Analog Transmit Signal Optimization for Undersampled Delay-Doppler Estimation Andreas Lenz∗, Manuel S. Stein†, A. Lee Swindlehurst‡ ∗Institute for Communications Engineering, Technische Universit¨at M¨unchen, Germany †Mathematics Department, Vrije Universiteit Brussel, Belgium ‡Henry Samueli School of Engineering, University of California, Irvine, USA E-Mail: [email protected], [email protected], [email protected] Abstract—In this work, the optimization of the analog transmit achievable sampling rate fs at the receiver restricts the band- waveform for joint delay-Doppler estimation under sub-Nyquist width B of the transmitter and therefore the overall system conditions is considered. Based on the Bayesian Cramer-Rao´ performance. Since the sampling rate forms a bottleneck with lower bound (BCRLB), we derive an estimation theoretic design rule for the Fourier coefficients of the analog transmit signal respect to power resources and hardware limitations [2], it when violating the sampling theorem at the receiver through a is necessary to find a trade-off between high performance wide analog pre-filtering bandwidth. For a wireless delay-Doppler and low complexity. Therefore we discuss how to design channel, we obtain a system optimization problem which can be the transmit signal for delay-Doppler estimation without the solved in compact form by using an Eigenvalue decomposition. commonly used restriction from the sampling theorem. The presented approach enables one to explore the Pareto region spanned by the optimized analog waveforms. Furthermore, we Delay-Doppler estimation has been discussed for decades demonstrate how the framework can be used to reduce the in the signal processing community [3]–[5]. In [3] a subspace sampling rate at the receiver while maintaining high estimation based algorithm for the estimation of multi-path delay-Doppler accuracy. Finally, we verify the practical impact by Monte-Carlo shifts is proposed and it is shown how the dimensionality of simulations of a channel estimation algorithm. the maximum likelihood (ML) estimator can be reduced by a Index Terms—Bayesian Cramer-Rao´ lower bound, compres- sive sensing, delay-Doppler estimation, signal optimization, sub- factor of two. In [4] a time-domain procedure for estimation Nyquist sampling, waveform design of delay-Doppler shifts and direction of arrival (DOA) is considered. Using prolate spheroidal wave (PSW) functions, I. INTRODUCTION the favorable transmit signal design with respect to time- delay accuracy is discussed in [6], while [7] considers such HANNEL parameter estimation enjoys significant atten- a technique for joint delay-Doppler estimation. Recent results tion in the signal processing literature and is key to C show that for wide-band transmit signals, analog receive filter applications, such as radar and mobile communication. Radar bandwidths which lead to violation of the sampling theorem systems use knowledge of the delay-Doppler shift to precisely can provide performance gains [8], [9]. Further, in [10] the determine the position and velocity of a target object, while optimization of receive filters in a compressed sensing frame- in wireless communication channel estimation is required for work has been investigated and improvements with respect to beamforming techniques and rate adaptation. matched filtering have been illustrated. In signal processing systems, the prevailing design paradigm for the bandwidth of the transmit and receive filter is com- Here we consider optimization of the transmit signal while pliance with the well-known sampling theorem, requiring a the receiver samples at a rate fs smaller than the Nyquist arXiv:1703.06858v2 [cs.IT] 20 Jun 2017 sufficiently high receive sampling rate. While this guarantees rate B. After introducing the system model for a single-input perfect signal reconstruction from the receive data, it stands single-output (SISO) delay-Doppler channel, we derive a com- in contrast to results from estimation theory, where high pact formulation of the transmit signal optimization problem bandwidths can be beneficial for parameter estimation, see e.g. in the frequency domain. We show how to solve the transmitter [1]. When the receive system is designed such that it satisfies design problem for B>fs by an Eigenvalue decomposition. the sampling theorem, i.e., the analog pre-filter bandlimits The potential Pareto-optimal region is visualized by optimizing the sensor signal to the analog-to-digital conversion rate, the the transmit waveform for different settings and comparing the results to conventional signal designs. We conclude the This work was supported by the EIKON e.V., the Heinrich and Lotte discussion with a performance verification via Monte-Carlo M¨uhlfenzl Foundation and the Institute for Advanced Study (IAS), Technis- simulations of a channel estimation algorithm. che Universit¨at M¨unchen (TUM), with funds from the German Excellence Initiative and the European Union’s Seventh Framework Program (FP7) under grant agreement no. 291763. This work was also supported by the II. SYSTEM MODEL German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF) and the People Program Consider the propagation of an analog, T0-periodic pilot (Marie Curie Actions) of the European Union’s Seventh Framework Program signal x˘(t) C through a wireless delay-Doppler channel. The (FP7) under REA grant agreement no. 605728 (P.R.I.M.E. - Postdoctoral ∈ Researchers International Mobility Experience). baseband signal at the receiver, which is perturbed by additive white Gaussian noise (AWGN) η˘(t) C with constant power estimation algorithm θˆ(y). The mean squared error (MSE) ∈ ˆ spectral density 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, ∈ Fisher information matrix (EFIM) random channel parameters. 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 covariance 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.