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The objective of this research is to track the phase changes in Binary Phase Shift Keying (BPSK) modulated signal in ZigBee communication systems using discrete Kalman Filter (KF). Therefore, Kalman Filtering is used to estimate and optimize the carrier phase of BPSK modulated signal, in the presence of Additive White Gaussian Noise (AWGN) channel, by minimizing the phase deviation error. Therefore, a simulation model, using MATLAB, will be created to demonstrate ZigBee transmission system with the impact of the integrated filter. The expected results will show that Kalman Filter tracks the phase of BPSK modulated signal correctly and the performance of tracking will be measured by Mean Square Error (MSE) with respect to Signal to Noise Ratio (SNR). This study proposes a new method of phase tracking in ZigBee receivers in the presence of AWGN channel which can be extended to Internet of Things (IoT) applications.

ZigBee is a low data rate wireless personal area networks (LR-WPAN) which is used for a short distance communication system. Therefore, it is a promising technology due to its low cost, low power consumption, and long battery life [

Kalman Filter is named after Rudolf Kalman in 1960 and it is a linear quadrature estimation used in system tracking, control, and communication systems. The Kalman Filter is used to predict and update the states of the filter; therefore, with a prior knowledge of the state dynamics behavior, Kalman can estimate the next state of the filter and correct the estimate [

The Costas PLL, which is a modified version of Phase Locked Loop (PLL), is a useful technique in both analog and digital communications systems. It can be used as a demodulator for BPSK to recover the transmitted data from the BPSK modulated signal [

The authors in [

The objective of this paper is to track BPSK modulated signal phase by discrete Kalman Filter in ZigBee receivers in the presence of Additive White Gaussian Noise (AWGN) channel. This paper has been organized as follows. Introduction is described in the Section 1. Section 2 describes the BPSK. Section 3 presents the AWGN channel. Section 4 illustrates the discrete Kalman Filter estimator. Section 5 shows the simulation and results. Section 6 presents the conclusion of our work.

ZigBee operated in the frequency band 868/950 MHz employs Binary Phase Shift Keying (BPSK) modulation technique for pulse shaping. For frequency range 868/950 MHz, ZigBee data rate is 20 Kb/sec and DSSS is its spreading method. ZigBee baseband chip is described by the raised cosine pulse shape with a roll-off factor = 1. Therefore, the cosine pulse shape g(t) is described by [

g ( t ) = { sin π t / T c π t / T c cos π t / T c 1 − 4 t 2 / T c 2 , t ≠ 0 1 , t = 0

The following shows the derivation of BPSK modulated signal starting from the phase-shift keying (PSK) modulation with M signal waveforms having an equal energy [

S m ( t ) = R e [ g ( t ) e i 2 π ( m − 1 ) M e i 2 π f c t ] , m = 1 , 2 , ⋯ , M = g ( t ) cos [ 2 π f c t + 2 π M ( m − 1 ) ] = g ( t ) cos ( θ m ) cos 2 π f c t − g ( t ) sin ( θ m ) sin 2 π f c t (1)

where,

g ( t ) is the signal pulse shape;

f c is the carrier frequency.

θ m = 2 π M ( m − 1 ) (2)

M is the number of possible carrier phases.

In BPSK there are two possible phases; so, M = 2 and m = 1 , 2 .

By substituting the values of M and m in (2) and (1) respectively:

θ 1 = 0 , for binary 1;

θ 2 = π , for binary 0.

Finally, BPSK can be represented with the two different possible phases as [

S 1 ( t ) = A c cos ( 2 π f c t ) , 0 ≤ t ≤ T b for binary 1;

S 0 ( t ) = A c cos ( 2 π f c t + π ) , 0 ≤ t ≤ T b for binary 0.

In general, BPSK signal is represented by [

S ( t ) = A c cos ( ω t + φ ( t ) ) (3)

where,

φ ( t ) is the phase change of BPSK signal;

ω = 2 π f c ;

T b is the bit duration;

A c is the amplitude of the carrier.

Obviously, the BPSK modulated signal has no Quadrature component, it has only In-phase component.

r ( t ) = S m ( t ) + n (t)

where r ( t ) represents the received signal from BPSK modulated signal S m ( t ) plus n ( t ) which is a waveform containing the white Gaussian noise. The white Gaussian has a zero mean and spectral density of N 0 / 2 [

In this paper, Kalman Filter is used to track the phase changes φ ( t ) in the BPSK modulated signal in Equation (3) and the following derivation is given by [

x ( t ) = [ cos ( ω t ) − ω sin ( ω t ) ] (4)

The system in (4) is continuous and it should be discretized which was done in [

x ( k ) = x ( k − 1 ) + Δ t x ˙ ( k ) (5)

where

k is a time step;

∆t is the sampling time.

The discrete state space transition model for estimating the phase change of the BPSK signal is defined as [

x ( k + 1 ) = F x ( k ) + w ( k ) (6)

y ( k ) = H x ( k ) + v ( k ) (7)

where,

x(k) represents the state vector of BPSK modulated signal at time step k;

F represents the state transition matrix;

y(k) BPSK signal’s phase at time step k;

H observation matrix;

v(k) measurement noise vector at time step k;

w(k) process noise vector at time step k.

The process and measurement noise have zero mean and they both are uncorrelated as done in [

E [ v ( k ) v T ( j ) ] = { R ( k ) , i = k 0 , i ≠ k k , j = 0 , 1 , 2 , ⋯

E [ w ( k ) w T ( j ) ] = { Q ( k ) , i = k 0 , i ≠ k k , j = 0 , 1 , 2 , ⋯

E [ w ( k ) v T ( j ) ] = E [ v ( j ) w T ( k ) ] = 0 k , j = 0 , 1 , 2 , ⋯

Kalman Filter AlgorithmThe Kalman Filter is used to predict and update the states of the filter; therefore, it can predict which phase change has occurred. It starts with some initial estimate x 0 and some initial update error covariance matrix P 0 . After that, Kalman gain G is applied to correct the prediction. The error terms proposed in [

The following describes the computation process which was done in [

1) Making some initializations for x 0 & P 0

2) Computing Kalman gain G with respect to a priori process covariance P ( k | k − 1 ) :

G ( k ) = P ( k | k − 1 ) ⋅ H T ⋅ [ H ⋅ P ( k | k − 1 ) ⋅ H T + R ] − 1

3) Calculating a posteriori covariance which expresses the update covariance of the process in terms of a priori covariance P ( k | k − 1 ) and Kalman gain G:

P ( k | k ) = [ I − G ( k ) ⋅ H ] ⋅ P ( k | k − 1 )

4) Calculating the Predicted Process Covariance:

P ( k + 1 ) = F ⋅ P ( k | k ) ⋅ F T + Q

Note: P ( k + 1 ) will be a previous estimate for next time step k.

5) By taking the measurement y ( k ) , we can define the current estimate x ^ ( k | k ) as:

x ^ ( k | k ) = x ^ ( k | k − 1 ) + G ( K ) ⋅ [ y ( k ) − H ⋅ x ^ ( k | k − 1 ) ]

where the previous estimate x ^ ( k | k − 1 ) can be defined as:

x ^ ( k | k − 1 ) = F x ^ ( k − 1 | k − 1 ) + Δ ⋅ u ( k − 1 )

6) Updating the estimate of the phase is done by choosing the minimum error value between the two error equations as follows:

e 1 = y ( k ) − H x ( k | k − 1 )

e 2 = y ( k ) + H x ( k | k − 1 )

ZigBee transmission model has been designed by MATLAB for the simulation in order to show the performance of Kalman filter in tracking the phase of BPSK modulated signal in ZigBee receivers under AWGN channel.

We successfully implemented a discrete Kalman Filter to track the phase changes in the BPSK in ZigBee communication systems in the presence of AWGN channel. The ZigBee transmission model was designed based on IEEE 802.15.4 standard. Kalman showed superiority of the tracking with respect to different SNR values. The findings of our study were attractive to extend the work to make a decision on the bit received and apply Kalman to different modulations schemes.

The authors declare no conflicts of interest regarding the publication of this paper.

Alqahtani, A. and Zohdy, M. (2019) ZigBee Signal Phase Tracking Using a Discrete Kalman Filter Estimator under AWGN Channel. Journal of Computer and Communications, 7, 10-17. https://doi.org/10.4236/jcc.2019.71002