Waveform Design for ISAC

The choice of waveform is the most fundamental design decision in an ISAC system. The waveform must simultaneously provide good radar sensing performance (sharp ambiguity function, low sidelobes) and efficient communication (high data rate, low bit error rate). This section analyses two dominant waveform families — OFDM and FMCW — from a dual-function perspective and discusses joint optimisation criteria.

Orthogonal frequency-division multiplexing (OFDM) is the dominant modulation for modern communication systems (4G LTE, 5G NR, WiFi). Its multicarrier structure also makes it a natural radar waveform, with a particularly elegant connection between the subcarrier and symbol dimensions and the range and Doppler domains.

Consider an OFDM frame with $K$ subcarriers, subcarrier spacing $\Delta f$, and $M$ OFDM symbols. The total bandwidth is $B = K \Delta f$ and the frame duration (excluding cyclic prefix overhead) is $T_{\text{frame}} = M T_{\text{sym}}$, where $T_{\text{sym}} = 1/\Delta f + T_{\text{CP}}$ is the total OFDM symbol duration including the cyclic prefix $T_{\text{CP}}$.

The transmitted baseband signal for the $m$-th OFDM symbol is:

$$s_m(t) = \sum_{k=0}^{K-1} X_{m,k} \, e^{j2\pi k \Delta f (t - m T_{\text{sym}})}$$

where $X_{m,k}$ is the data symbol on the $k$-th subcarrier of the $m$-th OFDM symbol. After reflection from a target at range $R$ with velocity $v$, the received signal on subcarrier $k$ of symbol $m$ (after CP removal and FFT demodulation) is:

$$Y_{m,k} = \alpha \, X_{m,k} \, e^{-j2\pi k \Delta f \tau} \, e^{j2\pi m T_{\text{sym}} f_d} + N_{m,k}$$

where $\tau = 2R/c$ and $f_d = 2v/\lambda$.

The frequency-modulated continuous wave (FMCW) waveform is a linear chirp that sweeps the instantaneous frequency linearly from $f_0 - B/2$ to $f_0 + B/2$ over the chirp duration $T_c$:

$$s(t) = e^{j\pi \mu t^2}, \quad 0 \leq t \leq T_c$$

where $\mu = B/T_c$ is the chirp rate (Hz/s). Upon reflection from a target at range $R$ with velocity $v$, the received signal is mixed with the transmitted signal (de-chirping), producing a beat signal:

$$y_b(t) = \alpha \, e^{j2\pi (f_b t + f_d t)}$$

where the beat frequency is:

$$f_b = \mu \tau = \frac{B}{T_c} \cdot \frac{2R}{c}$$

and $f_d = 2v f_0 / c$ is the Doppler shift. The range is extracted from the beat frequency via FFT:

$$R = \frac{f_b \cdot c \cdot T_c}{2B} = \frac{f_b \cdot c}{2\mu}$$

FMCW is the dominant waveform in automotive radar (77 GHz) due to its simple hardware implementation (a single mixer performs de-chirping) and its ability to achieve wide bandwidth with low-rate ADCs (the beat frequency is much lower than the signal bandwidth).

For ISAC systems, OFDM is generally preferred when the primary platform is a communication system (5G NR, WiFi) because the radar processing can be added as a software module with no hardware changes. FMCW is preferred when the primary platform is a radar system (automotive) and communication is a secondary function.

When designing a waveform to serve both radar and communication, two fundamental metrics characterise the sensing performance:

Mutual information (MI) for sensing: The radar channel between the transmitter and the target can be modelled as a Gaussian channel with a frequency-dependent response determined by the target's scattering characteristics. The mutual information between the transmitted waveform and the target impulse response is:

$$I_{\text{sense}} = \int_0^B \log_2\!\left(1 + \frac{|S(f)|^2 \sigma_h^2(f)}{\sigma_n^2}\right) df$$

where $|S(f)|^2$ is the power spectral density of the waveform, $\sigma_h^2(f)$ is the target's spectral response, and $\sigma_n^2$ is the noise power spectral density. Maximising $I_{\text{sense}}$ leads to water-filling over the target spectrum.

Cram'{e}r-Rao bound (CRB) for parameter estimation: The CRB provides a lower bound on the variance of any unbiased estimator of the target parameters $\boldsymbol{\eta} = [\tau, f_d, \theta]^T$. For delay estimation:

$$\mathrm{CRB}(\tau) = \frac{1}{8\pi^2 \cdot \text{SNR} \cdot \beta^2}$$

where $\beta^2 = \int f^2 |S(f)|^2 df / \int |S(f)|^2 df$ is the effective (RMS) bandwidth of the waveform. Minimising the CRB favours waveforms that spread energy to the spectral edges.

Communication rate: The standard Shannon capacity applies:

$$R_{\text{comm}} = \int_0^B \log_2\!\left(1 + \frac{|S(f)|^2 |H_c(f)|^2}{\sigma_n^2}\right) df$$

The joint optimisation problem is:

$$\max_{|S(f)|^2} \; R_{\text{comm}} \quad \text{s.t.} \quad \mathrm{CRB}(\tau) \leq \epsilon, \;\; \int |S(f)|^2 df \leq P$$

This formulation reveals a fundamental trade-off: the communication channel favours water-filling over $|H_c(f)|^2$ (allocating power to strong subcarriers), while the CRB favours maximising the RMS bandwidth (allocating power to edge subcarriers). The Pareto frontier of this trade-off characterises the achievable rate-accuracy region of the ISAC system.