ISAC Fundamentals: Why the Same Array?

Consider a base station equipped with a massive MIMO array of $N_t$ antennas. The transmitted signal over one block of $T$ symbols is the matrix $\mathbf{X} \in \mathbb{C}^{N_t \times T}$. The same signal simultaneously:

1. Serves communication users. For each user $k \in \{1, \dots, K\}$, the received signal is $$\mathbf{y}_k = \mathbf{H}_{k}^{H} \mathbf{X} + \mathbf{w}_{k},$$ where $\mathbf{H}_{k} \in \mathbb{C}^{N_t}$ is the downlink channel vector and $\mathbf{w}_{k} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I}_T)$.

2. Illuminates the scene. The signal reflected off a point target at angle $\theta$, range $r$, and Doppler $\nu$ returns to the same array after a round-trip delay. Under the colocated monostatic assumption, the received echo is $$\mathbf{Y}_r = \alpha \, \mathbf{a}(\theta)\mathbf{a}^H(\theta) \mathbf{X}(t - 2r/c) e^{j2\pi\nu t} + \mathbf{W}_r,$$ where $\alpha$ is the complex target reflectivity and $\mathbf{W}_r$ is the receiver noise at the sensing baseband.

The transmit signal $\mathbf{X}$ must be designed jointly for both tasks. In the monostatic geometry the sensing receiver is colocated with the transmitter; in the bistatic geometry the sensing receiver is a separate node (possibly another BS or a roadside unit).

Following the Liu–Masouros classification, ISAC systems are organized by how tightly the two functions share hardware and waveform:

• Coexisting. Two independent systems share the same band. Each treats the other as interference; mitigation is done through null steering, cognitive access, or spectrum-sharing MACs. No joint design.

• Cooperating. The two systems exchange side information (e.g., sensing results used to aid beam alignment, or communication CSI used for clutter rejection) but keep separate waveforms and hardware.

• Integrated. A single waveform, single precoder, and single hardware chain perform both functions. The radar-centric flavor embeds communication bits into a radar waveform (e.g., information embedding in chirps); the communication-centric flavor uses the downlink signal itself as the radar probe. Massive MIMO ISAC falls firmly in the communication-centric integrated camp: the comm signal already looks like a pseudo-random radar code with wide bandwidth and narrow angular mainlobe.

For a single range–angle cell, the sensing receiver faces a binary hypothesis test on the match-filtered output $y$:

• $\mathcal{H}_0$ : target absent, $y = w$, with $w \sim \mathcal{CN}(0, \sigma^2)$.
• $\mathcal{H}_1$ : target present, $y = \alpha s + w$, where $s$ is the deterministic reference signal and $\alpha$ the complex reflectivity.

The Neyman–Pearson detector compares $|y|^2$ against a threshold chosen to fix the false-alarm probability $P_{\text{fa}}$. The detection probability as a function of the integrated SNR $\rho = |\alpha|^2 |s|^2 / \sigma^2$ is $$P_d = Q_1\!\left(\sqrt{2\rho}, \sqrt{-2\ln P_{\text{fa}}}\right),$$ where $Q_1$ is the Marcum Q-function of order 1. Massive MIMO adds an array gain of $N_t$ to $\rho$ when the probing beam is aimed at the target.