**Abstract**

In this chapter, the fundamentals of distributed inference problem in wireless sensor networks (WSN) is addressed and the statistical theoretical foundations to several applications is provided. The chapter adopts a statistical signal processing perspective and focusses on distributed version of the binary-hypothesis test for detecting an event as correctly as possible. The fusion center is assumed to be equipped with multiple antennas collecting and processing the information. The inference problem that is solved, primarily concerns the robust detection of a phenomenon of interest (for example, environmental hazard, oil/gas leakage, forest fire). The presence of multiple antennas at both transmit and receive sides resembles a multiple-input-multiple-output (MIMO) system and allows for utilization of array processing techniques providing spectral efficiency, fading mitigation and low energy sensor adoption. The problem is referred to as MIMO decision fusion. Subsequently, both design and evaluation (simulated and experimental) of these fusion approaches is presented for this futuristic WSN set-up.

**Keywords:** MIMO, decision fusion, distributed MAC, statistical CSI, instantaneous CSI, large-scale WSN, environmental characterization, experimental validation

#### **1. Introduction**

This chapter addresses the fundamentals of distributed inference problems in wireless sensor networks (WSNs) and provides statistical theoretical foundations to several applications. It adopts a statistical signal processing perspective and focuses on the distributed version of the binary-hypothesis test for detecting an event as correctly as possible. The reference WSN scenario is described in Section 2 which consists of multiple transmit sensors and an information center equipped with multiple antennas for collecting and processing the information to arrive at a robust decision on an observed phenomenon of interest. The presence of multiple antennas at both transmit and receive side resembles a multiple-input-multiple-output (MIMO) system and the inference problem based on information fusion is referred to as MIMO decision fusion. Consequently, several channel-aware fusion rules guiding MIMO decision fusion (DF) is studied in Section 3. The practical implications of employing MIMO decision fusion for distributed inference in WSN is evaluated through an indoor-to-outdoor measurement campaign detailed in Section 4. Performance of the fusion rules over the measured environment is also compared with that over the simulated set-up in Section 4.

## **2. WSN scenario and system model**

In a WSN, decision fusion (DF) refers to the process of arriving at a final decision on an observed phenomenon by fusing local decisions transmitted by individual sensors on the said occurrence at a decision fusion center (DFC). In traditional WSNs, each sensor is allocated a dedicated orthogonal channel for transmitting their local observations, a communication scenario commonly referred to as parallel access channel (PAC) [1]. Sensor signals are transmitted over the PAC using time, frequency or code division multiple access. Over the recent years, largescale or massive WSNs are being deployed, that involves coexistence of multitude of sensors, and the bandwidth requirement increases linearly with the number of sensors. In such scenarios, all sensors transmit their decisions simultaneously over a multiple access channel (MAC), while suffering from intrinsic interference resulting from superposition of multiple sensor signals in time [2]. To alleviate fusion performance in presence of deep fading, shadowing and interference, a DFC equipped with multiple antennas is proposed in [3]. This choice demands only further complexity on DFC side and does not affect simplicity of sensors implementation. The result is a communication over a "virtual" MIMO channel between the sensors and the DFC, as shown in **Figure 1**.

We also assume that *PD*,*<sup>s</sup>* ≥*PF*,*s*, i.e. the decision taken by individual sensor always results in a receiver operating characteristics (ROC) higher than the decision threshold *γ*. The system probabilities of false alarm and correct detection is given by,

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

where Λ is the fusion statistics, *γ* is the decision threshold to which Λ is compared to, and *P A*ð Þ j*B* is the probability of event A conditioned on event B.

*bs*,*<sup>s</sup>*

**<sup>y</sup>** <sup>¼</sup> **<sup>H</sup>** ffiffiffi **B**

If the composite channel coefficient between the *s*th sensor and the *n*th receive

after sampling and matched filtering at the DFC, where **y**∈ *<sup>N</sup>* is the received

Gaussian noise (AWGN) vector, **H** ∈ *<sup>N</sup>*�*<sup>S</sup>* is the independent small scale fading matrix, **B**∈ *<sup>S</sup>*�*<sup>S</sup>* is the large scale attenuation and shadowing matrix with the *s*th

experienced by the *s*th sensor. Here, N ð Þ *λ*, Σ denotes circular symmetric complex normal distribution with mean vector *λ* and covariance matrix Σ respectively.

If the propagation channel is assumed to be Rician distributed, the fading vector

*<sup>Δ</sup> <sup>P</sup>*ð Þ <sup>Λ</sup>>*γ*jH<sup>0</sup> for False Alarm (1)

p *hn*,*<sup>s</sup>*, the received signal can be expressed as,

*<sup>w</sup>***I***<sup>N</sup>*

*<sup>Δ</sup>* diag *<sup>β</sup>*1, *<sup>β</sup>*2, … , *<sup>β</sup><sup>S</sup>* ½ �*<sup>T</sup>* � � accounting for pathloss and shadowing

ffiffiffiffiffiffiffiffiffiffiffiffi 1 � *κ*<sup>2</sup> *s*

**h** *^*

*<sup>s</sup>* � N *<sup>C</sup>*ð Þ **0***N*,**I***<sup>N</sup>* is the non-line-of-sight

ffiffiffiffiffiffiffiffiffiffiffiffi 1 � *κ ^*2 *s*

q

, Δ*<sup>s</sup>* is the shape factor for the fading distribution. For

is the Rician *K*-factor between *s*th

*:* (6)

q

1 <sup>2</sup>*<sup>π</sup>* **<sup>h</sup>** *^ s* ð<sup>2</sup>*<sup>π</sup>* 0

<sup>p</sup> **<sup>x</sup>** <sup>þ</sup> **<sup>w</sup>** (3)

� � is the additive white

*<sup>s</sup>* (4)

d*α* (5)

*<sup>Δ</sup> <sup>P</sup>*ð Þ <sup>Λ</sup>>*γ*jH<sup>1</sup> for Correct Detection*:* (2)

*PF*<sup>0</sup> ¼

*PD*<sup>0</sup> ¼

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

antenna at the DFC is denoted by ffiffiffiffiffiffi

signal, **<sup>x</sup>**<sup>∈</sup> *<sup>X</sup><sup>S</sup>* is the transmitted signal, **<sup>w</sup>** � N *<sup>C</sup>* **<sup>0</sup>***N*, *<sup>σ</sup>*<sup>2</sup>

**h**Rice

*<sup>s</sup>* ¼ *κs***u** *ϕ<sup>s</sup>* ð Þþ

*^*

ð<sup>2</sup>*<sup>π</sup>* 0 *κ ^ <sup>s</sup>* d*α* þ

**h**DR *<sup>s</sup>* <sup>¼</sup> <sup>Y</sup> 2

*<sup>Δ</sup>* ffiffiffiffiffiffiffiffi *Ks* 1þ*Ks* q

double-Rayleigh (DR) [7] distributed fading vector, the channel coefficients can be

*j*¼1 **h** *^ s j*

sensor and DFC. A Two Wave with Diffused Power (TWDP) [5, 6] distributed

*2.1.2 Signal model*

diagonal element **B** ¼

*2.1.3 Channel model*

where *κ ^ <sup>s</sup>* ¼

expressed as,

**193**

at the *s*th sensor can be modeled as,

where **u**ð Þ� is the steering vector, **h**

channel fading vector can be modeled by,

**h**TWDP

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Ks*½ � 1þΔ*<sup>s</sup>* cos *α* 1þ*Ks*½ � 1þΔ*<sup>s</sup>* cos *α*

q

*<sup>s</sup>* <sup>¼</sup> **<sup>u</sup>** *<sup>ϕ</sup><sup>s</sup>* ð Þ 2*π*

(NLOS) scattered component and *κ<sup>s</sup>* ¼

#### **2.1 System model for MIMO configuration**

#### *2.1.1 Sensing and local decision model*

A WSN consisting of *S* sensors and a DFC equipped with *N* receive antennas is considered for investigation in this section, where the *s*th sensor communicates its local decision, d*s*, about the presence or absence of a target, after being mapped on an On-and-Off Shift Keying (OOK) [4] modulated symbol, *xs* ∈*X* ¼ f g 0, 1 . Irrespective of the scenario and target, d*<sup>s</sup>* ¼ H*<sup>i</sup>* maps into *xs* ¼ *i*, *i* ∈f g 0, 1 , where H*<sup>i</sup>* ¼ *<sup>Δ</sup>* f g <sup>H</sup>0, <sup>H</sup><sup>1</sup> is the set of binary hypotheses with <sup>H</sup>0*=*H<sup>1</sup> representing the absence or presence of a specific target. The sensor decisions are assumed to be transported over a flat fading multiple access distributed (or virtual) MIMO channel with perfect synchronization at the receiver end.

The performance of WSN can be evaluated in terms of the conditional probability mass function (pmf) *P*ð Þ **x**jH*<sup>i</sup>* . Assuming conditionally independent and identically distributed (iid) decisions, we denote the probability of detection *PD*,*<sup>s</sup>* ¼ *P*1,*<sup>s</sup>* ð Þ or *PD* ¼ *P*<sup>1</sup> and false alarm *PF*,*<sup>s</sup>* ¼ *P*0,*<sup>s</sup>* ð Þ or *PF* ¼ *P*<sup>0</sup> at the *s*th sensor.

**Figure 1.** *Virtual MIMO based DF in WSN.*

*Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

We also assume that *PD*,*<sup>s</sup>* ≥*PF*,*s*, i.e. the decision taken by individual sensor always results in a receiver operating characteristics (ROC) higher than the decision threshold *γ*. The system probabilities of false alarm and correct detection is given by,

$$P\_{F\_0} \triangleq P(\Lambda > \gamma | \mathcal{H}\_0) \qquad \text{for False Alarm} \tag{1}$$

$$P\_{D\_0} \triangleq P(\Lambda > \gamma | \mathcal{H}\_1) \qquad \text{for Correct detection.} \tag{2}$$

where Λ is the fusion statistics, *γ* is the decision threshold to which Λ is compared to, and *P A*ð Þ j*B* is the probability of event A conditioned on event B.

#### *2.1.2 Signal model*

**2. WSN scenario and system model**

*Wireless Sensor Networks - Design, Deployment and Applications*

the sensors and the DFC, as shown in **Figure 1**.

**2.1 System model for MIMO configuration**

perfect synchronization at the receiver end.

*2.1.1 Sensing and local decision model*

H*<sup>i</sup>* ¼

**Figure 1.**

**192**

*Virtual MIMO based DF in WSN.*

In a WSN, decision fusion (DF) refers to the process of arriving at a final decision on an observed phenomenon by fusing local decisions transmitted by individual sensors on the said occurrence at a decision fusion center (DFC). In traditional WSNs, each sensor is allocated a dedicated orthogonal channel for transmitting their local observations, a communication scenario commonly referred to as parallel access channel (PAC) [1]. Sensor signals are transmitted over the PAC using time, frequency or code division multiple access. Over the recent years, largescale or massive WSNs are being deployed, that involves coexistence of multitude of sensors, and the bandwidth requirement increases linearly with the number of sensors. In such scenarios, all sensors transmit their decisions simultaneously over a

multiple access channel (MAC), while suffering from intrinsic interference resulting from superposition of multiple sensor signals in time [2]. To alleviate fusion performance in presence of deep fading, shadowing and interference, a DFC equipped with multiple antennas is proposed in [3]. This choice demands only further complexity on DFC side and does not affect simplicity of sensors implementation. The result is a communication over a "virtual" MIMO channel between

A WSN consisting of *S* sensors and a DFC equipped with *N* receive antennas is considered for investigation in this section, where the *s*th sensor communicates its local decision, d*s*, about the presence or absence of a target, after being mapped on

*<sup>Δ</sup>* f g <sup>H</sup>0, <sup>H</sup><sup>1</sup> is the set of binary hypotheses with <sup>H</sup>0*=*H<sup>1</sup> representing the absence or presence of a specific target. The sensor decisions are assumed to be transported over a flat fading multiple access distributed (or virtual) MIMO channel with

The performance of WSN can be evaluated in terms of the conditional proba-

bility mass function (pmf) *P*ð Þ **x**jH*<sup>i</sup>* . Assuming conditionally independent and identically distributed (iid) decisions, we denote the probability of detection *PD*,*<sup>s</sup>* ¼ *P*1,*<sup>s</sup>* ð Þ or *PD* ¼ *P*<sup>1</sup> and false alarm *PF*,*<sup>s</sup>* ¼ *P*0,*<sup>s</sup>* ð Þ or *PF* ¼ *P*<sup>0</sup> at the *s*th sensor.

an On-and-Off Shift Keying (OOK) [4] modulated symbol, *xs* ∈*X* ¼ f g 0, 1 . Irrespective of the scenario and target, d*<sup>s</sup>* ¼ H*<sup>i</sup>* maps into *xs* ¼ *i*, *i* ∈f g 0, 1 , where

If the composite channel coefficient between the *s*th sensor and the *n*th receive antenna at the DFC is denoted by ffiffiffiffiffiffi *bs*,*<sup>s</sup>* p *hn*,*<sup>s</sup>*, the received signal can be expressed as,

$$\mathbf{y} = \mathbf{H}\sqrt{\mathbf{B}}\mathbf{x} + \mathbf{w} \tag{3}$$

after sampling and matched filtering at the DFC, where **y**∈ *<sup>N</sup>* is the received signal, **<sup>x</sup>**<sup>∈</sup> *<sup>X</sup><sup>S</sup>* is the transmitted signal, **<sup>w</sup>** � N *<sup>C</sup>* **<sup>0</sup>***N*, *<sup>σ</sup>*<sup>2</sup> *<sup>w</sup>***I***<sup>N</sup>* � � is the additive white Gaussian noise (AWGN) vector, **H** ∈ *<sup>N</sup>*�*<sup>S</sup>* is the independent small scale fading matrix, **B**∈ *<sup>S</sup>*�*<sup>S</sup>* is the large scale attenuation and shadowing matrix with the *s*th diagonal element **B** ¼ *<sup>Δ</sup>* diag *<sup>β</sup>*1, *<sup>β</sup>*2, … , *<sup>β</sup><sup>S</sup>* ½ �*<sup>T</sup>* � � accounting for pathloss and shadowing experienced by the *s*th sensor. Here, N ð Þ *λ*, Σ denotes circular symmetric complex normal distribution with mean vector *λ* and covariance matrix Σ respectively.

#### *2.1.3 Channel model*

If the propagation channel is assumed to be Rician distributed, the fading vector at the *s*th sensor can be modeled as,

$$\mathbf{h}\_s^{\text{Ricce}} = \kappa\_s \mathbf{u}(\phi\_s) + \sqrt{\mathbf{1} - \kappa\_s^2} \,\breve{\mathbf{h}}\_s \tag{4}$$

where **u**ð Þ� is the steering vector, **h** *^ <sup>s</sup>* � N *<sup>C</sup>*ð Þ **0***N*,**I***<sup>N</sup>* is the non-line-of-sight (NLOS) scattered component and *κ<sup>s</sup>* ¼ *<sup>Δ</sup>* ffiffiffiffiffiffiffiffi *Ks* 1þ*Ks* q is the Rician *K*-factor between *s*th sensor and DFC. A Two Wave with Diffused Power (TWDP) [5, 6] distributed channel fading vector can be modeled by,

$$\mathbf{h}\_{\varepsilon}^{\text{TWDP}} = \frac{\mathbf{u}(\phi\_{\varepsilon})}{2\pi} \int\_{0}^{2\pi} \bar{\kappa}\_{\varepsilon} \, \mathbf{d}a + \frac{1}{2\pi} \left. \bar{\mathbf{h}}\_{\varepsilon} \right|\_{0}^{2\pi} \sqrt{1 - \bar{\kappa}\_{\varepsilon}^{2}} \, \mathbf{d}a \tag{5}$$

where *κ ^ <sup>s</sup>* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Ks*½ � 1þΔ*<sup>s</sup>* cos *α* 1þ*Ks*½ � 1þΔ*<sup>s</sup>* cos *α* q , Δ*<sup>s</sup>* is the shape factor for the fading distribution. For double-Rayleigh (DR) [7] distributed fading vector, the channel coefficients can be expressed as,

$$\mathbf{h}\_{\mathbf{s}}^{\text{DR}} = \prod\_{j=1}^{2} \breve{\mathbf{h}}\_{\mathbf{t}\_{j}}.\tag{6}$$

with *κ<sup>s</sup>* ¼ 0 and no line-of-sight (LOS) components existing between the sensors and the DFC.

The sensors are uniformly deployed with distances from the DFC varying between dmin and dmax, and large scale attenuation of *<sup>β</sup><sup>s</sup>* <sup>¼</sup> *<sup>ν</sup><sup>s</sup>* dmin d*s* � �*<sup>η</sup><sup>P</sup>* where *η<sup>P</sup>* is the pathloss exponent and *<sup>ν</sup><sup>s</sup>* is a log-normal variable such that 10 log <sup>10</sup>ð Þ�N *<sup>ν</sup><sup>s</sup> <sup>μ</sup>P*, *<sup>σ</sup>*<sup>2</sup> *P* � � with <sup>N</sup> ^*λ*, <sup>Σ</sup>^ � � representing normal distribution with mean vector ^*<sup>λ</sup>* and covariance matrix Σ^ respectively, d*<sup>s</sup>* is the distance of the *s*th sensor from the DFC, *μ<sup>P</sup>* and *σ<sup>P</sup>* are the mean and standard deviations in dBm respectively.

#### *2.1.4 Modified system model for non-coherent decision fusion*

Non-coherent decision fusion over MAC using the received-energy test has been investigated in [2, 8]. In such a scenario, if the probability of false alarm for any sensor decision is lower than the probability of detection, the received energy can prove to be optimal for arriving at the right decision about an observed phenomenon at the DFC for mutually independent and identically distributed (i.i.d.) sensor decisions.

In this case, let us consider that a group of sensors transmit their local decisions to the DFC equipped with *N* antennas over a Raleigh faded MAC with channel coefficients of equal mean power, thereby exploiting diversity combining, either in time, frequency, code or in polarization domain. Statistical channel state information (CSI) is assumed at the DFC, i.e. only the pdf of each fading coefficient is available.

Let us denote: *yn* the received signal at the *n*th diversity branch of the DFC after matched filtering and sampling; *hn*,*<sup>s</sup>* � N 0, *<sup>σ</sup>*<sup>2</sup> *h* � �, the fading coefficient between the *s*th sensor and the *n*th diversity branch of the DFC; *wn* the additive white Gaussian noise at the nth diversity branch of the DFC. The vector model at the DFC is the following:

$$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w} \tag{7}$$

Λopt ¼ ln

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

**x**∈ *<sup>S</sup>*�<sup>1</sup>

given by,

**195**

where <sup>Π</sup>^*<sup>s</sup>* <sup>¼</sup>

*<sup>Δ</sup> <sup>x</sup>*^*s*þ<sup>1</sup>

**<sup>x</sup>**^MMSE <sup>¼</sup> sign **<sup>x</sup>** <sup>þ</sup> **C H** ffiffiffi

where **x** ¼ Ef g**x** and **C** ¼

statistics for CV-ML and CV-MMSE rules.

P

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

rules [12], defined by the following test statistics,

ΛMax�Log ¼ min

<sup>Λ</sup>CV <sup>¼</sup> <sup>X</sup> *S*

*s*¼1

**B** � � p †

**x**∈*X<sup>S</sup>*

� min **x**∈*X<sup>S</sup>*

respectively, all assuming identical sensor performances.

<sup>Π</sup>^*<sup>s</sup>* ln *PD*,*<sup>s</sup> PF*,*<sup>s</sup>* � �

especially, the Maximum Likelihood (ML) detector [13] to obtain,

**x**^ML ¼ arg min **x**∈*X<sup>S</sup>*

and the Minimum Mean-Squared Error (MMSE) detector [14] to get,

of the transmit signal vector respectively. The estimated **x**^ from the above two detectors can be incorporated directly in the CV-rule of (12) to obtain the test

**H** ffiffiffi **B** � � <sup>p</sup> **C H** ffiffiffi

P

**<sup>x</sup>**∈*X<sup>S</sup>* exp � **<sup>y</sup>**�**<sup>H</sup>** ffiffi

**<sup>x</sup>**∈*X<sup>S</sup>* exp � **<sup>y</sup>**�**<sup>H</sup>** ffiffi

assuming conditional independence of **<sup>y</sup>** from <sup>H</sup>*i*, given *xs*, **<sup>H</sup>** <sup>∈</sup> *<sup>N</sup>*�*<sup>S</sup>* and

to this group are considered for this section to compensate for the asymptotically increasing computational complexity of the optimum rule. These rules are Maximal Ratio Combining (MRC) [10], Equal Gain Combining (EGC) [11] and Max-Log

ΛMRC ∝ ℜ **1***<sup>t</sup>*

<sup>Λ</sup>EGC <sup>¼</sup> <sup>ℜ</sup> *<sup>e</sup> <sup>j</sup>*�<sup>∠</sup> **<sup>H</sup>** ffiffi

**<sup>y</sup>** � **<sup>H</sup>** ffiffiffi **B** <sup>p</sup> **<sup>x</sup>** � � �

> *σ*2 *w*

**<sup>y</sup>** � **<sup>H</sup>** ffiffiffi **B** <sup>p</sup> **<sup>x</sup>** � � �

> *σ*2 *w*

The other set of fusion rules (Decode-then-Fuse) aims at concluding to a global decision after estimating the transmit signal from the received signal vector. Using Chair-Varshney (CV) rule, the test statistics for which over a noiseless channel is

<sup>þ</sup> <sup>1</sup> � <sup>Π</sup>^*<sup>s</sup>*

<sup>2</sup> . Two different detectors are considered under this umbrella,

**<sup>y</sup>** � **<sup>H</sup>** ffiffiffi **B** <sup>p</sup> **<sup>x</sup>**

**B** � � p †

� � " # (14)

� ��<sup>1</sup>

<sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>w</sup>***I***<sup>N</sup>*

*<sup>Δ</sup>* ð Þ **<sup>x</sup>** � **<sup>x</sup>** ð Þ **<sup>x</sup>** � **<sup>x</sup>** † � � are the mean and covariance matrix

� � �

. The test statistics for three different sub-optimum fusion rules belonging

*<sup>S</sup>* **<sup>H</sup>** ffiffiffi **B** � � p †

**B** <sup>p</sup> ð Þ **<sup>1</sup>***<sup>S</sup>* � �†

> � 2

" #

� 2 **y**

�<sup>X</sup> *S*

" #

*s*¼1

�<sup>X</sup> *S*

� � ln <sup>1</sup> � *PD*,*<sup>s</sup>*

� � � � (12)

� � � 2

1 � *PF*,*<sup>s</sup>*

**<sup>y</sup>** � **<sup>H</sup>** ffiffiffi **B** <sup>p</sup> **<sup>x</sup>**

*s*¼1

**y**

**B** <sup>p</sup> k k**<sup>x</sup>** <sup>2</sup> *σ*2 *w* � � <sup>Q</sup>

**B** <sup>p</sup> k k**<sup>x</sup>** <sup>2</sup> *σ*2 *w* � � <sup>Q</sup>

*S s*¼1

*S s*¼1 *P xs* ð Þ jH<sup>1</sup>

(8)

(11)

(13)

*P xs* ð Þ jH<sup>0</sup>

� � (9)

� � (10)

*P xs* ð Þ jH<sup>0</sup>

*P xs* ð Þ jH<sup>1</sup>

where **<sup>y</sup>**<sup>∈</sup> *<sup>N</sup>*, **<sup>x</sup>**<sup>∈</sup> *<sup>X</sup><sup>S</sup>* and **<sup>w</sup>** � N *<sup>C</sup>* **<sup>0</sup>***N*, *<sup>σ</sup>*<sup>2</sup> *<sup>w</sup>***I***<sup>N</sup>* � � are the received signal, transmitted signal and the AWGN vectors respectively. Finally, we define the random variable *<sup>l</sup>*≜*l x*ð Þ¼ <sup>P</sup>*<sup>S</sup> <sup>s</sup>*¼<sup>1</sup>*xs*, representing the number of active sensors and the set L≜f g 0, … , *S* of possible realizations of *l*. It is worth-mentioning here that it will be more practical to assume an asymmetric model for the statistics of the channel coefficients resulting in scenario-dependent analysis. Therefore, symmetric channel model is considered here to analyze performance with power control possibility depending on the application scenario.

#### **3. MIMO decision fusion**

#### **3.1 Instantaneous CSI**

Two types of fusion rules are considered and compared in this chapter in order to arrive at a reliable choice depending on the communication scenario. One set of rules (Decode-and-Fuse) uses the received signal directly to arrive at a decision on whether a target is present or absent, without taking any information from the transmitted signal into consideration, the optimum (opt) test statistics [9] for which is given by,

*Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

$$\Lambda\_{\text{opt}} = \ln \left[ \frac{\sum\_{\mathbf{x} \in X^{\mathcal{S}}} \exp \left( - \frac{\left\| \mathbf{y} - \mathbf{H} \sqrt{\mathbf{B}} \mathbf{x} \right\|^{2}}{\sigma\_{w}^{2}} \right) \prod\_{s=1}^{S} P(\mathbf{x}\_{s} | \mathcal{H}\_{1})} \right] \tag{8}$$
 
$$\left[ \frac{\sum\_{\mathbf{x} \in X^{\mathcal{S}}} \exp \left( - \frac{\left\| \mathbf{y} - \mathbf{H} \sqrt{\mathbf{B}} \mathbf{x} \right\|^{2}}{\sigma\_{w}^{2}} \right) \prod\_{s=1}^{S} P(\mathbf{x}\_{s} | \mathcal{H}\_{0})} \right] \tag{9}$$

assuming conditional independence of **<sup>y</sup>** from <sup>H</sup>*i*, given *xs*, **<sup>H</sup>** <sup>∈</sup> *<sup>N</sup>*�*<sup>S</sup>* and **x**∈ *<sup>S</sup>*�<sup>1</sup> . The test statistics for three different sub-optimum fusion rules belonging to this group are considered for this section to compensate for the asymptotically increasing computational complexity of the optimum rule. These rules are Maximal Ratio Combining (MRC) [10], Equal Gain Combining (EGC) [11] and Max-Log rules [12], defined by the following test statistics,

$$\Lambda\_{\rm MRC} \propto \Re \left( \mathbf{1}\_{\rm S}^{t} \left( \mathbf{H} \sqrt{\mathbf{B}} \right)^{\dagger} \mathbf{y} \right) \tag{9}$$

$$
\Lambda\_{\rm EGC} = \Re \left( \left( e^{j \cdot \omega \left( \mathbf{H} \sqrt{\mathbf{B}} \ \mathbf{1}\_{\mathbb{S}} \right)} \right)^{\dagger} \mathbf{y} \right) \tag{10}
$$

$$\begin{split} \Lambda\_{\text{Max-Log}} &= \min\_{\mathbf{x} \in X^{\mathcal{S}}} \left[ \frac{\left||\mathbf{y} - \mathbf{H}\sqrt{\mathbf{B}}\mathbf{x}\right||^{2}}{\sigma\_{w}^{2}} - \sum\_{s=1}^{S} P(\mathbf{x}\_{i}|\mathcal{H}\_{0}) \right] \\ &- \min\_{\mathbf{x} \in X^{\mathcal{S}}} \left[ \frac{\left||\mathbf{y} - \mathbf{H}\sqrt{\mathbf{B}}\mathbf{x}\right||^{2}}{\sigma\_{w}^{2}} - \sum\_{s=1}^{S} P(\mathbf{x}\_{i}|\mathcal{H}\_{1}) \right] \end{split} \tag{11}$$

respectively, all assuming identical sensor performances.

The other set of fusion rules (Decode-then-Fuse) aims at concluding to a global decision after estimating the transmit signal from the received signal vector. Using Chair-Varshney (CV) rule, the test statistics for which over a noiseless channel is given by,

$$\Lambda\_{\rm CV} = \sum\_{s=1}^{S} \left[ \hat{\Pi}\_{\rm t} \ln \left( \frac{P\_{D,\rm s}}{P\_{F,\rm s}} \right) + \left( \mathbf{1} - \hat{\Pi}\_{\rm t} \right) \ln \left( \frac{\mathbf{1} - P\_{D,\rm s}}{\mathbf{1} - P\_{F,\rm s}} \right) \right] \tag{12}$$

where <sup>Π</sup>^*<sup>s</sup>* <sup>¼</sup> *<sup>Δ</sup> <sup>x</sup>*^*s*þ<sup>1</sup> <sup>2</sup> . Two different detectors are considered under this umbrella, especially, the Maximum Likelihood (ML) detector [13] to obtain,

$$\hat{\mathbf{x}}\_{\text{ML}} = \underset{\mathbf{x} \in X^{S}}{\text{arg min}} \left\| \mathbf{y} - \mathbf{H} \sqrt{\mathbf{B}} \mathbf{x} \right\|^{2} \tag{13}$$

and the Minimum Mean-Squared Error (MMSE) detector [14] to get,

$$\hat{\mathbf{x}}\_{\text{MMSE}} = \text{sign}\left[\overline{\mathbf{x}} + \mathbf{C} \left(\mathbf{H}\sqrt{\mathbf{B}}\right)^{\dagger} \left(\left(\mathbf{H}\sqrt{\mathbf{B}}\right)\mathbf{C} \left(\mathbf{H}\sqrt{\mathbf{B}}\right)^{\dagger} + \sigma\_w^2 \mathbf{I}\_N\right)^{-1} \left(\mathbf{y} - \mathbf{H}\sqrt{\mathbf{B}}\overline{\mathbf{x}}\right)\right] \tag{14}$$

where **x** ¼ Ef g**x** and **C** ¼ *<sup>Δ</sup>* ð Þ **<sup>x</sup>** � **<sup>x</sup>** ð Þ **<sup>x</sup>** � **<sup>x</sup>** † � � are the mean and covariance matrix of the transmit signal vector respectively. The estimated **x**^ from the above two detectors can be incorporated directly in the CV-rule of (12) to obtain the test statistics for CV-ML and CV-MMSE rules.

with *κ<sup>s</sup>* ¼ 0 and no line-of-sight (LOS) components existing between the sensors

d*s* � �*<sup>η</sup><sup>P</sup>*

� �, the fading coefficient between

**y** ¼ **Hx** þ **w** (7)

� � are the received signal, transmit-

where *η<sup>P</sup>* is the

*P* � �

The sensors are uniformly deployed with distances from the DFC varying

pathloss exponent and *<sup>ν</sup><sup>s</sup>* is a log-normal variable such that 10 log <sup>10</sup>ð Þ�N *<sup>ν</sup><sup>s</sup> <sup>μ</sup>P*, *<sup>σ</sup>*<sup>2</sup>

with <sup>N</sup> ^*λ*, <sup>Σ</sup>^ � � representing normal distribution with mean vector ^*<sup>λ</sup>* and covariance matrix Σ^ respectively, d*<sup>s</sup>* is the distance of the *s*th sensor from the DFC, *μ<sup>P</sup>* and *σ<sup>P</sup>*

Non-coherent decision fusion over MAC using the received-energy test has been investigated in [2, 8]. In such a scenario, if the probability of false alarm for any sensor decision is lower than the probability of detection, the received energy can prove to be optimal for arriving at the right decision about an observed phenomenon at the DFC for mutually independent and identically distributed (i.i.d.) sensor

In this case, let us consider that a group of sensors transmit their local decisions

Let us denote: *yn* the received signal at the *n*th diversity branch of the DFC after

*h*

*<sup>w</sup>***I***<sup>N</sup>*

*<sup>s</sup>*¼<sup>1</sup>*xs*, representing the number of active sensors and the set

to the DFC equipped with *N* antennas over a Raleigh faded MAC with channel coefficients of equal mean power, thereby exploiting diversity combining, either in time, frequency, code or in polarization domain. Statistical channel state information (CSI) is assumed at the DFC, i.e. only the pdf of each fading coefficient is

the *s*th sensor and the *n*th diversity branch of the DFC; *wn* the additive white Gaussian noise at the nth diversity branch of the DFC. The vector model at the DFC

ted signal and the AWGN vectors respectively. Finally, we define the random

L≜f g 0, … , *S* of possible realizations of *l*. It is worth-mentioning here that it will be more practical to assume an asymmetric model for the statistics of the channel coefficients resulting in scenario-dependent analysis. Therefore, symmetric channel model is considered here to analyze performance with power control possibility

Two types of fusion rules are considered and compared in this chapter in order to arrive at a reliable choice depending on the communication scenario. One set of rules (Decode-and-Fuse) uses the received signal directly to arrive at a decision on whether a target is present or absent, without taking any information from the transmitted signal into consideration, the optimum (opt) test statistics [9] for

between dmin and dmax, and large scale attenuation of *<sup>β</sup><sup>s</sup>* <sup>¼</sup> *<sup>ν</sup><sup>s</sup>* dmin

are the mean and standard deviations in dBm respectively.

*Wireless Sensor Networks - Design, Deployment and Applications*

*2.1.4 Modified system model for non-coherent decision fusion*

matched filtering and sampling; *hn*,*<sup>s</sup>* � N 0, *<sup>σ</sup>*<sup>2</sup>

where **<sup>y</sup>**<sup>∈</sup> *<sup>N</sup>*, **<sup>x</sup>**<sup>∈</sup> *<sup>X</sup><sup>S</sup>* and **<sup>w</sup>** � N *<sup>C</sup>* **<sup>0</sup>***N*, *<sup>σ</sup>*<sup>2</sup>

depending on the application scenario.

**3. MIMO decision fusion**

**3.1 Instantaneous CSI**

which is given by,

**194**

and the DFC.

decisions.

available.

is the following:

variable *<sup>l</sup>*≜*l x*ð Þ¼ <sup>P</sup>*<sup>S</sup>*

#### **3.2 Statistical CSI**

If statistical CSI [15] is used for DF at the receiver instead of the instantaneous CSI extracted from the sensor signals received at the DFC, the optimal test statistics can be formulated as,

$$\Lambda\_{\text{opt}} \triangleq \ln \left[ \frac{p\left(\mathbf{y} | \mathcal{H}\_1\right)}{p\left(\mathbf{y} | \mathcal{H}\_0\right)} \right] \begin{array}{c} \hat{\mathcal{H}} = \mathcal{H}\_1 \\ > \\ < \\ \hat{\mathcal{H}} = \mathcal{H}\_0 \end{array} \tag{15}$$

of receive antennas at the DFC *N*) virtual MIMO system at 2.53 GHz with 20 MHz bandwidth and subcarrier spacing of around 0.15 MHz. The campaign is conducted with different spatial combinations of half-omnidirectional single transmit antennas representing the sensors, deployed in two different rooms of the Facility of Overthe-Air Research and Testing (FORTE) at Fraunhofer IIS in Ilmenau, Germany (Conference Room, C, located on the 1st floor and Instrumentation Room, I, located on the ground floor) and receive antennas mounted and co-located on an outside tower representing the DFC. The antennas emulating the sensors are deployed at different heights, namely, near the ground and ceiling and at heights of 1meter (m), 1.5 m and 2 m from the ground, sometimes on all 4 walls, sometimes on all 3 walls and sometimes only on 1 wall of each room at a time. The channel measurements are collected over a measurement set-up detailed in **Figure 2** and are

The dimensions of the two rooms selected are 8.45 m by 4.52 m by 2.75 m for the C room and 5.7 m by 3.5 m by 3 m for the I room. These rooms are chosen such that a variety of indoor environments is represented including I room with keyhole effect (no windows) and with no direct LOS communication, C room (smart office) and room cluttered with several noisy electrical and metering equipment (potential scenario for Industry 4.0). In both rooms, measurement set-up is repeated for stationary scenarios and scenarios with people moving around. Due to channel reciprocity conditions, it is assumed that channel estimates can be used for both uplink and downlink.

Large and small scale channel statistics are extracted from the channel impulse response (CIRs) and channel frequency responses (CFRs) recorded in the abovementioned campaign. In order to separate out large scale statistics, average received power and attenuation at each measurement location is calculated by averaging the recorded CIRs at that location. The pathloss exponent is determined from the slope of the best fit line to the logarithm of distances v/s logarithm of average attenuation plot. The probability density function (pdf) of deviation of each value of calculated attenuation from the best-fit line to the log–log plot yields the shadowing distribution. **Figures 3**–**5** demonstrates the log–log attenuation plots for three different measurement scenarios, static environment - Conference (SC), dynamic environment - Conference (DC) and static environment - Instrumentation (SI) rooms respectively, while **Table 1** summarizes the average values for the pathloss exponents (*ηP*) and mean and standard deviation (*μP*, *σP*) of the shadowing distributions

recorded using the MEDAV RUSK - HyE MIMO channel sounder.

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

in all the three above-mentioned measurement scenarios.

**4.2 Environment characterization**

**Figure 2.**

**197**

*Block diagram of measurement set-up.*

where *H*^ is the estimated hypothesis, Λopt is the Log-Likelihood-Ratio (LLR) of the optimal fusion rule and *γ* is the decision threshold to which Λopt is compared to. The threshold can be determined using either the Bayesian approach (i.e. the threshold is detected based on the one that minimizes the probability of error) or the Neyman-Pearson approach [16] (i.e. the threshold is detected based on the one that ensures fixed system false-alarm rate). An explicit expression of the LLR from Eq. (15) is given by,

$$\Lambda\_{\text{opt}} = \ln \left[ \frac{\sum\_{l=0}^{S} \frac{1}{\left(\sigma\_w^2 + \sigma\_h^2\right)^N} \exp\left(-\frac{\left\|\mathbf{y}\right\|^2}{\sigma\_w^2 + \sigma\_h^2}\right) P(l|\mathcal{H}\_1)}{\sum\_{l=0}^{S} \frac{1}{\left(\sigma\_w^2 + \sigma\_h^2\right)^N} \exp\left(-\frac{\left\|\mathbf{y}\right\|^2}{\sigma\_w^2 + \sigma\_h^2}\right) P(l|\mathcal{H}\_0)} \right] \tag{16}$$

where we have exploited the conditional independence of **y** from H*<sup>i</sup>* (given *l*).

In the case of conditionally (given H*i*) i.i.d. sensor decisions ( *PD*,*<sup>s</sup>* ð Þ¼ , *PF*,*<sup>s</sup>* ð Þ *PD*, *PF* , *s* ∈*S*) we have that *l*∣H<sup>1</sup> � **B**ð Þ *S*, *PD* and *l*∣H<sup>0</sup> � **B**ð Þ *S*, *PF* . Differently, when local sensor decisions are conditionally i.n.i.d. the pmfs *P l*ð Þ jH*<sup>i</sup>* are represented by the more general Poisson-Binomial distribution with expressions given by,

$$\begin{aligned} P(l|\mathcal{H}\_1) &= \sum\_{\mathbf{x}:\mathbf{x}(l)=l} \prod\_{\iota=1}^{\mathcal{S}} (P\_{D,\iota})^{\mathbf{x}\_{\iota}} \prod\_{u=1}^{\mathcal{S}} (P\_{D,u})^{(1-\mathbf{x}\_u)} \\ P(l|\mathcal{H}\_0) &= \sum\_{\mathbf{x}:\mathbf{x}(l)=l} \prod\_{\iota=1}^{\mathcal{S}} (P\_{F,\iota})^{\mathbf{x}\_{\iota}} \prod\_{u=1}^{\mathcal{S}} (P\_{F,u})^{(1-\mathbf{x}\_u)} \end{aligned} \tag{17}$$

It is to be noted here that calculating the sums in Eq. (17) practically becomes impossible with the increase in the number of sensors *S*. Several alternatives have been proposed across literature to tackle such exhaustive computations, which include fast convolution of individual Bernoulli probability mass functions (pmfs) [17], Discrete Fourier Transform (DFT) [18] based computation and recursionbased iterative approaches.

#### **4. Performance evaluation of MIMO decision fusion**

#### **4.1 Measurement campaigns**

An indoor-to-outdoor measurement campaign has been conducted in [19] for investigating propagation characteristics of an 8 � 8 (number of sensors *S* = number

#### *Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

of receive antennas at the DFC *N*) virtual MIMO system at 2.53 GHz with 20 MHz bandwidth and subcarrier spacing of around 0.15 MHz. The campaign is conducted with different spatial combinations of half-omnidirectional single transmit antennas representing the sensors, deployed in two different rooms of the Facility of Overthe-Air Research and Testing (FORTE) at Fraunhofer IIS in Ilmenau, Germany (Conference Room, C, located on the 1st floor and Instrumentation Room, I, located on the ground floor) and receive antennas mounted and co-located on an outside tower representing the DFC. The antennas emulating the sensors are deployed at different heights, namely, near the ground and ceiling and at heights of 1meter (m), 1.5 m and 2 m from the ground, sometimes on all 4 walls, sometimes on all 3 walls and sometimes only on 1 wall of each room at a time. The channel measurements are collected over a measurement set-up detailed in **Figure 2** and are recorded using the MEDAV RUSK - HyE MIMO channel sounder.

The dimensions of the two rooms selected are 8.45 m by 4.52 m by 2.75 m for the C room and 5.7 m by 3.5 m by 3 m for the I room. These rooms are chosen such that a variety of indoor environments is represented including I room with keyhole effect (no windows) and with no direct LOS communication, C room (smart office) and room cluttered with several noisy electrical and metering equipment (potential scenario for Industry 4.0). In both rooms, measurement set-up is repeated for stationary scenarios and scenarios with people moving around. Due to channel reciprocity conditions, it is assumed that channel estimates can be used for both uplink and downlink.

#### **4.2 Environment characterization**

**3.2 Statistical CSI**

can be formulated as,

Eq. (15) is given by,

given by,

based iterative approaches.

**4.1 Measurement campaigns**

**196**

Λopt ¼ ln

P*<sup>S</sup> l*¼0

*Wireless Sensor Networks - Design, Deployment and Applications*

*P l*ð Þ¼ jH<sup>1</sup>

*P l*ð Þ¼ jH<sup>0</sup>

**4. Performance evaluation of MIMO decision fusion**

P*<sup>S</sup> l*¼0

In the case of conditionally (given H*i*) i.i.d. sensor decisions

X *x*:*x l*ð Þ¼*l*

X *x*:*x l*ð Þ¼*l*

1 *σ*2 *w*þ*σ*<sup>2</sup>

1 *σ*2 *w*þ*σ*<sup>2</sup>

( *PD*,*<sup>s</sup>* ð Þ¼ , *PF*,*<sup>s</sup>* ð Þ *PD*, *PF* , *s* ∈*S*) we have that *l*∣H<sup>1</sup> � **B**ð Þ *S*, *PD* and *l*∣H<sup>0</sup> � **B**ð Þ *S*, *PF* . Differently, when local sensor decisions are conditionally i.n.i.d. the pmfs *P l*ð Þ jH*<sup>i</sup>* are represented by the more general Poisson-Binomial distribution with expressions

> Y *S*

ð Þ *PD*,*<sup>s</sup> xs*

ð Þ *PF*,*<sup>s</sup> xs*

It is to be noted here that calculating the sums in Eq. (17) practically becomes impossible with the increase in the number of sensors *S*. Several alternatives have been proposed across literature to tackle such exhaustive computations, which include fast convolution of individual Bernoulli probability mass functions (pmfs) [17], Discrete Fourier Transform (DFT) [18] based computation and recursion-

An indoor-to-outdoor measurement campaign has been conducted in [19] for investigating propagation characteristics of an 8 � 8 (number of sensors *S* = number

Y *S*

ð Þ *PD*,*<sup>u</sup>* ð Þ <sup>1</sup>�*xu*

ð Þ *PF*,*<sup>u</sup>* ð Þ <sup>1</sup>�*xu*

*u*¼1

Y *S*

*u*¼1

*s*¼1

Y *S*

*s*¼1

*<sup>h</sup>* ð Þ*<sup>N</sup>* exp � k k**<sup>y</sup>** <sup>2</sup>

*<sup>h</sup>* ð Þ*<sup>N</sup>* exp � k k**<sup>y</sup>** <sup>2</sup>

where we have exploited the conditional independence of **y** from H*<sup>i</sup>* (given *l*).

*σ*2 *w*þ*σ*<sup>2</sup> *h*

� �

*σ*2 *w*þ*σ*<sup>2</sup> *h*

� �

*P l*ð Þ jH<sup>1</sup>

*P l*ð Þ jH<sup>0</sup>

If statistical CSI [15] is used for DF at the receiver instead of the instantaneous CSI extracted from the sensor signals received at the DFC, the optimal test statistics

> � � *p* **y**jH<sup>0</sup> � � " #

where *H*^ is the estimated hypothesis, Λopt is the Log-Likelihood-Ratio (LLR) of the optimal fusion rule and *γ* is the decision threshold to which Λopt is compared to. The threshold can be determined using either the Bayesian approach (i.e. the threshold is detected based on the one that minimizes the probability of error) or the Neyman-Pearson approach [16] (i.e. the threshold is detected based on the one that ensures fixed system false-alarm rate). An explicit expression of the LLR from

H¼H ^ <sup>1</sup> > *γ* < H¼H ^ <sup>0</sup>

(15)

(16)

(17)

<sup>Λ</sup>opt <sup>≜</sup> ln *<sup>p</sup>* **<sup>y</sup>**jH<sup>1</sup>

Large and small scale channel statistics are extracted from the channel impulse response (CIRs) and channel frequency responses (CFRs) recorded in the abovementioned campaign. In order to separate out large scale statistics, average received power and attenuation at each measurement location is calculated by averaging the recorded CIRs at that location. The pathloss exponent is determined from the slope of the best fit line to the logarithm of distances v/s logarithm of average attenuation plot. The probability density function (pdf) of deviation of each value of calculated attenuation from the best-fit line to the log–log plot yields the shadowing distribution. **Figures 3**–**5** demonstrates the log–log attenuation plots for three different measurement scenarios, static environment - Conference (SC), dynamic environment - Conference (DC) and static environment - Instrumentation (SI) rooms respectively, while **Table 1** summarizes the average values for the pathloss exponents (*ηP*) and mean and standard deviation (*μP*, *σP*) of the shadowing distributions in all the three above-mentioned measurement scenarios.

**Figure 2.** *Block diagram of measurement set-up.*

**Figure 3.** *Log average attenuation versus log distance for conference - static environment.*

contribute constructively to reflected signal power than in an open indoor environ-

**Scenario** *η<sup>P</sup> μP*ð Þ dB *σP*ð Þ dB SC 2.72 1.22 2.4 SI 3.96 1.48 1.89 DC 2.56 1.77 3.6

To analyze the small scale channel statistics, the power delay profile (PDP) of the channel is drawn by averaging the power across all the delay bins for each sensor (**Table 2**). The average delay spread is the first moment and the root mean square (rms) delay spread is the square root of the second central moment of each sensor channel PDP respectively. The fading vector is obtained by concatenating CFRs at all the frequency points experienced over each sensor-DFC channel. The number of frequency points encountered is calculated by dividing the discrete bandwidth of the measured signal with the discrete coherence bandwidth of the sensor-DFC channel. The measurement is also used to deduce additional details like

**Scenario** *K* **Δ** *ξ*e**,**<sup>f</sup> **A** SC 2.5424 0 0.6511 7.64 SI 1.1217 0 0.5877 3.8158 DC 8.5287 0.6004 0.41 4.5227

ment (C room) stocked with wooden tables and chairs.

*Log average attenuation versus log distance for instrumentation - static environment.*

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

**Figure 5.**

**Table 1.**

**Table 2.**

**199**

*Small scale parameters [19].*

*Large scale parameters.*

**Figure 4.** *Log average attenuation versus log distance for conference - dynamic environment.*

The gamma distribution with mean of �1 � 6 dB and standard deviation of �5 � 5 dB offers a good approximation to the shadowing experienced in the campaign. The pathloss exponent varies between 2 to 4. Higher *η<sup>P</sup>* is experienced over a shorter distance direct link between the sensors and the DFC. Lower shadowing is observed in an environment (I room) cluttered with metallic surfaces that

*Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

**Figure 5.** *Log average attenuation versus log distance for instrumentation - static environment.*


**Table 1.** *Large scale parameters.*

contribute constructively to reflected signal power than in an open indoor environment (C room) stocked with wooden tables and chairs.

To analyze the small scale channel statistics, the power delay profile (PDP) of the channel is drawn by averaging the power across all the delay bins for each sensor (**Table 2**). The average delay spread is the first moment and the root mean square (rms) delay spread is the square root of the second central moment of each sensor channel PDP respectively. The fading vector is obtained by concatenating CFRs at all the frequency points experienced over each sensor-DFC channel. The number of frequency points encountered is calculated by dividing the discrete bandwidth of the measured signal with the discrete coherence bandwidth of the sensor-DFC channel. The measurement is also used to deduce additional details like


**Table 2.** *Small scale parameters [19].*

The gamma distribution with mean of �1 � 6 dB and standard deviation of �5 � 5 dB offers a good approximation to the shadowing experienced in the campaign. The pathloss exponent varies between 2 to 4. Higher *η<sup>P</sup>* is experienced over a shorter distance direct link between the sensors and the DFC. Lower shadowing is

observed in an environment (I room) cluttered with metallic surfaces that

*Log average attenuation versus log distance for conference - dynamic environment.*

*Log average attenuation versus log distance for conference - static environment.*

*Wireless Sensor Networks - Design, Deployment and Applications*

**Figure 3.**

**Figure 4.**

**198**

**Figure 6.** *CDF of K-factor for all environments.*

antenna correlation, determined from the correlation coefficients between each pair of fading vectors. The phase information from the complex CIRs is used to compute the steering vector for each transmit antenna.

The distribution of the derived fading vector fits the Rician distribution in most cases, with DR and TWDP distributions fitting the remaining few. **Figure 6** demonstrates the range of *K*-factor values for the Rician and TWDP distribution fitting and **Figure 7** plots the range of Δ values good for the TWDP distribution fitting where Δ-factor arises due to the presence of two strong interfering components and is given by <sup>Δ</sup> <sup>¼</sup> <sup>2</sup>*V*1*V*<sup>2</sup> *V*2 1þ*V*<sup>2</sup> 2 � 0 to 1, with *V*<sup>1</sup> and *V*<sup>2</sup> are the instantaneous amplitudes of the specular components. **Figures 8** and **9** depicts the range of antenna correlation coefficients and amount-of-fading (AF) values experienced over all the measurement scenarios, respectively. The average values for each of channel parameters, *K*, Δ, correlation coefficients, *ξ*e,<sup>f</sup> and AF (**A**) obtained after analysis of the measurement data over each measurement scenarios of SC, DC and SI.

A special scenario is observed in case of the I room where the propagation environment can be approximated by DR fading (*K* ¼ 0) distribution. As the I room is devoid of any windows, a rich scattering environment with 'keyhole' effect [20] is experienced with the existence of a waveguide propagation channel.

In the dynamic scenario, two sets of specular multipath components arrive at the receiver, one over the direct LOS link and the other due to reflection from the moving human body, thereby yielding en environment which can be accurately approximated by the TWDP fading distribution with *K* values ranging between 6 and 20 and Δ values varying between 0.1 and 0.9. Large distances between the transmit sensor and the DFC and nearness of most of the scattering surfaces to the sensors has resulted in similar AF values over all the measurement scenarios (refer to **Figure 9**).

summary, both fading and shadowing gets detrimental with the increase in intersensor distances. Separation between the sensors leads to very low coordination between them making the transmit signals vulnerable to noise, interference and fading, while a large number of different shadowing values are encountered resulting in higher shadowing variance and increased shadowing severity. For

*CDF of antenna correlation coefficient for all environments.*

*CDF of* Δ *for the dynamic environment in the conference room.*

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

**Figure 8.**

**201**

**Figure 7.**

Rich scattering and diffraction around the sensors in the windowless I room has resulted in low correlation between the transmitted signals, while a high correlation is observed among the sensor signals in the open environment of the C room. In

*Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

**Figure 7.** *CDF of* Δ *for the dynamic environment in the conference room.*

**Figure 8.** *CDF of antenna correlation coefficient for all environments.*

summary, both fading and shadowing gets detrimental with the increase in intersensor distances. Separation between the sensors leads to very low coordination between them making the transmit signals vulnerable to noise, interference and fading, while a large number of different shadowing values are encountered resulting in higher shadowing variance and increased shadowing severity. For

antenna correlation, determined from the correlation coefficients between each pair of fading vectors. The phase information from the complex CIRs is used to compute

The distribution of the derived fading vector fits the Rician distribution in most cases, with DR and TWDP distributions fitting the remaining few. **Figure 6** demonstrates the range of *K*-factor values for the Rician and TWDP distribution fitting and **Figure 7** plots the range of Δ values good for the TWDP distribution fitting where Δ-factor arises due to the presence of two strong interfering components and

the specular components. **Figures 8** and **9** depicts the range of antenna correlation coefficients and amount-of-fading (AF) values experienced over all the measurement scenarios, respectively. The average values for each of channel parameters, *K*, Δ, correlation coefficients, *ξ*e,<sup>f</sup> and AF (**A**) obtained after analysis of the measure-

A special scenario is observed in case of the I room where the propagation environment can be approximated by DR fading (*K* ¼ 0) distribution. As the I room is devoid of any windows, a rich scattering environment with 'keyhole' effect

In the dynamic scenario, two sets of specular multipath components arrive at the

Rich scattering and diffraction around the sensors in the windowless I room has resulted in low correlation between the transmitted signals, while a high correlation is observed among the sensor signals in the open environment of the C room. In

[20] is experienced with the existence of a waveguide propagation channel.

receiver, one over the direct LOS link and the other due to reflection from the moving human body, thereby yielding en environment which can be accurately approximated by the TWDP fading distribution with *K* values ranging between 6 and 20 and Δ values varying between 0.1 and 0.9. Large distances between the transmit sensor and the DFC and nearness of most of the scattering surfaces to the sensors has resulted in similar AF values over all the measurement scenarios (refer

ment data over each measurement scenarios of SC, DC and SI.

� 0 to 1, with *V*<sup>1</sup> and *V*<sup>2</sup> are the instantaneous amplitudes of

the steering vector for each transmit antenna.

*Wireless Sensor Networks - Design, Deployment and Applications*

is given by <sup>Δ</sup> <sup>¼</sup> <sup>2</sup>*V*1*V*<sup>2</sup>

**Figure 6.**

to **Figure 9**).

**200**

*V*2 1þ*V*<sup>2</sup> 2

*CDF of K-factor for all environments.*

over the direct LOS link is recorded to be equal to 40 dB yielding an equivalent

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

• *Impact of large scale channel parameters*: In order to analyze the impact of large scale channel effects, the small scale fading vectors are modeled to be Rayleigh

distributed (**h***<sup>n</sup>*,*<sup>s</sup>* � N ð Þ 0, 1 ). From the Decode-and-Fuse group of suboptimum fusion rules, MRC and Max-Log are used for DF over four different communication scenarios; No Shadowing ('Th'; *ηP*, *μP*, *σ<sup>P</sup>* = 1, 0 dB, 0 dB; **h***<sup>n</sup>*,*<sup>s</sup>* � N ð Þ 0, 1 ), SC (*ηP*, *μP*, *σ<sup>P</sup>* = 2.72, 1.22 dB, 2.4 dB), DC (*ηP*, *μP*, *σ<sup>P</sup>* = 2.56, 1.77 dB, 3.6 dB), SI (*ηP*, *μP*, *σ<sup>P</sup>* = 1.96, 1.48 dB, 1.89 dB), and the ROC

With increase in shadowing and pathloss, MRC outperforms Max-Log. Depen-

behind its poor performance in presence of rich shadowing. From the second group, CV-ML and CV-MMSE rules are used for DF over the above-mentioned four communication scenarios and the ROC performances are plotted in **Figure 11**. With increase in large scale channel effects, CV-MMSE outperforms CV-ML. The propagation environment has no impact on the performance of CV-ML owing to its dependence only on the SNR which is kept constant for the plots in **Figure 11**. In general, sub-optimum fusion rules perform better over SC scenario than over DC and over DC than over SI. With a strong LOS link existing in case of the SC scenario, it experiences the lowest pathloss. The DC scenario experiences higher pathloss due to penetration losses contributed by the moving human bodies.

• *Impact of small scale channel parameters*: In order to analyze the effect of the small scale channel effects, sub-optimum fusion rules are used for DF over four

*Comparative ROC for the first group of fusion rules for different measured large scale parameters (varying ηP, μ<sup>P</sup> and σP) with S* ¼ 8*, N* ¼ 8 *and Rayleigh distributed fading vector. Results for no shadowing condition,*

*<sup>w</sup>* is the principle reason

channel SNR of 40 ð Þ¼ � 20 20 dB.

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

performances are plotted in **Figure 10**.

different communication scenarios;

**Figure 10.**

**203**

*denoted by 'Th' are also plotted for comparison.*

dence of Max-Log rule on the noise spectral density *σ*<sup>2</sup>

**Figure 9.** *CDF of amount of fading (AF) for all environments.*


#### **Table 3.**

*Generalized range of values small scale parameters.*

indoor-to-outdoor virtual MIMO based communication scenario in WSNs, the propagation environment can experience a *K*-factor varying between 0 to 20 and a Δ-value varying between 0.1 to 0.9 in an open-concept smart office or home and in industry-like environments, as compiled in **Table 3**.

#### **4.3 Performance analysis**

The fusion performance of the formulated sub-optimum fusion rules is evaluated in this subsection, where the propagation environment is modeled using the accumulated measurements. Based on the observation in SC, DC and SI experimental scenarios, Ricean, TWDP and DR distributions are used to characterize the propagation channel.

#### *4.3.1 Receiver operating characteristics (ROC)*

For the different fusion rules of Section 3, probability of detection (*PD*<sup>0</sup> ) is plotted against probability of false alarm (*PF*<sup>0</sup> ) (commonly referred to as the Receiver Operating Characteristics (ROC)) for *S* ¼ 8 and *N* ¼ 8 in presence of a fixed channel SNR of 20 dB. The particular value of 20 dB is chosen for the plots, as the average attenuation **A**ð Þ*i* recorded for any measurement location *i* is approximately around 20 dB across all measurement environments. The measured SNR

*Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

over the direct LOS link is recorded to be equal to 40 dB yielding an equivalent channel SNR of 40 ð Þ¼ � 20 20 dB.

• *Impact of large scale channel parameters*: In order to analyze the impact of large scale channel effects, the small scale fading vectors are modeled to be Rayleigh distributed (**h***<sup>n</sup>*,*<sup>s</sup>* � N ð Þ 0, 1 ). From the Decode-and-Fuse group of suboptimum fusion rules, MRC and Max-Log are used for DF over four different communication scenarios; No Shadowing ('Th'; *ηP*, *μP*, *σ<sup>P</sup>* = 1, 0 dB, 0 dB; **h***<sup>n</sup>*,*<sup>s</sup>* � N ð Þ 0, 1 ), SC (*ηP*, *μP*, *σ<sup>P</sup>* = 2.72, 1.22 dB, 2.4 dB), DC (*ηP*, *μP*, *σ<sup>P</sup>* = 2.56, 1.77 dB, 3.6 dB), SI (*ηP*, *μP*, *σ<sup>P</sup>* = 1.96, 1.48 dB, 1.89 dB), and the ROC performances are plotted in **Figure 10**.

With increase in shadowing and pathloss, MRC outperforms Max-Log. Dependence of Max-Log rule on the noise spectral density *σ*<sup>2</sup> *<sup>w</sup>* is the principle reason behind its poor performance in presence of rich shadowing. From the second group, CV-ML and CV-MMSE rules are used for DF over the above-mentioned four communication scenarios and the ROC performances are plotted in **Figure 11**. With increase in large scale channel effects, CV-MMSE outperforms CV-ML. The propagation environment has no impact on the performance of CV-ML owing to its dependence only on the SNR which is kept constant for the plots in **Figure 11**. In general, sub-optimum fusion rules perform better over SC scenario than over DC and over DC than over SI. With a strong LOS link existing in case of the SC scenario, it experiences the lowest pathloss. The DC scenario experiences higher pathloss due to penetration losses contributed by the moving human bodies.

• *Impact of small scale channel parameters*: In order to analyze the effect of the small scale channel effects, sub-optimum fusion rules are used for DF over four different communication scenarios;

#### **Figure 10.**

*Comparative ROC for the first group of fusion rules for different measured large scale parameters (varying ηP, μ<sup>P</sup> and σP) with S* ¼ 8*, N* ¼ 8 *and Rayleigh distributed fading vector. Results for no shadowing condition, denoted by 'Th' are also plotted for comparison.*

indoor-to-outdoor virtual MIMO based communication scenario in WSNs, the propagation environment can experience a *K*-factor varying between 0 to 20 and a Δ-value varying between 0.1 to 0.9 in an open-concept smart office or home and in

**Scenario** *K* **Δ** SC 0.5 to 4 — SI 0 — DC 6 to 20 0.1 to 0.9

The fusion performance of the formulated sub-optimum fusion rules is evaluated in this subsection, where the propagation environment is modeled using the accumulated measurements. Based on the observation in SC, DC and SI experimental scenarios, Ricean, TWDP and DR distributions are used to characterize the

For the different fusion rules of Section 3, probability of detection (*PD*<sup>0</sup> ) is plotted against probability of false alarm (*PF*<sup>0</sup> ) (commonly referred to as the Receiver Operating Characteristics (ROC)) for *S* ¼ 8 and *N* ¼ 8 in presence of a fixed channel SNR of 20 dB. The particular value of 20 dB is chosen for the plots, as the average attenuation **A**ð Þ*i* recorded for any measurement location *i* is approximately around 20 dB across all measurement environments. The measured SNR

industry-like environments, as compiled in **Table 3**.

*Generalized range of values small scale parameters.*

*CDF of amount of fading (AF) for all environments.*

*Wireless Sensor Networks - Design, Deployment and Applications*

*4.3.1 Receiver operating characteristics (ROC)*

**4.3 Performance analysis**

**Table 3.**

**Figure 9.**

propagation channel.

**202**

3.DC scenario with TWDP fading vectors (*ηP*, *μP*, *σ<sup>P</sup>* = 2.56, 1.77 dB, 3.6 dB;

*<sup>s</sup>* with ½*Ks*, min , *Ks*, max � ¼ ½ � 6, 20 , ½Δ*s*, min , Δ*s*, max � ¼ ½ � 0*:*1, 0*:*9 ),

and the ROC performances are plotted in **Figures 12**–**14** respectively.

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

4.SI scenario with DR fading vectors (*ηP*, *μP*, *σ<sup>P</sup>* = 1.96, 1.48 dB, 1.89 dB;

If the small scale fading vectors are Rayleigh distributed, EGC performs best and CV-ML performs worst. CV-ML is worst under all considered propagation scenario. Max-Log performs a tad bit better than CV-ML over Rician, TWDP and DR fading channels. If the fading vectors are Rician and TWDP distributed, MRC, EGC and CV-MMSE perform almost equivalently. CV-MMSE however champions over MRC and EGC if the small scale channel effects follow DR distribution. Some analogies between performances under measured environment and simulated (as in [21]) can also be concluded from the results presented in **Figures 10–14**. In both cases ROC performance demonstrates that CV-MMSE performs better than CV-ML rule, CV-MMSE performs close to MRC/EGC rules, while CV-ML exhibits the worst

In **Figures 15** and **16**, we show system probabilities of detection, *PD*<sup>0</sup> with two groups of fusion rules as an interpolated function of the number of receive antennas

• *Impact of measurement environment*: If both large and small scale channel parameters are varied, the probability of detection *PD*<sup>0</sup> with MRC and CV-MMSE rules saturates with the increase in *N* over the SC scenario, but increases

*Comparative ROC for all fusion rules for the* DC *environment with S* ¼ 8*, N* ¼ 8 *in TWDP fading condition.*

*Results for Rayleigh fading-only condition ('Th') are plotted for comparison.*

**h**TWDP

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

*Ks* ¼ 0),

performance.

*4.3.2 PD*<sup>0</sup> v/s *N*

**Figure 13.**

**205**

*N* under *PF*<sup>0</sup> ≤0*:*01.

#### **Figure 11.**

*Comparative ROC for the second group of fusion rules for different measured large scale parameters (varying ηP, μ<sup>P</sup> and σP) with S* ¼ 8*, N* ¼ 8 *and Rayleigh distributed fading vector. Results for no shadowing condition, denoted by 'Th' are also plotted for comparison.*


#### **Figure 12.**

*Comparative ROC for all fusion rules for the* SC *environment with S* ¼ 8*, N* ¼ 8 *in Rician fading condition. Results for Rayleigh fading-only condition ('Th') are plotted for comparison.*


and the ROC performances are plotted in **Figures 12**–**14** respectively.

If the small scale fading vectors are Rayleigh distributed, EGC performs best and CV-ML performs worst. CV-ML is worst under all considered propagation scenario. Max-Log performs a tad bit better than CV-ML over Rician, TWDP and DR fading channels. If the fading vectors are Rician and TWDP distributed, MRC, EGC and CV-MMSE perform almost equivalently. CV-MMSE however champions over MRC and EGC if the small scale channel effects follow DR distribution. Some analogies between performances under measured environment and simulated (as in [21]) can also be concluded from the results presented in **Figures 10–14**. In both cases ROC performance demonstrates that CV-MMSE performs better than CV-ML rule, CV-MMSE performs close to MRC/EGC rules, while CV-ML exhibits the worst performance.
