# Physical Layer Design in Wireless Sensor Networks for Fading Mitigation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model of a System in the Presence of Gaussian Noise

_{jn}(k). To each symbol a spreading sequences c

_{in}(k) is assigned to obtain a chip sequence m(k) = b

_{jn}(k)c

_{in}(k) that comes to the input of a multiplexer. If a symbol is in binary form, b

_{jn}(k) = ±1, then the system represents a direct sequence spread spectrum system (DSSSS) and the receiver has one correlator as it was said before. If a combination of K bits represents a symbol then the number of sequences sent is N = 2

^{K}, which is equal to the number of required correlators at the receiver side. Thus, in the following sections, two cases, a single and N-correlator receiver, will be analyzed.

#### 2.1. Single-Correlator Receiver

_{1}at the receiver side. The chip sequence m(k) is split into in-phase and quadrature sequences using the demultiplexer block (DEMUX) in such a way that the even-indexed chip sequence m

_{I}(k) modulates the in-phase carrier and odd-indexed chip sequence m

_{Q}(k) modulates the quadrature carrier , where E

_{c}is the energy per chip and M is the number of interpolated samples contained in one chip interval. Therefore, the transmitted signal can be defined as

_{I}(k) and m

_{Q}(k) are in-phase and quadrature chip sequences expressed in discrete time domain respectively and Ω

_{c}is the frequency of the carrier. It is important to note that k is a discrete time variable.

_{N}=N

_{0}/2 and n

_{I}and n

_{Q}are in-phase and quadrature noise samples of zero mean and unit variance. The block schematic of this noise generator is presented inside Figure 1. The noise is expressed in this form to comply with the applied signal processing demodulation procedure of both the signal and noise at the receiver side. Namely, in simulation, it is important to achieve that the power of the noise in respect to the power of the signals are controlled at all times at the transmitter side.

_{R}(k)= s(k) = n(k) is demodulated using a correlator that consists of a multiplier and an adder. It is sufficient to present the processing of the signal in one branch of the demodulator because the processing in both branches is equivalent. The signal at output of the receiver multiplier is

_{i}

_{1}(k), i.e., m(k) = c

_{i}

_{1}(k), and a random sample z

_{i}of a random process Z

_{i}in I branch is obtained, i.e.,

_{i}

_{i}are samples of the in-phase and quadrature baseband noise having zero mean and unit variance. In the correlator block, a locally generated reference chip sequences (c

_{i}

_{1}, i 1,2,3,...,2β) is multiplied with the incoming z

_{i}random sequence and then the products are added inside the bit interval. The resulting sum for the first positive message bit sent is

_{1}can be approximated by the Gaussian random variable, with its mean

_{c}, we may have

_{1}can be expressed in this general form

_{1}is Gaussian and can be expressed as

_{N}= Mσ

^{2}. For variance σ

^{2}= BN

_{0}and the bandwidth B=1/2T

_{c}=1/2M, the energy is calculated to be E

_{N}= N

_{0}/2. If the source generates binary bits and the spreading sequence is in binary form, we may have

_{b }= 2βE

_{c}.

#### 2.2. N-Correlator Receiver

^{K}. The modulation and demodulation will be the same as in the case of a single-correlator receiver. However, the correlation for each sequence must be performed in its own receiver correlator. Therefore, at the receiver, a bank of N correlators needs to be implemented, as shown in Figure 1.

_{1}at the output of the first correlator is calculated in Equation (7) and the related mean, variance and probability of error in Equations (9), (12) and (16).

_{n}is a random sample of a random variable defined for the n-th correlator output. Due to the central limit theorem (CLT) this random variable can be approximated by the Gaussian random variable with zero mean

_{1}= w. The probability of correct decision is equal to the conditional probability of that all outputs are less than this threshold value for the given value w

_{1}= w, i.e.,

_{n}, this expression becomes

_{1}values is the probability of that the first symbol is correctly transmitted. It can be expressed as

^{K}. By inserting Equation (21) the bit error probability can be calculated as

## 3. Communication System Analysis in the Presence of Fading

#### 3.1. Single-Correlator Receiver

_{R}(k) can be expressed as [13]

_{1}(k) in Equation (7), can be expressed as

_{1}as a random function, as it was done in [15]. The probability of error then can be derived in this closed form

_{b}= 2βE

_{c}.

#### 3.2. N-Correlator Receiver

_{n}) and the mean value that depends on α (for w

_{1}). Following the procedure in Section 2.2, the probability of bit error can be expressed as

_{n}. Therefore the probability of error needs to be calculated as the mean value of this random function to get the probability of bit error for fading channel as

## 4. Interleaver Communication System Analysis in the Presence of Fading

#### 4.1. Single Correlator Receiver

_{i}as independent and identically distributed random variables, as it was done in [15]. The probability of error is derived in this closed form

#### 4.2. N-Correlator Receiver

## 5. Simulation Results and Discussions

_{N}. The fading is generated using a fading generator with the empirical density function which corresponds to the theoretical Rayleigh density function.

^{7}bits to achieve 99% confidence in BER estimates [14]. This BER value estimation was done for each signal to noise ratio that is defined as abscissa value in all Figures presented in this Section.

#### 5.1. Single-Correlator Receiver

**Figure 2.**BER curves for a single-correlator receiver in presence of additive white Gaussian noise channel (AWGN): theory (blue) and simulation (red).

**Figure 3.**BER curves for a single-correlator receiver in the presence of AWGN and fading: theoretical (blue), simulation (red) for fading, and theory for noise only (black).

**Figure 4.**BER curves for a single correlator receiver with interleavers in presence of AWGN and fading: theoretical (blue) for fading, simulated (red) for fading with interleavers and theoretical for noise only (black).

^{−2}to 1 × 10

^{−4}(more than two order of magnitude) for SNR = 10 dB, as can be seen in Figure 4. By using interleavers and deinterleavers the influence of fading inside each symbol is practically randomized and the BER curve is coming closer to the curve representing BER when noise only is present in the channel (black curve). The interleaver curve obtained by the simulation can be easily confirmed by the theoretical Equation (46).

#### 5.2. N-Correlator Receiver

**Figure 5.**BER curves for 16-correlator receiver in presence of AWGN: theoretical (black) chip error rate (CER), simulation CER (red) and BER on CER SNR scale (red dashed).

**Figure 8.**Estimated position of the BER curve when fading channel and interleavers and deinterleavers are present in communication system.

## 6. Conclusions

## Conflict of Interest

## References

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**MDPI and ACS Style**

Berber, S.; Chen, N. Physical Layer Design in Wireless Sensor Networks for Fading Mitigation. *J. Sens. Actuator Netw.* **2013**, *2*, 614-630.
https://doi.org/10.3390/jsan2030614

**AMA Style**

Berber S, Chen N. Physical Layer Design in Wireless Sensor Networks for Fading Mitigation. *Journal of Sensor and Actuator Networks*. 2013; 2(3):614-630.
https://doi.org/10.3390/jsan2030614

**Chicago/Turabian Style**

Berber, Stevan, and Nuo Chen. 2013. "Physical Layer Design in Wireless Sensor Networks for Fading Mitigation" *Journal of Sensor and Actuator Networks* 2, no. 3: 614-630.
https://doi.org/10.3390/jsan2030614