figure 2-4 Figure 2-4 shows the pilot design based on ECRB of channel coefficients. The number of pilots required to minimize the ECRB of channel coefficients is much bigger than the case for time-delay (13) since we have more unknowns comparing to the previous case. Page 18 (55)

19 Chapter 3: Joint Communication and Positioning by Asymptotic ECRB Page 19 (55)

20 1. Introduction The design of combined positioning and communications systems that can perform well in terms of high-data-rate transmission and delay estimation accuracy is a challenging problem. In general, the signals used for one application perform poorly in the other case. To design a signal which can be applied for both purposes, one needs to consider the system specifications for time-delay estimation accuracy and data-rate communications. To date, different approaches have been adopted to design pilot symbols that improve the performance of channel estimators [1] [3]. The results show that equi-spaced, equi-powered pilots are optimal in terms of mean squared error. Pilot designs based on carrier frequency offset (CFO) estimation [4], or joint channel and CFO estimation [5] are considered by others. However, pilot design based on time delay estimation has received little attention. A pilot design based on joint CRB of channel and time delay is proposed in [6]. However, since CRBs in [6] are functions of specific channel realizations, the resulting pilots cannot be guaranteed to be optimal for all instances of random channels. The problem has been solved by designing based on averaging the CRB over a set of channel realizations known as Expected CRB (ECRB) [7]. In [7], joint design data and pilot power allocations for the case of limited number of subcarriers N and channel taps L is investigated. However, increasing the number of subcarriers and channel taps leads to very complex and close to singular matrix inversions using non-asymptotic bounds. Specifically, applying matrix inversion algorithms such as Gaussian elimination requires the computational complexity of the order OO(LL 33 ) [8] for long channels which makes the proposed bounds in [7] complex and close to singular. A method based on the effect of increasing the number of subcarriers N and channel taps L on joint channel coefficients and clock offset estimation is proposed in [9]. However, the bounds are limited to a specific type of channel. In this paper, we consider the effect of increasing the number of subcarriers N and channel taps L on the joint expected Cramér-Rao bound of time-delay and channel coefficients. First we analyze the problem of finding Asymptotic expected CRB by assuming large number of subcarriers. Also, we investigate the effect of increasing the number of channel taps L, which is usual in wire-line applications (e.g., VDSL and PLT), such that the ratio between the number of channel taps and subcarriers LL/NN is sufficiently small. Increasing the number of subcarriers and channel taps leads to very complex and close to singular matrix inversions using non-asymptotic bounds. Here, we aim to reduce the complexity to OO(LL) by doing the inversion only at strong pilots and setting the rest of eigenvalues and their corresponding eigenvectors to zero. Finally, we compare joint design of data and pilot allocations based on asymptotic bounds with non-asymptotic bounds using several numerical examples. Page 2 (55)

21 2. System Model and Preliminaries In this section, we briefly present the system model for OFDM system and ECRB of joint timedelay and channel coefficients. 2.1 OFDM Signal Model the continuous-time received signal from a standard OFDM symbol passed through a frequency selective channel, after removing the guard interval is yy (aa) NN (tt) = dd NN,kk gg (aa) (tt kkkk) + νν (aa) (tt), (2.1) kkkkz where TT is the sampling period at the transmitter such that TT = NNNN is the observation window, is the total number of subcarriers, dd NN,kk represents the output from the inverse FFT (IFFT) block at the transmitter, and νν (aa) (tt) is additive zero-mean complex Gaussian noise. we assume that the impulse response gg (aa) (tt) is a delta function with the time limit of [, LLLL) where LL is the number of channel taps determined based on the RMS delay and channel model LL 1 gg (aa) (tt) = h ll δδ(tt llll ττ), ll= (2.2) where h ll is the channel coefficient of lltth path and ττ is the timing offset or equivalently the time- delay of first path. Finally, the discrete-time received signal yy NN [nn] = yy NN (aa) (nnnn). or its vector form yy NN can be written as yy NN = RR NN (ττ dd )hh + νν NN, (2.3) where yy NN = [yy NN [],, yy NN [NN 1]] TT, νν NN = [νν NN [],, νν NN [NN 1]] TT, hh the (nn, ll)th element of RR NN (ττ dd ) is NN 1 RR NN (ττ dd ) = 1 NN DD NN,nn eejj nn = 2ππ NN nn (nn ll ττ dd ), = [h,, h LL 1 ] TT, and (2.4) with DD NN,nn defined as pilot subcarrier at the nn th frequency, and ττ = ττ dd TT. Finally, taking the FFT of the output, we find YY NN = FF NN,NN RR NN (ττ dd )hh + VV NN. Also, one can obtain a similar model as in [8] as YY NN = DD NN ΓΓ(ττ dd )FF NN,LL hh + VV NN with DD NN an NN NN diagonal matrix of the input, FF NN,LL the first L columns of DFT matrix FF NN,NN, and ΓΓ(ττ dd ) an N N diagonal matrix with entries ee jj2ππ TT kkττ dd. Page 21 (55)

22 2.2 Expected Cramér-Rao Bound In this part, we briefly explain the ECRB of joint time-delay and channel coefficients. The related results can be found in [5]-[7]. Defining the parameter vector by θθ = [hh TT RR, hh TT II, ττ dd ] TT, the corresponding Fisher information matrix (FIM) can be written as JJ FF = NNNN[UU NN ] NNR[UU NN ] NN 2 JJ[VV NN hh], (2.5) 2 NNR[UU NN ] NNNN[UU NN ] NN 2 R[VV NN hh] σσ 2 NN 2 R[hh HH VV HH NN ] NN 2 JJ[hh HH VV HH NN ] NN 3 hh HH WW NN hh where UU NN = 1 NN RR NN HH (ττ dd )RR NN (ττ dd ), VV NN = 1 NN 2 RR NN HH (ττ dd )QQ NN (ττ dd ), WW NN = 1 NN 3 QQ NN HH (ττ dd )QQ NN (ττ dd ), being QQ NN HH (ττ dd ) = ddrr NN (ττ dd )/ddττ dd. Using the well known block inversion matrix lemma [1] and defining a new estimation parameter θθ = [hh TT, ττ dd ] TT, we find EE[ hh NN hh 2 ] σσ2 2NN (2tr(UU NN 1 ) + γγ NN 1 ββ NN 2 ), (2.6) EE[(ττ dd,nn ττ dd ) 2 ] σσ2 2NN 3 γγ NN, (2.7) where ββ NN = UU NN 1 VV NN hh, γγ NN = hh HH (WW NN VV NN HH UU NN 1 VV NN )hh. Finally, taking the expectation with respect to channel coefficients and using Jensen s inequality, the approximate expression for the ECRB of the timing offset and the channel coefficients would be ECRB h σσ2 2NN (2tr(UU NN 1 ) + γγ NN 1 ββ NN 2 ), (2.8) σσ2 ECRB ττdd 2NN 3, γγ NN (2.9) where ββ NN 2 = tttt([vv HH NN UU HH NN UU 1 NN VV NN ]RR h ), γγ NN = tr([ww NN VV HH NN UU 1 NN VV NN ]RR h ), where RR h is the channel covariance matrix. Note that the actual expected CRB is tighter than the above expressions due to Jensen s inequality. In the next section, we obtain the asymptotic ECRB for channel coefficients and timing offset where the number of subcarriers NN is assumed to be sufficiently large. Page 22 (55)

23 3. Asymptotic Expected Cramér-Rao Bound The main results for asymptotic ECRB of joint time-delay and channel coefficients is briefly discussed in this section. Obviously, the goal is not to present all the proofs and formulas. See [13], for more details on the asymptotic bounds and proofs. 3.1 Asymptotic Behavior for Finite Channels Using the asymptotic expressions of UU NN, VV NN, and WW NN defined as UU, VV, and WW respectively, we obtain asymptotic ECRB of channel coefficients and time-delay NECRB h as σσ2 2 (2tr(UU 1 ) + γγ 1 ββ 2 ), (3.1) NN 3 EEEEEEEE ττdd aaaa σσ2, 2γγ (3.2) where ββ 2 = tttt([vvuu HH UU 1 VV]RR h ), γγ NN = tr([ww VV HH UU 1 VV]RR h ). We have proved the asymptotic expressions of UU, VV, and WW as UU = FF LL,NN PPFF HH LL,NN, (3.3) VV = jjjjff LL,NN ffffff HH LL,NN, (3.4) WW = 4ππ2 FF 3 LL,NNff 2 PPFF HH LL,NN, (3.5) where FF LL,NN is the first LL rows of the discrete Fourier transform matrix FF NN,NN, and PP and ff are pilot power and derivative matrices defined as PP = diag{p,, P N 1 }, (3.6) ff = diag{ NN,, NN 1 NN }. (3.7) Page 23 (55)

24 3.2 Asymptotic Behavior for Long Channels In this part, the asymptotic expressions for the case of long channels (where L is sufficiently large such that the ratio between L and the number of subcarriers N, L/N is sufficiently small) is obtained. For this case, Fisher information matrix (FIM) is close to singular or badly conditioned. It is proved that instead of using the inverse of FIM to obtain the bounds, one needs to apply the pseudo-inverse [11]. So, the inverse term UU 1 in ββ 2 and γγ NN is replaced by its pseudo-inverse UU. Using the eigenvalue decomposition, we obtain NNECRB aaaa h σσ2 (2tr Σ γ 1 L 1 ββ L 1 2 ), (3.8) NN 3 ECRB aaaa ττdd σ2 2γ L 1, (3.9) where ββ L 1 2 = tr(σσ 2 2 ΣΣ 1 2 EE RR h EE H ), γγ LL 1 = tr ΣΣ 3 ΣΣ 2 2 ΣΣ 1 1 EE RR h EE H, EE is normalized eigenvectors while the eigenvalues are stored in diagonal matrices ΣΣ 1, ΣΣ 2 and ΣΣ 3. Note that by definition the pseudo-inverse of a matrix UU is the inverse of its nonzero eigenvalues that are greater than or equal to a threshold δδ stored in diagonal matrix ΣΣ 1 with the rest of entries are set to zero. Therefore, omitting the eigenvectors corresponding to the eigenvalues less than δδ, we obtain the normalized eigenvector reduced matrix as EE, and the corresponding eigenvalue reduced matrices as ΣΣ 1, ΣΣ 2 and ΣΣ 3. Considering the above discussion (3.8) can be further simplified by replacing the term γ 1 L 1 ββ L 1 2 as γ 1 L 1 ββ L 1 2 = σ 2 k HH (k)λ k [σ 3 (k) σ 2 1 (k) σ 1 (k) σ 2 (k)]λ k k HH, (3.1) where σσ ii (kk) is the kkth diagonal entry of ΣΣ ii, and HH represents the set of subcarriers where eigenvalues of EE RR h EE H (λ k ) are not zero. In other words, parts of the frequency region with strong channel frequency response are used for estimation. Page 24 (55)

25 4. Channel Capacity Rewriting the signal model YY NN = DD NN ΓΓ(ττ dd )FF NN,LL hh + VV NN as YY NN = HHDD NN + VV NN, (4.1) where DD NN = [D N,,, D N,N 1 ] T, HH = diag{h(),, H(N 1)}, HH(kk) = γγ ττ (kk)hh (kk), HH (kk) = FF NN,LL hh, γγ ττ (kk) = e j2π T kτ, and FF kk,ll is the kkth row of FF NN,LL. Replacing HH by HH + HH in (4.1), we obtain YY NN = HH DD NN + HH DD NN + VV NN, (4.2) where HH and HH represent estimated and error in the estimation values with kkth diagonal entries HH (kk) = γγ ττ (kk)ff NN,LL hh, and HH (kk) = γγ ττ (kk)ff NN,LL hh respectively. Note that channel coefficients hh are estimated in the receiver through known pilot symbols inserted at the transmitter. The receiver feeds the estimated channel coefficients back to the transmitter (Adaptive Modulation). However, the estimated coefficients at the transmitter are not error-free due to estimation error. Consequently, it can be shown that the lower bound of ergodic capacity CC llll (pp p, pp d ) for the so called partially known channel at the transmitter is of the form of CC llll(pp p, pp d ) = 1 NN EE[log 2 det(ii + PP drr 1 ye HH HH H )], (4.3) where pp p and pp d are pilot and data vectors respectively, PP d is an NN NN diagonal matrix of data with diagonal elements pp dd,kk for kkkkkk (where DD represents the set of subcarriers used for data transmission) and zero elsewhere, and RR yyyy defined as RR yyyy = PP d EE HH HH H + σ 2 II. (4.4) H Replacing HH HH by dddddddd{ff EE[hh hh HH kk,ll ]FF HH kk,ll } kkkkkk, and EE[hh hh HH ] by the vector form of its asymptotic expression, we find the ergodic capacity as CC llll pp p, pp d = EE C lb pp p, pp d, (4.5) where C lb pp p, pp d is the instantaneous capacity defined as C lb pp p, pp d = 1 NN log p d,k ζ k p d,k ζ k pp p + σ 2, (4.6) kϵdd where ζ k = FF kk,ll hh hh HH FF HH kk,ll, and ζ k pp p = FF kk,ll JJ hh (pp pp )FF HH with JJ h (pp pp ) being the matrix form of kk,ll asymptotic ECRB of channel coefficients. Page 25 (55)

26 5. Power Allocation Optimization In this section, we formulate the optimization problem used for pilot design for joint communication and positioning. To maximize the cost function which is the lower bound of ergodic capacity (4.5), one needs to solve the following optimization problem where KK is CC CCCC = EE max ppp,pp d KKC lb pp p, pp d, (5.1) KK = pp p, pp d JJ ττdd pp p εε, 11 TT pp p + 11 TT pp d PP TT, pp pp TT pp dd =, pp p, pp d }, (5.2) where CC CCCC is the channel capacity for the closed-loop (CL) system using partially known CSI at the transmitter side, KK is a set such that pp p and pp d satisfy some constraints, pp p and pp d represent pilot and data power vectors respectively, PP TT is the total power, JJ ττdd pp p represents the asymptotic bound for time-delay, and εε is the minimum accuracy in the estimation. Constraint 11 TT pp p + 11 TT pp d PP TT limits the total power in the design of data and pilots to PP TT, pp pp TT pp dd = means that subcarriers used for data transmission cannot be used as pilots which leads to a combinatorial optimization that is not convex. To solve this issue, we use the relaxation approach by omitting the constraint pp pp TT pp dd = and solving the relaxed problem. Solving the relaxed problem, one needs only a few subcarriers as pilots with higher amplitudes while the rest are set for data transmission. Finally, pp p, pp d emphasizes that pilots and data are non-negative values where is an element-wise operator. Page 26 (55)

27 Page 27 (55) 6. Simulation Results In this section we present the simulation results based on the asymptotic bounds. 6.1 Parameters Within this section we use the number of subcarriers N=48, total power for pilots and data P t =5, noise power σ 2 =.1, minimum accuracy in time-delay estimation is ϵ=.2, assuming Gaussian random channel with independent taps and consequently diagonal channel covariance matrices of sizes 4 4 with = Rh, 6 6 with = Rh, and 8 8 with = Rh.

28 6.2 Asymptotic Behavior of the ECRB aaaa Figure 6-1 shows the asymptotic behaviour of the ECRB of channel coefficients EEEEEEEE h comparing to the non-asymptotic bound EEEEEEEE h by increasing the number of subcarriers N. The result shows that even for the finite number of subcarriers (e.g. for NN = 48) the asymptotic bound converges to the non-asymptotic bound. Figure 6-1 Asymptotic behaviour of EEEEEEEE hh and EEEEEEEE hh aaaa versus number of subcarriers NN. Table 6-1 RMSE of the difference between asymptotic and non-asymptotic ECRB of time-delay Variables N RMSE [db] aaaa The asymptotic behaviour of the EEEEEEEE of time-delay EEEEEEEE ττdd comparing to the non-asymptotic bound EEEEEEEE ττ is shown in Table 6-2. This is to make the difference visible due to the fact that dd aaaa the convergence by increasing the number of subcarriers for EEEEEEEE ττdd is of the order NN 3 that is aaaa much faster than the convergence for EEEEEEEE h that is of the order of NN. The Root Mean Square aaaa Error (RMSE) between EEEEEEEE ττdd and EEEEEEEE ττ is of the order of -56.4dB to 16.13dddd for NN dd from 1 to 15 respectively. Page 28 (55)

29 6.3 Joint Design Figure 6-2 shows the joint design of pilots and data power allocations for IIIIIIII channel model with maximum number of paths L= 1 σ τ /Ts = 4 where σ τ is the RMS delay spread and TT ss represents the sampling period. Figure 6-2 Joint pilot and data allocation for NN = 4444 and LL = 44 with diagonal channel covariance. Figure 6-3 and Figure 6-4 show the joint design of pilots and data power allocations based on the proposed channel model with maximum number of paths LL = 6, and LL = 8, with diagonal channel covariance matrices, i.e. independent channel coefficients, defined in the simulation parameters, and number of subcarriers NN = 48. The results show that using the channel of length LL joint design of pilots and data power allocations requires LL + 2 pilots for estimation with the rest of subcarriers saved for data transmission. Obviously, increasing the number of taps from LL = 4 to LL = 8 reduces the capacity by around 2.3\% since the number of subcarriers for data transmission is reduced. Page 29 (55)

30 Figure 6-3 Joint pilot and data allocation for NN = 4444 and LL = 66 with diagonal channel covariance Figure 6-4 Joint pilot and data allocation for NN = 4444 and LL = 88 with diagonal channel covariance. Page 3 (55)

31 Figure 6-5 studies the trade-off between capacity and time-delay estimation accuracy. The dashed-line curves with upward and downward triangle marks represent the capacity achieved by solving the optimization problem without any restriction on the pilot distributions using non-asymptotic and asymptotic bounds respectively. For a given accuracy in the estimation of time-delay represented by εε the difference between capacity achieved by asymptotic and nonasymptotic bounds is negligible. However, using the asymptotic bounds simplify the joint design problem of data and pilot power allocations significantly specially by increasing the number of channel taps LL. Figure 6-5 Asymptotic behaviour of ECRB h as versus number of subcarriers N. The curves with square and circle marks represent maximum achievable capacity obtained by the solution to the optimization problem where subcarriers are allowed to be shared by pilot and data symbols for non-asymptotic and asymptotic bounds respectively. Note that by losing the time-delay accuracy εε down to.14 and higher arbitrary joint design problem based on asymptotic and non-asymptotic bounds converge to the same amount of capacity. We use the Page 31 (55)

32 arbitrary designs of data and pilot power allocations as the upper bound of the maximum achievable capacity. Figure 6-6 shows ergodic capacity for closed-loop system CC CCCC obtain based on the optimization problem versus SNR. From Figure 6-6, it is clear that by improving the accuracy in the estimation of time-delay CC CCCCis reduced considerably for SNR below 3dB while by increasing SNR the effect of time-delay estimation accuracy on CC CCCCis reduced. Figure 6-6 Ergodic channel capacity for the closed-loop (CL) system CC CCCC using partially known CSI at the transmitter.. Page 32 (55)

33 7. Conclusions Using asymptotic bounds one can reduce the computational complexity of the optimization problem specially by increasing the number of subcarriers NN and channel taps LL. The performance of near-optimal pilot and data power allocations for the case of asymptotic bounds is compared with the traditional non-asymptotic bounds. Results show that after a certain number of subcarriers which can be as low as NN = 48, asymptotic bounds converge to the nonasymptotic bounds. Further, the performance of joint data and pilot power allocations is only affected negligibly even for the limited number of subcarriers and channel taps. Page 33 (55)

34 8. References [1] R. Negi and J. Cioffi, Pilot tone selection for channel estimation in a mobile OFDM system, IEEE Transactions on Consumer Electronics, vol. 44, pp , Aug [2] I. Barhumi, G. Leus, and M. Moonen, Optimal training design for MIMO OFDM systems in mobile wireless channels, IEEE Transactions on Signal Processing, vol. 51, no. 6, pp , Jun. 23. [3] H. Minn and N. Al-Dhahir, Optimal training signals for MIMO-OFDM channel estimation, IEEE Transactions on Wireless Communications, vol. 5, no. 5, pp , May. 26. [4] H. Minn and S. Xing, An optimal training signal structure for frequency-offset estimation, IEEE Transactions on Communications, vol. 53, no. 2, pp , Feb. 25. [5] P. Stoica and O. Besson, Training sequence design for frequency offset and frequency-selective channel estimation, IEEE Transactions on Communications, vol. 51, no. 11, pp , Nov. 23. [6] M.D. Larsen, G. Seco-Granados, and A.L. Swindlehurst, Pilot optimization for timedelay and channel estimation in OFDM systems, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 211, pp , IEEE. [7] R. Montalban, J. A. Lopez-Salcedo, G. Seco-Granados, and A. L. Swindlehurst, Power allocation method based on the channel statistics for combined positioning and communications OFDM systems, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 213, pp , IEEE. [8] S. Delić and Ž Jurić, Some improvements of the gaussian elimination method for solving simultaneous linear equations, in IEEE International Convention on Information and Cummunication Technology Electronics and Microelectronics (MIPRO). 213, pp , IEEE. [9] S. Gault, W. Hachem, and P. Ciblat, Joint sampling clock offset and channel estimation for ofdm signals: Cramer-rao bound and algorithms, IEEE Transactions on Signal Processing, vol. 54, no. 5, pp , May. 26. [1] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, New York, NY, USA, 21. [11] P. Stoica and T. Marzetta, Parameter estimation problems with singular information matrices, IEEE Transactions on Signal Processing, vol. 49, no.1, pp. 87 9, Jan. 21. [12] B. Hassibi and B. M. Hochwald, How much training is needed in multiple-antenna wireless links, IEEE Transactions on Information Theory, vol. 49, no. 4, pp , Apr. 23. [13] A. Shahmansoori, R. Montalban, J. A. Lopez-Salcedo, G. Seco-Granados, Design of OFDM Sequences for Joint Communications and Positioning Based on the Asymptotic Expected CRB, in International Conference on Localization and GNSS (ICL-GNSS). 214 (Accepted). Page 34 (55)

35 Chapter 4: Effect of Channel Variability on Joint Design of Data and Pilot Page 35 (55)

36 1. Introduction Location awareness is one of the fundamental characteristics of cognitive radio (CR). Applications of location awareness can require different level of positioning accuracy. For instance, indoor positioning usually requires higher positioning accuracy. Also, one of the crucial aspects for developing a cooperative positioning system is to establish an accurate positioning method. On the other hand, one needs to keep the communications rate above a certain level at the same time. Therefore, designing a sequence that can satisfy both requirements simultaneously based on the system specifications for time-delay estimation (This can be converted to the distance measurements for positioning purposes) accuracy and channel capacity is of interest. Pilot symbols are widely used for estimation and synchronization in communications and navigation systems. For instance, pilot design in the estimation of time-delay can be used for ranging in global navigation satellite systems (GNSS). Since the choice of pilot design significantly affects estimation performance, it is of critical importance to find the optimal pilots that can improve the estimation accuracy. To date, different approaches have been adopted to design pilot symbols that improve the performance of channel estimators [1-3]. The results show that equi-spaced, equi-powered pilots are optimal in terms of minimum mean square error (MMSE). Pilot designs based on carrier frequency offset (CFO) estimation [4], or joint channel and CFO estimation [5] are considered by others. However, pilot design based on joint time-delay and channel estimation has received little attention. In [6], a pilot design for joint channel and time-delay based on hybrid CRB (CRB) is presented. However, the bound is not a tight bound comparing to the other bounds, e.g. ECRB. A Pilot design based on joint CRB of channel and time-delay is proposed in [7]. However, since CRBs in [7] are functions of specific channel realizations, the resulting pilots are not guaranteed to be optimal for all instances of random channels. The problem has been solved by designing based on averaging the CRB over a set of channel realizations known as ECRB. Note that in [7,8], it is assumed that the design is based on one OFDM symbol. In this paper, we use ECRB for joint time-delay and channel coefficients [8]. On the one hand, the effect of channel variations between different OFDM symbols is not captured by extending the model for one OFDM symbol to several symbols. On the other hand, channel variations Page 36 (55)

37 between different OFDM symbols can be modelled as variations of channel covariance and channel mean for a given power. Sequential methods for updating MMSE for different symbols (e.g., Kalman filters) provides insight to our approach of updating channel covariance and channel mean between different OFDM symbols. However, applying sequential methods of updating MMSE of time-delay and channel coefficients requires the estimation of time-delay as a non-linear parameter resulting the use of extended Kalman filters [9] which is beyond the scope of this paper. Consequently, we use an alternative approach by changing channel covariance and mean for a certain channel power to model variations between different OFDM symbols. A pilot design approach for optimization based on per-symbol signal to interference and noise ratio (SINR) for a given time-delay estimation accuracy is investigated. The method converts the general non-convex optimization to a relaxed optimization that can be solved by convex optimization tools. Results show a significant improvement in terms of accuracy in the estimation of time-delay and channel coefficients by reducing channel variability (Reducing channel covariance and increasing channel mean), and saving the total power for a given accuracy and capacity. Page 37 (55)

38 2. Signal Model and ECRB In this section the signal model and ECRB of time-delay ECRB and the trace of ECRB of channel coefficients tr{ EEEEEEEE h } for one OFDM symbol and MM OFDM symbols are briefly explained. 2.1 Signal Model We use the following OFDM channel model YY = ΩΩ(ττ dd )hh + WW, (2.1) where ΩΩ(ττ dd ) = XXXX(τ d )FF L, (2.2) and XX represents an NN NN diagonal matrix of the input with the kkth diagonal element XX[kk] representing the input at the kkth subcarrier for kk = NN 2 + 1,, NN 2 for even values of NN and kk = NN 1,, NN 1 for odd values of NN, ΓΓ(τ 2 2 d ) is an NN NN diagonal matrix with the kkth diagonal element eeeeee( jj2 ππ kkτ d) where τ d is the time-delay, TT ss is the OFDM symbol length, TTss FF L contains the first LL columns of a Discrete Fourier Transform (DFT) matrix centered around zero, hh is an LL 1 column vector representing channel coefficients, WW is zero-mean complex Gaussian noise distributed as CCCC(, σσ 2 ww II), and finally YY is an NN 1 vector representing the output signal. Page 38 (55)

39 2.2 ECRB for One OFDM Symbol Using the Fisher Information Matrix (FIM) for the joint estimation of time-delay τ d and channel coefficients hh [7], one can obtain the CRB. However, CRB depends on the channel coefficients hh. To find a bound that does not depend on the channel but channel statistics, i.e. channel mean μμ h and covariance ΣΣ h, one can take the expectation of CRB with respect to channel coefficients hh and use the Laplace approximation, i.e. EE[ XX EE[XX] ]~ YY EE[YY], to obtain the approximated EEEEEEEE as [8] ECRB τ d σσ 2 1 ww EEEEEEEE h 2 JJ τ d, JJ h (2.3) where JJ τd = tr{(mm 1 MM 2 ) ΣΣ h } + μμ h H (MM 1 MM 2 )μμ h, (2.4) and JJ h = 2tr{QQ 1 } 1 + JJ τd (tr MM q ΣΣ h + μμ H h MM q μμ h ), (2.5) with MM 1, MM 2, and MM q defined as MM 1 =FF L HH DD HH PPPPFF LL, MM 2 = FF L HH PPPPFF LL, QQ = FF L HH PPFF LL, and MM q = FF L HH PPPPFF LL QQ 11 QQ 11 FF L HH PPPPFF LL respectively. Also, PP = XX HH XX is an NN NN diagonal matrix representing pilot powers, DD = 2ππ diag{ NN + 1,, NN } for even values of NN TT ss 2 2 and DD = 2ππ diag{ NN 1,, NN 1 } for odd values of NN, μμ TT ss 2 2 h = EE[hh] is the channel mean, and ΣΣ h = EE[(hh μμ h )(hh μμ h ) HH ] is the channel covariance matrix. Throughout the paper we assume σσ ww 2 is set to one to simplify the notation. 2 Page 39 (55)

40 2.3 ECRB for M OFDM Symbols Using a simple extension to our model in (2.1) for MM OFDM symbols, we obtain ΩΩ 1 (ττ dd ) YY = hh + WW ΩΩ M (ττ dd ) (2.6) where ΩΩ i (ττ dd ) = XX i ΓΓ(τ d )FF L, (2.7) XX i is an NN NN diagonal matrix of input at the iith OFDM symbol with the same definition as in (2.2), hh is the channel coefficient vector as a result of column wise concatenation of channel coefficients of the iith OFDM symbol hh i assuming to be correlated with the channel coefficient vector of the jjth OFDM symbol hh j, and WW is the result of concatenating noise vectors WW i as a zero-mean complex Gaussian noise of the iith OFDM symbol, and assumed to be independent of WW j for different values of ii and jj. Applying the same approximation and assumption for the case of single OFDM symbol, we obtain the ECRB for MM OFDM symbols as ECRB τd σσ 2 ww JJ τd EEEEEEEE h 2 JJ h 1, (2.8) where JJ τd = tr MM 1 MM 2 ΣΣ h + μμ h H MM 1 MM 2 μμ h, (2.9) and JJ h = 2tr QQ JJ τd (tr MM q ΣΣ h + μμ H h MM q μμ h ), (2.1) with (ii) MM 1 = diag{mm MM (ii) 1 }ii=1, MM 2 = diag{mm MM (ii) 2 }ii=1, MM q = diag{mm MM qq }ii=1, μμ h = [μμ TT 1 μμ TT MM ] TT where μμ ii is the channel mean at iith OFDM symbol, and QQ = diag{qq ii } MM ii=1. MM 1 (i) =FFL HH DD HH PP i DDFF LL, MM 2 (ii) = FFL HH PP i DDFF LL, QQ i = FF L HH PP i FF LL, and MM q (ii) = FFL HH PP i DDFF LL QQ i 11 QQ i 11 FF L HH PP i DDFF LL respectively. Also, PP i = XX HH XX is an NN NN diagonal matrix representing pilot powers at the iith OFDM symbol. Page 4 (55)

41 Finally, ΣΣ h represents a Block-Toeplitz matrix [1] defined as ΣΣ HH ΣΣ 1 ΣΣ h = HH ΣΣ MM 1 ΣΣ 1 ΣΣ MM 1 ΣΣ MM 2, (2.11) HH ΣΣ ΣΣ MM 2 with (ΣΣ kk ) {kk=,,mm 1} are LL LL matrices (not necessarily Toeplitz) representing channel covariance matrix of symbol ii for kk = and cross correlation matrices between different OFDM symbols for kk. This is due to the fact that channel has the same covariance matrix for different OFDM symbols and cross correlation matrix between symbols ii and jj depends on ii jj. However, channel coefficients can have different means μμ {ii=1,,mm} for different OFDM symbols. The entries of each sub matrix ΣΣ kk are kk of the form $(ΣΣ kk ) {uu,vv} = (σσ) {uu,vv}, for uu, vv = 1,, LL. Consequently, (2.9) and (2.1) can be written as MM (i) (i) JJ τd = [tr MM 1 MM2 ΣΣ + μμ H (i) (i) i MM 1 MM2 μμi ], ii=1 (2.12) MM JJ h = 2tr QQ 1 i 1 (ii) + JJ τd (tr MM q ΣΣ + μμ H (ii) i MM q μμi ), ii=1 (2.13) From (2.12) and (2.13), it is clear that the effect of cross correlation matrices ΣΣ kk is not captured by the ECRB. Next, we propose an alternative approach to consider the effect of channel variations by changing channel covariance and mean for a given channel power and design the pilots based on the new model. Page 41 (55)

42 3. Pilot Design In this section, first a model for channel variability between different OFDM symbols for a given channel power is presented. Second, we propose a pilot design based on signal to interference and noise ratio (SINR) (the effect of channel estimation error is considered as interference) for a given accuracy in the estimation of time-delay. 3.1 Modelling the Channel Variability We propose a method to model channel variability between different OFDM symbols by scaling channel mean and covariance for a given power PP h (this is due to the fact that channel power between different OFDM symbols does not change) defined as EE[ hh 2 ], the operation. is the norm of a vector. A coefficient aa 2 is defined to change the channel covariance as aa 2 ΣΣ h, and a coefficient bb to change the channel mean as bbμμ h. Using the expected value of a quadratic function, the total channel power PP h is where PP h = aa 2 εε 2 + bb 2 μμ 2, (3.1) εε 2 = tr{σσ h } = EE[ hh 2 ]. (3.2) Actually, (3.1) means that we are scaling the covariance with the coefficient aa 2 and the mean squared with the coefficient bb 2 in such a way that their sum remains constant. From (3.1), for a fixed value of bb 2 the coefficient aa 2 can be written as aa 2 = PP h bb 2 μμ 2 εε 2. (3.3) The coefficients aa 2 and bb 2 are used in the next part to investigate the effects of channel variability on pilot design. Page 42 (55)