Analog Transmit Signal Optimization for Undersampled Delay-Doppler Estimation
aa r X i v : . [ c s . I T ] J un Analog Transmit Signal Optimization forUndersampled 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, USAE-Mail: [email protected], [email protected], [email protected]
Abstract —In this work, the optimization of the analog transmitwaveform for joint delay-Doppler estimation under sub-Nyquistconditions is considered. Based on the Bayesian Cram´er-Raolower bound (BCRLB), we derive an estimation theoretic designrule for the Fourier coefficients of the analog transmit signalwhen violating the sampling theorem at the receiver through awide analog pre-filtering bandwidth. For a wireless delay-Dopplerchannel, we obtain a system optimization problem which can besolved in compact form by using an Eigenvalue decomposition.The presented approach enables one to explore the Pareto regionspanned by the optimized analog waveforms. Furthermore, wedemonstrate how the framework can be used to reduce thesampling rate at the receiver while maintaining high estimationaccuracy. Finally, we verify the practical impact by Monte-Carlosimulations of a channel estimation algorithm.
Index Terms —Bayesian Cram´er-Rao lower bound, compres-sive sensing, delay-Doppler estimation, signal optimization, sub-Nyquist sampling, waveform design
I. I
NTRODUCTION C HANNEL parameter estimation enjoys significant atten-tion in the signal processing literature and is key toapplications, such as radar and mobile communication. Radarsystems use knowledge of the delay-Doppler shift to preciselydetermine the position and velocity of a target object, whilein wireless communication channel estimation is required forbeamforming techniques and rate adaptation.In signal processing systems, the prevailing design paradigmfor the bandwidth of the transmit and receive filter is com-pliance with the well-known sampling theorem, requiring asufficiently high receive sampling rate. While this guaranteesperfect signal reconstruction from the receive data, it standsin contrast to results from estimation theory, where highbandwidths can be beneficial for parameter estimation, see e.g.[1]. When the receive system is designed such that it satisfiesthe sampling theorem, i.e., the analog pre-filter bandlimitsthe sensor signal to the analog-to-digital conversion rate, the
This work was supported by the EIKON e.V., the Heinrich and LotteM¨uhlfenzl Foundation and the Institute for Advanced Study (IAS), Technis-che Universit¨at M¨unchen (TUM), with funds from the German ExcellenceInitiative and the European Union’s Seventh Framework Program (FP7)under grant agreement no. 291763. This work was also supported by theGerman Academic Exchange Service (DAAD) with funds from the GermanFederal Ministry of Education and Research (BMBF) and the People Program(Marie Curie Actions) of the European Union’s Seventh Framework Program(FP7) under REA grant agreement no. 605728 (P.R.I.M.E. - PostdoctoralResearchers International Mobility Experience). achievable sampling rate f s at the receiver restricts the band-width B of the transmitter and therefore the overall systemperformance. Since the sampling rate forms a bottleneck withrespect to power resources and hardware limitations [2], itis necessary to find a trade-off between high performanceand low complexity. Therefore we discuss how to designthe transmit signal for delay-Doppler estimation without thecommonly used restriction from the sampling theorem.Delay-Doppler estimation has been discussed for decadesin the signal processing community [3]–[5]. In [3] a subspacebased algorithm for the estimation of multi-path delay-Dopplershifts is proposed and it is shown how the dimensionality ofthe maximum likelihood (ML) estimator can be reduced by afactor of two. In [4] a time-domain procedure for estimationof delay-Doppler shifts and direction of arrival (DOA) isconsidered. Using prolate spheroidal wave (PSW) functions,the favorable transmit signal design with respect to time-delay accuracy is discussed in [6], while [7] considers sucha technique for joint delay-Doppler estimation. Recent resultsshow that for wide-band transmit signals, analog receive filterbandwidths which lead to violation of the sampling theoremcan provide performance gains [8], [9]. Further, in [10] theoptimization of receive filters in a compressed sensing frame-work has been investigated and improvements with respect tomatched filtering have been illustrated.Here we consider optimization of the transmit signal whilethe receiver samples at a rate f s smaller than the Nyquistrate B . After introducing the system model for a single-inputsingle-output (SISO) delay-Doppler channel, we derive a com-pact formulation of the transmit signal optimization problemin the frequency domain. We show how to solve the transmitterdesign problem for B > f s by an Eigenvalue decomposition.The potential Pareto-optimal region is visualized by optimizingthe transmit waveform for different settings and comparingthe results to conventional signal designs. We conclude thediscussion with a performance verification via Monte-Carlosimulations of a channel estimation algorithm.II. S YSTEM M ODEL
Consider the propagation of an analog, T -periodic pilotsignal ˘ x ( t ) ∈ C through a wireless delay-Doppler channel. Thebaseband signal at the receiver, which is perturbed by additivehite Gaussian noise (AWGN) ˘ η ( t ) ∈ C with constant powerspectral density N , can be denoted as ˘ y ( t ) = γ ˘ x ( t − τ )e j2 πνt + ˘ η ( t ) (1)with channel coefficient γ ∈ C , time-delay τ ∈ R and Dopplershift ν ∈ R . The signal ˘ y ( t ) ∈ C is filtered by a linear receivefilter h ( t ) ∈ C , such that the final analog receive signal y ( t ) = (cid:0) γ ˘ x ( t − τ )e j2 πνt + ˘ η ( t ) (cid:1) ∗ h ( t )= v ( t ; θ ) + η ( t ) (2)is obtained, where θ = (cid:0) τ ν (cid:1) T ∈ R denotes the unknown,random channel parameters. For the duration T , the signal y ( t ) ∈ C is sampled in intervals of T s = f s , resulting in aneven number of N = T T s ∈ N samples y = v ( θ ) + η , (3)with the receive vectors y , v ( θ ) , η ∈ C N defined as [ y ] i = y (cid:18)(cid:18) i − N − (cid:19) T s (cid:19) , (4) [ v ( θ )] i = v (cid:18)(cid:18) i − N − (cid:19) T s , θ (cid:19) , (5) [ η ] i = η (cid:18)(cid:18) i − N − (cid:19) T s (cid:19) . (6)We use positive integers as indices for vectors and matricesand thus i ∈ { , , . . . , N } . The noise samples η in (3) followa zero-mean Gaussian distribution with covariance matrix R η = E η [ ηη H ] ∈ C N × N . (7)Note that R η depends on the receive filter h ( t ) and thesampling rate f s and thus is not necessarily a scaled identitymatrix. The unknown parameters θ are considered to be Gaus-sian distributed p ( θ ) ∼ N ( , R θ ) with known covariance R θ = (cid:18) σ τ σ ν (cid:19) . (8)Here we assume that the channel γ is known at the re-ceiver, which simplifies the formulation of the transmit signaloptimization problem. However, when testing the optimizedwaveforms for a practical scenario in the last section we willtreat γ to be a deterministic unknown. For the derivation, wefirst assume a fixed sampling rate f s at the receiver while theperiodic transmit signal ˘ x ( t ) is band-limited with two-sidedbandwidth B . Then we consider the case of a variable rate f s . In contrast to the sampling theorem assumption B ≤ f s ,in our setup we allow B > f s . Note that at the receiver, wealways use an ideal low-pass filter h ( t ) featuring the samebandwidth B as the transmit signal.III. C HANNEL E STIMATION P ROBLEM
Under the assumption that γ is known, the task of thereceiver is to infer the unknown channel parameters θ basedon the digital receive data y using an appropriate channel estimation algorithm ˆ θ ( y ) . The mean squared error (MSE)of the estimator ˆ θ ( y ) is defined as R ǫ = E y , θ h(cid:0) ˆ θ ( y ) − θ (cid:1)(cid:0) ˆ θ ( y ) − θ (cid:1) T i . (9)A fundamental limit for the estimation accuracy (9) is the Bayesian Cram´er-Rao lower bound (BCRLB) [11, p. 5] R ǫ (cid:23) J − B , (10)where J B is the Bayesian information matrix (BIM) J B = J D + J P . (11)The first summand of the BIM (11) represents the expectedFisher information matrix (EFIM) J D = E θ (cid:2) J F ( θ ) (cid:3) , (12)with the Fisher information matrix (FIM) exhibiting entries [ J F ( θ )] ij = − E y | θ " ∂ ln p ( y | θ ) ∂ [ θ ] i ∂ [ θ ] j . (13)For the signal model (3), the FIM entries (13) are [ J F ( θ )] ij = 2Re ((cid:18) ∂ v ( θ ) ∂ [ θ ] i (cid:19) H R − η (cid:18) ∂ v ( θ ) ∂ [ θ ] j (cid:19)) . (14)The second summand in (11) denotes the prior informationmatrix (PIM) J P with entries [ J P ] ij = − E θ " ∂ ln p ( θ ) ∂ [ θ ] i ∂ [ θ ] j . (15)IV. T RANSMITTER O PTIMIZATION P ROBLEM
The design problem of finding a transmit signal ˘ x ⋆ ( t ) thatminimizes the MSE (9) of the estimation algorithm ˆ θ ( y ) undera particular positive semi-definite weighting M ∈ R × ,subject to a transmit power constraint P , can be phrased as ˘ x ⋆ ( t ) = arg min ˘ x ( t ) tr( M R ǫ ) , s . t . T Z T | ˘ x ( t ) | d t ≤ P. (16)Although the BCRLB (10) can be achieved with equality onlyunder special conditions [11, p. 5], it closely characterizesthe estimation performance trend (see Sec. VII-C). It is hencepossible to formulate (16) based on the BIM (11) ˘ x ⋆ ( t ) = arg min ˘ x ( t ) tr( M J − B ) , s . t . T Z T | ˘ x ( t ) | d t ≤ P. (17)In order to avoid optimization with respect to J − B in (17), weconsider an alternative maximization problem ˘ x ⋆ ( t ) = arg max ˘ x ( t ) tr( M ′ J B ) , s . t . T Z T | ˘ x ( t ) | d t ≤ P. (18)It can been shown that if ˘ x ⋆ ( t ) is a solution of the max-imization problem (18) with M ′ , there exists a weightingmatrix M (not necessarily equal to M ′ ) for which the originalminimization problem (17) has the same solution ˘ x ⋆ ( t ) [12].Since J P is independent of ˘ x ( t ) , (18) then simplifies to ˘ x ⋆ ( t ) = arg max ˘ x ( t ) tr( M ′ J D ) , s . t . T Z T | ˘ x ( t ) | d t ≤ P. (19). E STIMATION T HEORETIC P ERFORMANCE M EASURE
Solving the optimization problem (18) requires an analyticalcharacterization of the EFIM (12). A frequency-domain rep-resentation enables a compact notation of the receive signalmodel [9] and thus provides further insights on the FIMentries (14). Note that a frequency-domain approach naturallyembodies the bandwidth restriction required in practice bylimiting the number of Fourier coefficients.
A. Signal Frequency Domain Representation
Due to periodicity, the transmit waveform ˘ x ( t ) can berepresented by its Fourier series ˘ x ( t ) = K − X k = − K X k e j kω t , (20)where ω = πT = 2 πf and K = ⌈ πBω ⌉ ∈ N is the totalnumber of harmonics. X k denotes the k -th Fourier coefficientof the transmit signal. Inserting expression (20) into (2) andapplying the filtering operation in (2), we obtain v ( t ; θ ) = γ K − X k = − K X k (cid:16) e j kω ( t − τ ) e j2 πνt (cid:17) ∗ h ( t )= γ e j2 πνt K − X k = − K e j kω t e − j kω τ H ( kω + 2 πν ) X k , (21)where H ( ω ) is the Fourier transform of the receive filter h ( t ) .Evaluating v ( t ; θ ) at instants nT s , n = − N , . . . , N − yields v ( nT s ; θ ) = γ K − X k = − K e j2 πνnT s e j2 π knN e − j kω τ H ( kω + 2 πν ) X k = K − X k = − K [ C ( θ )] n + N +1 ,k + K +1 X k , (22)with the channel matrix C ( θ ) ∈ C N × K , defined by C ( θ ) = γ √ N D ( ν ) W H T ( τ ) H ( ν ) . (23)The indices of C ( θ ) in (22) stem from the fact that we usepositive integers as indices for vectors and matrices. Here D ( ν ) ∈ C N × N stands for a diagonal matrix [ D ( ν )] ii = e j2 π ( i − N − ) νT s , (24)which represents the Doppler frequency-shift. Further W ∈ C K × N is a tall discrete Fourier transform (DFT) matrix [ W ] ij = 1 √ N e − j2 π ( i − K − )( j − N − ) N , (25)and T ( τ ) ∈ C K × K denotes the diagonal time-delay matrix [ T ( τ )] ii = e − j ( i − K − ) ω τ . (26) The diagonal matrix H ( ν ) ∈ C K × K in (23) characterizes thefrequency shifted receive filter spectrum and has elements [ H ( ν )] ii = H (cid:18) i − K − (cid:19) ω + 2 πν ! . (27)Note that the channel matrix (23) describes the propagation of ˜ x through the channel and its transformation from the spectralto the time domain. Further note that the aliasing effect dueto bandwidths B higher than the sampling frequency f s isautomatically included by the wide IDFT matrix W H .Stacking the entries of v ( nT s ; θ ) (22) into one vector yields v ( θ ) = C ( θ ) ˜ x , (28)with the transmit filter spectrum vector ˜ x ∈ C K formed bythe Fourier coefficients [ ˜ x ] i = X i − K − . (29) B. Fisher Information of the Delay-Doppler Channel
In order to compute the FIM elements (14), it is necessary tocompute the derivatives of v ( θ ) with respect to the parameters θ . Using the frequency domain representation (28), we obtain ∂∂ [ θ ] i v ( θ ) = ∂ C ( θ ) ∂ [ θ ] i ˜ x . (30)The derivatives of the channel matrix are ∂ C ( θ ) ∂τ = γ √ N D ( ν ) W H ∂T ( τ ) H ( ν ) , (31) ∂ C ( θ ) ∂ν = γ √ N (cid:16) ∂D ( ν ) W H T ( τ ) H ( ν )+ D ( ν ) W H T ( τ ) ∂H ( ν ) (cid:17) , (32)with the partial derivatives [ ∂D ( ν )] ii = j2 π (cid:18) i − N − (cid:19) T s e j2 π ( i − N − ) νT s , (33) [ ∂T ( τ )] ii = − j (cid:18) i − K − (cid:19) ω e − j ( i − K − ) ω τ , (34) [ ∂H ( ν )] ii = ∂∂ν H (cid:18)(cid:18) i − K − (cid:19) ω + 2 πν (cid:19) . (35)Inserting (31) and (32) into (14), the FIM entries can beexpressed as quadratic terms [ J F ( θ )] ij = 2Re ( ˜ x H ∂ C H ( θ ) ∂ [ θ ] i R − η ∂ C ( θ ) ∂ [ θ ] j ˜ x ) . (36)The elements of the expected Fisher information matrix(EFIM) (12) are then obtained by [ J D ] ij = 2Re ( ˜ x H E θ " ∂ C H ( θ ) ∂ [ θ ] i R − η ∂ C ( θ ) ∂ [ θ ] j ˜ x ) = ˜ x H ( Γ ij + Γ ji ) ˜ x , (37)with the channel sensitivity matrix Γ ij ∈ C K × K Γ ij = E θ " ∂ C H ( θ ) ∂ [ θ ] i R − η ∂ C ( θ ) ∂ [ θ ] j . (38)I. T RANSMIT S IGNAL O PTIMIZATION
In the following we solve the transceiver design problem(18) using the EFIM expressions (37). With the frequencydomain representation (29) of the transmit signal, the opti-mization problem (18) becomes a maximization with respectto the transmit Fourier coefficients ˜ x ˜ x ⋆ = arg max ˜ x tr (cid:0) M ′ J D (cid:1) s . t . ˜ x H ˜ x ≤ P. (39)Expanding the trace operation, the objective function becomes tr (cid:0) M ′ J D (cid:1) = X i =1 2 X j =1 [ M ′ ] ji [ J D ] ij = ˜ x H Γ˜ x , (40)with the weighted channel sensitivity matrix Γ = X i =1 2 X j =1 [ M ′ ] ji ( Γ ij + Γ ji ) . (41)The solution to the problem (39) is the Eigenvector γ of thematrix Γ corresponding to its largest Eigenvalue.VII. R ESULTS
There exists a trade-off between the estimation of delay andDoppler-shift. By solving the optimization problem (39) for allpositive semi-definite weightings M ′ , we are able to approx-imate the Pareto-optimal region. This region is characterizedby the set of transmit waveforms for which the estimation ofone parameter cannot be improved by changing the transmitsignal without reducing the accuracy of the other parameter.For visualization, we define the relative measures χ τ/ν = 10 log (cid:2) J − D | ˜ x rect (cid:3) / (cid:2) J − D | ˜ x (cid:3) / ! , (42)with respect to a rectangular pulse ˜ x rect of bandwidth B = f s ,as it is used in Global Navigation Satellite Systems (GNSS).For the following results, the expectation (38) with respect to p ( θ ) is computed using Hermite-Gaussian quadrature . − − χ τ [dB] χ ν [ d B ] ρ = 1 ρ = 2 ρ = 1 ( ˜ x rect ) Fig. 1. Pareto regions for bandwidths B = ρf s with f s = 10MHz A. Pareto-Optimal Region - Fixed Sampling Rate
For a setting where T = 10 µ s , f s = 10MHz , σ ν = 5kHz and σ τ = 10ns , Fig. 1 shows the Pareto-optimal regions fordifferent bandwidths B = ρf s . Note that here for all systemsthe same sampling frequency f s has been used. The resultsindicate that a potential performance gain of roughly for delay estimation and for Doppler estimation can beobtained when optimizing the transmit system for ρ = 2 . Notethat when increasing the transceive bandwidth B from ρ = 1 to ρ = 2 , two main effects affect the estimation performance.First, a larger transmit bandwidth is beneficial for the delayestimation due to high-frequency signal parts. On the otherhand, a higher receive filter bandwidth results in a largernoise power at the receiver and therefore in a lower Dopplerestimation accuracy. However, the optimized system is ableto compensate this effect by efficiently using the availabletransmit spectrum, which leads to a moderate loss. B. Pareto-Optimal Region - Fixed Bandwidth
In the previous section, we have seen that optimized wave-forms have the potential to increase the accuracy of delay-Doppler estimation methods. We now investigate the estima-tion performance for a fixed transmit bandwidth B = 10MHz ,a signal period T = 10 µ s and different sampling frequencies f s = Bκ . In order to focus on the case with undersampling weconsider setups where κ > . Fig. 2 shows the Pareto regions − − ˜ x ⋆ χ τ [dB] χ ν [ d B ] κ = 1 κ = 2 κ = 4 κ = 1 ( ˜ x rect ) Fig. 2. Pareto regions for rates f s = Bκ with B = 10MHz of the optimized waveforms with respect to a rectangularsignal. Note that the sampling rate for the reference system isheld constant, while the sampling rate of the optimized systemdecreases with increasing κ . This indicates that although lowersampling rates are used, the optimized waveform design stillbears the potential to provide high estimation accuracy. C. Simulation Results
To verify that the optimization based on the EFIM yieldssubstantial performance gains for practical scenarios, we con-duct Monte-Carlo simulations with randomly generated noise η and channel parameters θ . As the channel γ is in general notnown to the receiver, we use the hybrid maximum likelihood-maximum a posteriori (ML-MAP) estimator [11, p. 12] (cid:18) ˆ γ ML ( y ) ˆ θ MAP ( y ) (cid:19) = arg max θ ,γ (cid:0) ln p ( y | θ , γ ) + ln p ( θ ) (cid:1) . (43)For simulations we use T = 10 µ s and B = 10MHz .We compare the MSE of a rectangular pulse signal with f s = 10MHz and the optimized transmit signal ˜ x ⋆ with f s = 5MHz , i.e., κ = 2 . The transmitter design ˜ x ⋆ used forthe simulations corresponds to the point of the Pareto-regionin Fig. 2 with largest distance to the origin. Fig. 3 and Fig. 4show the normalized empirical mean squared error (NMSE) NMSE ˆ τ/ ˆ ν = MSE ˆ τ/ ˆ ν σ τ/ν (44)of the hybrid ML-MAP estimator for both systems, where MSE ˆ τ/ ˆ ν represents the diagonal elements of (9), empiricallyevaluated based on the results of the estimation algorithm (43).The signal-to-noise ratio (SNR) is given by SNR =
PBN . (45)It is observed that for low SNR the MSE saturates at σ τ,ν , −
10 0 10 20 − −
100 SNR [dB] N M S E ˆ τ [ d B ] BCRLB τ ( ˜ x ⋆ )BCRLB τ ( ˜ x rect )NMSE ˆ τ ( ˜ x ⋆ )NMSE ˆ τ ( ˜ x rect ) Fig. 3. MSE and BCRLB - Time-delay τ −
10 0 10 20 − −
100 SNR [dB] N M S E ˆ ν [ d B ] BCRLB ν ( ˜ x ⋆ )BCRLB ν ( ˜ x rect )NMSE ˆ ν ( ˜ x ⋆ )NMSE ˆ ν ( ˜ x rect ) Fig. 4. MSE and BCRLB - Doppler-shift ν since in this case the estimation merely relies on the priorinformation p ( θ ) . In the high SNR regime, the MSE of thehybrid ML-MAP estimator shows close correspondence withthe BCRLB and the estimator benefits from the waveform op-timization. For moderate to high SNR values the performancegain is roughly . for the estimation of the time-delayand . for the Doppler-shift estimation. This correspondsto the findings from the Pareto-region in Fig. 2.VIII. C ONCLUSION
We have derived an optimization framework for the transmitwaveform of an undersampled pilot-based channel estimationsystem. By employing the BCRLB, the transmitter designproblem was reformulated as a maximization problem withrespect to the expected Fisher information matrix. A frequencydomain representation of the receive signal allows one tofind an analytical solution to the maximization problem viaan Eigenvalue decomposition. The BCRLB of the optimizedwaveforms can be used to approximately characterize thePareto-optimal design region with respect to other delay-Doppler estimation methods. Further, our results show thatusing optimized transmit waveforms enables the receiver tooperate significantly below the Nyquist sampling rate whilemaintaining high delay-Doppler estimation accuracy. Finally,Monte-Carlo simulations support the practical impact of theconsidered transmit design problem.R
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