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6.1 Introduction

Cell site optimization involves live air data collection as a function of distance and statistical analysis of the data for adjusting RF coverage footprints. Since multipath propagation is fuzzy owing to numerous RF barriers, uneven terrain, hills, trees, buildings, etc., there is a large variation of received signal level (RSL) at the receiver [1,2,3,4,5].

Figure 6.1 shows an example to illustrate this scenario. Here, the received signal level (RSL) is measured in dBm and the distance is measured in km. Notice that the signal strength decays logarithmically as a function of distance where the distance is plotted in the linear scale. Also notice that we have plotted a solid line, which is known as the regression line or the best fit. The regression line has a special significance since 50% data are above the regression line and 50% data are below the regression line. Furthermore, at a given distance, the distribution of data has a shape known as Gaussian (bell shaped). We shall discuss these points along with their attributes later in this chapter. Today, a large number of PC-based data collection tools are commercially available. These tools are capable of importing measurement data and of generating statistical outputs such as mean error, standard deviation, max., min., etc.

Fig. 6.1
figure 1

Received signal strength as a function of distance in semilogarithmic scale

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These data collection tools are capable of importing measurement data and generating statistical outputs such as mean error, standard deviation, max., min., etc.

In this chapter, we will see how these tools are used to collect live air data and perform statistical analysis to optimize the cell site. Let’s take a closer look!

6.2 Live Air Data Collection

Figure 6.2 shows the basic concept of RF data collection technique. It is PC based and uses a cellular radio, MS Excel, a GIS (Geographic Information Services) software, and a GPS (Global Positioning System) receiver. The GPS receiver is used to collect the coordinates (longitude and latitude) of each sampling point. The outcome is a pair of long/lat corresponding to each RSL value. Since the cell site location is fixed and has a unique long/lat value, the distance of each sampling point with respect to the cell site is readily available as an output.

Fig. 6.2
figure 2

Drive test and data collection technique. The received signal level (RSL) is measured as a function of distance

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Table 6.1 shows an output file, which was obtained by means of drive test. Notice that the received signal level (RSL) is measured in dBm as a function of distance, where the distance is in meters. We can now perform statistical analysis to find the following parameters:

  • Mean

  • Standard deviation

  • Minimum RSL value

  • Maximum RSL value and

  • The regression line

Table 6.1 Output file obtained from drive test
Full size table

The above statistical parameters were calculated in Excel and also presented in Table 6.1 at the end.

The above data was collected from a cell site in a typical urban environment. Figure 6.3 shows the received signal level (RSL) as a function of distance in a semilogarithmic scale. RSL is measured in dBm and the distance is measured in meters. Here, we see that the signal strength decays logarithmically as a function of distance where the distance is plotted in the linear scale. The rate of decay depends on the propagation environment.

Fig. 6.3
figure 3

The received signal level (RSL) as a function of distance in a semilogarithmic scale. RSL is measured in dBm and the distance is measured in meters

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Our analysis indicates that the cell site exhibits following performance characteristics:

  • Mean RSL = −69.8 dBm

  • Standard Dev. = 10.08 dB

  • Max. RSL = −42 dBm

  • Mi. RSL = −92 dBm

These values are typical in urban environment and the cell site is healthy. The 10 dB standard deviation has a special significance in designing reliable cell sites, which we shall see next.

6.3 Statistical Analysis and Optimization

Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data [6,7,8,9,10,11]. For radio-frequency (RF) engineering, it involves live air data collection as a function of distance. Since multipath propagation is fuzzy owing to numerous RF barriers, uneven terrain, hills, trees, buildings, etc., there is a large variation of received signal level (RSL) at the receiver [5].

Now we consider a set of random variables RSLi having n sample values where i = 1, 2, …, n. The distribution or the density of such a set of random numbers is generally approximated by a continuous curve known as normal distribution. The equation that describes a normal distribution is given by [9, 10]:

$$ f(RSL)=\frac{1}{\sigma \sqrt{2\pi }}\exp {\left(-0.5\left[\left(\frac{RSL-\overline{RSL}}{\sigma}\right)\right]\right)}^2 $$
(6.1)

where the mean is given by:

$$ \overline{RSL}=\frac{{\mathrm{RSL}}_1+{\mathrm{RSL}}_2+\dots +{\mathrm{RSL}}_{\mathrm{n}}\;}{\mathrm{n}} $$
(6.2)

and the variance is given by:

$$ {\sigma}^2=\frac{{\left({\mathrm{RSL}}_1-\overline{RSL}\right)}^2+{\left({\mathrm{RSL}}_2-\overline{RSL}\right)}^2+\dots +{\left({\mathrm{RSL}}_{\mathrm{n}}-\overline{RSL}\right)}^2}{\mathrm{n}-1} $$
(6.3)

σ being the standard deviation.

The curve of Fig. 6.4 is also known as the Gaussian distribution or a bell-shaped curve which is symmetric with respect to the mean whose peak at \( \overline{RSL}=0 \) increases as σ decreases.

Fig. 6.4
figure 4

Normal distribution with zero mean (RSL = 0) and variable standard deviation

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Figure 6.5 shows the distribution curve for \( \overline{RSL}\ne 0 \). We notice that for a positive mean, the curve has the same shape but is shifted to the right and, for a negative mean, it is shifted to the left. This illustrates the fact that the variance is the average dispersion from the mean.

Fig. 6.5
figure 5

Normal distribution with RSL = 1 and RSL = −1, s = variable

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The probability density function is generally obtained from the standard table called standard normal distribution or by means of a curve called cumulative distribution function as shown in Fig. 6.6. Both are based on the following probability distribution function:

Fig. 6.6
figure 6

Cumulative probability distribution as a function of normalized standard deviation(z)

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$$ \mathrm{F}\left(\mathrm{z}\right)=\frac{1}{\sigma \sqrt{2\pi }}\underset{-\infty }{\overset{z}{\int }}\exp {\left(-0.5\left[\left(\frac{RSL-\overline{RSL}}{\sigma}\right)\right]\right)}^2d(RSL) $$
(6.4)

with \( \overline{RSL}=0 \) and σ = 1. Then the random variable (RSL) can be estimated from the following normalized standard deviation z where σ is the measured standard deviation [5]:

$$ \mathrm{z}=\left(\frac{RSL-\overline{RSL}}{\sigma}\right) $$
(6.5)

or

$$ \mathrm{RSL}=\sigma \mathrm{z}+\overline{RSL} $$
(6.6)

The above cumulative distribution function (Fig. 6.6) along with Eq. (6.6) forms the basis of cell site optimization technique. To illustrate the concept, here is an example:

Given:

  • Desired received signal at the cell edge: RSL = −80 dBm

  • Measured standard deviation: σ = 8 dB

  • Required confidence level is 80%

Find:

  • The minimum received signal level at the cell edge RSL to satisfy the above requirements

Solution:

For 80% confidence level, using the curve in Fig. 6.6, we have:

$$ \mathrm{z}(0.8)=0.842. $$

Therefore, the minimum RSL can be computed as:

$$ {\displaystyle \begin{array}{c}\mathrm{RSL}=\sigma \mathrm{z}+\overline{RSL}\\ {}\kern0.62em =\left(8\times 0.842\right)-80\kern0.62em \mathrm{dBm}\\ {}\kern0.62em =-63.26\kern0.62em \mathrm{dBm}.\end{array}} $$

  • This is the optimized signal strength at the cell edge, which is stronger than the specified RSL.

  • It ensures that 80% of the data will fall within the interval −σ and +σ, i.e., within +8 dB.

  • This interval is called the confidence interval and the probability (80%) is called the confidence level.

6.4 Conclusions

  • Drive test, live air data collection, and data analysis techniques were presented.

  • Reviewed statistical analysis and showed that random data such as received signal level (RSL) can be predicted for cell site optimization with confidence.