Signal Strength and Distance Correlation: Principles and Formulas

Today, distance estimation and positioning have become popular topics, especially with the spread of 5G technology and Internet of Things (IoT) applications. Although Received Signal Strength Indicator (RSSI) based distance estimation is widely used because of its simple hardware requirements and low energy consumption, it suffers from several limitations in terms of accuracy. In this article, I will discuss the basic physical principles behind this method and explain how the calculations are performed.

Starting from the basics, most of us encounter signal strength indicators in our daily lives, even if we do not pay much attention to them. On our phones, tablets, or laptops, we usually judge GSM or Wi-Fi signal quality by looking at the number of bars on the screen. As we move farther away from the modem, the signal becomes weaker. I tried to illustrate this situation in Figure 1. As the distance increases, the number of signal bars decreases. But are these distances evenly spaced? Does the signal completely disappear when we move a little farther, say by a distance d? Let us try to understand this.

Figure 1 - Relationship Between Signal Strength Icon and Distance on Mobile Devices

A point light source (S) that radiates its energy uniformly in all directions is illustrated in Figure 2. In the formula shown in the figure, 4πr² represents the surface area of the sphere. Accordingly, the light intensity per unit area on the surface of the sphere can be calculated as S/4πr². In this way, the light intensity or power per unit area (P) is obtained.

At this point, the answer starts to become clear. As shown in the figure, the light distributed over an area A at a distance r spreads over an area of 4A at a distance of 2r and 9A at a distance of 3r. This indicates a quadratic increase in the illuminated area. This phenomenon is known as the Inverse Square Law and also applies to gravity, electric fields, and sound propagation [1].

Figure 2 - Distribution of Light over Surface Area

It is important to remember that light is an electromagnetic wave and that radio signals, such as Wi-Fi, behave in a similar way. The spherical model explained earlier is simplified using a flashlight example in Figure 3. Here, distance is represented by d instead of the radius r. In the previous explanation, it was shown that the illuminated area increases proportionally to the square of the distance. Therefore, light intensity or power decreases inversely with the square of the distance.

Figure 3 - Simple Flashlight Model Showing the Decrease of Light Intensity with Distance

Now, let us move beyond the concept of light and continue with radio signals. As discussed earlier, signal strength decreases inversely with distance. This relationship can be expressed by Equation 1 and forms the basis of the RSSI distance model.

Most people have heard the term deciBel at some point in their lives, especially when buying headphones or dealing with sound related topics. Although deciBels are commonly used to measure quantities such as sound intensity, they are fundamentally ratio based units and are not limited to acoustics. In fact, a deciBel is one tenth of a Bel, meaning that it is obtained by multiplying the Bel unit by ten (Equation 2). Technically, deciBels are used to calculate transmission losses, for example, to represent how much a signal has weakened. A change of 1 Bel corresponds to a tenfold change in power. Therefore, the Bel is not used as an absolute unit, but rather to describe ratios.

As mentioned earlier, the Bel essentially represents a ratio. However, when an actual measurement is required, a reference point of 0 dBm = 1 mW is used. The term dBm stands for “deciBel milliwatt” and expresses signal power in watts relative to one milliwatt. A scale showing the relationship between dBm and mW is presented in Figure 4. As illustrated in the figure, a tenfold increase or decrease in power corresponds to a 10 dB change.

The main reason for using such a scale is to avoid dealing with power values that contain many zeros. Instead of expressing very small power levels such as 10⁻⁸ mW, a more readable representation like −80 dBm is used. With this notation, it becomes easy to understand that there is a 10 dB difference between 20 dBm and 30 dBm. In this context, dB represents a ratio, whereas dBm is a referenced power measurement.

Figure 4 - Logarithmic Relationship Between dBm and mW

As explained in Equation 1 and Figure 4, the expression log(1/d²) represents the inverse square relationship between power and distance in logarithmic form. In Equation 3, this variation is defined as loss. In Equation 4, the expression is multiplied by 10 to convert it into deciBel units. In Equation 5, the exponent −2 is moved to the front using the power rule of logarithms. Assuming that the transmitted signal power is 0 dBm, the signal strength (in dBm) at any distance d can be calculated using Equation 6 [2].

If the transmitted signal power is unknown or different from 0 dBm, Equation 7 can be used instead. Here, the RSSI_d₀ value represents the measurement taken at a reference distance d₀ selected by the user. Using this reference value, RSSI can be estimated at any given distance. In addition, Equation 8 (the inverse form) can be applied for distance estimation.

It should be kept in mind that this equation usually does not produce reliable results directly in real world environments. Materials, walls, human presence, and atmospheric conditions can significantly affect the measurements. For this reason, more advanced approaches such as filtering techniques, statistical models, or machine learning methods are often recommended. These methods are not discussed in this article; however, interested readers are encouraged to explore more comprehensive models such as the ITU R P.1238 model or the Friis equation [3,4].

Figure 5 - Logarithmic Decrease of Signal Strength

The question “Are these distances equally spaced?” raised in Figure 1 cannot actually be answered using the signal icon alone. My intention was to start with a familiar visual and demonstrate that signal strength decreases in a logarithmic manner. This behavior is summarized in Figure 5. In future articles, I will discuss distance estimation and positioning methods based on Time of Arrival (ToA). If you notice any incorrect information or have suggestions, please feel free to contact me.

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References

[1] Voudoukis, N. F., & Oikonomidis, S. (2017). Inverse square law for light and radiation: A unifying educational approach. European Journal of Engineering and Technology Research, 2(11), 23–27. https://doi.org/10.24018/ejeng.2017.2.11.517

[2] C. -H. Huang, L. -H. Lee, C. C. Ho, L. -L. Wu and Z. -H. Lai, “Real-Time RFID Indoor Positioning System Based on Kalman-Filter Drift Removal and Heron-Bilateration Location Estimation,” in IEEE Transactions on Instrumentation and Measurement, vol. 64, no. 3, pp. 728–739, March 2015, doi: 10.1109/TIM.2014.2347691

[3] K. F. Warnick, F. Broydé, L. Jelinek, M. Capek and E. Clavelier, “Generalized Friis Transmission Formula Using Active Antenna Available Power and Unnamed Power Gain,” in IEEE Transactions on Antennas and Propagation, vol. 72, no. 8, pp. 6321–6331, Aug. 2024, doi: 10.1109/TAP.2024.3427400

[4] C. Pyo, H. Sawada and T. Matsumura, “A Deep Learning-Based Indoor Radio Estimation Method Driven by 2.4 GHz Ray-Tracing Data,” in IEEE Access, vol. 11, pp. 138215–138228, 2023, doi: 10.1109/ACCESS.2023.3340204