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Optimizing the estimates of received power signals is important as it can improve the process of transferring an active call from one base station in a cellular network to another base station without any interruptions to the call. The lack of effective techniques for estimation of shadow power in fading mobile wireless communication channels motivated the use of Kalman Filtering (KF) as an effective alternative. In our research, linear second-order state space Kalman Filtering was further investigated and tested for applicability. We first created simulation models for two KF-based estimators designed to estimate local mean (shadow) power in mobile communications corrupted by multipath noise. Simulations were used extensively in the initial stage of this research to validate the proposed method. The next challenge was to determine if the models would work with real data. Therefore, in [1] we presented a new technique to experimentally characterize the wireless small-scale fading channel taking into consideration real environmental conditions. The two-dimensional measurement technique enabled us to perform indoor experiments and collect real data. Measurements from these experiments were then used to validate simulation models for both estimators. Based on the indoor experiments, we presented new results in [2], where we concluded that the second-order KF-based estimator is more accurate in predicting local shadow power profiles than the first-order KF-based estimator, even in channels with imposed non-Gaussian measurement noise. In the present paper, we extend experiments to the outdoor environment to include higher speeds, larger distances, and distant large objects, such as tall buildings. Comparison was performed to see if the system is able to operate without a failure under a variety of conditions, which demonstrates model robustness and further investigates the effectiveness of this method in optimization of the received signals. Outdoor experimental results are provided. Findings demonstrate that the second-order Kalman filter outperforms the first-order Kalman filter.

Because wireless technology and smart cell phones are experiencing dramatic growth, the accurate estimation of local mean (shadow) power in a cell phone is becoming a popular area of challenge for engineers in both industry and academia. Researchers are encouraged to find ways to enhance device performance in power control and handoff, particularly to address mobility-induced fading in metropolitan areas.

Wireless cell phones operate by transferring information over a distance between two or more stations that are not connected by cables. Instead, cell phones use radio waves to carry information, such as sound, by systematically modulating some property of electromagnetic energy waves transmitted through space, such as their amplitude, frequency, phase, or pulse width [

Two significant forms of fading in cellular communications are multipath and shadow fading. Since cell phone users tend to move a lot, received signal strength fluctuates with these two multiplicative forms of fading [

Having an accurate estimate of the shadowing component of a received power signal will allow the mobile communication system to efficiently compensate for the signal degradation that will occur. As a result, it can help the system perform handoff at the most effective times (predict when and where to handoff user). In [

Kalman filtering is a very effective algorithm that uses a series of measured observations and produces optimal estimates of states as explained in [

The next challenge was to find a way to measure cell phone signal strength outside of the lab environment and to test our second-order KF-based estimator. In this paper, we looked at mobility-induced fading and present experimental results from the outdoor environment that further confirmed validation of the proposed method. We will explain how Kalman filter method can be applied in optimization of received signals in mobile communications. The system was able to operate without a failure under a variety of conditions, which demonstrates model robustness. In subsequent sections, it will be demonstrated that the second-order KF based estimator we designed exceeds the performance of the first-order KF-based estimator, even in the outdoor environment where parameters for mobile velocity varied.

The description of Shadow Power Signal and its models as they pertain to our problem are presented in this section. In a wireless cellular radio environment, a model for an instantaneous received power signal l(t) at a cell phone is given in

To solve the problem, we start with the multipath model shown in Equation (2).

where:

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Equation (3) shows a widely accepted first-order state space model for the shadow process given by [

First shadow power coefficient

where,

Finally, system noise covariance is given in Equation (8), where term

Kalman Filter theory was developed and introduced in 1960 by Dr. Rudolf Kalman. This led to the use of the Kalman filter during the Apollo program, carried out by NASA, which accomplished landing the first humans on the Moon. Since then, his contributions and thoughts educated and inspired inventors across many disciplines. As a result, the Kalman filter has been the subject of extensive research and application, especially in digital computing.

In this research, we applied the Kalman filter algorithm to estimate power signal in a mobile communication corrupted by multipath noise. The Kalman filter is a form of a linear algorithm for optimal recursive estimation of a system state with a specific set of output equations. The estimates are calculated every time a new measurement is received. Data received is processed sequentially, so it is not necessary to store the complete data set or to reprocess existing data when new measurement data is received.

This section derives the equations of the discrete-time Kalman filter. This filter is applied as a recursive solution to the estimation problem studied in this research. To use the Kalman filter to estimate signal of interest, one must first create matrices to fix the system model into a Kalman filter format. The following sets of equations describe the format of the linear discrete-time system:

The Kalman filter is a great tool, but its computation is complex and requires some explanation. An optimal value for

The k’s on the subscripts are states and can be treated as discrete time intervals. In general, when applying the Kalman Filter, the goal is to estimate state

Each system has to have initial values. Notation

This algorithm takes into account the measurement noise, process noise, and the previous estimated output values so that it can minimize the prediction error upon a continuous cycle of prediction and filtering. It looks at the error between the true state and the estimated state. Therefore, the next step is to derive an error equation. Equations (14) and (15) define an priori estimate error and an posteriori estimate error, respectively.

Then, Equations (14) and (15) are used to compute covariance of the estimation error, which is denoted as

After the measurement at time k − 1 is processed, the estimate of the

To begin the estimation process, initial values of the system determined in Equation (13) must be initialized. Then, with

From time

Next step is to derive time-update equation for the covariance of the state estimation error. The term

Final step requires derivation of measurement-update equations for

where

The matrix

The random variable

In outdoor experiments, the process noise covariance matrix

The Kalman Filter is a form of a linear algorithm for optimal recursive estimation of a system state with a specific set of output equations. To build a simulator, understanding of system model and its dynamic behaviors is necessary. Then, the system must be represented in the state space format to be able to apply Kalman filtering. In other words, we need to mathematically model its states and parameters. This section presents set of equations used to create first-order KF-based estimator. References [

To build an estimation model in MATLAB, we started with equations introduced in Section 4.1 and substituted suitable entries from this problem to reflect the linear channel model [

where:

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After initializing Kalman filter using initial values

where

Similarly, Equations (37), (38) and (39) belong to the “Measurement Update” state of the linear Kalman Filter, also known as the “Correction State.” Equations (37), (38) and (39) are derived by substituting suitable entries from this problem into Equations (27), (28) and (29). Here we adjust the projected estimate by an actual measurement at time k. Equation (37) was derived by substituting environment noise covariance

Equation (37) computes the Kalman Gain, Equation (38) adjusts the projected estimate by an actual measurement

If

The next step is to represent these estimates over a period of sufficient time. The output estimate in the previous step will be the input estimate in the next step. The main goal is to find an optimal value for

In this research, we assumed that the first-order state space model can be used to model Shadow Power. To extend the first-order state space model equations [

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Equation (40) shows

Next set of equations present prediction states for second-order linear Kalman filter. Equation (34) can be rewritten as Equation (41). Therefore, Equation (41) in this section projects the state ahead, and Equation (43) projects the error covariance ahead. For the second-order KF, the matrices are defined as follows:

B is input matrix that relates the control input to the state

Parameter Q in Equation (42) represents the predicted process noise. Term

Equation (43) can then be expressed in the following state space format:

Next set of equations present correction states. Equation (44) was used to compute Kalman gain, which takes into consideration measurement noise due to multipath.

Equation (45) updates the estimate via

It is assumed that channel variation is mainly due to the changing mobile velocity and the correlation distance. Therefore, only the variation of the shadow process coefficient is considered. The smaller the sample period, the closer the shadow process coefficient is to one.

When the channel is nonlinear, the Unscented Kalman Filter also can be applied to the state space model optimize the shadow power presented in this section. In our problem, distribution of multipath is non-Gaussian. However, even when the white Gaussian noise assumption is not valid, the linear Kalman filter is still the optimal LMMSE estimator if the driving and measurement noises are white [

In [

A cell phone or portable phone uses radio waves to establish connection with its base station. Radio waves can travel long distances, but they easily get interrupted. As the transmitted signals travel from tower station to cell phone, they penetrate the atmosphere, and some signals are scattered, reflected, or observed. Objects obstructing the propagation path between the transmitter and receiver can cause variations in the received signal. All this can have a significant impact on signal strength in the cell phone device.

In this experiment, a mobile phone signal refers to signal strength received by a mobile antenna from a cellular network. There are several ways to measure mo- bile signal strength. The two most common units of measurement used in radio signals are dBm (decibels) and RSSI (Received Signal Strength Indicator). RSSI is

a measurement of the power present in a received radio signal. The higher the RSSI number, the stronger the signal. These values allow users to know when they are receiving a stronger signal or a weaker signal.

Area or a region can impact signal strength or path loss. Therefore, as part of the experiment, we collected data in suburban and urban areas. Measurements have been conducted in two different environments while the user was driving a vehicle at different speeds:

1) Suburban environment, Oakland University campus in Auburn Hills, Michigan.

2) Urban environment, downtown Detroit, Michigan.

Valid research experiment must meet the certain criteria. To satisfy terrain requirement, an experiment must be conducted in an area that has good wireless cell phone coverage. Presence of large obstructing objects such as tall building structures are essential for forming a fading channel. Finally, equipment required for power signal data acquisition and processing include: base station, mobile station, vehicle, telecom tool, and laptop with built-in Bluetooth model and MATLAB software.

Cell phones work by communicating via radio waves using a system of cell towers that send and receive calls. A base station, also known as a cell tower, is placed on a big metal pole about 300 ft. high. Cell towers have triangular platforms on the top of the pole for cellular providers to keep their equipment. The process of a cell phone tower transmission requires the following equipment: radios, antennas for receiving and transmitting radio frequency signals, computerized switching control equipment, GPS receivers, power sources, and some kind of protective cover. In this experiment, Verizon was the cellular provider and the location of the base station is shown in

A mobile station consists of the physical equipment (radio transceiver, display and digital signal processors) and software package needed for communication with a mobile network. In this experiment, the Samsung Galaxy S5 smartphone was used as a mobile station. Any type of legal vehicle is acceptable to perform a driving test on public roads. As cell phone user moves around while using a cell phone, tall buildings will shadow the radio signal, which can result in a power drop at the receiver input. In this research, initial experiments were performed next to large buildings on the Oakland University campus to create a shadow fading phenomena in the outdoor environment. Supplementary experiments were conducted in downtown Detroit.

Telecom tool that was released to the market by Wylisis in March 2017 is recommended for recording captured data (

This section presents outdoor experiment results for shadow process estimation and pertinent performance analysis. The purpose of these experiments was to study and analyze output results of the first-order state space model and to compare them to the second-order state space model while applying a Kalman Filter technique to determine shadow power signal in mobile communications from measurements that have impinged Rayleigh fast fading noise. As stated before, we were able to validate this concept through laboratory experiments with data from real scenarios, but those experiments performed in the indoor environment were limited by lower speed and obstacle contribution. The outdoor experiment allowed us to conduct tests that include higher mobile velocity, exact shadow variance values, and large-scale fading configurations.

Measurements have been conducted outside while the cell phone user was driving a vehicle at different speeds, which caused variation in default parameters, such as mobile receiver velocity, shadow variance, and effective correlation distance. Multiple experiment trials were performed to collect sufficient amount of data, but in this paper we include results from driving the vehicle at 36 mph in urban area as shown in

The plots of outdoor experiment results supported by the field data are shown in Figures 8-10. These plots show results of the actual shadow power signal and estimations with Kalman Filtering. In

the second-order Kalman filter output has less lag from the actual shadow power.

Authors in [

In this work, a second-order KF-based estimator has been further investigated in the outdoor environment, which is able to estimate local mean shadow power in mobile communications corrupted by multipath noise. In our experiments, we mainly explored how the second-order KF-based estimator compares to the first-order KF-based estimator. Based on our results from the indoor experiments of small-scale fading presented in [

In this paper, we presented results from outdoor experiments which further confirmed validation of the proposed method and the theoretical analysis. The results supported by field data are provided in Figures 8-10. These plots clearly show that the second-order Kalman filter tracks the actual shadow power more accurately than the first-order Kalman filter. The system was able to operate without a failure under variety of conditions, which demonstrates model robustness. With MATLAB software, we were able to efficiently explore, analyze, and visualize measured data from the outdoor experiment. Comparison analysis was performed as explained in [

Math Works currently offers some basic examples of Kalman Filter theory. Therefore, we will most likely share our code for a first-order KF-based estimator and second-order KF-based estimator by deploying an Application with MATLAB, so others can use it too. According to MathWorks’ web site, there is a wide range of options for deploying and sharing an application that was developed in MATLAB. As future work, we will look into these options.

When the channel is nonlinear, the Unscented Kalman filter also can be applied to the state space model to further improve and optimize the shadow power presented in this paper. The Unscented Kalman filter is popular due to its superiority in approximating and estimating nonlinear systems and its ability to handle non-Gaussian noise environments [

As future work, we also are considering designing a third-order KF-based estimator. When the order of the filter is higher, we predict that there will be better noise repair. However, there is a tradeoff between three things: order of filter, computational difficulty of filter, and accuracy. Therefore, we need to look at these to determine if higher order estimators are practical.

The authors would like to express appreciation to Eric Yaharmatter from Autoliv Inc. for his initial thoughts on this subject.

The authors declare that they have no conflict of interest.

Kapetanovic, A., Zohdy, M.A. and Mawari, R. (2017) Fading Channels Parametric Data Simulation Su- pported by Real Data from Outdoor Experiments. Journal of Signal and Information Processing, 8, 113-131. https://doi.org/10.4236/jsip.2017.83008