# Radio Wave Propagation

## Contents

## Radio Wave Propagation and the Wireless Fading Channel

Nature is seldom kind. This is true for the typical wireless radio channel as well, which is due to the effect of wave propagation and noise resulting in an unpredictable evolution of its characteristic parameters. The resulting complexity of the channel is easy to evaluate but might be hard to predict which is known as the concept of Physical Unclonable Function (PUF)s for cost-efficient, strong security mechanism ^{[1]}. In this chapter, we introduce fundamentals of the wireless channel, which are important for later analyses and development of WPLS primitives.
In this section, we give an overview of the properties of a wireless radio channel. We refer to literature of Rappaport ^{[2]}, Goldsmith ^{[3]}, and Jakes ^{[4]} for further details and elaborate here on the issue of building a WPLS system rather than a robust communication system.

**Essentials**. The wireless radio channel is a complex shape. It is time variant, environment dependent, carrier frequency dependent, and due to motion a time/space uncertainty is introduced. The measurement accuracy is dependent on receiver and transmitter hardware, of the transmitted waveform, as well as of the Analog to Digital Converter (ADC). Nevertheless, it can be measured and characterized by the changes and affections a signal takes by traveling through it. Radio waves sent through a physical medium can be affected by the following channel distortions ^{[5]}:

- Attenuation over distance
- Multipath propagation
- Shadowing and absorption
- Reflections at large obstacles
- Refraction at medium crossing
- Scattering at small obstacles
- Diffraction at edges

- Interference with other transmissions
- Random noise

The channel between two antennas, e.g., of two transceivers, is of interest. In practice, channels are bandwidth limited. This is due to physical limitations of the transceiver’s hardware, e.g., number of antennas, antenna sizes and design, power amplifier, and low-noise amplifier, etc. Additionally, a frequency-selective wireless radio channel can be time-variant. That meant that due to changes in the environment, e.g., temperature variations, motion of scatters, or motion of one or both antennas, the channel also changes. The connection between physical changes of the environment and the resulting channel changes can be extremely complex, e.g., in a rich multipath environment. However, it can also be trivial, e.g., in free-space only the distance between both antennas matters. Summarizing, a wireless radio channel is a shared, bandwidth restricted resource which has a characteristic that can vary in time, spectrum and space/environment. The detailed trace of a channel can be infinitely complex, but it mainly depends on the measurement procedure.

**Details**. Details of the wireless channel are given in the section about Key Extraction Principles.

### Free Space Propagation and Log-Normal

In the absence of multipath propagation effects, interferences, and noise, the radio wave propa- gation can be simulated using the free space propagation model in Equation (2.1). The power of the received signal at the antenna is:

Equation2.1:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_r = P_SG_SG_R(\frac{\lambda}{4\pi d})^2}

Here **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle G_r}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle G_s}**
are the antenna gains of the receiver and sender, espectively. **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_s}**
is the transmission power of the sender, and *d* is the distance between both antennas.
When we represent **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_r}**
in dBm:

Equation2.2:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_R[dBm] = P_0 - 20log(\frac{d}{d_0})}

Here **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_0}**
is the received power of the signal at sender distance **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle d_0}**
.
The log-normal shadowing model^{[6]} (cf. [157]) extended the free space propagation model by adding a Gaussian random variable **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle X_\delta}**
with zero mean and standard deviation **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \delta}**
for simulating noise:

Equation2.3:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_{r, log-normal}[dBm] = P_0 - 10 \alpha log( \frac d {d_0}) + X_\delta}

Here **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \alpha}**
is the path loss exponent. It depends on the specific propagation environment, i.e.,
type of construction material, architecture, and location within a building. The values of a
range from 1.2 (Waveguide effect) to 8 ^{[7]}. In free space, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \alpha}**
is 2.

## Multipath Propagation

We adopt the multipath propagation model from Goldsmith [72, Chapter 3], which is widely
applied for Ultra Wide Band (UWB) applications. The propagation phenomenon multipath
is given if two or more time-delayed, phase-shifted, and attenuated copies of the transmitted
signal reach the receiving antenna. Multipath signal propagation causes multipath interference
including constructive and destructive interference, and phase shifting of the signal.
Given **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle s(t) = \Re\{u(t)e^{ j(2\pi f c t+\phi_0)}\}}**
the transmitted bandpass signal s(t) at carrier frequency **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle f_c}**
with its complex envelope **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle u(t) = x(t) + jy(t)}**
. The bandwidth of s(t) (and u(t)) is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle B_u}**
and the power is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle P_u}**
. The corresponding received signal r(t) is obtained by upconverting the convolution of u(t) with c(**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \tau}**
,t), where h(**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \tau}**
,t) represents the equivalent lowpass time-varying Channel Impulse Response (CIR) at the moment τ. A channel is characterized in a fashion how it modifies a signal traveling through it. Figure 2.5 illustrates how an impulse based signal influenced by channel effects looks like. We can write the received signal as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle r(t) = \Re\{[h(\tau, t-v) * u(t) + n(t)]e^{j2\pi f_ct}\}}**

with the CIR h(**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \tau}**
,t) (including the sum of all multipath components according to the tapped-
delay-line model), **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \upsilon}**
is the unknown propagation delay (corresponding to line-of-sight propaga-
tion), ∗ denotes convolution, and n(t) accounts for thermal noise assumed to be additive white
Gaussian Additive White Gaussian Noise (AWGN),
**Calculation 2.5:**

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle h(\tau,t) = \sum_{n=1}^{N(t)}\alpha_nt(e)^{−j\phi _n(t)}\delta(t − \tau_n(t)) }

and
**Calculation 2.6:**

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \phi_n(t) = 2\pi f_c\tau_n(t) − \phiD_n(t) − \phi_0}

If the channel is time-invariant then the time-varying parameters in h(

**Calculation 2.7:**

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle h(\tau) = \sum_{n_1}^N\alpha_ne^{−j\phi_n}\delta(t − \tau_n(t))}

Figure 2.6 shows an example of multipath effects on a flattened impulse illustrated in the time domain.

## References

- ↑ Pim Tuyls, BSkori´ c, Sjoerd Stallinga, Anton HM Akkermans, and Wil Ophey. Information-theoretic security analysis of physical uncloneable functions. In Financial Cryptography and Data Security, pages 141–155. Springer, 2005.
- ↑ Theodore S. Rappaport et al. Wireless communications: principles and practice, volume 2. Prentice Hall PTR New Jersey, 1996.
- ↑ Andrea Goldsmith. Wireless Communications. Cambridge university press, 2005.
- ↑ William C. Jakes. Microwave Mobile Communications. John Wiley and Sons Inc., 1994.
- ↑ Sanja Sain. Modelling and Characterization of Wireless Channels in Harsh Environments. PhD thesis, School of Innocation, Design and Engineering, Mälardalen University, 2011.
- ↑ Theodore S. Rappaport et al. Wireless communications: principles and practice, volume 2.Prentice Hall PTR New Jersey, 1996
- ↑ Aleksandar Neskovic, Natasa Neskovic, and George Paunovic. Modern approaches in modeling of mobile radio systems propagation environment. IEEE Communications Surveys and Tutorials, 3(3):2–12, 2000.