Abstract
A sensor array or just "array" is a collection of sensors that performs spatial sampling of signals arriving from various directions. In general, the signals can be EM waves, sound waves, or other types of signals. In this chapter, we concentrate on plane monochromatic EM waves arriving from different directions and impinging on a sensor array. Arrays typically processes the signals received at the sensors to estimate the directions of arrival or DOAs. In some cases the arrays operate in the beamforming mode, by providing large gain in certain directions of arrival, compared to other directions. An array is said to be linear if the sensors are on a straight line, and this is what we consider in this introductory chapter. More general arrays are discussed in Chapter 5 in the book. The most commonly used arrays are uniform linear arrays or ULAs, which have adjacent sensors separated by ∕2 where is the wavelength of the impinging monochromatic waves. But there are also other types of arrays where a number of adjacent sensor pairs have larger separation. These are called sparse arrays, and they have a number of advantages. In this chapter, we will introduce sparse arrays and discuss their advantages. In the world of array signal processing, such arrays have been known for many decades, especially in the context of imaging and beamforming. Sparse arrays received more attention in the last decade because of the introduction of nested and coprime arrays, which have certain advantages compared to theoretically optimal arrays that have been well-known for much longer. We begin by reviewing the fundamentals of Array Signal Processing in Section 1.2, including DOA estimation and beamforming. Sparse arrays are introduced in Section 1.3. A unique property of some sparse arrays is that the number D of DOAs that can be identified can exceed the number N of sensors in the array. The reason for this counterintuitive property is explained in Section 1.4, and in greater detail in Section 1.5. The difference coarray, which is central to this property, is also introduced. Specific algorithms for DOA estimation with sparse arrays, such as coarray MUSIC, are introduced in Section 1.6. Well-known classes of sparse arrays such as nested arrays and coprime arrays are introduced in Section 1.7, and their properties are discussed. Then, in Section 1.8, we review classical optimal sparse arrays such as the minimum redundancy arrays or MRAs, minimum hole arrays or MHAs (also called Golomb rulers), and perfect arrays. Section 1.9 explains how coprime arrays can be used to develop a novel type of beamformer, where the number of nonoverlapping spatial beams can be much larger than the number of sensors. This is another unique advantage of some sparse arrays. In Section 1.10, we conclude the chapter by providing directions for further reading. Notations We use bold face letters for matrices and vectors, as in A and v, where usually uppercase letters denote matrices and lowercase letters denote column vectors. The notations A T , A * , and A H denote, respectively, transpose, conjugate, and transpose-conjugate. The notation (n 1 , n 2 , … , n N-1 ) stands for the greatest common divisor (gcd) of the integers n 1 , n 2 , … , n N-1 .
Showing the abstract — retrieve the full paper via the Exa API.