Trade-offs among signal correlation, SNR, and number of Trade-offs among signal correlation, SNR, and number of snapshots for direction of arrival estimation using sparse arrays snapshots for direction of arrival estimation using sparse arrays

Arrays can be designed as a full array of sensors or a sparse array of sensors with an equivalent aperture. Reductions in cost, size, weight and power consumption are important factors to consider when designing an array of sensors. Sparse arrays can provide a reduction in these factors. It is important to observe the performance of sparse arrays in relation to full arrays to see if they are able to achieve similar performance. The ability for the full and sparse arrays of sensors to accurately estimate the direction of arrival (DOA) of plane waves impinging them is observed in this thesis. The Mean-squared error (MSE) of the DOA estimation is measured against the Creamer-Rao bound (CRB) as a benchmark for optimal performance for an unbiased estimator. We derive an optimal way to remove sensors for one signal impinging on an array by ﬁnding the minimum CRB. This method considers all the unique combinations a sparse array could be assembled for a given number of sensors and aperture.


ABSTRACT
Arrays can be designed as a full array of sensors or a sparse array of sensors with an equivalent aperture.Reductions in cost, size, weight and power consumption are important factors to consider when designing an array of sensors.Sparse arrays can provide a reduction in these factors.It is important to observe the performance of sparse arrays in relation to full arrays to see if they are able to achieve similar performance.The ability for the full and sparse arrays of sensors to accurately estimate the direction of arrival (DOA) of plane waves impinging them is observed in this thesis.The Mean-squared error (MSE) of the DOA estimation is measured against the Creamer-Rao bound (CRB) as a benchmark for optimal performance for an unbiased estimator.We derive an optimal way to remove sensors for one signal impinging on an array by finding the minimum CRB.This method considers all the unique combinations a sparse array could be assembled for a given number of sensors and aperture.
Parameters such as number of sensors, sensor locations, signal-to-noise ratio (SNR), number of snapshots, and signal correlation affect the MSE and CRB.
Through observing trends in the MSE as a result of changing these parameters we can come to conclusions to how full and sparse arrays perform as a result of adjusting these parameters.The MSE of the DOA of the full and sparse arrays is obtained through a covariance estimation from subspace-based DOA algorithms.
The sample covariance and the diagonal-averaged sample covariance matrix performances are compared.
The sample covariance matrix for a full array provides the best estimations at high SNR while the diagonal-averaging method provides a more accurate estimation at low SNR.The sparse array diagonal-averaged covariance matrix estimation performs similarly to the diagonal-averaged covariance matrix of the full array, but is still slightly worse due to missing information.How sparsity affects performance was also observed.As sparsity increases for an array, the performance change is non-linear for the MSE and CRB.Increases in signal correlation cause an increase in the overall MSE and a decrease in the overall CRB for both sparse and full arrays.test the accuracy and evaluated the performance of the estimation algorithms.For the case of one plane wave impinging on a sensor array, we derive an analytical method to find the sparse array that has the minimum CRB among all sparse arrays with an equal number of sensors and aperture.
The CRB and MSE depend on the parameters such as number of sensors, sensor locations, signal-to-noise ratio, number of snapshots, and signal covariance.
Moreover, the MSE also depends on the estimation algorithm.We analyze the MSE trends of full and sparse arrays when using different covariance matrix estimates in subspace-based DOA estimation algorithms.For a full array, using a sample covariance matrix yields more accurate results at high SNR while using a diagonalaveraged covariance matrix leads to lower MSEs at low SNR.The MSE trend of a sparse array for MUSIC or MNM is similar to the MSE pattern of a full array using a diagonal-averaged covariance matrix.An analysis of the variation of MSEs with the array sparsity reveals the non-linear dependence of the CRB and MSE on the number of sensors in a sparse array.

Introduction
When a plane wave impinges on an array of sensors, the data sampled by the sensors can be used to estimate the DOA (direction of arrival) of the plane wave.
DOA estimation of plane waves finds applications in many areas including sonar and radar fields.Sensor arrays that have an average intersensor spacing of more than half the wavelength of the impinging plane waves are called sparse arrays.
Although arrays with narrow mainlobe width, low peak sidelobe height, small sidelobe area, or hole-free coarrays provide more accurate estimates of DOAs than their counterparts, these arrays do not guarantee optimality in DOA estimation.The CRB (Cramer-Rao bound) provides a benchmark against which the performances of estimation algorithms can be compared [30,1].For any unbiased estimator, the CRB provides a lower bound on the variance of the estimate.Thus, having an MSE (mean-squared error) that is equal to the CRB or is close to the CRB is a desired feature in DOA estimation.Thus, in this work, we derive an analytical method to design a sparse array that has the minimum CRB among all sparse arrays with an equal aperture and number of sensors.This method is for estimating the DOA of a single plane wave in white Gaussian noise.
The factors that affect the CRB and MSE include SNR (signal-to-noise ratio), number of snapshots, number of sensors in the array, sensor locations, and signal correlation.Additionally, the MSE also depends on the estimation algorithm.The size of ι is L × 1, where (L − 1)λ/2 is the array aperture.For the array in Fig. 1, The ι for a full array with L sensors is an L × 1 vector of all ones.
We define the sensor location indicator function as ι[l] for l = 0, 1, . .., L − 1, where ι[l] is equal to the l th element of ι.The coarray corresponding to the sensor The coarray is non-zero from l = −(L − 1) to l = L − 1, i.e. the support set of the coarray is −(L − 1) to L − 1.We represent the coarray indicator function by for sparse arrays, some values of κ[l] may be zero.We refer to the array that has κ[l] as its sensor location indicator function as a virtual array [18].Sparse arrays whose virtual arrays do not have any missing sensors are called fully augmentable arrays [16,31].Sparse arrays whose virtual arrays have missing sensors are called partially augmentable arrays.
The k th snapshot received by an array with aperture (L − 1)λ/2 is modeled as where the L × 1 vector n k ∼ CN (0, σ 2 n I) represents the noise, µ is the number of plane waves impinging on the array, Fig. 1 depicts a scenario where there are µ = 2 plane waves impinging on the array with angles θ 1 and θ 2 , respectively.The received signal vector x k can also be written as where V = v(u 1 ) v(u 2 ) . . .v(u µ ) is the array manifold matrix and The ensemble covariance matrix of a snapshot is For a wide-sense stationary (WSS) signal, R is a Toeplitz matrix.The top panel of Fig. 2 illustrates a full array with 9 sensors and the corresponding virtual array.
The virtual array sensors are indicated with unfilled circles.The bottom panel of Fig. 2 depicts the corresponding covariance matrix R. The element r[l] in the covariance matrix represents the signal autocorrelation function at lag l, i.e.

Covariance Matrix Estimation
The ensemble covariance matrix is, in general, not available and needs to be estimated from the array measurements.When K independent snapshots are available, the L × K data matrix is given by The L × L sample covariance matrix corresponding to data matrix X is The sample covariance matrix is the maximum likelihood estimate of the ensemble covariance matrix [1].Fig. 2 depicts a 9-sensor full ULA and the corresponding virtual array.The physical sensors are indicated by filled green circles whereas the virtual sensors are indicted by unfilled green circles.
For a sparse array, R contains all-zero columns and rows based on where sensors are missing.In fully augmentable sparse arrays, assuming the received signals are WSS, the L × L covariance matrix estimate can be obtained from R with diagonal averaging.Diagonal-averaging in fully augmentable arrays replaces any zero element in R by the average of all elements along the diagonal in which the zero element resides.The top panel of Fig. 3 illustrates a fully augmentable sparse array and its virtual array.The bottom panel shows the covariance matrix that can be estimated from the data collected by the sparse array.In partially augmentable sparse arrays, all elements along one or more diagonals of the sample covariance matrix might be zero.Thus, an L × L covariance matrix estimate cannot be obtained with diagonal averaging in such arrays.The top panel of Fig. 4 shows a partially augmentable array and its virtual array.The unfilled green circles represent virtual sensors whereas the unfilled black circles represent missing virtual sensors.If the hole-free virtual array ranges from −(H − 1) to H − 1, the covariance matrix that can be estimated with the sparse array has size H × H and H < L. For partially augmentable arrays, the right L − H columns and the bottom L − H rows of R are removed before diagonal averaging.As an example, the bottom panel of Fig. 4 depicts the 7 × 7 covariance matrix that can be estimated for the partially augmentable array in the figure.We denote the covariance matrix estimate corresponding to a sparse array by R.

Optimal Sensor Locations
For an unbiased estimator, the CRB provides a lower bound on the MSE [30].
For a given aperture and the number of sensors, this section considers finding the sensor locations that correspond to the minimum CRB.

Covariance Matrix
Sparse Array and its Virtual Array We utilize the compact data matrix obtained by removing the all-zero rows from X.We denote the compact data matrix by Y.For a full array, Y = X but not for a sparse array.The size of Y is L × K.Note that L is the number of sensors.For full arrays L = L whereas for sparse arrays L < L. The ensemble covariance matrix for both full and sparse arrays can be written as R = E{YY H }.
When the signals and noise are uncorrelated and the noise is white, we have where where P ⊥ V is the projection matrix onto the noise subspace given by The matrix D is the derivative matrix defined as For the single signal case, the Fisher information is If L is odd, the manifold vector v(u 1 ) can be expressed as v(u 1 ) = v c (u 1 ) exp(jπ0.5(L −1)), where the conjugate symmetric vector is the array manifold vector of an L-element ULA whose phase center is at the origin.The vector d(u 1 ) can be expressed as where d c (u 1 ) = ∂v c (u)/∂u| u=u 1 .The term that is dependent on the sensor locations in (11) is , the only term in ( 14) that is dependent on the sensor locations is Thus, minimization of the CRB is equivalent to the maximization of J 2 , which is the sum of the squares of the elements of the following vector where i is the index vector.
When the number of plane waves impinging on the array is more than 1, the sparse array that minimizes the CRB cannot be found analytically.

Analysis of MSE for Full and Sparse Arrays
This section analyzes the MSEs in DOA estimation using full and sparse arrays.We consider the two prominent subspace-based DOA estimation algorithms: MUSIC and MNM.
The first example analyzes the trade-offs in using the sample covariance matrix R and the diagonal-averaged covariance matrix R for a full ULA.We consider a scenario where L = 12 and there are two closely located sources with direction cosines u 1 = −0.5∆uand u 2 = 0.5∆u, where ∆u = 0.2165 × 4/ L is the half-power beamwidth of the array.For each value of SNR, we evaluated MSEs for MUSIC and MNM using 500 independent trials.The top panel of Fig. 6 plots the MSEs for different values of SNR.When SNR is low (less than −20 dB in the figure), the MSEs for both MUSIC and MNM are constant.[1] notes that in this region the DOA estimates have a uniform distribution, i.e. u i ∼ U(−1, 1).When the SNR increases to −10 dB, the MSEs for both algorithms begin to descend and become approximately equal to the CRB.The SNR at which the MSE becomes approximately equal to the CRB is called threshold SNR and the region below the threshold SNR is called subspace-swap region [32,33].The threshold SNR in Fig. 6 is 4 dB.When the SNR further increases to 37 dB, the MSEs deviate from the CRB again.Note that [1] analyzes these MSEs only up to SNR of 20 dB and assumes incorrectly that the MSEs follow the CRB for higher values of SNR.The bottom panel of Fig. 6 plots the bias-squared and variance for both algorithms.
The MSE is equal to the sum of bias-squared and variance [30].The figure depicts the trade-offs between bias and variance.When the SNR is below the threshold SNR, the bias and variance are commensurate.In the region where the SNR is above the threshold SNR and the MSEs are close to the CRB, the biases are negligible compared to the variances.Consequently, the MSEs are almost equal to the respective variances.At very high SNR, when the MSEs deviate from the CRB, the variances are negligible compared to the biases.Thus, the MSEs in this region are contributed almost entirely by the biases.Fig. 7 plots the MSEs, biases, and variances for the 12-sensor full ULA using R with the MUSIC and MNM algorithms.The threshold SNR is for the two algorithms is −2 dB.Thus, the subspace-swap region is shorter and ends at a lower value of SNR compared to Fig. 6.However, the MSEs start deviating from the CRB at only 10 dB.Thus, for SNR less than 10 dB, the MUSIC and MNM algorithms are more accurate with R. For SNR higher than 10 dB, the algorithms are more accurate with R. The comparison between the bias and variances in the bottom panel of Fig. 7 illustrates that below the threshold SNR, the bias and variances are comparable.The bias is almost negligible and the MSE is approximately equal to the variances when the SNR is higher than the threshold SNR.The array gain or detection gain of a ULA for white noise is equal to the number of sensors [1].For sparse arrays, the detection gain is approximately linear with the number of sensors [34,35].To examine if the MSE and CRB are linear with the number of sensors for a linear array with a fixed aperture, we consider an array with 30λ/2 aperture.Fig. 9  Similarly, the bottom panel plots the MNM MSEs and CRBs.For this case also, the CRBs and MSEs do not exhibit linear dependence on the number of sensors.

Conclusion
For a given number of sensors and aperture, we derived an analytical technique to select M sensors out of L sensors while minimizing the CRB.The result is valid for the single plane wave case.We verified the result using a combinatorial search for selecting 10 sensors out of 15 sensors.For a full array, we compared MSEs, biases, and variances of MUSIC and MNM algorithms using sample covariance matrix and diagonal-averaged covariance matrix.The MSE trends when using the two different covariance estimates were observed to be similar.The MSE patterns of a fully augmentable sparse array for the MUSIC and MNM alogrithms were similar to the full array case using the diagonal-averaged covariance matrix.
However, as expected, the MSEs were higher for a sparse array than for a full array with an equal aperture due the reduction in the number of sensors.The analysis of the MUSIC and MNM MSEs for a sparse array revealed non-linear dependence on the number of sensors.
d ) is the complex amplitude of the d th plane wave for snapshot k, u d = cos(θ d ) is the direction cosine corresponding to the d th plane wave, θ d is the angle made by the d th plane wave with the array axis, and v(u d ) is the steering vectors associated with direction cosine u d and it is given by [1] v(u d ) = e 0πu d e 1πu d . . .e (L−1)πu d T .

Figure 2 .
Figure 2. Top: A 9-sensor full ULA and the corresponding virtual array.Bottom:The covariance matrix that can be estimated using the data collected by the full array.

Figure 4 .
Figure 4. Top: A partially augmentable sparse array and the corresponding virtual array.Bottom: The covariance matrix that can be estimated using the data collected by the sparse array.
To maximize ||πi|| 2 while choosing only M sensors, the top half and bottom half elements of i should be chosen.If M is odd, we can either pick (M − 1)/2 top indices and (M + 1)/2 bottom indices or (M − 1)/2 bottom indices and (M + 1)/2 top indices.An analogous derivation and thus conclusion follow if L is even.To verify this optimal selection algorithm, we consider an array with L = 15 and M = 10.We found the CRB corresponding to each 10-sensor sparse array.There are 3003 possible sparse arrays.The CRBs are plotted in Fig. 5.The minimum CRB value is also indicated in the figure.The sensor combination number corresponding to the minimum CRB is 252, which is

Figure 5 .
Figure 5.The CRB values for 3003 different sparse arrays.The total number of sensors in the full array is L = 15.The total number of sensors in each sparse array is L = 10.

Figure 6 .Figure 7 .Figure 8 .
Figure 6.Performance analysis of MUSIC and MNM for a 12-sensor full ULA using R. Top: MSEs.Bottom: Bias and Variance.
plots the MSEs and CRBs for the array with increasing sparsity.The sensor locations are depicted in the top panel of Fig. 9.The fixed sensors are indicated by green circles while the removable sensors are indicated by red squares.Since the sensors numbered 20, 30, and 0 to 10 are kept fixed, the array is always fully augmentable.The removable sensors are discarded one at a time, starting from the right end and moving towards left.The middle panel plots the MUSIC MSEs and CRBs at fixed SNR with increasing sparsity.SNR values of 0 dB (purple dash-dot line) and 10 dB (blue dash line) are considered.The CRBs and MSEs do not vary linearly with the number of sensors.

Figure 9 .
Figure 9. Variation of MSE and CRB with the array sparsity.The SNR values are 0 dB (purple dash-dot line) and 10 dB (blue dash line).Top: Sensor locations.Middle: MUSIC.Bottom: MNM.
3Top: A fully augmentable sparse array and the corresponding virtual array.Bottom: The covariance matrix that can be estimated using the data collected by the sparse array. . . . . ...4Top: A partially augmentable sparse array and the corresponding virtual array.Bottom: The covariance matrix that can be estimated using the data collected by the sparse array. . . . . . 5 The CRB values for 3003 different sparse arrays.The total number of sensors in the full array is L = 15.The total number of sensors in each sparse array is L = 10. . . . . . . . . . . . . .6 Performance analysis of MUSIC and MNM for a 12-sensor full ULA using R. Top: MSEs.Bottom: Bias and Variance. . . . .7 Performance analysis of MUSIC and MNM for a 12-sensor full ULA using R. Top: MSEs.Bottom: Bias and Variance. . . . .8 Performance analysis of MUSIC and MNM for a 10-sensor sparse array using R. Top: MSEs.Bottom: Bias and Variance. . . . .9 Variation of MSE and CRB with the array sparsity.The SNR values are 0 dB (purple dash-dot line) and 10 dB (blue dash line).Top: Sensor locations.Middle: MUSIC.Bottom: MNM. .