# Atmospheric and Fog Effects on Ultra-Wide Band Radar Operating at Extremely High Frequencies
## 2 Propagation of a Linear FM Signal in a Dielectric Medium  
We start with a general review of the description of electromagnetic radiation in the frequency domain. The fundamental principles of FMCW radars are also given here, mainly for convenience, defining the terms and notations used in the following discussion. The formulation we have developed is applied for the study of a linear FM (\"chirp\") signal detected by a product detector after propagation and reflection from a target. We demonstrate the approach on a FMCW radar operating at millimeter and sub-millimeter wavelengths under fog conditions. We assume that the response of the components within the system is linear; this is a reasonable assumption as, though typical components have  
Figure 1: Atmospheric propagation factor in the EHF band: (**a**) attenuation coefficient \(20\text{log}(e)\cdot\alpha\left(f\right)\) in (dB/km); (**b**) group delay increment \(\Delta\tau_{d}=N\left(f\right)\cdot d/c\) in (ps/km). non-linear characteristics, it is possible in practice to use an equalizer to reduce the significance of this problem. In a FMCW radar (see Figure 2), a linear frequency-modulated signal (\"chirp\") is transmitted toward the target. The instantaneous frequency of the transmission is changing linearly according to:  
\[f\left(t\right)=f_{c}+\frac{\Delta f}{T_{sweep}}t \tag{1}\]  
where the carrier frequency is \(f_{c}\) and the frequency span \(\Delta f\) divided by the sweep time \(T_{sweep}\) is the compression ratio of the linear FM signal. The transmitted signal can be presented as a carrier wave at frequency \(f_{c}\) modulated by a wideband signal:  
\[E_{T}(t)=\text{Re}\left\{A_{T}\left(t\right)\cdot e^{\left|2\pi f_{c}t\right. \right\} \tag{2}\]  
Here \(A_{T}\left(t\right)\) is a complex envelope, representing the baseband modulating signal. In the case of linear FM, the complex amplitude of the \"chirp\" is:  
\[A_{T}\left(t\right)=\exp\left(j\pi\frac{\Delta f}{T_{sweep}}t^{2}\right) \tag{3}\]  
Fourier transform of the transmitted field Equation (2) yields:  
\[\mathbf{E}_{T}(f)=\frac{1}{2}\mathbf{A}_{T}\left(f-f_{c}\right)+\frac{1}{2} \mathbf{A}_{T}{}^{*}\left[-\left(f+f_{c}\right)\right] \tag{4}\]  
where \(\mathbf{A}_{T}\left(f\right)\) is the Fourier transform of the complex envelope \(A_{T}\left(t\right)\). The transmitted signal directed to the target located at a distance \(d\) is scattered back and arrives at the receiver after propagating a total distance of \(2d\) in the dielectric medium, and the \"phasor-like\" field presentation in the positive frequency domain is given by:  
\[\widetilde{\mathbf{E}}_{R}(f)=\mathbf{A}_{T}\left(f-f_{c}\right)\cdot e^{- \left|2k_{z}\left(f\right)\cdot d\right.} \tag{5}\]  
where \(k_{z}\left(f\right)\) is the complex propagation factor of the electromagnetic field in the medium. The Fourier transform of the field is:  
\[\mathbf{E}\left(r,f\right)=\int\limits_{-\infty}^{+\infty}E\left(r,t\right) \cdot e^{-\left|2\pi f\right|}dt \tag{6}\]  
Figure 2: Block diagram of the linear FM radar. an ultra-wideband signal. Since the frequency sweep in the FMCW signal is very large, one should consider the non-linearity that may emerge due to the inappropriate design of the radar RF chain [23]. A linear-phase frequency response is required to keep the linearity of the chirp along its whole temporal duration. Also, non-linear characteristics may cause a distortion in the modulated wave, resulting in further degradation in the sensitivity and thus an accuracy reduction in distance measurements. To overcome this issue, multi-stage amplification should be applied including limitation of the power gain to stay within the back-off regime of the amplifiers. In order to focus the study only on the atmospheric propagation effects, a completely linear chirp is assumed to be transmitted. It is assumed that the FMCW signal is transmitted and received by a linear system in order to explore the impact of the atmospheric medium.