OFDM-Based Sensing

Consider an OFDM frame with $N_c$ subcarriers, subcarrier spacing $\Delta f$, and $M$ symbols over the coherent processing interval (CPI). The transmitted baseband signal for OFDM symbol $m$ is

$$s_m(t) = \sum_{n=0}^{N_c - 1} d[n, m] \, e^{j2\pi n \Delta f (t - m T_{\mathrm{sym}})}$$

where $d[n, m]$ is the data/pilot symbol on subcarrier $n$ of OFDM symbol $m$ and $T_{\mathrm{sym}} = 1/\Delta f + T_{\mathrm{cp}}$ is the total symbol duration including the cyclic prefix of length $T_{\mathrm{cp}}$.

For a scene with $K$ point targets at delays $\{\tau_k\}$ and Doppler shifts $\{\nu_k\}$ with complex reflectivities $\{\alpha_k\}$, the received signal after OFDM demodulation (DFT at the receiver) is

$$Y[n, m] = \sum_{k=1}^{K} \alpha_k \, e^{-j2\pi n \Delta f \tau_k} \, e^{j2\pi m T_{\mathrm{sym}} \nu_k} \, d[n, m] + W[n, m]$$

where $W[n, m] \sim \mathcal{CN}(0, \sigma^2)$ is additive noise.

After element-wise division by the known data/pilot symbols $d[n,m]$, the compensated signal is

$$Z[n, m] = \frac{Y[n, m]}{d[n, m]} = \sum_{k=1}^{K} \alpha_k \, e^{-j2\pi n \Delta f \tau_k} \, e^{j2\pi m T_{\mathrm{sym}} \nu_k} + \tilde{W}[n, m]$$

The range-Doppler map (RDM) is obtained by a 2D-DFT of $Z[n,m]$:

$$\mathrm{RDM}[p, q] = \sum_{n=0}^{N_c-1}\sum_{m=0}^{M-1} Z[n, m] \, e^{-j2\pi n p / N_c} \, e^{-j2\pi m q / M}$$

This is a 2D periodogram estimator of the delay-Doppler scene. The range and velocity axes are

$$R_p = \frac{p \, c}{2 N_c \Delta f}, \qquad v_q = \frac{q \, \lambda}{2 M T_{\mathrm{sym}}}$$

and the resolutions are

$$\Delta R = \frac{c}{2 W}, \qquad \Delta v = \frac{\lambda}{2 M T_{\mathrm{sym}}}$$

where $W = N_c \Delta f$ is the total bandwidth and $\lambda$ is the carrier wavelength.