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This paper presents the theoretical analysis, simulation results and suggests design in digital technology of a physical layer for wireless sensor networks. The proposed design is able to mitigate fading inside communication channel. To mitigate fading the chip interleaving technique is proposed. For the proposed theoretical model of physical layer, a rigorous mathematical analysis is conducted, where all signals are presented and processed in discrete time domain form which is suitable for further direct processing necessary for devices design in digital technology. Three different channels are used to investigate characteristics of the physical layer: additive white Gaussian noise channel (AWGN), AWG noise and flat fading channel and AWG noise and flat fading channel with interleaver and deinterleaver blocks in the receiver and transmitter respectively. Firstly, the mathematical model of communication system representing physical layer is developed based on the discrete time domain signal representation and processing. In the existing theory, these signals and their processing are represented in continuous time form, which is not suitable for direct implementation in digital technology. Secondly, the expressions for the probability of chip, symbol and bit error are derived. Thirdly, the communication system simulators are developed in MATLAB. The simulation results confirmed theoretical findings.

Wireless sensor networks based on IEEE Standard 802.15.4 [

There are not too many references talking exclusively about physical layer design. A paper related to the modulator and demodulator design is presented in [

The processing in the physical layer includes techniques applied in direct sequence spread spectrum (DSSS) systems and code division multiple access systems (CDMA). It was shown that the chip interleaving techniques can reduce the influence of fading in DSSS and CDMA systems [

The contributions of this paper are as follows. Firstly, the mathematical model of communication system representing physical layer is based on the discrete time domain signal representation and processing. In the existing theory, these signals and their processing are represented in continuous time form. However, signals in analog form are not suitable for direct implementation of the system mathematical model into digital technology; thus, all signals in this paper are represented in discrete time form. Secondly, in contrast to published work in [

The paper contains six Sections starting with this Introduction. In the second Section a theoretical structure of the system is presented and procedure of signal processing in each block is demonstrated when AWG noise is present in the channel. The complete analysis is done for a single-correlator and

_{jn}_{in}_{jn}_{in}_{jn}^{K}

In this case the spreader is represented by a multiplier, as shown on the left hand upper side of _{1} at the receiver side. The chip sequence _{I}_{Q}_{c} is the energy per chip and _{I}_{Q}_{c}

The noise is to be generated at discrete time instants defined by _{N} =_{0}/2 and _{I}_{Q}

Block schematic of communication system.

The received noisy signal _{R}

The samples of this signal are added in the chip interval (corresponds to integration in continuous time systems). Because, in the case of a single-correlator receiver, the source generates binary signal the output of the transmitting spreader was the first spreading sequence _{i}_{1}(_{i}_{1}(_{i}_{i}

A multiplexer (MUX) is used to combine in-phase and quadrature sequences back into a 2_{i}_{i}_{i}_{1}, _{i}_{1} can be approximated by the Gaussian random variable, with its mean
_{c}

The powers for all chips are equal and the power of noise is equal to the noise variance. Therefore, the variance of _{1} can be expressed in this general form
_{1} is Gaussian and can be expressed as
_{N}^{2}. For variance σ^{2} = _{0} and the bandwidth _{c}_{N}_{0}_{b }_{c}

In this case, the source can generate ^{K}

The receiver for binary symbols transmission is analyzed in previous section. In that case it was sufficient to have the first correlator in the receiver and the symbol to sequence conversion is performed by a multiplier as shown separately in _{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).

If the source generates

Suppose that the first sequence is sent. Following the procedure of a single-correlator receiver modeling, the output of the

The first term is inter-sequence interference and the second term is noise term. The value _{n}

Suppose the threshold value inside the decision circuit is _{1} = _{1} =

Inserting Equation (21), and having in mind statistical independence between variables _{n}_{1} values is the probability of that the first symbol is correctly transmitted. It can be expressed as
^{K}

The flat fading channel results in multiplicative distortion of the transmitted signal _{R}_{1}(

This expression was derived by finding the mean values and variance for α as a random variable and analyzing _{1} as a random function, as it was done in [_{b}_{c}

Following the procedure explained in

From the expressions for a single and _{n}_{1}). Following the procedure in

In this expression the probability of bit error depends on the fading coefficient α, which is inside the expression for the variance of _{n}

The fading increases significantly BER inside communications systems. It is assumed that a particular fading coefficient affects each sequence/symbol transmitted. If interleaver/deinterleaver blocks are included into the transceiver structure, as shown in

The received chip samples are applied to the chip sequence correlator input. The output sample of the first correlator is

The probability of bit error can be calculated as

If the source generates binary symbols and the spreading sequence is in binary form, we may have

This probability depends on a random function which is a sum of 2

This expression was derived by finding the mean values and variance for α_{i}

As can be seen from this expression, the probability of error will decrease as the spreading factor 2

Following the procedure explained in

In this expression, the probability of bit error depends on the sum of fading coefficient and their squared values. Therefore, the probability of error needs to be calculated as the mean value of this random function.

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

The simulation was conducted according to the following procedure. The sequence of message bits is generated at the input of the receiver. The spreading sequence is assigned to each bit or the symbol, depending on the systems structure. Then the spreading sequence modulated the carrier using OQPSK modulation, which is defined by the Standard for WSNs [

The detected bits are compared with the message bits generated at the input of the system and BER rate is calculated as the ratio of the number of errors detected and the number of bits transmitted. The Chebyshev’s inequality method for accurate estimation of each BER value was used. According to this method it was needed to transmit at least ^{7}

Two sets of simulators are designed. The first set included simulators with one correlator receiver. In this case the bits were directly related to the spreading sequence as defined by the Standard for WSNs [

In the following simulations, the spreading sequences have been chosen according to the standard for sensor networks [

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

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).

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).

As can be seen from ^{−2} to 1 × 10^{−4} (more than two order of magnitude) for SNR = 10 dB, as can be seen in

Further improvement could be achieved if the spreading factor is increased. Therefore, the chip interleaving is efficient technique to mitigate fading influence and contributes directly to the power saving in wireless sensor networks. Namely, for the required BER a smaller value of SNR is required if the system includes interleaver and deinterleaver blocks.

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).

Rayleigh fading and AWGN are present in the channel.

Fading channel and interleaver and deinterleaver are present on the transceiver.

The expected gain is between 8 and 9 dB. To estimate the gain in BER values the BER curve was shifted for this amount as shown in

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

The paper contains analytical model of a communication system (transmitter, receiver and channel) for physical layer of wireless sensor networks in the case when all signals are represented and processed in discrete time domain suitable for direct design in digital technology. The expressions for BER rate for a single and an

The authors declare no conflict of interest.