An OFDM frame carrying complex modulation symbols can be represented by an $N\times M$ matrix \begin{equation} \mathbf{F}= \begin{bmatrix} c_{0,0}&\cdots&c_{0,M-1}\\\vdots&\ddots&\vdots\\ c_{N-1,0}&\cdots&c_{N-1,M-1}\\ \end{bmatrix} \in \mathbb{A}^{N\times M} \end{equation} where the number of subcarriers $N$ and the number of symbols $M$ correspond to the number of rows and columns, respectively [4].

The changes from the original signal can be explained by following effects:

Under the assumption of one receive antenna and a total of H reflecting objects, the backscattered signal of the sent signal $s(t)$ can be represented as \begin{equation} r(t)=\sum_{h=0}^{H-1}{b_hs(t-\tau_h)e^{j2\pi f_{D,h}t}e^{j{\bar{\varphi}}_h}+\widetilde{z}(t)}. \end{equation}

• The magnitude of the received reflections will be attenuated by a factor ${b}_{h}$. This factor depends on the distance ${d}_{h}$ to an object and the radar cross section of the object.
• A time delay of the reflections $\tau$, which is the time the signal needs to travel to an object and back. For an object $h$ at distance ${d}_h$ and the speed of light $c_0$, $\tau_h$ can be determined to be $\tau_h=2\frac{{d}_h}{c_0}$
• A change in frequency caused by the Doppler effect. The received frequency is dependent on the relative velocity of an object $v_{\mathrm{rel},h}$ and carrier frequency $f_\mathrm{C}$. The frequency shift of the Doppler is $f_{\mathrm{D},h}=2\frac{v_{\mathrm{rel},h}}{c_0}f_\mathrm{C}$.
• A random phase ${\bar{\varphi}}_h$ adds rotation to the signal.
• Lastly, the signal is distorted by additive white Gaussian noise $\widetilde{z}(t)$.

First, the effects of the frequency change can be determined. Each subcarrier can be regarded as an individual discrete-time signal. Each symbol has a duration of $T_\mathrm{O}$ and will experience a change based on the $f_\mathrm{D}$. For an OFDM signal represented as a matrix this means that each element of a row is modulated by $e^{j2\pi f_\mathrm{D}T_\mathrm{O}l}, l=0...M-1.$

The time delay causes a phase shift for each individual transmitted symbol. The phase shift value depends on the frequency and is therefore different for every subcarrier. With the matrix representation and a delay $\tau$ the phase shift can be translated to $e^{-j2\pi (k\Delta f+f_0)\tau},k = 0...N-1$.