## 1. Introduction

Signal detection has become an integral part of applications in many areas, including wireless communications, optical telecommunications, automatic control system, magnetic resonance imaging, underwater acoustics, and radio astronomy. Typical digital communication environments involve transmission of analog pulses over a dispersive medium, inevitably corrupting the received signal by inter‐symbol interference (ISI). The present sequence estimation approaches, for example, maximum‐likelihood sequence estimation (MLSE) [1] and ‘Bussgang’ [2–4], usually estimate first the communication channel impulse response and then the transmitted sequence by an optimum method. MLSE is a useful approach for equalization of ISI but requires knowledge of the possibly time‐variant channel impulse response, and its complexity increases exponentially with the length of the channel delay spread. Consequently, adaptive channel estimation algorithms have to be employed [1]. “Bussgang” and its modified algorithms [2–4] are well‐known channel estimation algorithms but are costly and indirect due to their necessity for a long block of data to achieve algorithm convergence since they exploit implicit (embedded) statistical knowledge. Direct signal detection (DSD) is a new emerging method in communication systems, which can directly estimate the input sequence without estimating the channel impulse response [5–7].

In this work, we propose a new method that merges the functional network (FN) [8–9] models into a DSD technique for removing ISI without the training sequence. FN is a very useful general framework for modeling a wide range of probabilistic, statistical, mathematical, and engineering problems [10]. For instance, Iglesias et al. [11] adopted FN to improve the results of a determined artificial neural networks (ANN) application for the estimation of fishing possibilities, Zhou et al. [12] employed FN to solve classification problem, Emad et al. [13] found that FN (separable and generalized associativity) architecture with polynomial basis was accurate, was reliable, and outperforms most of the existing predictive data mining modeling approaches for the issue of permeability prediction, and Alonso‐Betanzos et al. [14] applied FN to predict the failure shear effort in concrete beams. As mentioned above, FN has been successfully applied in many fields, but so far methods of DSD using FN have not been reported.

The chapter is organized as follows. In Section 2, a system model is described. Then the network structure of MIMOFN to solve the special issues of DSD is constructed in Section 3. A multiple‐input multiple‐output FN (MIMOFN), in which the initial input vector is devised via QR decomposition of receiving signal matrix, is constructed to solve the special issues of DSD. The design method of the neural function of this special MIMOFN is proposed. Then the learning rule for the parameters of neural functions is trained and updated by back‐propagation (BP) algorithm in Section 4. In Section 5, simulation results are shown to verify the new approach's correctness and effectiveness, and some special simulation phenomena of this algorithm are given out followed by Conclusion and Discussion in Section 6.

## 2. The system model and basic assumptions

This system model is a linear single‐input multiple‐output (SIMO) (see **Figure 1**) finite impulse response channel with *N* output [15]. The following basic assumptions are adopted [16].

A1: Channel order

*L*is assumed to be known as a priori._{h}A2: The input signal

**s**(*t*) is zero‐mean and temporally independent and identically distributed (i.i.d).A3: Additive noise is spatially and temporally white noise and is statistically independent of the source.

For the simplification of the presentation of the proposed DSD without loss of generality, the *i*‐th sub‐channel output of an SIMO channel system in a noise‐free environment is expressed as

where *i‐*th system impulse response of length

where *q* and *T* denotes transpose of a matrix, and

and

Thus, the received data matrix can be formulated as

where

and *N* and *L _{w}* are the source signal length and length of the equalizer, respectively, and Toeplitz matrix

## 3. The FN structure for DSD

FN is a generalization of neural networks achieved by using multi‐argument and learnable functions, that is, in these models, transfer functions, associated with neurons, are not fixed but learned from data. There is a need to include weights to ponder links among neurons since their effect is subsumed by the neural functions. **Figure 2** shows an example of a general FN topology, where the input layer consists of the units *P*, *G*, *N*, *Q*; the second layer contains neurons *J*, *K*, *F*, *L*; and the output layer contains *P*, *G*, *N*, *Q, J*, *K*, *F*, *L,* is represented as a linear combination of the known functions of a given family such as polynomials, trigonometric functions, and Fourier expansions and is estimated during the learning process. Generally, instead of fixed functions, FNs extend the standard neural networks by allowing neuron functions *P*, *G*, *N*, *Q , J*, *K*, *F*, *L* to be not only true multi‐argument and multi‐variate functions but also different and learnable. Two types of learning methods exist: structural learning and parametric learning. The latter estimates the activation functions with the consideration of the combination of “basis” functions such as the least square, steepest descent, and mini‐max methods [13]. In this chapter, the least square method for estimating sequence is used.

** Definition 1:** Assume there exists a function set

*n‐*dimensional input vector into an

*m*‐dimensional output vector in a complex regime

** Definition 2:** Let

Generally, FNs are driven by specific problems, that is, the initial architecture is designed based on problems in hand. Here, an MIMOFN, shown in **Figure 3**, is designed for the special issue of DSD. In this topology, **w**_{1}, **w**_{2},…, **w**_{N} is the input vector, and *f*(.), g(.), and *h*^{‐1}(.) denote the neuron functions of the first, second, and output layer, respectively. Re(.) and Im(.) denote the real part and imaginary part, respectively.

In this case,

(7) |

where all of *f*(.), *g*(.), and *h*^{‐1}(.) are arbitrary continuous and strictly monotonic functions, and *j* is the imaginary unit.

## 4. DSD using MIMOFN

### 4.1. The input vector of MIMOFN

Our objective is to detect the source sequence directly from the observed output data by utilizing MIMOFN without training sequence and estimating the channel impulse response.

Since

where

For the DSD issues, there is a well‐defined gap in the singular values of

In order to obtain the full QR factorization, we proceed with SVD and extend

where

Thus, the input matrix of MIMOFN

where superscript *H* denotes conjugate transpose of a matrix. Clearly,

Since the column vectors of

### 4.2. Cost function

The next issue to be considered is construction of a cost function. The cost function of MIMOFN can be formulated as

where

### 4.3. Design of neural functions

Neural function *g*(.) must be easy to compute and complement, and hence needs to satisfy the following conditions:

*g*(.) should be simple, easy to calculate, continuously differentiable, and bounded.The derivative of

*g*(.) should be simple too, and it is the ordinary transformation of*g*(.).A priori knowledge of special issue must be considered, which makes the network easy to be trained with strong generalization ability in a smaller scale.

Since both the in‐phase and quadrature‐part of QAM signals belong to *f* (.) as

where *g*(.) can be calculated by

**Figure 4** shows the curves of *g*(.) neural function and its derivatives g'(.) for 16QAM with *a* = 5) will accelerate the convergence rate of algorithm.

## 5. The learning rule for the parameters of neural functions

Herein, the input vector and the cost function of MIMOFN are determined, and the following mission is design of a learning strategy of the MIMOFN's neural function.

(1) The learning strategy of the parameters of *f*(.).

A back‐propagation(BP) algorithm is adopted to update the parameters of *f*(.). Since the error function has been shown in Eq. (11), we can obtain the following equation

where

Let us look into the *i‐*th neuron. At the first iteration, the neural functions

where *f*(.) and

At the *n‐*th iteration, we have

and

To quicken the learning rate, an adaptive BP training algorithm with a momentum term is adopted, has shown in Eq. (20), is used

(2) The learning strategy of the parameters of *g*(.).

The attenuation factor *g*(.) function is not fixed but adjusted in accordance with the value of error function. First, when the value of error function is large,

where *B* and *C* are constants, and *A* is the iteration times of *J*(**c**_{A+1}) *<J*(**c**_{A}. The curve of *A* = 10, *B* = 10, and *C* = 5 is shown in **Figure 5**.

## 6. Simulation and results

In this section, simulation results are provided to illustrate and verify the developed theory. Unless noted otherwise, these experiments are based on a multi‐path channel **Figure 6**). And the order *f*(.) is [5, 18]. The signal modulation type is 16QAM and satisfies block‐fading feature. Results are averaged over 500 Monte Carlo runs.

**Figure 7** shows the average BER curves of DSD using an MIMOFN approach with the learning rate *N* = 500. This improved performance results from the decrease of **Figure 7** also illustrates that the MIMOFN approach can work well even if the signal‐to‐noise ratio (SNR) = 12 dB, which is hard to reach for most existing DSD.

**Figure 8** illustrates the curves of the average cost function *J* of the MIMOFN approach with several different lengths of data *N* = (100, 150, 200, 250, 500), when

**Figure 9** and **Table 1** illustrate the rate of correct recognition *N* = (100, 150 ,…, 500) for SNR = 15 dB.

where *N*.

We compare our method (MIMOFN) with the classical subspace algorithm (SSA), linear prediction algorithm (LPA), and outer‐product decomposition algorithm (OPDA) [15], which are shown in **Figure 10**. It is found that the MIMOFN approach is superior to those of the above‐mentioned DSD in performance.

The following experiments are based on 16QAM, SNR = 15 dB, *N* = 200, and

In **Figure 11**, the solid circles and hollow circles denote ideal 16QAM signal constellation points and the positions of the signal points with the iteration, respectively. **Figure 12** illustrates the phase trajectories of signal points using the MIMOFN approach. All lines denote the phase trajectories of signal points with the iteration, respectively, and the eight bold lines express the trajectories of the given different signal points, respectively. We can see that the phase trajectories are different and irregular, but all of them will reach their respective true signal points when the algorithm is convergent. In addition, the iteration is only about 150 times even for 16QAM input case.

## 7. Conclusions

In this chapter, a unified approach based on MIMOFN to solve DSD issues, even if the sequence is short and the training sequence is absent, is shown. The proposed method can be applied to those cases where the constellation of the source signal is dense and the data frame is short. The structure of MIMOFN not only is suitable to the DSD of square‐QAM (e.g. 16QAM) issues but also can be extended to the cross‐QAM (e.g. 8QAM, 32QAM) cases.