Parameter Calculation Method

In this section, we will formally describe the method for radio model parameter calculation from a set of signal strength measurements. We represent the problem as a linear optimization problem and solve it using linear least squares optimization with parameter bounds and inequality constraints. This method is used for both infrastructure-based measurement and localization-based measurements.

The radio propagation model has the following equations for estimating the received signal strength at a receiver:

• Case 1: when the path from the transmitter to the receiver (T-R path) passes only one environment type, we use the original model (equation 4.1).

• Case 2: when the T-R path passes multiple environment types, we use our extended application of the original model (equation 4.2).

In the case of infrastructure-based measurements, the transmitter and the receiver are the base stations. In the case of localization-based measurements, the radio signal strength is measured at the base stations; so the transmitter is a mobile station and the receiver is a base station. The reverse is also possible, if the ARSS is measured at the mobile stations.

The model parameters that have to be determined are:

• The path loss exponents n_j and the standard deviations σ_j for every environment type j ∈ [1...|N|]. N is a vector of environment types, |N| is the number of environments (number of elements inN).

• The standard deviation for the case that the T-R path passes multiple environments σ_C

4.3. Parameter Calculation Method

In order to determine these parameters, our method requires a set of reference signal strength measurements. Every measurement contains the following information:

• the coordinates of the transmitter

• the coordinates of the receiver

• measurement result

– if the receiver received frames from the transmitter for some time period, then this is the average radio signal strength. These results are stored in a vectorV. – if the receiver did not receive any frames from the transmitter, the result is

“non-covered”. These results are stored at the vectorQ.

Our approach for model calibration is to first determine the path loss exponents in such a way that the model results and the measurement values inV are as close as possible.

For this purpose, we define a system of linear equations from the measurement values inV, the model, and the path loss exponents in N. The variables in this system are the path loss exponentsn_j forn_j ∈ N. This system is overdetermined, since the measured values are more than the environment types (|V| > |N|). This is because in general, there are multiple measurement values for each environment type. Since this system is overdetermined we solve it by using a least squared method. This means that the solution minimizes the sum of squared differences between the measured values and the model predictions. The residual is the difference between the measured values and the model predictions. We determine the standard deviations of the model from the variation of the residual. In addition, we use the vectorQin order to define constraints to the least squares solution. These constraints allow the model to preserve the measured non-coverage situations from the real system.

We derive the following system of linear equations from all measurements inV:

P(d) = V (4.13)

When P(d)for every measurement is replaced by the respective model equation, the linear system is transformed to:

P(d₀)−CN =V (4.14)

which is transformed to:

CN =P(d₀)−V (4.15)

The linear system 4.15 contains an equation for every measured value in the vector v_i ∈V. The left-hand side of the equation describes the model-predicted path loss which is the difference between the reference signal strengthP(d₀)next to the transmitter and

4. Automatic Radio Model Calibration

the measured signal strength at the receiver in V. C is a matrix of model-dependent constants. The unknown parameters inN affect only the path loss. The model-predicted path loss (left-hand side) is put equal to the measured path loss (right-hand side). Every equation in 4.15 has the following form:

c_i,1n₁+c_i,2n₂+...+ci,|N|n|N|=P(d₀)_i−v_i (4.16) In this equation the constantsci,j(i = [1...|V|], j = [1...|N|])are the model-dependent constants based on the distance. We determine them from the model, using information about the transmitter and receiver coordinates and the number and type of ray-segments along the path from the transmitter to the receiver. For the calculations of ci,j, we have two cases that depend on the used model equation:

• Case 1: for rays passing a single environment typen_j, the constantc_i,jis calculated as:

c_i,j = 10 log₁₀(d d0

) (4.17)

which follows from equation 4.1. All other constants are zero, since the other environment types do not influence the considered measurement.

• Case 2: in the general case, the ray passes multiple environment types, c_i,j is the sum of model constants over all subareas with environment typen_j (from equation 4.2):

c_i,j =X

10 log₁₀( d_l

d_l−1) (4.18)

P(d₀)is a known constant for every T-R combination (see section 4.2).

We apply the following constraints to the solution of this system:

• Parameter bounds for keeping the path loss exponents in a realistic range:

n_low ≤n_j ≤n_up (4.19)

These constraints ensure that the radio propagation model has realistic parameters.

The values of n_low and n_up have to be determined by an expert during the network deployment. nlow = 2is the value for radio propagation in vacuum (no obstacles). Measurements in different environments have resulted in values for n_up= [3.3...5][108][10].

• Inequality constraints for preserving the non-coverage situations from reality in the model (inequality 4.22, derived from 4.20 and 4.21). For all measurement results inQwe have:

4.3. Parameter Calculation Method

P(d)< Pmin (4.20)

P(d₀)−CN < P_min (4.21)

CN < P(d₀)−P_min (4.22) P_min is the minimum signal strength value that can be measured by the wireless adapter of the receiver. This constraint specifies that the measured non-coverage situations are represented by the model. It results in a system of linear inequalities.

The left-hand side is equal to the left-hand side of equation 4.15. The right-hand side expresses the measured maximum path loss.

In this way, we transform the radio model calibration problem into a least squares problem (equation 4.15) with constraints (equations 4.19 and 4.22). We solve this problem by applying an active-set optimization method originally published in [68] and extended within MATLAB [96] for equality and inequality constraints. The method operates in two phases. In the first phase, it finds an initial feasible point (a solution that satisfies all constraints). In the second phase it iteratively generates a sequence of feasible points which converge to the solution of the problem.

The solution of the least squares problem is the set of path loss exponents. The next step is to determine the standard deviations of the propagation model. We determine the standard deviations of the model from the variation of the residual. The residual is the difference between the measured values and the model predictions which results from the fact that the linear system is overdetermined. For a given path loss exponentn_j, the residual is given as:

Residual =Cn_j −V_j (4.23) which is over all measured signal strength values V_j within environment type n_j. We estimate the parameters of a normal distribution (mean and standard deviation) which fit these values inResidual. For this purpose we use a normal distribution parameter fitting function. The model parameterσ_j is the obtained standard deviation of this distribution, since it describes the variation of the model predictions about the distance dependent mean. We estimate the standard deviation for multiple environment types σC from the residual from all measurements, including multiple environment types.

Optimization method discussion In the general case the solver finds a solution of the defined optimization problem. The lower bound constraints and the inequality constraints are feasible, since they regulate the solution in the same direction (see figure 4.5). The only possible case of infeasibility is when some inequality constraint is not

4. Automatic Radio Model Calibration

Path loss exponent : n

Path loss (dB)

Lower bound

Upper bound Inequality

constraint Model-predicted

Measured Calibrated

Figure 4.5.: Radio model calibration

consistent with the upper bound. In this case the following solutions are possible from which the user chooses at design time:

1. The upper bound is increased so that it is consistent with the inequality constraint.

This is the pessimistic choice and the default choice.

2. The measurement leading to inconsistency is considered an outlier and is not used for the calibration. This is an optimistic choice.

3. Expert mode: the user is asked to decide between 1 and 2 at runtime.

Since the model is linear in terms of the unknowns n_j ∈ N (see equation 4.16), the minimum found by the linear least squares minimization is always a global minimum.