MIMO Beamspace Channel Matrix (Hb)

How Beamspace Fits into the Original MIMO Equation

1. Original MIMO Model

In a standard MIMO system, the received signal at the antennas is given by:

y = Hx + n

where:

• y: Received signal vector (size N_r×1)
• H: Channel matrix (size N_r×N_t)
• x: Transmitted signal vector (size N_t×1)
• n: Noise vector

This is the antenna-domain channel representation — it tells how each transmit antenna affects each receive antenna.

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2. Why We Need Beamspace

At mmWave frequencies, the signal propagation has only a few dominant paths (directions of arrival and departure). That means the channel is sparse in the angular domain, but not in antenna space. To expose that sparsity, we convert the channel into beamspace.

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3. Beamspace Representation

We express the physical channel as:

H = A_r H_b A_t^H

where:

• A_t – Transmit array response (DFT matrix mapping antennas → transmit beams)
• A_r – Receive array response (DFT matrix mapping antennas → receive beams)
• H_b – Beamspace channel matrix (sparse in angular domain)

Thus, the beamspace channel is related to the physical channel as:

H_b = A_r^H H A_t

This transformation simply changes the basis from antenna domain to beam (directional) domain.

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4. Substitute into the Original Equation

Substitute H = A_r H_b A_t^H into y = Hx + n:

y = A_r H_b A_t^H x + n

Now multiply both sides by A_r^H (applying a receive beamformer):

A_r^H y = H_b A_t^H x + A_r^H n

Define:

y_b = A_r^H y,   x_b = A_t^H x,   n_b = A_r^H n

Then the MIMO equation in beamspace becomes:

y_b = H_b x_b + n_b

This is the beamspace MIMO model — the same channel, now expressed in terms of transmit and receive beams.

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5. Adding Beamformers (Practical mmWave Case)

In real systems, we use analog/digital beamformers at both ends:

y = W^H H F x + n

Substitute the beamspace form of H:

y = W^H (A_r H_b A_t^H) F x + n

Simplify:

y = (W^H A_r) H_b (A_t^H F) x + n

Here, W^H A_r and A_t^H F act as the receive and transmit beam selectors.

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6. Compressive Sensing Form (OMP Model)

When multiple pilot signals are sent, and the received measurements are stacked, we can vectorize the model:

vec(Y) = (X^T ⊗ W^H) (A_t^* ⊗ A_r) vec(H_b) + n

Define:

y = vec(Y),   Q = (X^T ⊗ W^H)(A_t^* ⊗ A_r),   h_b = vec(H_b)

Then the final OMP equation becomes:

y = Q h_b + n

This is the compressed sensing model used by OMP for sparse channel estimation.

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7. Summary

Step	Equation	Description
1	y = Hx + n	Original MIMO channel (antenna domain)
2	H = Ar Hb AtH	Channel expressed in beamspace
3	yb = Hb xb + nb	MIMO model in beamspace domain
4	y = WH H F x + n	Practical beamformed MIMO model
5	y = Q hb + n	Compressed sensing form used in OMP

Thus, the beamspace transformation fits naturally into the MIMO model by representing the channel in terms of transmit and receive beams instead of antenna elements. This reveals the inherent sparsity of the mmWave channel, enabling efficient estimation using OMP.

Further Reading

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