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We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

A new array geometry, which is capable of significantly increasing the degrees of freedom of linear arrays, is proposed. This structure is obtained by systematically nesting two or more uniform linear arrays and can provide O(N2) degrees of freedom using only N physical sensors when the second-order statistics of the received data is used. The concept of nesting is shown to be easily extensible to multiple stages and the structure of the optimally nested array is found analytically. It is possible to provide closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors. This cannot be done for existing classes of arrays like minimum redundancy arrays which have been used earlier for detecting more sources than the number of physical sensors. In minimum-input-minimum-output (MIMO) radar, the degrees of freedom are increased by constructing a longer virtual array through active sensing. The method proposed here, however, does not require active sensing and is capable of providing increased degrees of freedom in a completely passive setting. To utilize the degrees of freedom of the nested co-array, a novel spatial smoothing based approach to DOA estimation is also proposed, which does not require the inherent assumptions of the traditional techniques based on fourth-order cumulants or quasi stationary signals. As another potential application of the nested array, a new approach to beamforming based on a nonlinear preprocessing is also introduced, which can effectively utilize the degrees of freedom offered by the nested arrays. The usefulness of all the proposed methods is verified through extensive computer simulations.

/pdf/nested-arrays-a-novel-approach-to-array-processing-with-4nooc8fgu7.pdf

Nested Arrays: A Novel Approach to Array Processing With Enhanced Degrees of Freedom

Although Cramer–Rao Bounds (CRB) for direction-of-arrival (DOA) estimation have been extensively studied for decades, existing results are mainly applicable when there are fewer sources than sensors. In contrast, this letter considers an underdetermined signal model (more sources than sensors) and investigates conditions under which CRB exist. Necessary and sufficient conditions are derived for the associated Fisher information matrix to be nonsingular, which in turn, leads to closed-form expressions for the CRBs for underdetermined DOA estimation. These conditions highlight crucial roles played by the array geometry, as well as the correlation between source signals. The CRB for different array geometries are numerically compared both in the overdetermined and underdetermined settings.

Cramér–Rao Bounds for Underdetermined Source Localization

Coprime sampling has, in the past, been used for signal processing applications such as range and doppler improvement in radar, and for identifying sinusoids in noise. This paper considers a coprime pair of samplers in space or time from the point of view of the difference coarray, which is key to the increased freedom available for processing with such arrays. First, two uniform linear arrays with N and M elements in space are considered. With the interelement spacings in the two arrays given by Mλ/2 and Nλ/2 where M and N are coprime integers, it is shown that the difference coarray has O(MN) freedoms, so that the number of freedoms available for DOA estimation and beamforming is O(MN). This implies in particular that a passive array can identify O(MN) sources using only M + N sensor elements. It is then shown that by using an N-band DFT filter bank and an M-band DFT filter bank in conjunction with the two arrays, it is possible to create a virtual filter bank with MN bands (i.e., MN beams). The increased number of freedoms can be exploited either in a passive setting (using time-domain averaging) or in an active setting as in radar and sonar.1

Sparse sensing with coprime arrays

Sparse support recovery techniques guarantee successful recovery of sparse solutions to linear underdetermined systems provided the measurement matrix satisfies certain conditions. The maximum level of sparsity that can be recovered with existing algorithms is O(M) where M denotes the size of the measurement vector. This paper shows how this can be improved to O(M2) by assuming certain prior knowledge about the correlation structure of the measurements. The theory for correlation aware framework for support recovery is developed, which involves the Khatri-Rao (KR) product of the measurement matrix. Necessary and sufficient conditions for unique recovery of the sparse support are provided for this new framework which outperforms the more traditional CS techniques in terms of required size of the measurement vector. It also gives rise to interesting questions of constructing classes of measurement matrices which can exploit the prior correlation knowledge in an effective way. 1

Correlation-aware techniques for sparse support recovery

This paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank Toeplitz covariance matrix. Certain families of structured deterministic samplers are shown to efficiently compress a high-dimensional Toeplitz matrix of size $N\times N$ , producing a compressed sketch of size $O(\sqrt{r})\times O(\sqrt{r})$.The reconstruction problem can be cast as that of line spectrum estimation, whereby, in absence of noise, Toeplitz matrices of any size $N$ can be exactly recovered from compressive sketches of size $O(\sqrt{r})\times O(\sqrt{r})$, no matter how large $N$ is. In the presence of noise and finite data, the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite (PSD) Toeplitz constraint to denoise the compressed sketch using a simple least-squares minimization framework. A major advantage of the algorithm is that it does not require any regularization parameter. It also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low-rank Toeplitz matrix. Explicit bounds on the reconstruction error are established and it is shown that the PSD constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size $O(\sqrt{r})\times O(\sqrt{r})$. Extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear-norm-based regularizers.

Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach

Nested and coprime arrays are sparse arrays which can identify O(m2) sources using only m sensors. Systematic algorithms have recently been developed for such identification. These algorithms are traditionally implemented by performing MUSIC or a similar algorithm in the difference-coarray domain. This paper considers the use of nested and coprime arrays for the case where the number of sources is less than m. It will be demonstrated that there are some important advantages even in this case. With the number of sources limited like this, it is possible to use MUSIC directly on the nested or coprime array rather than in the coarray domain. But owing to array sparsity, the unambiguous identifiability property has to be revisited. This paper first mentions two results for such identifiability. Second, the improvement in the Cramer-Rao bound (over uniform linear arrays or ULAs) is analyzed. One conclusion is that the CRB improvements of nested and coprime arrays are comparable to those of other known sparse arrays such as MRAs. It is also observed that nested and coprime arrays have higher resolvability than the ULA, for a fixed number of sensors.1

Piya Pal

Papers

Cramér–Rao Bounds for Underdetermined Source Localization

Sparse sensing with coprime arrays

Correlation-aware techniques for sparse support recovery

Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach

Direct-MUSIC on sparse arrays