mmWave MIMO Channel Estimation

This tutorial walks through the derivation of the Equivalent Sensing Matrix and how the mmWave MIMO channel estimation problem can be expressed as a compressed sensing model. A numerical example is included for clarity.

The mmWave MIMO System Model

In a hybrid beamforming setup, the received signal is:

\[ \mathbf{y} = \sqrt{P}\, \mathbf{W}^H \mathbf{H} \mathbf{F} \mathbf{s} + \tilde{\mathbf{n}} \]

Where:

Symbol	Meaning
\( P \)	Transmit power
\( \mathbf{W} = \mathbf{W}_{RF}\mathbf{W}_{BB} \)	Receiver hybrid combiner
\( \mathbf{F} = \mathbf{F}_{RF}\mathbf{F}_{BB} \)	Transmitter hybrid precoder
\( \mathbf{s} \)	Training / pilot vector
\( \tilde{\mathbf{n}} \)	Noise vector
\( \mathbf{H} \)	mmWave channel matrix

Sparse Channel Representation

The physical mmWave channel can be expressed using array response matrices:

\[ \mathbf{H} = \mathbf{A}_R \mathbf{H}_b \mathbf{A}_T^* \]

Where:

\( \mathbf{A}_R \): receive array response matrix
\( \mathbf{A}_T \): transmit array response matrix
\( \mathbf{H}_b \): beamspace (sparse) channel matrix

The vectorized form is:

\[ \text{vec}(\mathbf{H}) = (\mathbf{A}_T^* \otimes \mathbf{A}_R)\mathbf{h}_b \]

where \( \mathbf{h}_b = \text{vec}(\mathbf{H}_b) \).

Substituting the Channel into the System Model

\[ \mathbf{y} = \sqrt{P}\, \mathbf{W}^H \mathbf{A}_R \mathbf{H}_b \mathbf{A}_T^* \mathbf{F} \mathbf{s} + \tilde{\mathbf{n}} \]

Since \( \mathbf{s} \) is known during training, we absorb it into \( \mathbf{F} \).

Vectorization using the Kronecker Product

We use the identity:

\[ \text{vec}(\mathbf{A}\mathbf{X}\mathbf{B}) = (\mathbf{B}^T \otimes \mathbf{A})\text{vec}(\mathbf{X}) \]

Applying this gives:

\[ \text{vec}(\mathbf{W}^H \mathbf{A}_R \mathbf{H}_b \mathbf{A}_T^* \mathbf{F}) = ((\mathbf{A}_T^* \mathbf{F})^T \otimes \mathbf{W}^H \mathbf{A}_R)\, \mathbf{h}_b \]

Therefore:

\[ \mathbf{y} = \sqrt{P}\, ((\mathbf{A}_T^* \mathbf{F})^T \otimes \mathbf{W}^H \mathbf{A}_R)\, \mathbf{h}_b + \tilde{\mathbf{n}} \]

Substitute Hybrid Beamformers

Using \( \mathbf{F} = \mathbf{F}_{RF}\mathbf{F}_{BB} \) and \( \mathbf{W} = \mathbf{W}_{RF}\mathbf{W}_{BB} \):

\[ \mathbf{y} = \sqrt{P}\, ((\mathbf{A}_T^* \mathbf{F}_{RF}\mathbf{F}_{BB})^T \otimes \mathbf{W}_{BB}^H \mathbf{W}_{RF}^H \mathbf{A}_R)\, \mathbf{h}_b + \tilde{\mathbf{n}} \]

Simplifying the transpose:

\[ (\mathbf{A}_T^* \mathbf{F}_{RF}\mathbf{F}_{BB})^T = \mathbf{F}_{BB}^T \mathbf{F}_{RF}^T \mathbf{A}_T^* \]

So:

\[ \boxed{ \mathbf{y} = \sqrt{P}\, (\mathbf{F}_{BB}^T \mathbf{F}_{RF}^T \mathbf{A}_T^* \otimes \mathbf{W}_{BB}^H \mathbf{W}_{RF}^H \mathbf{A}_R)\mathbf{h}_b + \tilde{\mathbf{n}} } \]

Defining the Equivalent Sensing Matrix

We define:

\[ \boxed{ \mathbf{Q} = \sqrt{P} \, (\mathbf{F}_{BB}^T \mathbf{F}_{RF}^T \mathbf{A}_T^*) \otimes (\mathbf{W}_{BB}^H \mathbf{W}_{RF}^H \mathbf{A}_R) } \]

Then the model simplifies beautifully to:

\[ \boxed{\mathbf{y} = \mathbf{Q}\mathbf{h}_b + \tilde{\mathbf{n}}} \]

---

Dimensions

In a practical example (from Prof. Jagannatham’s slides):

Parameter	Value
\( N_T^{Beam} = N_R^{Beam} = 24 \)	\( 576 \) measurements
\( G = 32 \)	\( G^2 = 1024 \) unknowns

So \( \mathbf{Q} \in \mathbb{C}^{576 \times 1024} \) — an underdetermined system.

Numerical Example (2×2 System)

Setup

Quantity	Value
Transmit antennas \( N_T \)	2
Receive antennas \( N_R \)	2
Grid size \( G \)	2
Power \( P \)	1

Channel Basis

\[ \mathbf{A}_T = \mathbf{A}_R = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \]

Beamspace Channel

\[ \mathbf{H}_b = \begin{bmatrix} 1 & 0 \\ 0 & 0.5 \end{bmatrix}, \quad \mathbf{h}_b = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0.5 \end{bmatrix} \]

RF/Baseband Matrices

\[ \mathbf{F}_{RF} = \mathbf{W}_{RF} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \quad \mathbf{F}_{BB} = \mathbf{W}_{BB} = \mathbf{I}_2 \]

Compute Q

The transmit and receive terms each simplify to \( \sqrt{2}\mathbf{I}_2 \), giving:

\[ \mathbf{Q} = (\sqrt{2}\mathbf{I}_2) \otimes (\sqrt{2}\mathbf{I}_2) = 2\mathbf{I}_4 \]

Compute Output

\[ \mathbf{y} = \mathbf{Q}\mathbf{h}_b = 2\mathbf{h}_b = \begin{bmatrix} 2 \\ 0 \\ 0 \\ 1 \end{bmatrix} \]

If noise is added: \[ \mathbf{y} = \begin{bmatrix} 2 \\ 0 \\ 0 \\ 1 \end{bmatrix} + \tilde{\mathbf{n}} \]

Sparse Recovery

Since the system is underdetermined (\( 576 < 1024 \)), we use Compressed Sensing techniques to estimate \( \mathbf{h}_b \):

\[ \min_{\mathbf{h}_b} \|\mathbf{y} - \mathbf{Q}\mathbf{h}_b\|_2^2 \quad \text{s.t. } \mathbf{h}_b \text{ is sparse} \]

Algorithms:

OMP (Orthogonal Matching Pursuit)
Basis Pursuit / LASSO
Sparse Bayesian Learning

Finally, reconstruct the full channel:

\[ \mathbf{H} = \mathbf{A}_R \, \text{mat}(\hat{\mathbf{h}}_b) \, \mathbf{A}_T^* \]

Summary

Step	Description
1	Start with \( \mathbf{y} = \sqrt{P}\mathbf{W}^H \mathbf{H}\mathbf{F}\mathbf{s} + \tilde{\mathbf{n}} \)
2	Express \( \mathbf{H} = \mathbf{A}_R \mathbf{H}_b \mathbf{A}_T^* \)
3	Vectorize using the Kronecker identity
4	Define the equivalent sensing matrix \( \mathbf{Q} \)
5	Obtain \( \mathbf{y} = \mathbf{Q}\mathbf{h}_b + \tilde{\mathbf{n}} \)
6	Estimate sparse \( \mathbf{h}_b \) using compressed sensing
7	Reconstruct full channel \( \mathbf{H} \)

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