**3. Properties of linear sparse arrays**

#### **3.1 Sparse array terminology**

## *3.1.1 Sparse array notations*

In general, sparse arrays such as MRAs and MHAs are represented in the f g *:a:b:c:d:* format or simply f g *a*, *b*,*c*, *d* without the dots. This format has *n* � 1 entries for a *n*�element array. A sparse array configuration of f g *a*, *b*,*c*, *d* means that the array has five sensors at respective locations 0, f *a*, *a* þ *b*, *a* þ *b* þ *c*, *a* þ *b* þ *c* þ *d*g. For example, an 8-element MRA may be denoted as {.1.3.6.6.2.3.2.} or {1, 3, 6, 6, 2, 3, 2} or {1, 3, 6<sup>2</sup> , 2, 3, 2}. The power indicates the number of times the given spacing should be repeated. Therefore, as per the formulation, the above 8-element MRA has sensors at {0, 1, 4, 10, 16, 18, 21, 23}.

Another common representation is in the form of a binary string of 1 s and 0 s which represent the presence or absence of a sensor element on the respective grid point. For example, in the representation f g *:a:b:c:d:* , the dots indicate the presence of sensors and hence have to be written as 1. Accordingly, the binary string for the above sparse array would be {1, (a-1) zeroes, 1, (b-1) zeroes, 1, (c-1) zeroes, 1, (d-1) zeroes, 1}. Considering the 8-element MRA {.1.3.6.6.2.3.2.} given above, the binary string would be {1, (1-1) zeroes, 1, (3-1) zeroes and so on}. This gives the binary string {1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1}. Zero indicates the absence of a sensor at the respective grid point. The grid positions of all sensors for which the binary entry is one gives the sensor locations of the sparse array i.e., {0, 1, 4, 10, 16, 18, 21, 23} in the case of the 8-element MRA given above. The reader has to be comfortable in changing from one form to another as different papers and textbooks follow different notations [4, 41, 42]. The auto-correlation of the above binary string gives the weight function (explained shortly) in the coarray domain.

## *3.1.2 Difference set and the difference co-array*

For a sparse array with sensors at ¼ *z*1, *z*2, … , *zNs* f g, the difference set is defined as

$$\mathbb{H} = \left\{ \mathbf{z}\_i - \mathbf{z}\_j; i, j = \mathbf{1}, \mathbf{2}, \dots, \mathbf{N}\_s \right\} \tag{1}$$

The distinct entries (*d*Þ of the difference set form the difference co-array (DCA) or simply co-array of the physical sparse array. The DCA is symmetric, that is, every *p* ∈ *d*∃ � *p* ∈ *<sup>d</sup>* .

In simple terms, the difference set is obtained by subtracting all possible sensor positions in the given sparse array. This gives rise to spatial lags. The DCA is then formed by considering only the non-repeating (distinct) spatial lags. For example, a sparse array with sensors at {0, 1, 4, 6} can generate all spatial lags (differences) between 0 and 6, resulting in a difference set of {0, �1, �4, �6, 1, 0, �3, �5, 4, 3, 0, �2, 6, 5, 2, 0}. The DCA {�6, �5, �4, �3, �2, �1, 0, 1, 2, 3, 4, 5, 6} is obtained by sorting the numbers and retaining only the distinct elements. Note that the repeating lag of zero is considered only once.

The number of unique lags in the DCA of a sparse array gives the number of source angles that can be detected during DOA estimation (often known as the degrees of freedom (DOFs) offered by the sparse array). Unique lags are often used in the analysis of sparse arrays with holes in the DCA. The usefulness of arrays that have holes in the DCA is limited by the span of the central continuous portion of the DCA. Though there are methods [43, 44] that can extend the continuous portion of
