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arXiv:1703.06858v2 [cs.IT] 20 Jun 2017 F7 ne E rn gemn o 078(.....-Pos - (P.R.I.M.E. 605728 no. Experience). agreement Mobility grant International Frame Researchers Seventh REA Union’s under European Pe the (FP7) the support of and the Actions) (BMBF) also from Curie Research was (Marie and funds Education with work Prog of (DAAD) Ministry This Framework Federal Service 291763. Exchange Seventh Academic no. Union’s German agreement European German grant the the under from and funds StudyInitiative with Advanced (TUM), M¨unchen Universit¨at for Institute che the and Foundation M¨uhlfenzl yus apig aeomdesign optimizat waveform sampling, Nyquist estimation, delay-Doppler sensing, sive eosrt o h rmwr a eue ordc the algorithm. estimation reduce Monte- channel by to estima a impact high of practical used maintaining simulations the verify while be we receiver Finally, can the accuracy. we at framework Furthermore, rate the waveforms. sampling analog how reg optimized Pareto demonstrate decomposition the the Eigenvalue explore by to an one spanned using b enables approach by can presented which form The problem compact optimization in system sign solved through a obtain transmit receiver delay-D we the wireless analog a channel, at For the bandwidth. theorem pre-filtering of analog sampling wide coefficients the desi violating Fourier theoretic when the estimation an for derive rule we (BCRLB), bound lower odtosi osdrd ae nteBysa Cram Bayesian the on Based considered. sub-Nyq under is estimation conditions delay-Doppler joint for waveform h esrsga oteaao-odgtlcneso rate bandlimit conversion pre-filter analog-to-digital the analog to the satisfie signal it sensor i.e., that the high theorem, such designed where sampling is system the theory, receive e the see estimation estimation, When from [1]. for sta beneficial results be it can data, bandwidths to receive contrast the guarant from in this reconstruction a com- While signal requiring is rate. perfect theorem, filter sampling receive receive sampling high and well-known sufficiently transmit the the with of pliance bandwidth the for emomn ehiusadrt adaptation. f whil rate required object, and is target techniques estimation a beamforming channel of communication velocity wireless and in precise to position shift the delay-Doppler determine the Rada of communication. knowledge mobile use and systems as such applications, C hswr a upre yteEKNeV,teHirc n Lo and Heinrich the e.V., EIKON the by supported was work This ne Terms Index Abstract nsga rcsigsses h rviigdsg paradi design prevailing the systems, processing signal In ini h inlpoesn ieaueadi e to key is and literature processing signal the in atten- significant tion enjoys estimation parameter HANNEL I hswr,teotmzto fteaao transmit analog the of optimization the work, this —In nlgTasi inlOtmzto for Optimization Signal Transmit Analog nesmldDlyDplrEstimation Delay-Doppler Undersampled Bysa Cram —Bayesian ∗ nttt o omnctosEgneig ehiceUni Technische Engineering, Communications for Institute .I I. ‡ -al nra.ezmtmd,[email protected] [email protected], E-Mail: er aul colo niern,Uiest fCalifor of University Engineering, of School Samueli Henry NTRODUCTION rRolwrbud compres- bound, lower er-Rao † ´ nra Lenz Andreas ahmtc eatet rj nvrietBusl Belg Brussel, Universiteit Vrije Department, Mathematics ∗ aulS Stein S. 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[2], a low limitations find hardware and to and necessary resources wit is bottleneck power a to forms rate respect sampling the Since performance. width civbesmln rate sampling achievable osdrtepoaaino nanalog, an of propagation the Consider eew osdrotmzto ftetasi inlwhile signal transmit the of optimization consider we Here decades for discussed been has estimation Delay-Doppler † .LeSwindlehurst Lee A. , B x fe nrdcn h ytmmdlfrasingle-input a for model system the introducing After . ˘ B ( t ) ftetasitradteeoeteoealsystem overall the therefore and transmitter the of ∈ C esta ¨nhn Germany versit¨at M¨unchen, hog ieesdlyDplrcanl The channel. delay-Doppler wireless a through [email protected] , f > B I S II. i,Ivn,USA Irvine, nia, f YSTEM ium s s ya ievledecomposition. Eigenvalue an by ttercie etit h band- the restricts receiver the at ‡ f M s ODEL mle hnteNyquist the than smaller T 0 proi pilot -periodic izing pler ults em ive m- ter he e- h e , white Gaussian (AWGN) η˘(t) C with constant power estimation algorithm θˆ(y). The squared error (MSE) ∈ ˆ N0, can be denoted as of the estimator θ(y) is defined as

j2πνt ˆ ˆ T y˘(t)= γx˘(t τ)e +˘η(t) (1) Rǫ =Ey,θ θ(y) θ θ(y) θ . (9) − − − with channel coefficient γ C, time-delay τ R and Doppler A fundamental limit forh the estimation accuracy i (9) is the shift ν R. The signal y˘(t∈) C is filtered by∈ a linear receive Bayesian Cramer-Rao´ lower bound (BCRLB) [11, p. 5] ∈ C ∈ −1 filter h(t) , such that the final analog receive signal Rǫ J , (10) ∈  B j2πνt y(t)= γx˘(t τ)e +˘η(t) h(t) where J B is the Bayesian information matrix (BIM) − ∗ = v(t; θ)+ η(t)  (2) J B = J D + J P . (11) T The first summand of the BIM (11) represents the expected is obtained, where θ = τ ν R2 denotes the unknown, ∈ matrix (EFIM) random channel . For the duration T0, the signal  1 y(t) C is sampled in intervals of Ts = , resulting in an J D =Eθ J F (θ) , (12) ∈ fs even number of N = T0 2N samples Ts ∈ with the Fisher information matrix (FIM) exhibiting entries y = v(θ)+ η, (3) ∂2 ln p(y θ) [J (θ)] = E | . (13) F ij − y|θ ∂[θ] ∂[θ] with the receive vectors y, v(θ), η CN defined as " i j # ∈ For the signal model (3), the FIM entries (13) are N [y]i = y i 1 Ts , (4) ∂v(θ) H ∂v(θ) − 2 − [J (θ)] = 2Re R−1 . (14)    F ij ∂[θ] η ∂[θ] N ( i   j ) [v(θ)]i = v i 1 Ts, θ , (5) − 2 − The second summand in (11) denotes the prior information    N matrix (PIM) J P with entries [η]i = η i 1 Ts . (6) − 2 − 2    ∂ ln p(θ) [J P ]ij = Eθ . (15) We use positive integers as indices for vectors and matrices − " ∂[θ]i∂[θ]j # and thus i 1, 2,...,N . The noise samples η in (3) follow IV. TRANSMITTER OPTIMIZATION PROBLEM a zero-mean∈{ Gaussian distribution} with matrix The design problem of finding a transmit signal x˘⋆(t) that H N×N ˆ Rη =Eη[ηη ] C . (7) minimizes the MSE (9) of the estimation algorithm θ(y) under ∈ a particular positive semi-definite weighting M R2×2, Note that Rη depends on the receive filter h(t) and the subject to a transmit power constraint P , can be phrased∈ as sampling rate fs and thus is not necessarily a scaled identity ⋆ 1 2 matrix. The unknown parameters θ are considered to be Gaus- x˘ (t) = arg min tr(MRǫ), s.t. x˘(t) dt P. (16) x˘(t) T0 T0 | | ≤ sian distributed p(θ) (0, Rθ) with known covariance Z ∼ N Although the BCRLB (10) can be achieved with equality only σ2 0 R = τ . (8) under special conditions [11, p. 5], it closely characterizes θ 0 σ2  ν  the estimation performance trend (see Sec. VII-C). It is hence Here we assume that the channel γ is known at the re- possible to formulate (16) based on the BIM (11) ceiver, which simplifies the formulation of the transmit signal 1 x˘⋆(t) = arg min tr(MJ −1), s.t. x˘(t) 2dt P. (17) optimization problem. However, when testing the optimized B T | | ≤ x˘(t) 0 ZT0 waveforms for a practical scenario in the last section we will −1 treat γ to be a deterministic unknown. For the derivation, we In order to avoid optimization with respect to J B in (17), we consider an alternative maximization problem first assume a fixed sampling rate fs at the receiver while the ′ 1 periodic transmit signal x˘(t) is band-limited with two-sided x˘⋆(t) = argmax tr(M J ), s.t. x˘(t) 2dt P. bandwidth B. Then we consider the case of a variable rate B T | | ≤ x˘(t) 0 ZT0 fs. In contrast to the sampling theorem assumption B fs, (18) ≤ ⋆ in our setup we allow B>fs. Note that at the receiver, we It can been shown that if x˘ (t) is a solution of the max- ′ always use an ideal low-pass filter h(t) featuring the same imization problem (18) with M , there exists a weighting bandwidth B as the transmit signal. matrix M (not necessarily equal to M ′) for which the original minimization problem (17) has the same solution x˘⋆(t) [12]. III. CHANNEL ESTIMATION PROBLEM Since J P is independent of x˘(t), (18) then simplifies to

Under the assumption that γ is known, the task of the ⋆ ′ 1 2 x˘ (t) = argmax tr(M J D), s.t. x˘(t) dt P. receiver is to infer the unknown channel parameters θ based x˘(t) T0 T0 | | ≤ on the digital receive data y using an appropriate channel Z (19) V. ESTIMATION THEORETIC PERFORMANCE MEASURE The diagonal matrix H(ν) CK×K in (23) characterizes the ∈ Solving the optimization problem (18) requires an analytical frequency shifted receive filter spectrum and has elements characterization of the EFIM (12). A frequency-domain rep- K resentation enables a compact notation of the receive signal [H(ν)]ii = H i 1 ω0 +2πν . (27) − 2 − model [9] and thus provides further insights on the FIM   ! entries (14). Note that a frequency-domain approach naturally Note that the channel matrix (23) describes the propagation of embodies the bandwidth restriction required in practice by x˜ through the channel and its transformation from the spectral limiting the number of Fourier coefficients. to the time domain. Further note that the effect due to bandwidths B higher than the sampling frequency fs is A. Signal Frequency Domain Representation automatically included by the wide IDFT matrix W H. Due to periodicity, the transmit waveform x˘(t) can be Stacking the entries of v(nTs; θ) (22) into one vector yields represented by its Fourier series v(θ)= C(θ)x˜, (28) K 2 −1 with the transmit filter spectrum vector x˜ CK formed by x˘(t)= X ejkω0t, (20) k the Fourier coefficients ∈ K k=X− 2 [x˜]i = X K . (29) where ω = 2π = 2πf and K = 2πB 2N is the total i− 2 −1 0 T0 0 ⌈ ω0 ⌉ ∈ number of harmonics. Xk denotes the k-th Fourier coefficient B. Fisher Information of the Delay-Doppler Channel of the transmit signal. Inserting expression (20) into (2) and In order to compute the FIM elements (14), it is necessary to applying the filtering operation in (2), we obtain compute the derivatives of v(θ) with respect to the parameters

K θ. Using the frequency domain representation (28), we obtain 2 −1 v(t; θ)= γ X ejkω0(t−τ)ej2πνt h(t) ∂ ∂C(θ) k ∗ v(θ)= x˜. (30) k=− K ∂[θ] ∂[θ] X2   i i K 2 −1 The derivatives of the channel matrix are = γej2πνt ejkω0te−jkω0τ H(kω +2πν)X , 0 k ∂C(θ) H K =γ√ND(ν)W ∂T (τ)H(ν), (31) k=X− 2 ∂τ (21) ∂C(θ) =γ√N ∂D(ν)W HT (τ)H(ν)+ where H(ω) is the Fourier transform of the receive filter h(t). ∂ν  H Evaluating v(t; θ) at instants nT ,n = N ,..., N 1 yields D(ν)W T (τ)∂H(ν) , (32) s − 2 2 − K  2 −1 with the partial derivatives kn j2πνnTs j2π N −jkω0τ v(nTs; θ)=γ e e e H(kω0 +2πν)Xk N N j2π(i− 2 −1)νTs K [∂D(ν)]ii = j2π i 1 Tse , (33) k=− 2 − 2 − X   K −1 2 K K −j(i− 2 −1)ω0τ [∂T (τ)]ii = j i 1 ω0e , (34) = [C(θ)] N K X , (22) n+ 2 +1,k+ 2 +1 k − − 2 − K   k=X− 2 ∂ K [∂H(ν)]ii = H i 1 ω0 +2πν . (35) N×K ∂ν − 2 − with the channel matrix C(θ) C , defined by    ∈ Inserting (31) and (32) into (14), the FIM entries can be √ H C(θ)= γ ND(ν)W T (τ)H(ν). (23) expressed as quadratic terms The indices of C(θ) in (22) stem from the fact that we use ∂CH(θ) ∂C(θ) positive integers as indices for vectors and matrices. Here [J (θ)] = 2Re x˜H R−1 x˜ . (36) F ij ∂[θ] η ∂[θ] D(ν) CN×N stands for a diagonal matrix ( i j ) ∈ N The elements of the expected Fisher information matrix j2π(i− 2 −1)νTs [D(ν)]ii =e , (24) (EFIM) (12) are then obtained by which represents the Doppler frequency-shift. Further W ∂CH(θ) ∂C(θ) K×N ∈ ˜H −1 ˜ C is a tall discrete Fourier transform (DFT) matrix [J D]ij = 2Re x Eθ Rη x ( " ∂[θ]i ∂[θ]j # ) − K −1 − N −1 (i 2 )(j 2 ) 1 −j2π ˜H Γ Γ ˜ [W ] = e N , (25) = x ( ij + ji) x, (37) ij √N K×K with the channel sensitivity matrix Γij C and T (τ) CK×K denotes the diagonal time-delay matrix ∈ ∈ H K ∂C (θ) −1 ∂C(θ) −j(i− 2 −1)ω0τ Γ =E R . (38) [T (τ)]ii =e . (26) ij θ η " ∂[θ]i ∂[θ]j # VI. TRANSMIT SIGNAL OPTIMIZATION A. Pareto-Optimal Region - Fixed Sampling Rate

In the following we solve the transceiver design problem For a setting where T0 = 10µs, fs = 10MHz, σν = 5kHz (18) using the EFIM expressions (37). With the frequency and στ = 10ns, Fig. 1 shows the Pareto-optimal regions for domain representation (29) of the transmit signal, the opti- different bandwidths B = ρfs. Note that here for all systems mization problem (18) becomes a maximization with respect the same sampling frequency fs has been used. The results to the transmit Fourier coefficients x˜ indicate that a potential performance gain of roughly 12dB for delay estimation and 4dB for Doppler estimation can be ˜⋆ ′ ˜H ˜ x = argmax tr M J D s.t. x x P. (39) obtained when optimizing the transmit system for ρ =2. Note ˜ ≤ x that when increasing the transceive bandwidth B from ρ =1  Expanding the trace operation, the objective function becomes to ρ =2, two main effects affect the estimation performance.

2 2 First, a larger transmit bandwidth is beneficial for the delay ′ ′ H estimation due to high-frequency signal parts. On the other tr M J D = [M ]ji[J D]ij = x˜ Γ˜x, (40) i=1 j=1 hand, a higher receive filter bandwidth results in a larger  X X noise power at the receiver and therefore in a lower Doppler with the weighted channel sensitivity matrix estimation accuracy. However, the optimized system is able 2 2 to compensate this effect by efficiently using the available ′ Γ = [M ]ji (Γij + Γji) . (41) transmit spectrum, which leads to a moderate loss. i=1 j=1 X X B. Pareto-Optimal Region - Fixed Bandwidth The solution to the problem (39) is the Eigenvector γ of the 1 In the previous section, we have seen that optimized wave- matrix Γ corresponding to its largest Eigenvalue. forms have the potential to increase the accuracy of delay- Doppler estimation methods. We now investigate the estima- VII. RESULTS tion performance for a fixed transmit bandwidth B = 10MHz, There exists a trade-off between the estimation of delay and a signal period T0 = 10µs and different sampling frequencies B Doppler-shift. By solving the optimization problem (39) for all fs = . In order to focus on the case with we ′ κ positive semi-definite weightings M , we are able to approx- consider setups where κ> 1. Fig. 2 shows the Pareto regions imate the Pareto-optimal region. This region is characterized by the set of transmit waveforms for which the estimation of one parameter cannot be improved by changing the transmit 5 signal without reducing the accuracy of the other parameter. x˜⋆ For visualization, we define the relative measures

−1 J ˜ D xrect 11/22

| [dB] χτ/ν = 10 log −1 , (42) ν 0  J D x˜  ! | 11/22 χ κ =1 with respect to a rectangular pulse x˜rect of bandwidth B = fs, κ =2 as it is used in Global Navigation Satellite Systems (GNSS). κ =4 For the following results, the expectation (38) with respect to κ = 1 (x˜rect) p(θ) is computed using Hermite-Gaussian quadrature. 5 − 5 0 5 10 − 5 χτ [dB]

= B = 10MHz Fig. 2. Pareto regions for rates fs κ with B

of the optimized waveforms with respect to a rectangular [dB]

ν 0 signal. Note that the sampling rate for the reference system is χ held constant, while the sampling rate of the optimized system ρ =1 decreases with increasing κ. This indicates that although lower ρ =2 sampling rates are used, the optimized waveform design still ρ = 1 (x˜rect) bears the potential to provide high estimation accuracy. 5 − 5 0 5 10 C. Simulation Results − χτ [dB] To verify that the optimization based on the EFIM yields substantial performance gains for practical scenarios, we con- Fig. 1. Pareto regions for bandwidths B = ρfs with fs = 10MHz duct Monte-Carlo simulations with randomly generated noise η and channel parameters θ. As the channel γ is in general not known to the receiver, we use the hybrid maximum likelihood- since in this case the estimation merely relies on the prior maximum a posteriori (ML-MAP) estimator [11, p. 12] information p(θ). In the high SNR regime, the MSE of the hybrid ML-MAP estimator shows close correspondence with γˆML(y) ˆ = argmax ln p(y θ,γ) + ln p(θ) . (43) the BCRLB and the estimator benefits from the waveform op- θMAP(y) θ,γ |   timization. For moderate to high SNR values the performance  For simulations we use T0 = 10µs and B = 10MHz. gain is roughly 4.5dB for the estimation of the time-delay We compare the MSE of a rectangular pulse signal with and 3.5dB for the Doppler-shift estimation. This corresponds ⋆ fs = 10MHz and the optimized transmit signal x˜ with to the findings from the Pareto-region in Fig. 2. ˜⋆ fs = 5MHz, i.e., κ = 2. The transmitter design x used for VIII. CONCLUSION the simulations corresponds to the point of the Pareto-region We have derived an optimization framework for the transmit in Fig. 2 with largest distance to the origin. Fig. 3 and Fig. 4 waveform of an undersampled pilot-based channel estimation show the normalized empirical mean squared error (NMSE) system. By employing the BCRLB, the transmitter design MSEτ/ˆ νˆ problem was reformulated as a maximization problem with NMSE = (44) τ/ˆ νˆ 2 respect to the expected Fisher information matrix. A frequency στ/ν domain representation of the receive signal allows one to of the hybrid ML-MAP estimator for both systems, where find an analytical solution to the maximization problem via represents the diagonal elements of (9), empirically MSEτ/ˆ νˆ an Eigenvalue decomposition. The BCRLB of the optimized evaluated based on the results of the estimation algorithm (43). waveforms can be used to approximately characterize the The signal-to-noise ratio (SNR) is given by Pareto- region with respect to other delay- P Doppler estimation methods. Further, our results show that SNR = . (45) BN0 using optimized transmit waveforms enables the receiver to operate significantly below the Nyquist sampling rate while It is observed that for low SNR the MSE saturates at σ2 , τ,ν maintaining high delay-Doppler estimation accuracy. Finally, Monte-Carlo simulations support the practical impact of the 0 considered transmit design problem. REFERENCES [1] B. Sadler and R. Kozick, “A survey of time delay estimation performance 10 bounds,” in Fourth IEEE Workshop Sensor Array and Multichannel [dB] − Processing, Waltham, MA, 2006, pp. 282-288. ˆ τ [2] M. Verhelst and A. Bahai, “Where analog meets digital: Analog-to- information conversion and beyond,” IEEE Solid State Circuits Mag., ⋆ BCRLBτ (x˜ ) vol. 7, no. 3, pp. 67-80, Sep. 2015.

NMSE [3] A. Jakobsson, A. L. Swindlehurst, and P. Stoica, “Subspace-based 20 BCRLBτ (x˜rect) − ⋆ estimation of time delays and Doppler shifts,” IEEE Trans. Signal NMSEτˆ(x˜ ) Process., vol. 46, no. 9, pp. 2472–2483, Sep. 1998. [4] A. Jakobsson and A. L. Swindlehurst “A time domain method for joint NMSEτˆ(x˜rect) estimation of time delays, Doppler shifts and spatial signatures,” in 10 0 10 20 Ninth IEEE Workshop on Statistical Signal and Array − Processing, Portland, OR, 1998, pp. 388-391. SNR [dB] [5] B. Friedlander, “On the Cram´er-Rao bound for time delay and Doppler estimation,” IEEE Trans. Inf. Theory, vol. 30, no. 3, pp. 575-580, May 1984. Fig. 3. MSE and BCRLB - Time-delay τ [6] F. Antreich, “Array processing and signal design for timing synchro- nization,” Ph.D. dissertation, NWS, TUM, M¨unchen, 2011. [7] Q. Jin, K. Wong and Z. Luo, “The estimation of time delay and Doppler 0 stretch of wideband signals,” IEEE Trans. Sig, Proc., vol. 43, no. 4, pp. 904-916, Apr. 1995. [8] M. Stein, A. Lenz, A. Mezghani, and J. Nossek, “Optimum analog re- ceive filters for detection and inference under a sampling rate constraint,” in IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP), 10 Florence, 2014, pp. 1827–1831. [dB] − [9] A. Lenz, M. Stein, and J. A. Nossek, “Signal parameter estimation ˆ ν performance under a sampling rate constraint,” in 49th Asilomar Conf. ˜⋆ Signals, Systems and Computers, Pacific Grove, CA, 2015, pp. 503–507. BCRLBν (x ) [10] M. Khayambashi and A. L. Swindlehurst, “Filter design for a compres- NMSE 20 BCRLB (x˜ ) sive sensing delay and Doppler estimation framework,” in 48th Asilomar − ν rect NMSE (x˜⋆) Conf. Signals, Systems and Computers, Pacific Grove, CA, 2014, pp. νˆ 627–631. NMSEνˆ(x˜rect) [11] H. L. Van Trees and K. L. Bell, Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking. Piscataway, NJ: Wiley- 10 0 10 20 IEEE Press, 2007. − [12] M. Stein, M. Castaneda, A. Mezghani, and J. Nossek, “Information- SNR [dB] preserving transformations for signal parameter estimation,” IEEE Signal Process. Lett., vol. 21, no. 7, pp. 866–870, July 2014. Fig. 4. MSE and BCRLB - Doppler-shift ν