**8. Conclusions and summary**

simple descents that can be stuck in local optimums from several starting points. Geometrical detectors use a real-valued model and the singular value decomposition (SVD) rather than the QR one [34]. Let us rewrite the objective function by introducing the SVD of **<sup>H</sup>** <sup>¼</sup> **UDV***<sup>T</sup>* with **<sup>U</sup>** and **<sup>V</sup>** two orthogonal matrices and **<sup>D</sup>** the diagonal matrix containing the singular values *λ<sup>i</sup>* f g : 1≤*i* ≤*n* in ascending order.

*Advanced Radio Frequency Antennas for Modern Communication and Medical Systems*

� �*<sup>T</sup>*

using **x**<sup>0</sup> from Eq. (3). As the vectors of **V**, named **v***<sup>i</sup>* f g : 1≤ *i*≤ *n* , constitute a basis, we can define *α<sup>i</sup>* f g : 1≤*i* ≤*n* the coordinates of **x** � **x**<sup>0</sup> on this basis. Using the

Let Δ*<sup>i</sup>* be the straight line passing through *x*<sup>0</sup> and directed by **v***i*. One can note that Eq. (20) highlights that the objective function grows more slowly along the first Δ*<sup>i</sup>* rather than along the last ones so that promising points must be around these first straight lines. Then the solution is most likely to be found along this line. The geometrical exploration step is then performed, selecting some points near the first Δ*i*. Then a straightforward descent algorithm is performed by looking for the best

A soft-output version of this algorithm is possible using the max-log approximation and the list approach detailed in [35], Section 5.2. A field-programmable gate array (FPGA) implementation has recently been proposed. This groundwork points out that geometrical detectors may achieve good performance in the future

ZF Very poor Very simple Best linear detector regarding SNR criterion MMSE Poor Simple Best linear detector regarding SINR criterion SIC/OSIC Good Rather complex Best when there is a clear ranking in the quality of

PIC Good Rather complex Best when all data streams have the same quality level

Breadth-first Good Rather complex Possible trade-off between BER and complexity via

Deep neural Good Rather complex Possible trade-off between BER and complexity via

Bioinspired Good Very complex Resilient to imperfect CSI and channel correlation Geometrical Rather good Rather complex Possible trade-off between BER and complexity via

each data stream

the number of surviving paths

the number of layers

the number of descents

*i*¼1 *α*2 *i λ*2

<sup>∥</sup>**<sup>y</sup>** � **Hx**∥<sup>2</sup> <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

**DU***<sup>T</sup>***UD V***<sup>T</sup>*ð Þ **<sup>x</sup>** � **<sup>x</sup>**<sup>0</sup>

� � (19)

*<sup>i</sup> :* (20)

Consequently, the objective function can be rewritten as

orthogonality of **U** and **V** and the diagonality of **D**, Eq. (19) leads to

<sup>∥</sup>**<sup>y</sup>** � **Hx**∥<sup>2</sup> <sup>¼</sup> **<sup>V</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>** � **<sup>x</sup>**<sup>0</sup>

point in the close neighborhood until convergence.

**Detector BER Complexity Comment**

complex

yet being far from mature at that point [36].

ML Optimal Dramatically

Depth-first Optimal Very complex

Best-first Good Less complex

*Summary of all detectors described in this chapter.*

**Table 1.**

**216**

MIMO detection is a well-studied problem that has been tackled from several perspectives. The mathematical interpretation, as a combinatorial optimization problem, leads to the optimal and linear detectors. From the signal processing perspective, detecting a signal means improving the SNR or SINR so that the direct answer is to cancel the interference and to remove the noise. From an algorithmic perspective, the detection problem is the search for the best path in a weighted tree that relies on some well-known algorithms. Other sources of inspiration, such as nature or geometry, provide some interesting perspectives. These paradigms and the associated detectors are summed up in **Table 1**, and we compare all of them according to the BER-complexity trade-off.

All these perspectives shed a different light on the problem, leading to fruitful experimentation. Indeed, some methods take inspiration from others to keep on improving. Therefore, some improvement axes remain open, for instance, the permanent decrease of complexity with equal performance, the development for efficient hardware implementations, or the optimization of the interaction with decoders to exploit channel codings better.
