Ferkans — Interactive Telecom Tutor

Connecting Arrays to Propagation

Sections 7.1-7.3 developed antenna and array theory in isolation; Chapter 6 characterised the wireless channel without arrays. Now we unite both: the spatial channel model describes how multipath propagation interacts with transmit and receive arrays. The steering vectors from Section 7.2 become the bridge between the physical angles-of-arrival/departure (AoA/AoD) and the algebraic MIMO channel matrix $\mathbf{H}$ .

Definition:
Array Steering Vector (General Form)

For an arbitrary array with $N$ elements at positions $\mathbf{p}_0, \mathbf{p}_1, \ldots, \mathbf{p}_{N-1}$ , the steering vector for a plane wave arriving from direction $\hat{\mathbf{k}}$ (unit vector along the wave propagation) is

$[\mathbf{a}(\hat{\mathbf{k}})]_n = e^{jk\,\hat{\mathbf{k}} \cdot \mathbf{p}_n}$

For a ULA along the $z$ -axis with spacing $d$ and direction parametrised by elevation angle $\theta$ :

$[\mathbf{a}(\theta)]_n = e^{jnkd\sin\theta}, \quad n = 0, \ldots, N-1$

The steering vector has unit-magnitude entries and $\|\mathbf{a}(\theta)\|^2 = N$ .

Theorem: MIMO Channel Matrix from Multipath Clusters

For a narrowband MIMO system with an $N_r$ -element receive array and an $N_t$ -element transmit array, the channel matrix $\mathbf{H} \in \mathbb{C}^{N_r \times N_t}$ is

$\mathbf{H} = \sum_{l=1}^{L} \alpha_l\, \mathbf{a}_r(\theta_l)\, \mathbf{a}_t^H(\phi_l)$

where for the $l$ -th multipath cluster:

$\alpha_l \in \mathbb{C}$ is the complex path gain
$\theta_l$ is the angle of arrival (AoA) at the receiver
$\phi_l$ is the angle of departure (AoD) at the transmitter
$\mathbf{a}_r(\theta_l) \in \mathbb{C}^{N_r}$ is the receive steering vector
$\mathbf{a}_t(\phi_l) \in \mathbb{C}^{N_t}$ is the transmit steering vector

Each multipath cluster contributes a rank-1 matrix $\alpha_l\,\mathbf{a}_r(\theta_l)\,\mathbf{a}_t^H(\phi_l)$ . The channel is the sum of $L$ such rank-1 components. The rank of $\mathbf{H}$ is at most $\min(L, N_r, N_t)$ : MIMO spatial multiplexing requires multiple resolvable clusters (rich scattering), not just many antennas.

Proof

Single-path derivation

A single plane wave departing at angle $\phi$ from the transmit array arrives at angle $\theta$ at the receive array with complex gain $\alpha$ . The transmit array applies spatial signature $\mathbf{a}_t(\phi)$ ; the receive array observes the signal with spatial signature $\mathbf{a}_r(\theta)$ .

The received signal vector is $\mathbf{r} = \alpha\,\mathbf{a}_r(\theta)\,\mathbf{a}_t^H(\phi)\,\mathbf{x}$ ,

so the single-path channel is $\mathbf{H}_{l} = \alpha_l\,\mathbf{a}_r(\theta_l)\,\mathbf{a}_t^H(\phi_l)$ .

Superposition

By linearity, the total channel is

$\mathbf{H} = \sum_{l=1}^{L} \mathbf{H}_{l} = \sum_{l=1}^{L} \alpha_l\,\mathbf{a}_r(\theta_l)\,\mathbf{a}_t^H(\phi_l)$ .

This can be written compactly as $\mathbf{H} = \mathbf{A}_r\,\mathrm{diag}(\boldsymbol{\alpha})\,\mathbf{A}_t^H$

where $\mathbf{A}_r = [\mathbf{a}_r(\theta_1), \ldots, \mathbf{a}_r(\theta_L)]$ and $\mathbf{A}_t = [\mathbf{a}_t(\phi_1), \ldots, \mathbf{a}_t(\phi_L)]$ . $\blacksquare$

,

Building the MIMO Channel Matrix

Watch the MIMO channel matrix

\mathbf{H}

being constructed one multipath cluster at a time. Each cluster contributes a rank-1 outer product

\alpha_l\,\mathbf{a}_r(\theta_l)\,\mathbf{a}_t^H(\phi_l)

. The channel rank increases with each new cluster until reaching

\min(L, N_r, N_t)

.

Progressive construction of the MIMO channel from 3 multipath clusters. The heatmap shows

|\mathbf{H}|

gaining structure as each rank-1 component is added.

Spatial Channel with Multipath Clusters

Visualize the MIMO channel matrix formed by superposition of multipath clusters. Each cluster contributes a rank-1 outer product of steering vectors. Observe how the singular values of $\mathbf{H}$ depend on the number of clusters and the angular separation between them. The left panel shows the channel magnitude $|\mathbf{H}|$ ; the right panel shows the singular value distribution.

Parameters

Receive elements

N_r

8

Transmit elements

N_t

4

Number of clusters

L

3

Spacing

d/\lambda

0.5

Example: Rank of a Pure LOS MIMO Channel

Consider a MIMO system with $N_t = 4$ transmit and $N_r = 8$ receive antennas (both ULAs, $d = \lambda/2$ ). The channel has a single LOS path with AoD $\phi = 20^\circ$ and AoA $\theta = -10^\circ$ .

(a) Write the channel matrix $\mathbf{H}$ .

(b) What is the rank of $\mathbf{H}$ ?

(c) How many independent data streams can be spatially multiplexed?

Solution

Channel matrix

With a single path ( $L = 1$ ):

$\mathbf{H} = \alpha_1\,\mathbf{a}_r(-10^\circ)\,\mathbf{a}_t^H(20^\circ)$

where $[\mathbf{a}_t(20^\circ)]_n = e^{jn\pi\sin(20^\circ)}$ for $n = 0, \ldots, 3$ and $[\mathbf{a}_r(-10^\circ)]_m = e^{jm\pi\sin(-10^\circ)}$ for $m = 0, \ldots, 7$ .

$\mathbf{H} \in \mathbb{C}^{8 \times 4}$ is the outer product of an 8-vector and a 4-vector.

Channel rank

The outer product $\mathbf{a}_r\,\mathbf{a}_t^H$ is a rank-1 matrix. Therefore $\text{rank}(\mathbf{H}) = 1$ .

Only one non-zero singular value exists: $\sigma_1 = |\alpha_1| \cdot \|\mathbf{a}_r\| \cdot \|\mathbf{a}_t\| = |\alpha_1| \sqrt{N_r N_t} = |\alpha_1|\sqrt{32}$ .

Multiplexing capability

With rank 1, only one independent data stream can be transmitted, despite having 4 transmit and 8 receive antennas.

The system achieves a beamforming gain of $N_r N_t = 32$ ( $15.1$ dB) but no spatial multiplexing gain. This is the fundamental limitation of LOS MIMO without scattering. $\blacksquare$

Definition:
Angular Spread

The angular spread $\sigma_\theta$ measures the dispersion of multipath arrivals around a mean angle:

$\sigma_\theta = \sqrt{\frac{\sum_l P_l\,(\theta_l - \bar\theta)^2}{\sum_l P_l}}$

where $P_l = |\alpha_l|^2$ is the power of cluster $l$ and $\bar\theta = \sum_l P_l\theta_l / \sum_l P_l$ is the power-weighted mean AoA.

Angular spread determines the effective rank of the MIMO channel:

Small $\sigma_\theta$ (LOS, rural): low rank, beamforming gain only
Large $\sigma_\theta$ (rich scattering, urban NLOS): high rank, spatial multiplexing possible

Why This Matters: Massive MIMO and Channel Hardening

In massive MIMO ( $N_r \gg N_t$ , typically $N_r = 64$ - $256$ at the base station), the channel matrix columns $\mathbf{h}_1, \ldots, \mathbf{h}_{N_t}$ become approximately orthogonal as $N_r \to \infty$ :

$\frac{1}{N_r}\mathbf{H}^{H}\mathbf{H} \to \mathbf{I}$

This channel hardening phenomenon means the channel becomes nearly deterministic — small-scale fading effectively averages out over the many receive antennas. Simple matched filtering $\mathbf{w} = \mathbf{h}_k$ achieves near-optimal per-user performance. This is a direct consequence of the spatial channel model: with many array elements, the projections of different users' steering vectors become orthogonal.

Common Mistake: Expecting MIMO Gains in LOS Channels

Mistake:

Deploying massive MIMO expecting spatial multiplexing gains in a strong LOS environment with negligible scattering.

Correction:

A pure LOS channel has rank 1 regardless of the number of antennas. In LOS-dominated scenarios, MIMO provides only beamforming gain ( $\sim N_r N_t$ ) but not multiplexing gain. For multiplexing, one needs either:

Rich scattering (NLOS) to create multiple resolvable paths
Widely spaced arrays so that even the direct path creates multiple resolvable angular bins
Dual-polarized antennas to double the rank

The 3GPP model captures this through the Ricean K-factor: high- $K$ channels have near-rank-1 behaviour.

Quick Check

A MIMO system has $N_t = 4$ , $N_r = 4$ , and the channel has $L = 6$ clusters with distinct AoA/AoD. What is the rank of $\mathbf{H}$ ?

6

4

1

2

Correction:

4

Correct. $\text{rank}(\mathbf{H}) \leq \min(L, N_r, N_t) = \min(6, 4, 4) = 4$ . With 6 well-separated clusters, the channel is generically full rank (rank 4).

Why This Matters: Steering Vectors in RF Imaging

The steering vector $\mathbf{a}(\theta)$ developed here reappears as the core building block of the RF imaging forward model (Book RFI, Chapters 3-5). In RF imaging, the sensing matrix $\mathbf{A}$ is constructed from products of transmit and receive steering vectors evaluated at each voxel position, and the image is reconstructed by inverting $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ . The array geometry (ULA, UPA, or distributed) directly determines the imaging resolution — the same beamwidth formula $\lambda/(Nd)$ governs both communication beamforming and radar/imaging angular resolution.

Why This Matters: Advanced MIMO Theory

The spatial channel model $\mathbf{H} = \sum_l \alpha_l\, \mathbf{a}_r(\theta_l)\,\mathbf{a}_t^H(\phi_l)$ is the starting point for the entire MIMO book (Book MIMO). There, the theory extends to: massive MIMO with hundreds of antennas and channel hardening (Ch. 3-5), Joint Spatial Division and Multiplexing (JSDM) exploiting the covariance structure of steering vectors (Ch. 10), cell-free distributed MIMO (Ch. 14), and near-field/ XL-MIMO where the plane-wave assumption breaks down (Ch. 20).

Angular Spread

RMS spread of multipath arrival (or departure) angles around the mean direction. Determines the effective rank and spatial correlation of the MIMO channel.

Channel Hardening

The phenomenon where $\frac{1}{N}\mathbf{H}^{H}\mathbf{H} \to \mathbf{I}$ as $N \to \infty$ , causing the effective channel gain to become nearly deterministic. The hallmark of massive MIMO.

Spatial Channel Model with Arrays