ISAC Waveform Design

In OFDM-ISAC, the OFDM communication signal serves dual purposes. The transmitted signal on subcarrier $k$ is:

$$X[n, k] = S_d[n, k] + S_p[n, k]$$

where $S_d[n, k]$ is the data symbol and $S_p[n, k]$ is a dedicated sensing pilot (zero on most subcarriers).

The received echo from a target at range $R$ and velocity $v$ is:

$$Y_s[n, k] = \alpha \, e^{-j2\pi f_k \tau} \, e^{j2\pi n T_s \nu_D} \, X[n, k] + W[n, k]$$

where $\tau = 2R/c$ is the round-trip delay and $\nu_D = 2vf_0/c$ is the Doppler shift.

Sensing processing: After data-symbol compensation ($Z[n,k] = Y_s[n,k]/X[n,k]$), apply 2D-FFT to extract the range-Doppler map (as in Chapter 10).

The sensing performance of OFDM-ISAC depends critically on the pilot pattern. For $N_p$ pilot subcarriers placed at indices $\mathcal{K}_p \subset \{0, \ldots, N_c - 1\}$:

Range ambiguity function: $$\chi_R(\tau) = \sum_{k \in \mathcal{K}_p} e^{-j2\pi k \Delta f \tau}$$

Design criterion: Minimise the peak sidelobe level (PSL) of $|\chi_R(\tau)|$ outside the mainlobe:

$$\min_{\mathcal{K}_p} \; \max_{\tau > \tau_{\mathrm{res}}} \frac{|\chi_R(\tau)|}{|\chi_R(0)|}$$

Uniform placement ($k \in \{0, L, 2L, \ldots\}$) minimises PSL at $-13.2$ dB (Dirichlet kernel) but creates range ambiguity at $R_{\max} = c/(2L\Delta f)$. Random placement reduces PSL to $\sim -N_p/2$ dB (in expectation) but with higher variance.

In sensing-centric ISAC, communication data is embedded in an FMCW chirp. The baseband chirp signal is:

$$s(t) = e^{j\pi \mu t^2}$$

where $\mu = W/T_c$ is the chirp rate. Data embedding methods:

1. Phase modulation: Modulate the chirp's initial phase per sweep: $s_n(t) = e^{j\phi_n} e^{j\pi\mu t^2}$ where $\phi_n$ carries one PSK symbol per chirp.

2. Index modulation: Select one of $M$ chirp rates $\{\mu_1, \ldots, \mu_M\}$ to convey $\log_2 M$ bits.

3. OFDM-chirp hybrid: Partition the bandwidth into FMCW (for sensing) and OFDM (for data).

Data rate is typically low ($< 1$ Mbit/s) because each chirp carries only 1--4 bits.

OTFS (Orthogonal Time Frequency Space) places symbols on a delay-Doppler grid via the inverse symplectic finite Fourier transform (ISFFT). The transmit signal in the DD domain is:

$$X[\ell, k] = X_d[\ell, k] + X_p[\ell, k]$$

where $\ell = 0, \ldots, M-1$ is the delay index and $k = 0, \ldots, N-1$ is the Doppler index. The DD-domain input-output relation for a target with delay $\tau_i$ and Doppler $\nu_i$ is:

$$Y[\ell, k] = \sum_i h_i \, X[(\ell - \ell_i)_M, (k - k_i)_N] + W[\ell, k]$$

where $\ell_i = \lfloor \tau_i / (1/M\Delta f) \rfloor$ and $k_i = \lfloor \nu_i / (1/NT_s) \rfloor$ are the quantised delay and Doppler indices.

Key advantage for ISAC: In the DD domain, the channel is (approximately) a 2D convolution with a sparse kernel. Each target creates a single peak at $(\ell_i, k_i)$, making target detection equivalent to reading off the channel taps.