Ferkans — Interactive Telecom Tutor

From Phased Arrays to MIMO Radar

In Chapter 7 we saw that a conventional phased-array radar transmits the same waveform from every antenna, forming a narrow beam. The angular resolution is set by the physical aperture $D$ : $\Delta\theta \approx \lambda/D$ . To halve the beamwidth, you must double the array size --- an expensive proposition.

MIMO radar breaks this barrier. By transmitting orthogonal waveforms from $N_t$ antennas, the receiver can separate each transmit signal and thereby observe $N_t N_r$ independent spatial channels from only $N_t + N_r$ physical elements. The result is a virtual aperture whose resolution matches that of an $N_t N_r$ -element phased array --- a quadratic gain.

This is the key mechanism that makes multi-view RF imaging practical with a manageable number of antennas.

Historical Note: The Birth of MIMO Radar

2003--2007

The MIMO radar concept emerged independently from two communities around 2003--2007. Fishler, Haimovich, Blum, and Cimini introduced the statistical MIMO radar with widely separated antennas, exploiting spatial diversity for detection. Simultaneously, Li and Stoica formalised MIMO radar with co-located antennas, showing the virtual aperture gain. The connection to MIMO communications --- both exploit the spatial degrees of freedom created by multiple independent channels --- was immediately recognised and accelerated cross-fertilisation between the radar and wireless communities.

,

Definition:
Waveform Diversity

A MIMO radar with $N_t$ transmit antennas emits mutually orthogonal waveforms $\{s_i(t)\}_{i=1}^{N_t}$ satisfying the orthogonality condition:

$\int_0^{T_{\mathrm{CPI}}} s_i(t)\,s_{i'}^*(t)\,dt = E_s\,\delta_{ii'}$

where $T_{\mathrm{CPI}}$ is the coherent processing interval and $E_s$ is the per-waveform energy. At each of the $N_r$ receive antennas, matched filtering with $s_i^*(-t)$ extracts the echo corresponding to the $i$ -th transmitter, yielding $N_t N_r$ separated channels.

The orthogonality need not be exact --- practical implementations use TDM, FDM, CDM, or DDM schemes that approximate it. Section DWaveform Orthogonality Methods discusses the trade-offs.

Definition:
Virtual Array and Virtual Aperture

After matched filtering and waveform separation, the combined spatial response for the $(i, j)$ transmit-receive pair to a far-field scatterer at angle $\theta$ is:

$h_{ij}(\theta) = [\mathbf{a}(\theta)]_i \cdot [\hat{\mathbf{a}}(\theta)]_j = e^{j \kappa d_i^{\mathrm{tx}} \sin\theta} \cdot e^{j \kappa d_j^{\mathrm{rx}} \sin\theta} = e^{j \kappa (d_i^{\mathrm{tx}} + d_j^{\mathrm{rx}}) \sin\theta}$

This is equivalent to a single antenna element at the virtual position $d_v^{(i,j)} = d_i^{\mathrm{tx}} + d_j^{\mathrm{rx}}$ . The full set of virtual positions is:

$\mathcal{D}_v = \bigl\{d_i^{\mathrm{tx}} + d_j^{\mathrm{rx}} : i = 1, \ldots, N_t,\; j = 1, \ldots, N_r\bigr\}.$

The virtual array steering vector is the Kronecker product:

$\mathbf{a}_V(\theta) = \mathbf{a}(\theta) \otimes \hat{\mathbf{a}}(\theta) \in \mathbb{C}^{N_t N_r}.$

For a ULA with $N_t$ transmit elements spaced $N_r d$ apart and $N_r$ receive elements spaced $d$ apart (where $d = \lambda/2$ ), the virtual array $\mathcal{D}_v$ is a filled ULA of $N_t N_r$ elements at spacing $d$ . This is the most common MIMO radar configuration.

Theorem: Virtual Aperture Theorem

Consider a MIMO radar with $N_t$ transmit and $N_r$ receive elements arranged so that the virtual array $\mathcal{D}_v$ forms a filled ULA with element spacing $d$ . Then:

The number of virtual elements is $N_v = N_t N_r$ , achieved with $N_t + N_r$ physical elements.
The angular resolution equals that of a phased array with $N_v$ elements: $\Delta\theta \approx \frac{\lambda}{N_v \, d}.$
The aperture efficiency (virtual-to-physical element ratio) is: $\eta = \frac{N_t N_r}{N_t + N_r}$ which is maximised at $N_t = N_r$ , giving $\eta = N/4$ where $N = N_t + N_r$ is the total number of physical elements.

Each transmit antenna illuminates the entire scene with a unique waveform. Each receive antenna captures echoes from all transmit waveforms. After waveform separation, every Tx-Rx pair provides an independent spatial sample, filling the virtual aperture. The point is that the Kronecker product structure generates $N_t N_r$ distinct virtual positions from only $N_t + N_r$ physical ones --- and this is where the quadratic gain comes from.

Proof

Virtual positions form a filled ULA

Place Tx elements at $d_i^{\mathrm{tx}} = (i-1) N_r d$ for $i = 1, \ldots, N_t$ and Rx elements at $d_j^{\mathrm{rx}} = (j-1) d$ for $j = 1, \ldots, N_r$ . Then: $d_v^{(i,j)} = (i-1)N_r d + (j-1) d = \bigl[(i-1)N_r + (j-1)\bigr] d.$ As $i$ ranges over $\{1, \ldots, N_t\}$ and $j$ over $\{1, \ldots, N_r\}$ , the index $(i-1)N_r + (j-1)$ takes every value in $\{0, 1, \ldots, N_tN_r - 1\}$ exactly once. Hence $\mathcal{D}_v = \{0, d, 2d, \ldots, (N_v - 1)d\}$ is a filled ULA with $N_v = N_tN_r$ elements.

Angular resolution

The angular resolution of a filled ULA with $N_v$ elements at spacing $d$ is $\Delta\theta \approx \lambda / (N_v d)$ (first null of the array factor). Since the virtual array is such a ULA, the MIMO radar inherits this resolution.

Aperture efficiency

The ratio $\eta = N_v / (N_t + N_r) = N_tN_r / (N_t + N_r)$ . By AM-GM, $N_t + N_r \geq 2\sqrt{N_tN_r}$ , so $\eta \leq \sqrt{N_tN_r} / 2 \leq N/4$ with equality when $N_t = N_r = N/2$ . $\blacksquare$

,

MIMO Virtual Array Geometry

Visualises the physical Tx/Rx arrays and the resulting virtual array.

Top panel: Physical Tx positions (red triangles) and Rx positions (blue circles).

Middle panel: Virtual array positions (purple diamonds). For standard MIMO (Tx spacing $= N_r d$ ), the virtual array is a filled ULA.

Bottom panel: Virtual array beampattern $|B(\theta)|^2$ , showing angular resolution $\Delta\theta \approx \lambda/(N_tN_r\,d)$ .

Experiment with sparse, nested, and coprime configurations to see how they trade aperture size against grating lobes.

Parameters

N_t

(Tx antennas)4

N_r

(Rx antennas)8

Tx array geometry

Example: Designing a MIMO ULA for Automotive Radar

Design a MIMO radar at $f_0 = 77$ GHz ( $\lambda = 3.9$ mm) with $N_t = 3$ transmit and $N_r = 4$ receive antennas to create a filled virtual ULA. Specify the element spacings and compute the angular resolution.

Solution

Receive array

The receive array is a ULA with $N_r = 4$ elements at half-wavelength spacing: $d = \lambda/2 = 1.95$ mm.

Transmit array

For a filled virtual array, the transmit spacing must be $N_r d = 4 \times 1.95 = 7.8$ mm. The Tx array is a ULA with $N_t = 3$ elements at this spacing.

Virtual array

Virtual positions: $\{0, d, 2d, \ldots, 11d\} = \{0, 1.95, \ldots, 21.45\}$ mm. This is a filled 12-element ULA at $\lambda/2$ spacing.

Angular resolution

$\Delta\theta = \frac{\lambda}{N_v\, d} = \frac{3.9\text{ mm}}{12 \times 1.95\text{ mm}} \approx 0.167\text{ rad} \approx 9.5°.$ $With only$ N_t + N_r = 7$ physical elements, we achieve the resolution of a 12-element phased array.

Definition:
Waveform Orthogonality Methods

Several methods achieve (approximately) orthogonal MIMO waveforms:

Method	Orthogonality domain	Pros	Cons
TDM (time-division)	Time	Simple; any waveform	Reduced dwell time per Tx
FDM (frequency-division)	Frequency	No cross-talk	Reduced bandwidth per Tx
CDM (code-division)	Code space	Full bandwidth + dwell	Cross-correlation sidelobes
DDM (Doppler-division)	Doppler	Continuous illumination	Doppler ambiguity
Random	Statistical	Good for CS	Non-zero cross-correlation

TDM-MIMO (common in automotive radar) cycles through transmitters, creating a phase progression across the slow-time dimension that must be compensated for velocity estimation.

The choice of waveform orthogonality method directly affects the structure of the sensing matrix $\mathbf{A}$ . TDM creates a block-diagonal structure; CDM creates a more uniform structure that is better conditioned but harder to implement.

,

Degrees of Freedom: Phased Array vs. MIMO

The fundamental advantage of MIMO radar is the multiplicative increase in spatial degrees of freedom:

Phased array ( $N$ elements, same waveform): $N$ spatial measurements, beamwidth $\lambda/D$ .
MIMO ( $N_t$ Tx, $N_r$ Rx, orthogonal waveforms): $N_tN_r$ spatial measurements, beamwidth $\lambda/(N_tN_r\,d)$ .

For $N_t = N_r = N/2$ : MIMO provides $N^2/4$ virtual elements vs. $N$ for the phased array --- a quadratic gain in spatial samples. This translates directly to better conditioning of $\mathbf{A}$ and improved imaging resolution.

Quick Check

A MIMO radar has $N_t = 4$ and $N_r = 8$ . How many virtual array elements does it have?

12

32

24

64

Correction:

32

Correct. $N_v = N_t \times N_r = 4 \times 8 = 32$ . This is the quadratic gain of MIMO radar.

Quick Check

For a MIMO ULA with $N_r = 6$ receive elements at spacing $d = \lambda/2$ , what should the Tx element spacing be for a filled virtual array?

$d$

$N_r\, d = 3\lambda$

$N_t\, d$

$\lambda$

Correction:

N_r\, d = 3\lambda

Correct. The Tx spacing of $N_r\,d$ ensures the virtual array tiles without gaps.

Common Mistake: Overlapping Virtual Elements

Mistake:

Setting Tx and Rx spacings independently without checking whether the virtual positions $d_i^{\mathrm{tx}} + d_j^{\mathrm{rx}}$ produce duplicate values.

Correction:

Duplicate virtual positions waste degrees of freedom --- two physical antenna pairs provide only one independent spatial sample at the same virtual location. For a filled virtual ULA, the Tx spacing must be exactly $N_r\,d$ (or, symmetrically, the Rx spacing must be $N_t\,d$ ). For sparse/nested/coprime designs, the position sets must be verified to have the desired difference co-array structure.

Virtual Aperture

The set of effective antenna positions created by a MIMO radar through waveform diversity. Each Tx-Rx pair contributes a virtual element at position $d_v = d^{\mathrm{tx}} + d^{\mathrm{rx}}$ . The virtual aperture has $N_tN_r$ elements from $N_t + N_r$ physical antennas.

Waveform Diversity

The use of orthogonal or quasi-orthogonal transmit waveforms across MIMO radar antennas, enabling the receiver to separate the contribution of each transmitter via matched filtering.

Coherent Processing Interval (CPI)

The time duration over which a radar collects data coherently, i.e., maintaining phase relationships. In MIMO radar, the CPI determines the Doppler resolution: $\Delta f_D = 1 / T_{\mathrm{CPI}}$ .

Related: Waveform Diversity

Key Takeaway

MIMO radar transmits orthogonal waveforms to create a virtual array of $N_tN_r$ elements from $N_t + N_r$ physical antennas. The virtual array steering vector is the Kronecker product $\mathbf{a}_V(\theta) = \mathbf{a}(\theta) \otimes \hat{\mathbf{a}}(\theta)$ , and the quadratic aperture gain is the fundamental reason MIMO radar is attractive for RF imaging.

MIMO Radar Concept

From Phased Arrays to MIMO Radar

Historical Note: The Birth of MIMO Radar

Definition: Waveform Diversity

Definition: Virtual Array and Virtual Aperture

Theorem: Virtual Aperture Theorem

Virtual positions form a filled ULA

Angular resolution

Aperture efficiency

MIMO Virtual Array Geometry

Parameters

Example: Designing a MIMO ULA for Automotive Radar

Receive array

Transmit array

Virtual array

Angular resolution

Definition: Waveform Orthogonality Methods

Degrees of Freedom: Phased Array vs. MIMO

Quick Check

Quick Check

Common Mistake: Overlapping Virtual Elements

Virtual Aperture

Waveform Diversity

Coherent Processing Interval (CPI)

Key Takeaway

Definition:
Waveform Diversity

Definition:
Virtual Array and Virtual Aperture

Definition:
Waveform Orthogonality Methods