Range and velocity estimation

Distance estimation

The distance of an object can be determined by analyzing the time delay $\tau$. As previously shown, the time delay on the $k$-th subcarrier will result in a phase shift $e^{-jk2\pi \Delta f\tau}$ of the subcarrier symbols. This can be interpreted as a subcarrier dependant phase shift with the complex sinusoid frequency with $\hat{\Omega}_\mathrm{d}=\hat{\tau}(2\pi )\Delta f$. The distance can be attained by translating the time delay or complex sinusoid frequency to $$\hat{d}=\frac{1}{2}\hat{\tau}c_0=\frac{\hat{\Omega}_\mathrm{d}c_0}{2(2\pi )\Delta f}.$$

Velocity estimation

The relative velocity of an object to the radar can be attained from the frequency shift $e^{j2\pi f_\mathrm{D}T_\mathrm{O}l}$ on the $l$-th OFDM symbol. Again, a complex sinusoid frequency can be introduced as $\hat{\Omega}_\mathrm{v}=\hat{f}_\mathrm{D}(2\pi )f_\mathrm{C}T_\mathrm{O}$. The relative velocity $\hat{v}_\mathrm{rel}$ can therfore also be estimated from the complex sinusoid frequency $$\label{eqn:Velocity} \hat{v}_\mathrm{rel}=\frac{\hat{f}_\mathrm{D}c_0}{2f_\mathrm{C}}=\frac{\hat{\Omega}_\mathrm{v}c_0}{2(2\pi )f_\mathrm{C}T_\mathrm{O}}.$$

Distance velocity periodogram

The periodogram is an algorithm for estimating the spectral density of a discrete-time signal. In the case of complex sinusoids, the amplitude of the periodogram will have a peak at the sinusoids frequency. A convenient and commonly used way to calculate this is to use the Fast Fourier Transform (FFT) and quantize the frequency in regular intervals, \begin{align} Per_{s(k)}(f)&=\frac{1}{N}\left|\sum_{k=0}^{N-1}{s(k)e^{-j2\pi \frac{nk}{N_{Per}}}}\right|^2 \nonumber \\ &=\frac{1}{N}\left|FFT_{N_{Per}}[s(k)]\right|^2. \end{align}

The periodogram needs to be extended by one more dimension two allow for the estimation of $\hat{\Omega}_d$ and $\hat{\Omega}_v$ [4]. This leads to: \begin{align} \label{eqn:DistanceVelocityPeriodogram} Per_{s(k)}(n,m)&=\frac{1}{NM}\left|\sum_{k=0}^{N_\mathrm{Per}-1}{\left(\sum_{l=0}^{M_\mathrm{Per}-1}{(\mathbf{F})_{k,l}e^{-j2\pi \frac{lm}{M_\mathrm{Per}}}}\right)e^{j2\pi \frac{nk}{N_\mathrm{Per}}}}\right|^2 \nonumber \\ &=\frac{1}{NM}\left|CPer_{\mathbf F}(n,m)\right|^2. \end{align} $CPer_{\mathbf F}(n,m)$ is the resul of a FFT of length $M_\mathrm{Per}$ on each row and an IFFT of length $N_\mathrm{Per}$ on each column of $(\mathbf{F})_{k,l}$. The result of $Per_{s(k)}(f)$ is a matrix of size $N_\mathrm{Per}\times M_\mathrm{Per}$. The $\hat{n}$-th row and $\hat{m}$-th column correspond to the frequencies $\hat{\Omega}_\mathrm{d}=2 \pi \hat{n}/N_\mathrm{Per}$ and $\hat{\Omega}_\mathrm{v}=2 \pi \hat{m}/M_\mathrm{Per}$ of the complex sinusoids. A peak in the periodogram can therefore be translated to a target's distance $\hat{d}$ and velocity $\hat{v}$. The distance and velocity can be determined by inserting $\hat{\Omega}_\mathrm{d}$ and $\hat{\Omega}_\mathrm{v}$ into the distance and velocity equations, resulting in \begin{align} \hat{d}&=\frac{\hat{n}c_0}{2\Delta f N_\mathrm{Per}},\\ \hat{v}&=\frac{\hat{m}c_0}{2f_\mathrm{C}T_\mathrm{O}M_\mathrm{Per}}. \end{align} The indices $n$ and $m$ are chosen to be $$n=0,...,N_\mathrm{Per}-1 \quad\text{and}\quad m=\left\lfloor{\frac{-M_\mathrm{Per}}{2}}\right\rfloor,...,\left\lfloor{\frac{M_\mathrm{Per}}{2}}\right\rfloor-1.$$ This enables the full use of the unambiguous ranges. Therefore, the estimation is possible for the ranges $0\le d\le d_\mathrm{unamb}$ and $-v_\mathrm{rel, unamb}\le\hat{v}_\mathrm{rel}\le v_\mathrm{rel, unamb}$. Negative velocities correspond to targets moving away from the radar. Negative distances have no physical meaning and do not need to be accounted for by the choice of indices.