DOA Parameter Estimation with 1-Bit Quantization Bounds, Methods And

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Josef and cnsh Universität Germany München, echnische ,VieUiesti rse,Belgium Brussel, Universiteit Vrije ), 1 btsse n nielrcie ihinfinite with receiver ideal an and system -bit 1 [email protected] btQuantization -bit ∞ 1 btACrslto hsi associated is this resolution ADC -bit btls 6.T nlz hseffect, this analyze To [6]. loss -bit 1 btaiheis[] Therefore, [2]. arithmetics -bit 2 /π or 1 btqatzto.This quantization. -bit − 1 1 . btACi hta that is ADC -bit 96 B ntelow the in dB) with rto er near ion, tant nal rd- ter on ce ts. ce 1 of at al if o h c e - - l II. MOTIVATION B. Low-Complexity Receiver (1-bit ADC) In order to clearly outline the estimation theoretic motiva- The situation changes fundamentally if a nonlinear transfor- tion behind the presented work, we review efficient covariance- mation based estimation with infinite ADC resolution and outline the z = f(y) (10) challenges arising when treating a receiver with 1-bit ADC. is involved. Then the derivation of an exact representation of A. Ideal Receiver (∞-bit ADC) the likelihood p(z; θ), and therefore processing the data with the maximum-likelihood approach (5), can become difficult. If RM Consider a digital receive signal y ∈ which is well we assume 1-bit ADC and model the converters by an element- represented by a multivariate Gaussian random variable with wise hard-limiter which discards the amplitude information probability density function z = sign (y), (11) 1 1 TΣ−1 p(y; θ)= N exp − y y (θ)y (1) 2 Σ 2 where the element-wise signum function is defined by (2π) det y(θ) with a single parameterp θ ∈ Θ modulating the covariance +1 if xn ≥ 0 [sign (x)]n = (12) (−1 if xn < 0, T Σy(θ)=Ey;θ yy (2) then the likelihood function for one output constellation is and vanishing mean found by evaluating the integral

Ey;θ [y]= 0, ∀θ. (3) p(z; θ)= p(y; θ)dy. (13) ZY(z) Given N independent data snapshots Note, that Y(z) is the subset of Y which is mapped to the output signal z. Computing such an integral requires the y y y RM×N Y = 1 2 ... N ∈ , (4) orthant probability of a multivariate Gaussian variable (multivariate version of the Q-function). Unfortunately, a general the optimum asymptotically unbiased estimator θˆ(Y ) is ob- closed-form expression for the orthant probability is an open tained by maximizing the likelihood [9] mathematical problem. Only for the cases M ≤ 4 solutions θˆ(Y ) = argmax ln p(Y ; θ) are provided in literature [10] [11]. The problem becomes θ∈Θ even worse, if one is interested in analytically evaluating N the estimation performance of the 1-bit receive system. The = argmax ln p(yn; θ) associated Fisher information measure θ∈Θ n=1 2 X Σ Σ¯ Σ−1 ∂ ln p(z; θ) = arg min ln det y(θ) + Tr y(Y ) y (θ) , Fz(θ)= dz θ∈Θ Z ∂θ (5) Z ∂ ln p(z; θ) 2 = (14) where the receive covariance is given by ∂θ Z X N is computed by summing the squared score function over the Σ¯ 1 T y(Y )= ynyn . (6) Z M N discrete support of z. As contains 2 possible receive n=1 X constellations, direct computation of Fz(θ) is prohibitively As the maximum-likelihood estimator is consistent and effi- complex when M is large. cient, it is possible to characterize its asymptotic performance III. RELATED WORK AND OUTLINE in an analytical way through the Cramér-Rao lower bound [9] Due to the outlined problems (13) and (14), the literature 2 1 on analytic performance bounds and maximum-likelihood EY ;θ θ − θˆ(Y ) ≥ , (7) NFy(θ) algorithms for parametric covariance estimation with 1-bit h i quantization is limited. While [5] [3] are classical references where the Fisher information is defined for signal processing with 1-bit quantizer, more recently ∂ ln p(y; θ) 2 [2] covers the problem of signal parameter estimation from Fy(θ)= dy. (8) coarsely quantized data with uncorrelated noise. The work Y ∂θ Z [12] is concerned with 1-bit DOA estimation, but has to For the multivariate Gaussian model (1), we obtain [9, p. 47] restrict the analytical discussion to K =2 sensors due to the outlined problem (14) and resort to empirical methods of high 1 ∂Σ (θ) ∂Σ (θ) F (θ)= Tr Σ−1(θ) y Σ−1(θ) y . (9) computational complexity for K > 2. In contrast [13] studies y 2 y ∂θ y ∂θ location parameter estimation with a multivariate model and dithered 1-bit sampling while [14] discusses inference of the V. 1-BIT DOA ESTIMATION - SYSTEM MODEL autocorrelation function from hard-limited Gaussian signals. For the application of these results to blind DOA parameter Here we review the approach of exponential replacement estimation with 1-bit ADC, we assume a uniform linear which forms the basis for a generalized Fisher information array (ULA) with K sensors, where the spacing between the lower bound [8]. For the application of DOA estimation antennas is equal to half the wavelength. With a signal source with 1-bit quantized data this allows to derive a pessimistic T 2 Cramèr-Rao performance bound and to analyze the achievable x = xI xQ ∈ R (24) estimation accuracy with an arbitrary number of sensors K. Further, the exponential replacement enables to state a consisting of independent zero-mean Gaussian in-phase and conservative version of the maximum-likelihood estimator quadrature components with unit covariance which asymptotically performs equivalent to the presented T Ex xx = I (25) pessimistic performance bound [8]. 2 and under a narrowband assumption, the unquantized receive IV. THE EXPONENTIAL REPLACEMENT signal of size M =2K Consider an intractable parametric probabilistic model T T T RM p(z; θ). Choose the vector y = yI yQ ∈ , (26) T φ(z)= φ1(z) φ2(z) ... φL(z) (15) can be written in a real-valued notation [15] to be a set of L arbitrary transformations y = γA(θ)x + η, (27) RM R φl(z): → (16) where θ is the direction under which the transmit signal x im- RM of the output variable z and assume existence and access to pinges on the receive array. Note, that η ∈ is independent the two moments zero-mean additive Gaussian noise with unit variance T Eη ηη = IM . (28) µφ(θ)=Ez;θ [φ(z)] (17) and The full array steering matrix [15] T T T T T RM×2 Rφ(θ)=Ez;θ φ(z)φ (z) − µφ(θ)µφ(θ). (18) A(θ)= AI (θ) AQ(θ) ∈ , (29) Then replace the originalh model pi(z; θ) by an equivalent is modulated by the DOA parameter θ ∈ R and consists of an model p˜(z; θ) within the exponential family, i.e., a model with in-phase steering matrix sufficient statistics φ(z) and equivalent moments (17) and (18) ξ (θ) ψ (θ) for which the score function factorizes 1 1 ξ (θ) ψ (θ) 2 2 RK×2 ∂ lnp ˜(z; θ) T AI (θ)=  . .  ∈ (30) = β (θ)φ(z) − α(θ). (19) . . ∂θ   ξK (θ) ψK (θ) After optimizing the weights β RL and R it can   (θ) ∈ α(θ) ∈   be shown that the Fisher information measure of the original and a quadrature steering matrix model p(z; θ) is in general lower bounded by [8] −ψ1(θ) ξ1(θ) T ∂µ (θ) ∂µ (θ) −ψ2(θ) ξ2(θ) φ −1 φ   RK×2 Fz(θ) ≥ Rφ (θ) . (20) AQ(θ)= . . ∈ , (31) ∂θ ∂θ . .   Further, with independent output samples −ψK(θ) ξK (θ) N     Z = z1 z2 ... zN , (21) with entries forming the sample mean ξk(θ) = cos (k − 1)π sin (θ) N ψ (θ) = sin (k − 1)π sin (θ) . (32) 1 k φ˜ = φ(z ) (22) N n Therefore, the parametric covariance of the receive signal is n=1 X T and solving the equation Ey;θ yy = Σy(θ) T 2A AT I (33) ∂µφ(θ) − = γ (θ) (θ)+ 2K R 1(θ) φ˜ − µ (θ) =0 (23) ∂θ φ φ and the 1-bit receive signal can be modeled ˆ for θ, results in a consistent estimate θ(Z) with asymptotic z = sign {y} (34) variance equal to the inverse of N times the right-hand side of the Fisher information bound (20) [8]. as discussed in section II. VI. 1-BIT DOA ESTIMATION - PERFORMANCE ANALYSIS For the second moment of the auxiliary statistics (18)

A. 1-bit Exponential Replacement T T T T Ez;θ φ(z)φ (z) =Ez;θ vech zz vech zz (45) In order to apply the pessimistic approximation of the Fisher h i h i information (20) for DOA estimation after 1-bit hard-limiting, is required. This implies to evaluate the expected value we use the auxiliary statistics E[zizjzkzl] , i,j,k,l ∈{1,...,M}. (46) φ(z) = vech zzT , (35) For the cases i = j = k = l or i = j 6= k = l, we obtain where vech (B) denotes the half-vectorization of the symmet- E[z z z z ]=E z4 ric matrix B, i.e., the vectorization of the lower triangular part i j k l i 2 2 of B. The required mean (17) is given by =E zi zk =1. (47) µφ(θ)=Ez;θ [φ(z)] T If i = j = k 6= l, the arcsine law results in =Ez;θ vech zz T 3 = vech Ez;θ zz E[zizj zkzl]=E zi zl Σ = vech ( z(θ)) , (36) = E [zizl] 2 where by the arcsine law [16, pp. 284] = arcsin (ρ (θ)) , (48) π il 2 1 Σ (θ)= arcsin Σ (θ) . (37) like in the case i = j 6= k 6= l, where z π γ2 +1 y 2 For the derivative of the mean we ﬁnd E[zizjzkzl]=E zi zkzl = E [zkzl] ∂µφ(θ) ∂Σz(θ) = vech , (38) 2 ∂θ ∂θ = arcsin (ρ (θ)) . (49) π kl where the derivative of the quantized covariance matrix is The case i 6= j 6= k 6= l requires special care, as Σ 2 ∂ y (θ) ∂Σ (θ) ∂θ E[zizj zkzl]=Pr {zizjzkzl =1} z = ij (39) h i 2 ∂θ ij 2 1 Σ − Pr {zizjzkzl = −1} (50) π(γ + 1) 1 − (γ2+1)2 [ y(θ)]ij 4 with the derivative of the unquantizedq covariance being involves the evaluation of the 2 = 16 orthant probabilities ¯ T Φq(Σ(θ)) = Pr {±zi > 0, ±zj > 0, ±zk > 0, ±zl > 0} ∂Σy(θ) ∂A(θ) ∂A (θ) = γ2 AT(θ)+ A(θ) . (40) (51) ∂θ ∂θ ∂θ ! of a quadrivariate Gaussian variable with correlation matrix The derivative of the steering matrix is 1 ρij (θ) ρik(θ) ρil(θ) T T T ∂A(θ) ∂A θ ∂A (θ) = I ( ) Q , (41) ¯ ρij (θ) 1 ρjk(θ) ρjl(θ) ∂θ ∂θ ∂θ Σ(θ)=   . (52) ρik(θ) ρjk(θ) 1 ρkl(θ) with the in-phase componenth i ρil(θ) ρjl(θ) ρkl(θ) 1    ∂ξ1(θ) ∂ψ1(θ)   ∂θ ∂θ A closed-form solution for this problem, requiring calculation ∂ξ2(θ) ∂ψ2(θ) of four one-dimensional integrals, is given in [11]. ∂AI (θ)  ∂θ ∂θ  = . . (42) ∂θ . .  . .  B. 1-bit Quantization Loss  ∂ξK (θ) ∂ψK (θ)    Using result (20), we can evaluate the quantization loss for  ∂θ ∂θ  and a quadrature component  DOA parameter estimation for receivers with K > 2 in a pessimistic manner by forming the ratio − ∂ψ1(θ) ∂ξ1(θ) ∂θ ∂θ ∂µφ(θ) T −1 ∂µφ(θ) ∂ψ2(θ) ∂ξ2(θ) ∂θ Rφ (θ) ∂θ ∂AQ(θ)  − ∂θ ∂θ  χ(θ)= . (53) = . . , (43) ∂θ . . Fy(θ)  . .   ∂ψK (θ) ∂ξK (θ)  In Fig. 1 we plot the performance loss (53) versus the signal- −   ∂θ ∂θ  to-noise ratio while the individual entries are  SNR = γ2 (54) ∂ξ (θ) k = −(k − 1)π cos(θ) sin (k − 1)π sin (θ) ∂θ for two different DOA setups (θ = 10◦ and θ = 70◦). It can ∂ψk(θ) be observed that the quantization loss χ(θ) becomes smaller = (k − 1)π cos(θ)cos (k − 1)π sin (θ) . (44) ∂θ for arrays with a larger number of antennas K. In particular, −2 −1

−2 −4 in dB ◦ in dB −3 ) K = 2 (θ = 10 ) ) θ ◦ θ ( K = 2 (θ = 70 ) ( χ χ −6 K = 4 (θ = 10◦) θ =0◦ (SNR = −3 dB) K = 4 (θ = 70◦) −4 θ =0◦ (SNR = −15 dB) K = 8 (θ = 10◦) θ = 45◦ (SNR = −3 dB) K = 8 (θ = 70◦) θ = 45◦ (SNR = −15 dB) −8 −5 −20 −15 −10 −5 0 5 10 5 10 15 20 25 30 35 SNR in dB K

Fig. 1. 1-bit DOA Estimation - Quantization Loss vs. SNR Fig. 3. 1-bit DOA Estimation - Quantization Loss vs. Array Elements the array size plays a beneﬁcial role in the SNR range of −10 C. 1-bit CMLE Algorithm to 0 dB, which is a regime of high practical relevance for In order to demonstrate that the framework of exponential energy-efﬁcient broadband mobile communication systems. replacement also provides a useful guideline how to achieve However, for situations where SNR > 5 dB the quantization the guaranteed performance loss becomes pronounced. In Fig. 2 the performance loss is depicted as a function of the DOA parameter θ for three 2 1 EZ;θ θ − θˆ(Z) ≈ different array sizes (K = 2, 4, 8). It is visible that the ∂µ (θ) T ∂µ (θ) N φ R−1(θ) φ h i ∂θ φ ∂θ −2 = PCRLB , (55)

in Fig. 4 we plot the accuracy (RMSE) of the conservative −3 maximum-likelihood estimation (CMLE) algorithm (23) for an array size of K =4, a DOA parameter θ =5◦ and N = 1000 samples averaged over 10000 runs. It can be observed that the

in dB −4 −2 ) ·10 θ ( χ 1.4 RMSE −5 PCRLB

1.2 K =2 K =4 K =8 ◦ −6 0 20 40 60 80 1 θ in ◦ RMSE in Fig. 2. 1-bit DOA Estimation - Quantization Loss vs. DOA (SNR= 0 dB) 0.8 quantization loss χ(θ) becomes less dependent on the DOA parameter θ for large arrays. Finally, in Fig. 3 the quantization 0.6 loss χ(θ) is shown for a growing number of antennas K. For a low SNR setup (SNR = −15 dB) the gap between −6 −5 −4 −3 −2 −1 0 the quantized receiver and the unquantized receiver vanishes SNR in dB linearly with the array size K, while for a medium SNR ◦ scenario (SNR = −3 dB) the relative performance of the 1-bit Fig. 4. 1-bit DOA Estimation - CMLE Performance (K = 4, θ = 5 ) receive system strongly improves by increasing the amount of receive sensors K. CMLE performs close to the pessimistic version of the CRLB. VII. CONCLUSION [6] M. Stein, S. Theiler, and J. A. Nossek, “Overdemodulation for high- performance receivers with low-resolution ADC,” IEEE Wireless Com- We have discussed the method of exponential replacement mun. Lett., vol. 4, no. 2, pp. 169–172, Apr. 2015. [8] in the context of 1-bit DOA estimation with a single [7] M. Stein, A. Mezghani, and J. A. Nossek, “A lower bound for the Fisher signal source and a receive array of K sensors. The associated information measure,” IEEE Signal Process. Lett., vol. 21, no. 7, pp. 796–799, July 2014. pessimistic approximation for the Fisher information measure [8] M. Stein, J. A. Nossek, and K. Barbé, “Fisher information allows to analyze the achievable DOA estimation accuracy for bounds with applications in nonlinear learning, compression arrays with K > 2 in a conservative manner. Additionally, the and inference,” submitted to IEEE Trans. Signal Process., 2015, http://arxiv.org/abs/1512.03473. framework provides a guideline how to achieve the accuracy [9] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation guaranteed by the pessimistic CRLB. The performance analy- Theory. Upper Saddler River, NJ: Prentice Hall, 1993. sis shows that in the medium SNR regime DOA estimation [10] B. Kedem, Binary Time Series (Lecture Notes in Pure and Applied Mathematics). New York: Marcel Dekker, 1980. with 1-bit ADC can be performed at high accuracy if the [11] M. Sinn and K. Keller, “Covariances of zero crossings in Gaussian number of array elements K is large. This result supports processes,” Theory Probab. Appl., vol. 55, no. 3, pp. 485–504, 2011. the current discussion on future wireless systems which use [12] O. Bar-Shalom and A. J. Weiss, “DOA estimation using one-bit quantized measurements,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. a large number of low-complexity sensors, i.e., 1-bit massive 3, pp. 868–884, 2002. MIMO communication systems [17] [18]. [13] O. Dabeer and E. Masry, “Multivariate signal parameter estimation under dependent noise from 1-bit dithered quantized data,” IEEE Trans. Inf. REFERENCES Theory, vol. 54, no. 4, pp.1637–1654, 2008. [14] G. Jacovitti and A. Neri, “Estimation of the autocorrelation function of [1] R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE complex Gaussian stationary processes by amplitude clipped signals,” J. Sel. Areas Commun., vol. 17, no. 4, pp. 539–550, 1999. IEEE Trans. Inf. Theory, vol. 40, no. 1, pp. 239 – 245 , 1994. [2] A. Host-Madsen and P. Handel, “Effects of sampling and quantization [15] M. Stein, M. Castañeda, and J. A. Nossek, “Information-preserving spa- on single-tone frequency estimation,” IEEE Trans. Signal Process., vol. tial filtering for direction-of-arrival estimation,” 18th Int. ITG Workshop 48, no. 3, pp. 650–662, 2000. on Smart Antennas (WSA), 2014. [3] J. H. Van Vleck and D. Middleton, “The spectrum of clipped noise,” [16] J. B. Thomas, Introduction to Statistical Communication Theory. Hobo- Proc. IEEE, vol. 54, no. 1, pp. 2–19, 1966. ken, NJ: John Wiley & Sons, 1969. [4] M. Stein, A. Kürzl, A. Mezghani, and J. A. Nossek, “Asymptotic Param- [17] C. Risi, D. Persson, and E. G. Larsson, “Massive MIMO with 1-bit eter Tracking Performance with Measurement Data of 1-bit Resolution,” ADC,” preprint: http://arxiv.org/abs/1404.7736, 2014. IEEE Trans. Signal Process., vol. 63, no. 22, pp. 6086 – 6095, Nov. [18] S. Jacobsson, G. Durisi, M. Coldrey, U. Gustavsson, and C. Studer, 2015. “One-bit massive MIMO: Channel estimation and high-order modula- [5] W. R. Bennett, “Spectra of quantized signals”, Bell Syst. Tech. J., vol. tions,” preprint: http://arxiv.org/abs/1504.04540, 2015. 27, pp. 446–472, 1948.