*Sparse Linear Antenna Arrays: A Review DOI: http://dx.doi.org/10.5772/intechopen.99444*

arrays which can offer reliable and smooth operation even in the presence of a single-element failure. RMRAs ensure that each spatial lag is generated at least twice [85]. That is, there are at least two separate sensor pairs that can generate a given spatial lag. Even if a single sensor fails, the remaining sensors can generate all the necessary spatial lags. The RMRA is, therefore, preferred when array reliability is the foremost design concern. ULAs and RMRAs have only two essential sensors (the first and the last to preserve the aperture). Therefore, both these arrays have a fragility of 2/N. ULAs are the most robust and least fragile as they have many redundant sensors. RMRAs have been designed with the specific aim of achieving the least fragility in a sparse array while retaining the hole-free coarray properties of MRA.

As an example, a 10-element RMRA has sensors at {0, 1, 2, 6, 7, 11, 15, 16, 18, 19} [85]. The weight function of the RMRA is plotted in **Figure 5**. It can be seen that all the spatial lags from 0, f g �1, �2, … , �18 have a weight of two or more. That means there are at least two sensor pairs that can generate the required lags from 0 to 18. RMRAs are very much similar in concept to two-fold redundancy arrays (TFRAs). TFRAs are based on double difference sets [86]. However, RMRAs have a much broader scope and their design is more elegant. It can be observed from **Figure 5** that the weight *w*<sup>1</sup> is five which indicates that there are five sensor pairs with unit spacing. As per the empirical relation between the weight function and mutual coupling, it is easy to predict that the array is heavily prone to mutual coupling. Nevertheless, the array is robust to sensor failures.

Consider a situation where a particular element in the above RMRA fails (say the element at position 11). The weight function of the RMRA with failed sensor is shown in **Figure 6**. It can be observed that the weights of a few spatial lags fall down to one but none of them becomes zero. As long as there is just a single-element failure in RMRAs, the weight of any given spatial lag never becomes zero, meaning that holes would never occur in the DCA. This justifies the robustness of RMRAs.

In recent years, sparse arrays based on fractal geometries have been proposed. Such arrays use a small sparse array as a base (called generator) to obtain larger sparse arrays through recursive formulations. Examples include the Cantor arrays proposed by Liu and Vaidyanathan and the generalized fractal sparse arrays proposed by Cohen and Eldar [35, 87].

**Figure 5.** *Weight function of the 10-element RMRA described above.*

**Figure 6.** *Weight function of the RMRA with failed sensor at position 11.*

## **4.3 Sparse multiple input multiple output (MIMO) radar arrays**

Any review of linear sparse arrays would be incomplete without a mention of their usage in the design of sparse MIMO radar arrays. MIMO radar arrays make use of co-located transmit and receive arrays. These transmit and receive arrays work in tandem and produce the effect of a large virtual array. The transmitting array consists of *M* antennas and the receiving array contains *N* antennas. Hence, the total number of physical antennas in the MIMO array is *K* ¼ *M* þ *N*. Each transmit antenna emits an independent waveform. At each receiving antenna, there are *M* matched filters that are used to extract the reflected signals. As there are *N* receive elements, each equipped with *M* matched filters, the total number of extracted signals is *MN*. It has been shown that the matched filter output is equivalent to the signals received by an array of *MN* elements. This gives the effect of a large virtual array with *MN* elements [88], whereas physically there are only *M* þ *N* elements. The virtual array is also known as the sum co-array as it is obtained by adding all possible element positions in the transmit and receive arrays.

Following the introduction of the minimum redundancy MIMO radar [45] in 2008, several other sparse MIMO configurations have been proposed in the literature [89–93]. However, a thorough review of the properties of sparse MIMO arrays is beyond the scope of this article. However, it is easy to foresee that if the transmit and receive arrays are maximally sparse with hole-free co-arrays (like MRAs), the resulting MIMO radar array would also be highly sparse and, therefore, capable of providing the largest virtual array aperture for a given number of sensors and hence the highest DOFs as reported recently [94]. Several novel designs of sparse MIMO radar arrays are being proposed. Another research area which has gained traction in recent years is the co-existent MIMO radar and MIMO communications.
