Signal Processing for Line Spectra: New Sensor Arrays, Algorithms, and Theoretical Results

Citation

Kulkarni, Pranav Dhananjay (2025) Signal Processing for Line Spectra: New Sensor Arrays, Algorithms, and Theoretical Results. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/n6dp-p089. https://resolver.caltech.edu/CaltechThesis:05292025-205908563

Abstract

Line spectrum signals appear in diverse application areas such as molecular dynamics, power electronics, speech processing, and target localization. They are composed of sums of complex exponentials with distinct frequencies. Identifying the parameters of these constituent complex exponentials has been a prominent research topic in signal processing for over four decades. In this thesis, we focus on two specific applications involving line spectrum signals: direction of arrival (DOA) estimation using sensor arrays, and denoising of discrete-time periodic signals.

The main contribution of this thesis on the topic of DOA estimation is to propose unconventional sensor array geometries and algorithms in the presence of aperture constraints. In the first part, we demonstrate that under an aperture constraint, the traditional integer arrays (defined as arrays with sensors placed at integer multiples of the half-wavelength distance λ/2) can perform only suboptimally because of the restrictive sensor placement at integer locations. To address this, we propose to use 'rational arrays' that can have sensors located at rational multiples of λ/2. This offers greater flexibility in sensor placement under aperture constraints. In particular, we propose rational coprime arrays that can approach the Cramér-Rao bound (CRB) even at low signal-to-noise ratio (SNR) and with a limited number of snapshots, and can outperform the integer arrays. Numerical simulations show that rational arrays are also better equipped to resolve closely separated DOAs. To enable the derivation of the theoretical results and identifiability guarantees for rational coprime arrays, we extend the number-theoretic concepts such as greatest common divisor and coprimality to the case of rational numbers, and prove several number-theoretic properties. Rational arrays are also demonstrated to have important advantages when the DOAs are known to lie in a sector of the space, and for identifying O(N²) uncorrelated sources using N sensors under aperture constraint.

In the second part of the thesis, we propose modifications to the traditionally used sparse (integer) array design criteria. These modifications are aimed at mitigating the impact of mutual coupling on DOA estimation and reducing the required aperture. To reduce the impact of mutual coupling, we propose two types of sparse arrays that have either double or triple minimum inter-element spacing compared to the traditionally used λ/2 spacing. This introduces 'holes' at lags 1 and 2 in the difference coarrays (defined as the set of differences in sensor locations). The first type of arrays, called weight-constrained sparse arrays, have O(N) aperture, making them suitable when the available aperture is constrained and the number of DOAs is small. A general array construction, to further reduce the weights at other coarray lags, is also proposed. The second type of arrays, called weight-constrained nested arrays, have O(N²) degrees of freedom and are suitable when there are no aperture restrictions. Extensive Monte-Carlo simulations demonstrate that the proposed arrays have significantly smaller DOA estimation errors compared to the well-known sparse arrays from the literature, in the presence of high mutual coupling.

Because of the central holes in the difference coarrays of the weight-constrained arrays, there are two segments of consecutive entries in their coarrays: one on the positive side and the other on the negative side. To leverage these both, we propose to use an augmented coarray covariance matrix for the subspace-based algorithms such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance (ESPRIT). This further reduces the DOA estimation error for the weight-constrained arrays, and the computation time of augmented-MUSIC is significantly less than that of optimization-based methods, such as coarray interpolation and dictionary-based methods. We also develop methods to algorithmically interpolate the missing entries in the coarray at lags 1 and 2, to generate a larger coarray matrix. This approach demonstrates the capability to identify up to twice as many DOAs compared to what can be achieved using only the one-sided segment of consecutive lags in the coarray. This mitigates the main disadvantage of having central holes in the coarrays of weight-constrained arrays, while still benefiting from their advantage in reducing the impact of mutual coupling.

One major drawback of using coarray-MUSIC for DOA estimation is its inefficiency (i.e., the mean squared error (MSE) does not approach CRB, even asymptotically). We conduct several experiments to provide new insights into the complex relationship of coarray-MUSIC MSE on several parameters, such as array geometry, DOA separation, and accuracy of the estimated array output correlations. Furthermore, we demonstrate that an alternative way of constructing the Toeplitz covariance matrix can greatly improve the MSE compared to coarray-MUSIC, and can lead to efficient DOA estimation. This approach is based on solving an optimization problem whose objective is derived using the asymptotic error distribution of the known entries from the covariance matrix. We also propose a modification to the Toeplitz covariance matrix construction approach to account for the presence of mutual coupling and provide simulations with different sparse arrays.

The third part of the thesis is focused on developing a periodicity-aware signal denoising framework using Capon-optimized Ramanujan filter banks and pruned Ramanujan dictionaries. The signal reconstruction (synthesis) is done by solving a regularized optimization problem, based on the outputs of the analysis filter bank. This hybrid analysis-synthesis framework ensures that the denoised output is necessarily composed of discrete-time periodic components. Capon beamforming principles from array signal processing are utilized to optimize the Ramanujan filters to the incoming data. A computationally efficient way of obtaining the inverses of the required autocorrelation matrices is derived using Levinson’s recursion. The proposed denoising method is observed to be effective even when the signal length is small and demonstrates a high SNR gain across a wide range of input signal SNRs. Furthermore, we derive several decimation properties of Ramanujan subspace signals, which help in reducing the required computations by appropriately downsampling the filter outputs without any loss of information.

Towards the end of the thesis, we theoretically investigate the locations of zeros of Ramanujan filters. Additionally, we propose an ideal interpolation filter model for Ramanujan subspace signals, which has potential application in developing a synthesis filter bank counterpart to the Ramanujan analysis filter bank for perfect signal reconstruction. We also explore the use of dictionary learning to represent periodic signals, and adapt a convolutional neural network based DOA estimation method to sparse arrays.

Thesis Files