ISAC Fundamentals

A Dual-Function Radar-Communication (DFRC) system uses a single transmitted signal $\mathbf{x}(t)$ to simultaneously:

1. Communicate data to $K_u$ users via downlink beamforming.
2. Sense the environment by processing the echoes of the same signal reflected from $K_t$ targets.

The transmitter with $N_t$ antennas sends:

$$\mathbf{x}(t) = \mathbf{W}_c \, \mathbf{s}_c(t) + \mathbf{W}_s \, \mathbf{s}_s(t)$$

where $\mathbf{W}_c \in \mathbb{C}^{N_t \times K_u}$ is the communication precoder, $\mathbf{s}_c(t)$ carries data, $\mathbf{W}_s \in \mathbb{C}^{N_t \times K_t}$ is the sensing precoder, and $\mathbf{s}_s(t)$ is a dedicated sensing waveform (which may be zero in fully communication-centric designs).

The total power constraint is:

$$\operatorname{tr}(\mathbf{W}_c\mathbf{W}_c^H + \mathbf{W}_s\mathbf{W}_s^H) \leq P_t.$$

ISAC waveform design falls into three paradigms:

1. Communication-centric: Start with a standard communication waveform (e.g., OFDM) and extract sensing information from the echoes. The waveform is optimised for data throughput; sensing is opportunistic. This is the most practical paradigm for 5G/6G.

2. Sensing-centric: Start with a radar waveform (e.g., FMCW chirp) and embed communication data in its parameters (phase modulation, index modulation). Sensing performance is prioritised.

3. Joint design: Optimise a single waveform to balance both functions:

$$\min_{\mathbf{R}_x} \; D_s(\mathbf{R}_x) \quad \text{s.t.} \quad R_c(\mathbf{R}_x) \geq R_{\min}, \; \operatorname{tr}(\mathbf{R}_x) \leq P_t$$

where $D_s$ is a sensing distortion metric and $R_c$ is the communication rate.

Communication metric: Sum-rate for $K_u$ users:

$$R_c = \sum_{k=1}^{K_u} \log_2\!\left(1 + \frac{|\mathbf{h}_k^H \mathbf{v}_{c,k}|^2}{\sum_{j \neq k}|\mathbf{h}_k^H \mathbf{v}_{c,j}|^2 + \sigma^2}\right)$$

where $\mathbf{h}_k$ is the channel to user $k$ and $\mathbf{v}_{c,k}$ is the $k$-th communication beamforming vector.

Sensing metric (imaging): Mutual information between the scene reflectivity $\mathbf{c}$ and the received echo:

$$\mathrm{MI}_{\mathrm{sens}} = \log\det\!\left(\mathbf{I} + \frac{1}{\sigma^2}\mathbf{A}\mathbf{R}_x\mathbf{A}^{H}\right)$$

Sensing metric (tracking): Cramer-Rao bound for target parameter estimation:

$$\mathrm{CRB}(\theta_k) = \left[\mathbf{J}^{-1}(\boldsymbol{\theta})\right]_{kk}$$

where $\mathbf{J}$ is the Fisher information matrix.

The ISAC spectral efficiency gain over separate systems is:

Separate: Communication uses bandwidth $W_{c}$, sensing uses $W_{s}$. Total: $W_{c} + W_{s}$.

ISAC: Both share bandwidth $W_{\mathrm{ISAC}}$. The gain is:

$$G_{\mathrm{SE}} = \frac{W_{c} + W_{s}}{W_{\mathrm{ISAC}}} \leq 2$$

with equality when $W_{c} = W_{s} = W_{\mathrm{ISAC}}$.

Beyond spectrum, ISAC saves hardware (shared array, RF chains, baseband) and energy (single transmitter, no dedicated radar).