**3.2 Types of sparse arrays**

It is necessary to understand the structure of a ULA and the normalized sensor positions in order to comprehend the properties of grid-based sparse arrays.

### *3.2.1 Uniform linear array (ULA)*

As per definition, there would be no missing sensors in the ULA. Hence, the sensor positions are given by the set *ULA* ¼ f g 0, 1, 2, … , *N* � 1 . All sensor spacings are normalized to half wavelength.

### *3.2.2 Minimum redundancy arrays (MRAs)*

MRAs are synthesized from a ULA by eliminating selected sensors such that the sensors thus retained are capable of generating all possible spacings between zero and a specified number [41]. A ULA has many sensor combinations that provide a given spatial lag (e.g., considering the sensor positions from 0 to 9, a spatial lag of 4 can be obtained using any of the redundant sensor pairs {9, 5}, {8, 4}, {7, 3}, {6, 2}, {5, 1} and {4, 0}). MRAs minimize this redundancy by carefully removing select sensors. An optimum MRA has sensors at positions that are just enough to provide all the spatial lags from 0 up to a maximum number *L:* A zero redundancy MRA is one which generates each spatial lag exactly once. The 4-element MRA {0, 1, 4, 6} generates each spatial lag from 1 to 6 exactly once and is a zero redundancy MRA.

For arrays with more than four elements, it is not possible to get rid of the redundancy completely if one were to ensure all possible spatial lags. For example, in a 5-element MRA whose sensors are located at {0, 2, 5, 8, 9}, it can be seen that the available sensor pairs can generate all the spacings between 0 to 9. But in particular, a spacing of 3 can be obtained using the sensor pairs {5, 2} and {8, 5}. This leads to redundancy but is inevitable for large arrays.

Consider an *N*-element MRA with an aperture of *L* ≤ *N N*ð Þ �<sup>1</sup> 2 . The redundancy of the MRA is given by

$$R = \frac{N(N-1)}{2L} \tag{2}$$

A value of *R* ¼ 1 indicates zero redundancy i.e., there are no redundant sensor pairs and that each spatial lag is generated exactly once. However, in practice, *R*>1, indicating that the array contains certain redundant sensor pairs that can generate a given spatial lag more than once. Since zero redundancy cannot be achieved in arrays with more than four sensors, MRAs are optimized to achieve the configuration that provides the minimum redundancy possible for a given number of sensors.

MRAs do not have closed-form expressions to determine the optimum sensor positions and have to be synthesized using exhaustive search mechanisms. The optimization problem to find sensor positions in MRAs is given by

$$\max \text{s.t.} \, L(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \text{ s.t.} \\ h(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = \mathbf{0}, \\ \mathbf{x}\_i \in [\mathbf{0}, L] \tag{3}$$

where *xi* indicate the positions of the array elements in the ascending order; the constraint *h x*ð Þ¼ 1, *x*2, … , *xn* 0 ensures that there are no missing spatial lags within the segment 0, ½ � *L* . The optimization wishes to maximize the segment 0, ½ � *L* using *N* sensors without any missing lags [45].

## *3.2.3 Minimum hole arrays (MHAs)*

MHAs are obtained by optimizing the sensor positions such that a given spatial lag is obtained at most once. In other words, the sensors should be placed such that the spatial lags generated by them are unique. No two sensor pairs shall generate the same lag. Additionally, it is not bothersome in MHAs even if the available sensor pairs cannot generate all the spacings between 0 and *L*. For example, a 5-element MHA with sensors at {0, 1, 4, 9, 11}, can generate almost all the spatial lags between 0 and 11 but fails to generate a spacing of 6. It should also be noted that there should be no more than one sensor pair to generate a given spatial lag. A perfect MHA is one which can generate all the spacings between 0 and *L* exactly once. A 4-element MHA with sensor positions {0, 1, 4, 6} is perfect. MHAs are also referred to as Golomb arrays and the sensor positions represent the marks on the Golomb ruler. Perfect Golomb rulers or MHAs do not exist for arrays with more than four sensors and hence only optimum rulers can be designed for such cases. Optimal Golomb rulers up to 19 marks have been presented in the past [46]. Like MRAs, there are no closed-form expressions to determine the optimum placement of sensors in MHAs. MHAs are also known as non-redundant arrays.

Coincidentally, the definitions of a zero redundancy MRA and a perfect Golomb array (MHA) bear the same physical meaning. Arrays with fewer than four elements qualify both as zero redundancy MRAs as well as perfect MHAs (Eg: A 4-element array with sensors at {0, 1, 4, 6}). However, for arrays with more than four sensors, neither zero redundancy MRAs exist nor do perfect MHAs. MRAs and MHAs mean different things for arrays with more than four sensors.

The objective function to synthesize MHAs is described below. A Golomb ruler consists of a set of integers *A* ¼ f g *a*1, *a*2, … , *an ,* in ascending order, such that for each non-zero integer *x*, there is at most one solution to the equation *x* ¼ *a <sup>j</sup>* � *ai*∣ *ai*, *a <sup>j</sup>* ∈ *A.* The set of integers, *A* represents the positions of *n* marks on a ruler [47]. For MHAs, *a*<sup>1</sup> ¼ 0; *an* ¼ *L*. The largest known optimum ruler till date has 27 sensors and can provide an aperture of 553 [31]. Obtaining optimal Golomb rulers is a computationally hard problem [48].

## *3.2.4 Co-prime arrays*

The coprime array consists of two ULAs. It was one of the first sparse arrays introduced with closed-form expressions (CFE) for sensor positions. That is, the sensor locations or the sparse array configuration can be immediately obtained once the number of sensors is given without the need for any exhaustive search mechanisms. One ULA has 2*P* sensors with a spacing of *Q* units and another ULA of *Q* sensors spaced *P* units apart. *P* and *Q* are co-prime integers such that *P*< *Q* [27]. The first ULA is given by *S*<sup>1</sup> ¼ f g *qP*j*q* ¼ 0, 1, 2, … *:*, *Q* � 1 and the second ULA is given by *S*<sup>2</sup> ¼ f g *pQ*j*p* ¼ 0, 1, 2, … *:*, 2*P* � 1 .

The overall co-prime array is *Scp* ¼ *S*<sup>1</sup> ∪ *S*2*:* For example, considering *P* ¼ 2, *Q* ¼ 3; we have *S*<sup>1</sup> ¼ f g 0, 2, 4 , *S*<sup>2</sup> ¼ f g 0, 3, 6, 9 and the resulting co-prime array has sensors at {0, 2, 3, 4, 6, 9}. The drawback of co-prime arrays is that their DCAs are not hole-free.
