Quantization-Loss Reduction for Signal Parameter Estimation

QUANTIZATION-LOSS REDUCTION FOR SIGNAL PARAMETER ESTIMATION

Manuel Stein, Friederike Wendler, Amine Mezghani, Josef A. Nossek

Institute for Circuit Theory and Signal Processing Technische Universitat¨ Munchen,¨ Germany

ABSTRACT front-end. Coarse signal quantization introduces nonlinear effects which have to be taken into account in order to obtain Using coarse resolution analog-to-digital conversion (ADC) optimum system performance. Therefore, adapting existent offers the possibility to reduce the complexity of digital methods and developing new estimation algorithms, operat- receive systems but introduces a loss in effective signal-to- ing on quantized data, becomes more and more important. noise ratio (SNR) when comparing to ideal receivers with infinite resolution ADC. Therefore, here the problem of sig- An early work on the subject of estimating unknown pa- nal parameter estimation from a coarsely quantized receive rameters based on quantized signals can be found in [1]. In [2] signal is considered. In order to increase the system perfor- [3] [4] the authors studied channel parameter estimation based mance, we propose to adjust the analog radio front-end to on a single-bit quantizer assuming uncorrelated noise. The the quantization device in order to reduce the quantization- problem of signal processing with low resolution has been loss. By optimizing the bandwidth of the analog filter with considered in another line of work concerned with data trans- respect to a weighted form of the Cramer-Rao´ lower bound mission. It turns out that the well known reduction of low (CRLB), we show that for low SNR and a 1-bit hard-limiting SNR channel capacity by factor 2/π (−1.96 dB) due to 1-bit device it is possible to significantly reduce the quantization- quantization [5] holds also for the general MIMO case with loss of initially -1.96 dB. As application, joint carrier-phase uncorrelated noise. On the other hand it was shown by [6], and time-delay estimation for satellite-based positioning and that the channel capacity loss of the AWGN channel due to synchronization is discussed. Simulations of the maximum- 1-bit quantization at the receiver can be reduced by oversam- likelihood estimator (MLE) show that the optimum estimator pling the analog receive signal. [7] observed the possibility achieves the same quantization-loss reduction as predicted by to improve the correlator output SNR of a quantized receiver the performance bound of the optimized system. by using high sampling frequencies and reducing the receiver Index Terms— analog-to-digital conversion, parameter bandwidth. Recently, [8] derived a lower bound on the ca- estimation, analog filtering, satellite navigation systems. pacity of quantized MIMO channels with noise correlations showing that the loss can be lower than with white noise. 1. INTRODUCTION To the best of our knowledge, estimating multiple param- eters from 1-bit quantized signals with colored noise is still Most state-of-the-art signal parameter estimation techniques not covered by previous works. We provide new results on assume that the receiver (or sensing device) has access to the low SNR estimation performance under 1-bit quantization digital signals with arbitrary high precision. In practice this and noise correlation. More precisely, we derive the Cramer-´ implies the existence of analog-to-digital conversion (ADC) Rao lower bound (CRLB) for the estimation performance in with a resolution which is large enough to neglect the effect of the presence of temporally colored noise within the low SNR amplitude quantization. The push toward high-speed process- regime, taking into account the effects of quantization. It is ing and acquisition renders the assumption of having available observed, that the popular -1.96 dB loss [9] [10] is not valid high-resolution ADC unrealistic or even invalid. In fact, fast in the presence of noise correlation. In fact, the loss can be and high resolution ADC is expensive, power consuming and smaller for certain favorable noise correlation. Therefore, we therefore not appropriate for portable devices. Coarse reso- present a new approach aiming at optimizing the analog front- lution ADC may be a cost and energy-effective solution for end to reduce the quantization-loss. Surprisingly, even adapt- such applications. In particular, 1-bit ADC meets the require- ing the bandwidth of the analog filter is sufficient to signifi- ments for energy efficiency and allows to use a simple analog cantly improve the performance of 1-bit receivers. Note that such analog adjustments are highly attractive from the digi- The authors would like to express their gratitude to the German Fed- eral Ministry of Economics and Technology for supporting this work through tal signal processing point of view as they do not require any grant 50NA1110. Contact authors: {Stein, Nossek}@nws.ei.tum.de extra computational complexity during system operation. 2. SYSTEM MODEL as figure-of-merit where W is a positive semi-definite matrix. Without direct access to R(θ) optimization of the receive For the discussion, we consider a band-limited receive signal system can be carried out with respect to a lower bound (one-sided bandwidth B) sampled at a rate of 1/Ts MSEW(θ) ≥ tr WF −1(θ) . (9) N q y = s(θ) + n ∈ C , (1) The matrix F (θ) is the so called FI matrix [11] with the N q where s(θ) ∈ C is a signal of known structure modulated property [12, p. 3] K through θ ∈ R , the signal parameter that has to be esti- N −1 mated. The vector n ∈ C is additive Gaussian noise with R(θ) F q (θ) (10) the covariance matrix entries for a receiver using q bits of resolution. For the considered R = 2BN sinc (2BT |i − j|) , ij 0 s (2) signal model (4) the entries of F q(θ) are given by where N0 is the spectral density of the noise. 0H 0 ∂s (θ) 0−1 ∂s (θ) F q,ij(θ) = 2 · Re R . (11) ∂θi ∂θj 2.1. Quantized Receivers For quantized receivers, we define the signal 4. 1-BIT HARD-LIMITING RECEIVER

q = Q(y), (3) For a hard-limiting ADC with 1-bit resolution the element- wise quantization operation Q(·) is defined by where for simplicity Q(·) is an element-wise operation, indi- vidually mapping each element yn ∈ C onto one point out of Q(x) = sign (Re {x}) + j sign (Im {x}). (12) a finite set Q ⊂ C. By the orthogonality principle [13, p.177] the output q of a general quantizer can be approximated [8] Fortunately, for this quantizer a low SNR gain-noise model [8] can be specified using the arcsine-law [13, p. 438] q = Gy + e = Gs(θ) + Gn + e = s0(θ) + n0, (4) 2 1 where R0 = arcsin R π 2BN0 H H −1 −1 r r G = E qy E yy = RqyRyy , (5) 2 1 G = I. (13) π 2BN and for scenarios with low SNR 0

0 0 0H H If the noise covariance R is a scaled identity matrix R = E n n ≈ E qq = Rqq. (6) H 1 ∂s (θ) −1 ∂s(θ) While (6) just specifies the second moment of the effective F 1-bit,ij(θ) = Re arcsin (I) noise probability density function, it can be shown that for the BN0 ∂θi ∂θj discussed signal model, multivariate additive Gaussian noise 2 = F ∞,ij(θ). (14) minimizes Fisher information (FI). Therefore, a nearly opti- π mum estimation is attained based on the distribution This verifies the established result [9] [10] that 1-bit hard- 1 limiting quantization at low SNR leads to a loss of p(q; θ) = exp −(q − s0(θ))H R0−1(q − s0(θ)). πN det R0 2 10 log = −1.96 dB (15) 3. SIGNAL PARAMETER ESTIMATION π in system performance when comparing to an ideal receive In the following the goal is to calculate an estimate θˆ(q) of the system with ADC of infinite resolution. signal parameter θ based on the quantized observation vector q. We restrict the discussion to unbiased estimators, define the conditional mean square error (MSE) matrix 5. SATELLITE-BASED POSITIONING

h ˆ ˆ T i R(θ) = Eq|θ (θ(q) − θ)(θ(q) − θ) , (7) In order to visualize the possible performance improvements, we use a GNSS signal parameter estimation scenario. Here and use the weighted sum of the individual estimation errors the analog receive signal is modeled by √ jφ MSEW(θ) = tr (WR(θ)) (8) y(t) = e Cx(t − τ) + n(t), (16) N×N with φ being a carrier-phase shift, C the carrier power and τ T (τ) ∈ C a diagonal matrix with entries a time-delay. As satellite signal GPS L1 C/A (satellite 1) [14] T (τ) = e− j(n−(N/2+1))ω0τ , (24) is used. For observation periods smaller than 20 ms [15] nn and x˜ the vector with the Fourier coefficients +∞ X " #T x(t) = bkh(t − kTc), (17) N N x˜ = X − ω0 ,...,X − 1 ω0 , (25) k=−∞ 2 2 with bk ∈ {−1, 1} being the k-th symbol of a periodic binary 2π ∂x(τ) with ω0 = . For the signal derivative such an ap- T0 ∂τ sequence b with 1023 elements and chip-duration of Tc = proach allows to write 977.52 ns such that x(t) repeats after T0 = 1 ms. The pulse ∂x(τ) " # = DT (τ)Ωx˜, (26) 1 Tc Tc ∂τ h(t) = √ Si 2πB t + − Si 2πB t − N×N π Tc 2 2 with Ω ∈ C being a diagonal matrix with entries is the band-limited version of a rectangular pulse filtered with Ωnn = − j ω0(n − (N/2 + 1)). (27) an ideal low-pass filter with bandwidth B where by definition Further, if N is sufficiently large any covariance matrix R has approximately the structure of a circulant Toeplitz matrix and Z t sin(x) Si(t) = dx. (18) is therefore diagonalized 0 x R ≈ DΛDH , (28) A discussion regarding quantized positioning with advanced N×N BOC pulse forms can be found in [16]. such that Λ ∈ R forms a diagonal matrix with entries 1 Λnn = ψ (n − (N/2 + 1))ω0 , (29) The GNSS signal spectrum can be written T0 X(ω) = B(ω)H(ω), (19) where ψ(ω) is the power-spectral density of the random Gaus- sian signal with covariance matrix R. For the information with B(ω) being the discrete spectrum of the satellite se- measure related to the phase-rotation φ quence b and H(ω) the Fourier transform of the pulse h(t). −1 C H 1 The sampled and quantized receive signal is F 1-bit,11 = · x (τ) arcsin R x(τ) BN0 2BN0 r r 2 C ≈ 2C · x˜H Λ−1 x˜ q = s0(θ) + n0 = ejφx(τ) + n0, (20) 1-bit π 2BN0 N −1 2 2 2C X |X(nω0)| T = = F1-bit,φ(B,Ts), (30) with θ = φ τ and xk(τ) = x(kTs − τ). T ψ (nω ) 0 N 1-bit 0 n=− 2 5.1. FI - Formulations and Asymptotic Analysis with k=∞ For the considered estimation problem the diagonal elements X − j ωkTs ψ1-bit(ω) = 2BN0Ts arcsin (sinc (2BkTs)) e . of the FI matrix F 1-bit(θ) are k=−∞ (31) C 1 −1 F = · xH (τ) arcsin R x(τ) 1-bit,11 Letting the time of one signal period go to infinity, T0 → ∞ BN0 2BN0 the FI asymptotically becomes C ∂xH (τ) 1 −1 ∂x(τ) F = · arcsin R , Z 2πB 2 1-bit,22 ¯ C |X(ω)| BN0 ∂τ 2BN0 ∂τ F1-bit,φ ≈ dω. (32) (21) π −2πB ψ1-bit(ω) For the FI measure of the time-delay τ and the off-diagonal elements F 1-bit,12 and F 1-bit,21 vanish. −1 Under the assumption that the band-limited signal x(t) is pe- C ∂xH (τ) 1 ∂x(τ) F 1-bit,22 = · arcsin R riodic the sampled signal vector x(τ) can be written BN0 ∂τ 2BN0 ∂τ H H −1 x(τ) = DT (τ)x˜, (22) ≈ 2C · x˜ Ω Λ1-bitΩx˜ N −1 2 2 2 D ∈ N×N 2C X (nω0) |X(nω0)| where C is a modified DFT matrix = = F1-bit,τ (B,Ts), T0 ψ1-bit(nω0) (n−(N/2+1))(i−(N/2+1)) n=− N 1 j 2π 2 Dni = √ e N , (23) N (33) with the asymptotic form is attained by plotting the values NMSEφ and NMSEτ after solving Z 2πB 2 2 ¯ C ω |X(ω)| F1-bit,τ ≈ dω. (34) ? π −2πB ψ1-bit(ω) ρ = arg min NMSEW(ρ, µ) s.t. 0 ≤ ρ ≤ 1 (39) ρ

5.2. System Optimization - Bandwidth and Sampling for fixed µ and different weightings α. It can be observed that without oversampling a compromise between the two pa- For system optimization we choose N = 2046, B = 1 , Tc rameters has to be made. Increasing for example the perfor- T = Tc and use an ideal system with infinite resolution as s 2 mance of the carrier-phase measurement results in a perfor- reference in order to formulate a normalized version of (9) mance loss for the time-delay. This is not the case with oversampling where the highest performance gain is attained by NMSEW(ρ, µ) ≥ α · NMSE (ρ, µ) + (1 − α) · NMSE (ρ, µ), φ τ doubling the sampling rate of the receiver. Further increasing (35) the sampling rate only gives marginal additional performance. with Also the performance gains for the maximum-likelihood estimator (MLE) on different points of the feasibility regions F∞,φ/τ (B,Ts) are plotted in Fig. 2. It can be observed that the estimator NMSEφ/τ (ρ, µ) = , (36) Ts achieves the same improvement through the analog front-end F1-bit,φ/τ (ρB, ) µ design as predicted by the optimized performance bound. where the bandwidth fraction satisfies 0 ≤ ρ ≤ 1 and the oversampling factor µ ≥ 1. Note, that with infinite resolution 1-bit (ρ = 1, µ = 1) 2.2 ? Ts 1-bit (ρ , µ = 1) F (ρB, ) ≤ F (B,T ) (37) ? ∞,φ/τ µ ∞,φ/τ s 1-bit (ρ , µ = 2) 2 1-bit (ρ?, µ ≈ ∞) for ρ ≤ 1, µ ≥ 1 which does not hold for the 1-bit case. In MLE Simulations fact, Fig. 1 visualizes the individual quantization-loss τ 1.8

χ (ρ, µ) = −10 log (ρ, µ) NMSE φ/τ NMSEφ/τ (38) 1.6 with respect to the bandwidth ρ. Without oversampling, i.e. 1.4

−1 1.2 1.2 1.4 1.6 1.8 2 2.2

−1.5 NMSEφ Fig. 2. Feasibility Region - Normalized MSE −2

in dB χφ(ρ, 1)

χ χ (ρ, 2) −2.5 φ χφ(ρ, ∞) 6. CONCLUSION χτ (ρ, 1) −3 χτ (ρ, 2) Simple ADC allows to reduce the complexity of analog re- χτ (ρ, ∞) ceive systems and to simplify digital signal processing. Here −3.5 it was proposed to optimize the analog radio front-end with 0.6 0.7 0.8 0.9 1 ρ respect to a theoretic performance measure behind an quantization device. For the extreme case of 1-bit quantization it Fig. 1. 1-bit Quantization-loss versus bandwidth ρ was shown that the well-known −1.96 dB performance loss is not valid in the context of low SNR signal parameter esti- µ = 1, the best performance (−1.6 dB) in φ is attained with a mation if appropriate noise correlation is present. Here these bandwidth fraction of ρ = 0.68. For τ the best conﬁguration correlations were produced in a controlled fashion through (−1.81 dB) is ρ = 0.86. With oversampling the tendency is the analog preprocessing chain leading to signiﬁcantly higher to use the full bandwidth (ρ = 1). Fig. 2 shows the feasi- system performances. In practice these improvements are bility region for different oversampling factors µ ≥ 1 which achieved by calculation of the optimum estimator. 7. REFERENCES [14] ”NAVSTAR GPS Space Segment/Navigation User In- terfaces,” IS-GPS-200, Rev. D, ARINC Engineering [1] R. Curry, ”Estimation and Control with Quantized Mea- Services, El Segundo, CA, 2004. surements”, M.I.T Press, 1970. [15] G. Seco-Granados, J.A. Lopez-Salcedo, D. Jimenez- [2] T. M. Lok, V. K. W. Wei, ”Channel Estimation with Banos, G. Lopez-Risueno, ”Challenges in Indoor Global Quantized Observations”, IEEE International Sympo- Navigation Satellite Systems: Unveiling its core fea- sium on Information Theory, pp. 333, August 1998. tures in signal processing”, IEEE Signal Processing Magazine, Vol. 29, No. 2, pp. 108 –131, 2012. [3] M. T. Ivrlac,ˇ J. A. Nossek, ”On Channel Estimation in Quantized MIMO Systems”, International ITG Work- [16] F. Wendler, M. Stein, A. Mezghani, J.A. Nossek, shop on Smart Antennas (WSA), 2007. ”Quantization-loss reduction for 1-bit BOC positioning”, Proceedings of the ION International Technical [4] A. Mezghani, F. Antreich, J.A. Nossek, ”Multiple pa- Meeting, January 2013. rameter estimation with quantized channel output”, In- ternational ITG Workshop on Smart Antennas (WSA), pp. 143 –150, 2010.

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