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Beamforming: Where ISAC Gets Spatial

Waveform design (Section 29.3) determines what you transmit in time-frequency. Beamforming determines where you point that energy in space. In ISAC, the beamformer must simultaneously form communication beams toward users and sensing beams toward targets --- and the interplay between these beams determines the rate-CRB tradeoff.

The good news: when $N_t \gg K_u + K_t$ , the spatial degrees of freedom are abundant and both functions can be served with minimal mutual interference. This is the regime of massive MIMO ISAC.

Definition:
ISAC Beamforming Problem

Design the beamforming matrix $\mathbf{W} = [\mathbf{v}_{1}, \ldots, \mathbf{v}_{K_u+K_t}]$ to simultaneously serve $K_u$ communication users and illuminate $K_t$ sensing targets:

$\min_{\mathbf{W}} \; \sum_{k=1}^{K_t} \mathrm{CRB}(\theta_k) \quad \text{s.t.} \quad \mathrm{SINR}_j \geq \gamma_j, \; j = 1, \ldots, K_u, \quad \|\mathbf{W}\|_F^2 \leq P_t$

where the SINR for communication user $j$ is:

$\mathrm{SINR}_j = \frac{|\mathbf{h}_j^H\mathbf{v}_{j}|^2}{\sum_{i \neq j}|\mathbf{h}_j^H\mathbf{v}_{i}|^2 + \sigma^2}.$

This is non-convex due to the CRB objective and SINR constraints. Common approaches: semidefinite relaxation (SDR), alternating optimisation, or successive convex approximation.

Definition:
SDR for ISAC Beamforming

The semidefinite relaxation replaces $\mathbf{v}_{k}\mathbf{v}_{k}^{H}$ with a PSD matrix $\mathbf{W}_k \succeq 0$ :

$\min_{\{\mathbf{W}_k\}} \; -\sum_{k=1}^{K_t} \operatorname{tr}(\mathbf{B}_k \mathbf{W}_k) \quad \text{s.t.} \quad \begin{cases} \operatorname{tr}(\mathbf{H}_j\mathbf{W}_j) \geq \gamma_j\left(\sum_{i \neq j}\operatorname{tr}(\mathbf{H}_j\mathbf{W}_i) + \sigma^2\right) \\ \sum_k \operatorname{tr}(\mathbf{W}_k) \leq P_t \\ \mathbf{W}_k \succeq 0 \end{cases}$

where $\mathbf{H}_j = \mathbf{h}_j\mathbf{h}_j^H$ and $\mathbf{B}_k$ encodes the CRB objective. If the solution has $\operatorname{rank}(\mathbf{W}_k) = 1$ for all $k$ , the relaxation is tight and beamforming vectors are extracted via eigendecomposition.

SDR is tight (rank-1 solutions) in many practical ISAC scenarios, particularly when $N_t$ is large relative to $K_u + K_t$ . When rank $> 1$ , Gaussian randomisation provides approximate solutions with bounded performance loss.

Null-Space Projection ISAC Beamforming

Complexity:

O(N_t^{2} K_u)

for ZF + projection (dominated by pseudo-inverse). Much cheaper than SDR (

O(N_t^{6.5})

).

Input: Communication channel

\mathbf{H} \in \mathbb{C}^{K_u \times N_t}

,

target steering vectors

\{\mathbf{a}(\theta_k)\}_{k=1}^{K_t}

,

power budget

P_t

, SINR targets

\{\gamma_j\}

Output: Joint beamforming matrix

\mathbf{W}

1. Communication beamforming (ZF):

\mathbf{W}_c = \mathbf{H}^H(\mathbf{H}\mathbf{H}^H)^{-1} \operatorname{diag}(\sqrt{p_1}, \ldots, \sqrt{p_{K_u}})

where

p_j

satisfies

\mathrm{SINR}_j = p_j / \sigma^2 \geq \gamma_j

2. Null-space projector:

\mathbf{P}_\perp = \mathbf{I} - \mathbf{H}^H(\mathbf{H}\mathbf{H}^H)^{-1}\mathbf{H}

3. Project sensing beams into null space:

For each target

k

:

\mathbf{v}_{s,k} = \mathbf{P}_\perp \mathbf{a}(\theta_k) / \|\mathbf{P}_\perp \mathbf{a}(\theta_k)\|

4. Power allocation:

P_c = \sum_j p_j

,

P_s = P_t - P_c

Distribute

P_s

among sensing beams (equal or CRB-optimal)

5. Return

\mathbf{W} = [\mathbf{W}_c, \sqrt{P_s/K_t} \, \mathbf{W}_s]

Null-space projection is suboptimal (it does not exploit communication beams for sensing) but provides a simple, closed-form solution. It is near-optimal when $N_t \gg K_u + K_t$ (massive MIMO regime).

Joint Communication-Sensing Beampattern

Visualise the ISAC beampattern showing communication beams (toward users) and sensing beams (toward targets). Compare null-space projection, naive power splitting, and joint SDR approaches. Observe how more antennas reduce the tradeoff.

Parameters

Number of Antennas16

Communication Users2

Sensing Targets1

Beamforming Method

Example: ISAC Beamforming for a 5G mmWave Base Station

A 5G ISAC base station with $N_t = 16$ antennas at 28 GHz serves $K_u = 3$ users at angles $\{-30^\circ, 10^\circ, 45^\circ\}$ and tracks $K_t = 2$ targets at $\{-60^\circ, 20^\circ\}$ . Design the beamforming using null-space projection.

Solution

DOF analysis

Total beams: $K_u + K_t = 5$ . Available DOF: $N_t = 16$ . The system is well under-determined ( $16 \gg 5$ ), so both functions can be served with margin for sidelobe control.

Communication beamforming

ZF: $\mathbf{W}_c = \mathbf{H}^H(\mathbf{H}\mathbf{H}^H)^{-1}\operatorname{diag}(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3})$ . This nulls inter-user interference using 3 DOF.

Sensing beamforming

Project sensing beams into the null space of $\mathbf{H}$ : $\mathbf{W}_s = \mathbf{P}_\perp[\mathbf{a}(-60^\circ), \mathbf{a}(20^\circ)]$ . Since target ( $20^\circ$ ) is near user ( $10^\circ$ ), the projection removes some energy. Target at $-60^\circ$ is well-separated, retaining $> 90\%$ beam energy.

Performance

With $P_t = 30$ dBm: communication SINR $\geq 15$ dB per user (64-QAM). Sensing beampattern: $\geq 20$ dBm at $-60^\circ$ , $\geq 17$ dBm at $20^\circ$ (3 dB loss from proximity to user). The null-space projection uses 13 of 16 DOF for sensing.

Definition:
RIS-Assisted ISAC

A Reconfigurable Intelligent Surface (RIS) with $N_{\mathrm{RIS}}$ elements assists ISAC by providing additional propagation paths:

$\mathbf{y}_k = (\mathbf{h}_{d,k} + \mathbf{H}_r \boldsymbol{\Phi} \mathbf{h}_{t,k})^H \mathbf{x} + w_k$

where $\mathbf{h}_{d,k}$ is the direct path, $\mathbf{H}_r$ is the RIS-to-user channel, $\boldsymbol{\Phi} = \operatorname{diag}(e^{j\phi_1}, \ldots, e^{j\phi_{N_{\mathrm{RIS}}}})$ is the RIS phase-shift matrix, and $\mathbf{h}_{t,k}$ is the BS-to-RIS channel.

For sensing: The RIS creates a virtual transmitter at the RIS location, providing additional viewing angles for imaging. The sensing matrix becomes:

$\mathbf{A}_{\mathrm{total}} = \begin{bmatrix} \mathbf{A}_{\mathrm{direct}} \\ \mathbf{A}_{\mathrm{RIS}}(\boldsymbol{\Phi}) \end{bmatrix}$

Joint optimisation of $\mathbf{W}$ (beamforming) and $\boldsymbol{\Phi}$ (RIS phases) enables simultaneous communication enhancement and sensing coverage extension.

RIS-ISAC is particularly valuable for NLOS sensing: the RIS can illuminate targets hidden behind obstacles, providing coverage that monostatic ISAC cannot achieve. The RIS also increases the effective aperture for imaging, improving angular resolution.

RIS-Assisted ISAC Sensing Coverage

Explore how a RIS extends the sensing coverage of an ISAC base station. The plot shows the combined sensing power map from direct and RIS-reflected paths. Move the RIS position to see how it illuminates NLOS regions.

Parameters

BS Antennas8

RIS Elements64

RIS Position X (m)30

RIS Position Y (m)50

Example: RIS-ISAC for NLOS Target Detection

An ISAC BS with $N_t = 8$ antennas is at the origin. A target is at position $(30, 40)$ m, blocked by a building. A RIS with $N_{\mathrm{RIS}} = 64$ elements is mounted on a wall at $(40, 0)$ m. Compute the RIS-reflected path gain relative to a hypothetical direct path.

Solution

Path geometry

BS to RIS: $d_1 = 40$ m. RIS to target: $d_2 = \sqrt{10^2 + 40^2} = 41.2$ m. Hypothetical direct path: $d_d = \sqrt{30^2 + 40^2} = 50$ m.

RIS path gain

RIS coherent gain: $N_{\mathrm{RIS}}^2 = 64^2 = 4096$ ( $36$ dB) when phases are perfectly aligned. Path loss ratio (RIS vs. direct, two-way for sensing): $G_{\mathrm{RIS}} = N_{\mathrm{RIS}}^2 \cdot (d_d^4) / (d_1^2 d_2^2 \cdot d_2^2 d_1^2)$ . Numerically: $4096 \times 50^4 / (40 \times 41.2)^4 \approx 4096 \times 6.25 \times 10^6 / 7.3 \times 10^6 \approx 3500$ . The RIS-reflected path provides $35$ dB gain over the (blocked) direct path --- enabling NLOS sensing.

Practical considerations

This gain assumes perfect CSI for RIS phase optimisation. In practice, CSI errors reduce the coherent gain by 3--6 dB. The RIS also introduces additional delay spread, which may require wider bandwidth for range resolution.

Common Mistake: Insufficient DOF for Joint Beamforming

Mistake:

Designing ISAC beamforming with $N_t < K_u + K_t$ , where there are not enough spatial degrees of freedom to serve both functions simultaneously.

Correction:

When $N_t < K_u + K_t$ , the communication and sensing beams cannot be formed independently. Options: (1) reduce $K_u$ or $K_t$ via user/target scheduling; (2) use time-division (alternate between communication and sensing frames); (3) accept a severe rate-CRB tradeoff. The "ISAC sweet spot" requires $N_t \geq 2(K_u + K_t)$ for comfortable operation.

⚠️Engineering Note

Hardware Sharing Challenges in ISAC

ISAC promises hardware savings by sharing the antenna array, RF chains, and baseband between communication and sensing. Practical challenges:

Self-interference: In monostatic ISAC, the transmitter and sensing receiver share the same array. TX-RX isolation of 80--100 dB is needed, requiring circulators, SIC, or full-duplex techniques.
Dynamic range: Communication signals have 40--60 dB dynamic range (near-far users); sensing echoes add another 60--80 dB. The ADC must handle 100--140 dB total.
Timing: Communication uses continuous transmission; pulsed radar needs quiet periods for echo reception. OFDM ISAC avoids this by using cyclic prefix as the "quiet period" for near-range targets.

Quick Check

In null-space projection ISAC beamforming, the sensing beam is projected into the null space of the communication channel. What happens when the target direction is close to a user direction?

The projected sensing beam loses energy toward the target

The communication beam automatically senses the target

The beamforming problem becomes infeasible

Correction:

The projected sensing beam loses energy toward the target

When the target steering vector has a large component in the communication channel's column space, the null-space projection removes that component, reducing the sensing beam gain.

Key Takeaway

ISAC beamforming jointly optimises communication SINR and sensing CRB. SDR provides near-optimal convex solutions; null-space projection offers a simpler alternative for massive MIMO. RIS extends ISAC sensing to NLOS regions by adding virtual viewing angles. With $N_t \geq 2(K_u + K_t)$ , both functions can be served with minimal mutual interference.

ISAC Beamforming

Beamforming: Where ISAC Gets Spatial

Definition: ISAC Beamforming Problem

Definition: SDR for ISAC Beamforming

Null-Space Projection ISAC Beamforming

Joint Communication-Sensing Beampattern

Parameters

Example: ISAC Beamforming for a 5G mmWave Base Station

DOF analysis

Communication beamforming

Sensing beamforming

Performance

Definition: RIS-Assisted ISAC

RIS-Assisted ISAC Sensing Coverage

Parameters

Example: RIS-ISAC for NLOS Target Detection

Path geometry

RIS path gain

Practical considerations

Common Mistake: Insufficient DOF for Joint Beamforming

Hardware Sharing Challenges in ISAC

Quick Check

Key Takeaway

Definition:
ISAC Beamforming Problem

Definition:
SDR for ISAC Beamforming

Definition:
RIS-Assisted ISAC