Ferkans — Interactive Telecom Tutor

Two Flavours of MIMO Radar

The MIMO radar concept bifurcates into two fundamentally different operating regimes depending on the antenna separation relative to the target range:

Co-located MIMO: All antennas are on the same platform (separation $\ll$ range). The target's scattering response is the same for all Tx-Rx pairs. The gain is in angular resolution via the virtual aperture.
Distributed MIMO: Antennas are at widely separated locations (separation $\sim$ range). Different Tx-Rx pairs see different scattering cross-sections. The gain is in spatial diversity and multi-view coverage.

For RF imaging, both regimes are relevant: co-located arrays provide fine angular resolution within each node, while distributed nodes provide the multi-view geometry needed for tomographic reconstruction.

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Definition:
Co-Located MIMO Radar

In a co-located MIMO radar, all $N_t + N_r$ antennas are physically close (inter-element spacing $\ll$ target range $R$ ). Under the far-field assumption, the target appears at the same angle $\theta$ from all antennas.

The received signal (after matched filtering) for the $(i,j)$ Tx-Rx pair is:

$y_{ij} = \sum_{q=1}^{Q} \mathbf{c}_{q} \, [\mathbf{a}(\theta_q)]_i \, [\hat{\mathbf{a}}(\theta_q)]_j + w_{ij}$

where $\mathbf{c}_{q}$ is the complex reflectivity of the $q$ -th scatterer at angle $\theta_q$ .

Stacking all $N_tN_r$ measurements into a vector:

$\mathbf{y} = \sum_{q=1}^{Q} \mathbf{c}_{q} \, \bigl(\mathbf{a}(\theta_q) \otimes \hat{\mathbf{a}}(\theta_q)\bigr) + \mathbf{w} = \mathbf{A}_V \, \mathbf{c} + \mathbf{w}$

where $\mathbf{A}_V = [\mathbf{a}_V(\theta_1), \ldots, \mathbf{a}_V(\theta_Q)]$ is the virtual array steering matrix.

Co-located MIMO creates a virtual aperture for improved angular resolution but does not provide diversity against target scintillation. All Tx-Rx pairs observe the same scattering cross-section --- they are correlated measurements.

Definition:
Distributed MIMO Radar

In a distributed MIMO radar, the $N_t$ transmitters and $N_r$ receivers are at widely separated locations $\{\mathbf{s}_{i}\}$ and $\{\mathbf{r}_{j}\}$ . Each Tx-Rx pair $(i,j)$ observes the target from a different bistatic angle:

$\alpha_{ij} = \angle(\mathbf{p} - \mathbf{s}_{i},\, \mathbf{p} - \mathbf{r}_{j}).$

The scattering cross-section $\sigma(\alpha_{ij})$ generally varies with bistatic angle, so the measurements carry independent information:

$y_{ij} = \sigma(\alpha_{ij}) \, e^{-j\kappa(\|\mathbf{p} - \mathbf{s}_{i}\| + \|\mathbf{p} - \mathbf{r}_{j}\|)} + w_{ij}.$

This spatial diversity improves detection probability (the target cannot be simultaneously at a scintillation null for all bistatic angles) and enables tomographic imaging.

The price of distributed MIMO is the need for synchronisation between nodes (time, frequency, phase) and higher system complexity. Section 11.3 develops the geometry in detail.

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Theorem: Detection Diversity Gain of Distributed MIMO

Consider a distributed MIMO radar with $K = N_tN_r$ Tx-Rx pairs detecting a Swerling-I target (single-scan Rayleigh-fading cross-section). If the $K$ bistatic scattering cross-sections are independent, the detection probability satisfies:

$P_D \geq 1 - (1 - P_{D,1})^K$

where $P_{D,1}$ is the single-pair detection probability. The diversity order is $K$ , meaning the miss probability decays as $\text{SNR}^{-K}$ at high SNR.

A monostatic radar fails to detect a Swerling-I target when the target happens to present a small cross-section towards the radar. With $K$ independent viewing angles, the probability that the target is simultaneously invisible from all angles decreases exponentially with $K$ . This is the radar analogue of receive diversity in communications --- the same $\text{SNR}^{-K}$ slope appears.

Proof

Independence assumption

Each Tx-Rx pair $(i,j)$ observes an independent complex scattering coefficient $h_{ij} \sim \mathcal{CN}(0, \sigma_{\mathrm{RCS}}^2)$ . The detection event $D_{ij} = \{|y_{ij}|^2 > \gamma\}$ is independent across pairs.

Diversity combining

Using a square-law combiner: $T = \sum_{ij} |y_{ij}|^2$ . Under $H_1$ , $T$ is the sum of $K$ independent non-central chi-squared variables. The miss probability is: $P_M = \prod_{ij} P(|y_{ij}|^2 \leq \gamma) = (1 - P_{D,1})^K.$ Hence $P_D = 1 - P_M \geq 1 - (1 - P_{D,1})^K$ . $\blacksquare$

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Definition:
The Phased-MIMO Continuum

Between the two extremes --- phased array (all Tx send the same waveform) and full MIMO (all Tx send orthogonal waveforms) --- lies a continuum parametrised by the number of independent waveforms $L$ , with $1 \leq L \leq N_t$ :

$L = 1$ (phased array): Maximum coherent gain ( $N_t^{2}$ power gain), narrowest beam, $N_r$ virtual elements.
$L = N_t$ (full MIMO): Maximum spatial diversity ( $N_tN_r$ virtual elements), widest illumination, lowest per-direction gain.
$1 < L < N_t$ (phased-MIMO): The $N_t$ antennas are partitioned into $L$ sub-arrays, each transmitting a common waveform. This yields $L\,N_r$ virtual elements and sub-array coherent gain of $(N_t/L)^2$ per sub-array.

The choice of $L$ trades beamforming gain (SNR in a given direction) against waveform diversity (number of virtual elements / angular coverage).

For RF imaging, we typically want $L = N_t$ (full MIMO) to maximise the number of independent measurements. For communication in ISAC systems, smaller $L$ provides higher SNR for data transmission while still enabling some sensing capability.

Phased Array vs. MIMO Radar

Property	Phased Array	Co-Located MIMO	Distributed MIMO
Waveform	Same from all Tx	Orthogonal per Tx	Orthogonal per Tx
Virtual elements	$N_r$	$N_tN_r$	$N_tN_r$
Coherent Tx gain	$N_t^{2}$	$1$ (omnidirectional)	$1$ (omnidirectional)
Angular resolution	$\lambda / (N_r\,d)$	$\lambda / (N_tN_r\,d)$	View-dependent
Diversity order	1	1	$N_tN_r$
Synchronisation	Trivial (shared LO)	Trivial (shared LO)	Critical challenge
Primary benefit	Max Tx gain	Virtual aperture	Spatial diversity
Imaging use	Narrow FoV, high SNR	Wide FoV, fine resolution	Multi-view tomography

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Phased Array vs. MIMO Beampattern

Compare the transmit beampatterns of a phased array (coherent gain, narrow beam) and MIMO radar (wide illumination, virtual aperture gain after receive processing).

The phased array focuses energy in one direction; MIMO illuminates the entire field of view. The "MIMO after processing" curve shows the effective angular resolution after virtual array beamforming.

Use the slider to explore the phased-MIMO continuum: as $L$ increases from 1 to $N_t$ , the beam widens and the number of virtual elements grows.

Parameters

N_t

8

N_r

8

L

(independent waveforms)1

Steering angle (deg)0

Example: Phased-MIMO Trade-off for ISAC

An ISAC base station has $N_t = 8$ Tx and $N_r = 8$ Rx antennas. Compare the following configurations: (a) Full phased array ( $L = 1$ ): all Tx coherently steer towards the communication user. (b) Phased-MIMO ( $L = 4$ ): 4 sub-arrays of 2 antennas each. (c) Full MIMO ( $L = 8$ ): all waveforms orthogonal. For each, compute the Tx gain towards the user at $\theta = 0°$ and the number of virtual elements for sensing.

Solution

Full phased array ($L = 1$)

Tx gain: $N_t^{2} = 64$ (coherent combining). Virtual elements: $1 \times N_r = 8$ . Sensing angular resolution: $\lambda/(8d)$ .

Phased-MIMO ($L = 4$)

Each sub-array has $N_t/L = 2$ elements. Tx gain per sub-array: $2^2 = 4$ . Total radiated power is the same, but spread across 4 beams. Virtual elements: $L \times N_r = 32$ . Sensing angular resolution: $\lambda/(32d)$ .

Full MIMO ($L = 8$)

Tx gain: $1$ (each antenna radiates independently). Virtual elements: $8 \times 8 = 64$ . Sensing angular resolution: $\lambda/(64d)$ .

Trade-off summary

Moving from phased array to full MIMO trades $18$ dB of Tx coherent gain for an $8\times$ increase in virtual elements (and correspondingly finer angular resolution). The phased-MIMO with $L = 4$ is a practical compromise for ISAC: it retains $6$ dB of coherent gain while providing $4\times$ the angular resolution of the phased array.

Common Mistake: Confusing Virtual Elements with SNR Gain

Mistake:

Assuming that $N_tN_r$ virtual elements provide $N_tN_r$ times the SNR of a single element.

Correction:

MIMO radar does not increase the total radiated power towards the target. Phased array has coherent Tx gain $N_t^{2}$ ; MIMO has Tx gain $1$ (omnidirectional illumination). The total power at the target is $N_t$ times lower for MIMO.

The MIMO advantage is in angular diversity, not SNR: more independent measurements enable better imaging and detection through processing gain, but the per-measurement SNR is lower. The net detection performance depends on the target model (Swerling 0 favours phased array; Swerling I/III favour MIMO diversity).

Historical Note: The Phased-MIMO Concept

2010

The phased-MIMO hybrid was introduced by Hassanien and Vorobyov in 2010, who recognised that partitioning the transmit array into sub-arrays provides a tuneable trade-off between coherent gain and waveform diversity. This concept proved particularly influential for ISAC systems, where the communication function demands directional gain while the sensing function benefits from wide illumination. The phased-MIMO continuum remains the standard framework for joint radar-communication waveform design.

Co-Located MIMO Radar

A MIMO radar where all transmit and receive antennas are on the same platform. The primary benefit is the virtual aperture for improved angular resolution.

Distributed MIMO Radar

A MIMO radar with antennas at widely separated locations, providing spatial diversity and multi-view coverage. The challenge is inter-node synchronisation.

Why This Matters: MIMO Communications and MIMO Radar

The mathematical structure of MIMO radar is strikingly similar to MIMO communications (Chapter 15). Both exploit the Kronecker product of transmit and receive steering vectors:

In communications, the MIMO channel matrix $\mathbf{H} \in \mathbb{C}^{N_r \times N_t}$ is estimated and used for spatial multiplexing.
In radar, the same matrix structure appears in the sensing matrix $\mathbf{A}$ , but the goal is to image the scene rather than decode data.

ISAC systems unify both: the same array and (partially) the same waveform serve both communication and sensing. The phased-MIMO continuum provides the design knob to balance the two functions.

See full treatment in Chapter 34، Section 1

Key Takeaway

Co-located MIMO creates virtual aperture for angular resolution; distributed MIMO provides spatial diversity with diversity order $N_tN_r$ . The phased-MIMO continuum parametrised by $L$ independent waveforms trades coherent gain against waveform diversity --- the key design knob for ISAC systems.

Co-Located vs. Distributed MIMO Radar