In this chapter, the basic model for direction finding in Chapter 3.1 and in [5] is adopted to describe the incoming signals, the array geometry and the redundant dictionary, D ∈ CL×N, containing the L-dimensional (complex-valued) array steering vectors for
N specific angles. It is assumed that there exist K point-like sources in the farfield, that emit monochromatic signals of wavelength λ0 and circular-symmetric complex
Gaussian distributed amplitudes with variance σk2, uk(t) ∼ CN (0, σk2), k = 1, . . . , K,
where t is the time index. A number of T snapshots are observed at time instances tν, ν = 1, . . . , T , and collected in a matrix Y = [y(t1), . . . , y(tT)]. For each snapshot, a
sparse vector, x(tν), and a noise vector, n(tν), are defined and collected in the matrices
X = [x(t1), . . . , x(tT)] ∈ CN ×T and N = [n(t1), . . . , n(tT)] ∈ CL×T, respectively. The
noise components are assumed to be zero-mean, circular-symmetric complex Gaussian distributed with variance σ2
n, i.e. nl(tν) ∼ CN (0, σn2), l = 1, ..., L, ν = 1, ..., T . Given
the dictionary, D, the true DOAs of the signals are assumed to appear at the same dictionary indices for all snapshots. These indices are collected in the set of significant sparse coefficients, S = {s1, . . . , sK}, with cardinality |S| = K. Ideally, the non-zero
components in x(tν) are equal to the signal amplitudes, i.e. xsk(tν) = uk∀ ν = 1, . . . , T ,
k = 1, . . . , K. Model errors due to gain and phase mismatches are considered in terms of uncertainty in the dictionary. The true (but unknown) dictionary is denoted by
˜
D = D + ∆D , (4.1)
where D is the dictionary derived under ideal conditions and ∆D is a correction term due to model errors. The extended signal model for T snapshots, including uncertainty, is given by
4.3 Problem Statement 33
It is assumed that the measurement noise, N, the source signals in X, and the model errors, ∆D, are mutually independent. The goal is to estimate the DOAs of the incom- ing signals at high resolution by solving a particular instance of the `1- minimization
problem [5], i.e. ˆ x = arg min x∈CN kx (`2)k 1 s.t. kY − DXk22 ≤ βreg, (4.3) where ∀ n = 1, . . . , N and ∀ ν = 1, . . . , T : x(`2) n = k [xn(t1), . . . , xn(tT)]>k2, (4.4) |xn(tν)| = p Re{xn(tν)}2+ Im{xn(tν)}2. (4.5)
Taking into account that the data is complex-valued, Equation (4.3) can be be re- written as a second-order cone program [5]:
min qaux s.t. 1>raux ≤ qaux, Y − DX (βreg− 1)/2 2 ≤ (βreg− 1)/2 , x(`2) n ≤ rn,aux, n = 1, . . . , N, (4.6)
where qaux and raux are auxiliary parameters. Efficient solvers such as SeDuMi [125] can
be used to solve Problem (4.6). However, this solution is sensitive to the regularization parameter, βreg, especially if the observed data cannot be exactly described by the
model due to model errors and uncertainty in the dictionary. Therefore, an appropriate sparse regularization method has to account for these effects. Later on, dictionary calibration is used to obtain a modified and calibrated dictionary, that better mimics the observed data in order to alleviate the regularization problem.
4.3.1
Probabilistic Models for Array Imperfections
Typical problems in practical systems are errors in the sensor gains and in the phase of the sensor signals. They may arise, for example, from inaccuracies in the fabrication process or external effects such as temperature fluctuations or erosion. Figure 4.1 shows a typical antenna pattern of an ideal and an impaired uniform linear array. It depicts the sensitivity of the array in dependence of the azimuth angle, ϑ. Two particular types of model errors are considered: sensor position errors and general gain and phase errors. For each type, a probabilistic model is developed.
The presented models have been previously published in [67]4, [66]5, [68]6. 4.3.1.1 Sensor Position Errors
Sensor position errors cause inaccuracies in the expected time delays, which leads to mismatches in the signal phase. The uniform linear array is assumed to be aligned with the x-axis and only errors in x-direction are considered. According to the model in Chapter 3.1, the ideal sensor spacing is given by ∆d = λ0/2. The position error,
δd, should be smaller than half of the sensor spacing, i.e. δd ≤ ∆d/2. Using (3.2), the corresponding absolute phase error at the l-th sensor, l = 1, . . . , L, for a signal with DOA ϑ is given by |∆φl(ϑ)| = ω0 cp [ (l − 1)∆d + δd ]cos(ϑ) − ω0 cp [ (l − 1)∆d ]cos(ϑ) = ω0 cp δd cos(ϑ) ≤ ω0 cp ∆d 2 = ω0 cp λ0 4 = π 2. (4.7)
For simplicity, it is assumed that all phase errors, ∆φl(ϑ), can be described by general i.i.d. zero-mean, Gaussian distributed variables with variance σ2
φ, i.e. ∆φ ∼ N (0, σφ2)
for any DOA ϑ ∈ [0◦, 180◦] and any index l = 1, . . . , L. Given the relation in (4.7), the standard deviation of the phase error can be defined by σφ = pφπ2, where pφ is a
parameter that controls the error.
4.3.1.2 General Gain and Phase Errors
In order to model general gain and phase errors, a complex-valued gain variable can be introduced for each sensor. It is assumed that mutual coupling between the sensors can be neglected, and that the complex-valued gain errors can be modeled as i.i.d. zero- mean, circular symmetric complex Gaussian random variables with variance σ2
g, i.e.
˜
gl ∼ CN (0, σg2), l = 1, ..., L. For each sensor, the ideal gain value in absence of phase
errors is denoted by g0 ∈ R+. Similar to the case of sensor position errors, the standard
deviation can be defined by σg = pgσg,max, where it is assumed that 0 ≤ pg ≤ 1 and
σg,max = g0/2.
4C. Weiss and A. M. Zoubir, “DOA Estimation in the Presence of Array Imperfections: A Sparse
Regularization Parameter Selection Problem,” in Proc. of the IEEE Workshop on Statistical Signal Processing (SSP), Gold Coast, Australia, June/July, 2014.
5C. Weiss A. M. Zoubir, “A Sparse Regularization Technique for Source Localization with Non-
uniform Sensor Gain,” in Proc. of the IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM), A Coru˜na, Spain, June, 2014.
6C. Weiss and A. M. Zoubir, “Robust High-Resolution DOA Estimation with Array Pre-
Calibration,” in Proc. of the 22nd European Signal Processing Conference (EUSIPCO), Lisbon, Portugal, September, 2014.
4.4 Sparse Regularization for Direction Finding With Sensor Position Errors 35