**5. Multiuser classification by means of m sequential hypothesis testing algorithms**

As it was established before, OMA signal is recovered through EKF filtering and then, NOMA users could be efficiently decoded by one of the following methods:


The advantages of these algorithms for binary testing are well known long ago from A. Wald [25], but the results of their generalizations for the case of m hypothesis testing are mainly addressed to its tiny mathematical problems [24–29] and very few to its implementation.

For practical implementation, their solutions seem rather complex except in a few special cases. Other options ("ad-hoc") so far are based on the heuristic extensions of the binary test to the case of the m hypothesis (see also below).

In this regard it was found reasonable to invoke the results from [29] together with the approach stated at [26] where the asymptotic approximates for the optimum Bayesian algorithms for testing are presented for the case of small probabilities of decoding errors and an expected rather long testing time, particularly for low SNR cases. Both those assumptions might be realistic for SIC design at Doubly Selective channels, applying the Gaussian statistics as well.

Here, it is time to remind, that NOMA transmission, as it was commented above, needs sufficiently different channel conditions for the users, including certainly multiusers as well. Though channel conditions for OMA and multiusers are assumed as significantly different, but between multiusers it is not always the case, i.e. hypothesis for multiusers might be "close", therefore testing time might be large!

The extractions from [26, 29] for the asymptotic characteristics of the algorithm are as following:

$$\langle N\_k \rangle = -\frac{\log A\_k}{\min\_{j: j \neq k} D \left( f\_k, f\_j \right)} \; a\_k = \pi\_k A\_k \gamma\_k \qquad a = \sum\_{k=0}^{m-1} \pi\_k A\_k \gamma\_k \tag{10}$$

Where *D f* ð Þ¼ , *<sup>g</sup>* <sup>Ð</sup> *f x*ð Þ*log f x*ð Þ *g x*ð Þ *dx* is the Kullback-Liebler distance between the PDFs f(x) and g(x); ⟨*NA*⟩ <sup>¼</sup> *<sup>N</sup>*� *<sup>A</sup>*is the average value of the first *n* ≥ 1 when the decision for the hypothesis is taken; {*πk*} a priori probabilities of hypothesis.ð Þ *<sup>k</sup>* <sup>¼</sup> 1,�*<sup>m</sup>* , {*γk*} and {*Ak*} are detailed below.

In the usual case when the value of the probability *α* is predefined, from (10) it follows:

$$\mathcal{L} = \frac{a}{\sum\_{k=0}^{m-1} \frac{\pi\_c}{\delta\_k}}, \qquad \delta\_k = \min\_{j \neq k} D\left(\left.f\_k, f\_j\right\vert\_j\right), \qquad A\_k = \frac{c}{\delta\_k \chi\_k} \tag{11}$$

where 0 <*γ<sup>k</sup>* <1 is tabulated in Table 3.1 at [26, 29].

For concrete data, using (10)–(11), the asymptotic characteristics for m-testing can be calculated. It should be noticed that the essence and benefits of any sequential algorithm strongly depend on the choice of the threshold which defines the decision region for testing [26, 29]; the data processing algorithm can be: coherent (incoherent), energy detection, etc. which are well known [26].

For the concrete case of a Gaussian channel for all multiusers, where the decoding problem for multiusers depends on their SNR, QoS, amplitudes, average power, etc. the formal scenario is as follows: let *<sup>k</sup>* <sup>¼</sup> 0, *<sup>m</sup>*�� 1 for hypothesis testing f g *Hk* and *x*1, *x*2, … is a sequence of independent Gaussian random variables with means f g *<sup>θ</sup><sup>k</sup>* and variances *<sup>σ</sup>*<sup>2</sup> *<sup>k</sup>*, which are known a priori (see also [21, 26]). If f g *xk* are multiuser amplitudes, the data processing algorithm is obvious: P*<sup>n</sup> <sup>i</sup>*¼<sup>0</sup>*xi*, and the results must be compared with the asymptotic thresholds for each "*k*" as follows [28]:

$$\frac{n(\theta\_k + \theta\_{k+1})}{2} + \frac{\log A\_k}{\theta\_{k+1} - \theta\_k} \frac{n(\theta\_k + \theta\_{k+1})}{2} - \frac{\log A\_{k+1}}{\theta\_{k+1} - \theta\_k} \tag{12}$$

Where *k* ¼ 0, *m* � 1.

It is interesting to express here a remarkable issue: these asymptotic algorithms correspond exactly to the set of simple binary algorithms for testing: H0 vs. H1; H1 vs. H2 and so forth, i.e. are "ad hoc"(see above). This corresponds totally with the intuitive processing of the m-tests as a straightforward generalization of the binary sequential test (called "ad-hoc" [25, 28, 29]) that was known long ago and shown at [29] as an asymptotically optimum for m-hypothesis testing.

Therefore, in the Gaussian case the m-decoding asymptotic algorithm is nothing else but a "block" of cells each for the sequential testing of the different binary hypotheses of the multiusers, see **Figure 3**.

The examples of numerical results, for this testing can be found at [25], which show that for such kinds of algorithms the number of samples (time of analysis) can be reduced 2 or 3 times compared with the optimum algorithm for the fixed time testing. The latter follows exactly from the convergence of the m testing to the (m-1) set of "binary" algorithms well known several decades ago and rigorously presented first by A. Wald.

It is worth mentioning that robust features to the multiuser statistics follow due to the asymptotic character of the proposed algorithms. But the latter requires some further investigation.
