*4.2.2 Fault tolerance in sparse arrays a.k.a. fragility*

The notions of robustness, fragility, essentialness etc., in relation to sparse sensor arrays have been introduced in recent years by Liu and Vaidyanathan [82–84]. The fragility of a sparse array gives a measure of how vulnerable the array is to its'sensor failures. Fragility is defined as the number of essential sensors to the total number of sensors in the sparse array. A sensor is said to be essential if its failure/absence alters the difference coarray or introduces holes into the coarray. Arrays in which all sensors are essential are known as maximally economic sparse arrays (MESA). MRAs, nested and super nested arrays are maximally economic as all their sensors are essential. For this reason, these arrays are highly fragile with a fragility of *<sup>N</sup> <sup>N</sup>* ¼ 1.

It is well-known in MRA theory that, in an array of *N* elements, the failure of a single-element can cause up to *N* � 1 missing spatial lags, thereby rendering the sparse array useless for parameter estimation in the coarray domain. Robust MRAs (RMRAs) have been recently proposed with the aim of designing resilient sparse
