Analog, Digital and Hybrid Precoding (Beamforming)

This document explains the structure and equations of hybrid analog–digital MIMO systems, emphasizing how digital beamforming works alongside analog beamforming for mmWave systems.

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1. Complete Hybrid MIMO Signal Model

In a hybrid MIMO transmitter, processing is split into two stages:

• Digital (baseband) precoder: \( \mathbf{F}_{\text{BB}} \)
• Analog (RF) precoder: \( \mathbf{F}_{\text{RF}} \)

The transmitted signal is:

\( \mathbf{x} = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} \)

Subject to the power constraint:

\( \| \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \|_F^2 = N_s \)

At the receiver, the signal is:

\( \mathbf{y} = \mathbf{H} \mathbf{x} + \mathbf{n} = \mathbf{H} \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} + \mathbf{n} \)

After analog and digital combining:

\( \mathbf{r} = \mathbf{W}_{\text{BB}}^H \mathbf{W}_{\text{RF}}^H \mathbf{H} \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} + \mathbf{W}_{\text{BB}}^H \mathbf{W}_{\text{RF}}^H \mathbf{n} \)

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2. Digital Beamforming (Baseband Processing)

Digital beamforming operates at baseband and manages inter-stream interference, amplitude/phase control, and power allocation.

Step 1 — Effective Channel

\( \tilde{\mathbf{H}} = \mathbf{W}_{\text{RF}}^H \mathbf{H} \mathbf{F}_{\text{RF}} \)

Step 2 — Baseband Precoder and Combiner Design

Using Singular Value Decomposition (SVD):

\( \tilde{\mathbf{H}} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^H \)

Then choose:

\( \mathbf{F}_{\text{BB}} = \mathbf{V}_1, \quad \mathbf{W}_{\text{BB}} = \mathbf{U}_1 \)

Step 3 — Normalization

\( \mathbf{F}_{\text{BB}} = \sqrt{P_t} \, \frac{\mathbf{F}_{\text{BB}}}{\| \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \|_F} \)

Step 4 — Equivalent Baseband Model

\( \mathbf{r} = \mathbf{W}_{\text{BB}}^H \tilde{\mathbf{H}} \mathbf{F}_{\text{BB}} \mathbf{s} + \mathbf{n}_{\text{eff}} \)

where \( \mathbf{n}_{\text{eff}} = \mathbf{W}_{\text{BB}}^H \mathbf{W}_{\text{RF}}^H \mathbf{n} \).

Step 5 — Achievable Rate

\( R = \log_2 \det \left( \mathbf{I}_{N_s} + \frac{1}{\sigma_n^2 N_s} (\mathbf{W}_{\text{BB}}^H \tilde{\mathbf{H}} \mathbf{F}_{\text{BB}})(\mathbf{W}_{\text{BB}}^H \tilde{\mathbf{H}} \mathbf{F}_{\text{BB}})^H \right) \)

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3. Comparison: Analog vs. Digital Beamforming

Feature	Analog Beamforming (\( \mathbf{F}_{\text{RF}}, \mathbf{W}_{\text{RF}} \))	Digital Beamforming (\( \mathbf{F}_{\text{BB}}, \mathbf{W}_{\text{BB}} \))
Domain	RF (Phase shifters)	Baseband (DSP)
Control	Only phase	Amplitude & phase
Flexibility	Limited	Fully flexible
Purpose	Beam steering	Interference management, power allocation
Complexity	Low	High

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4. Fully Digital Beamforming (Special Case)

If every antenna has its own RF chain:

\( \mathbf{F}_{\text{RF}} = \mathbf{I}_{N_t}, \quad \mathbf{W}_{\text{RF}} = \mathbf{I}_{N_r} \)

The digital beamforming then becomes optimal:

\( \mathbf{F}_{\text{BB}} = \mathbf{V}_1, \quad \mathbf{W}_{\text{BB}} = \mathbf{U}_1 \)

Achievable rate:

\( R_{\text{digital}} = \log_2 \det \left( \mathbf{I} + \frac{P_t}{\sigma_n^2 N_s} \mathbf{H} \mathbf{F}_{\text{BB}} \mathbf{F}_{\text{BB}}^H \mathbf{H}^H \right) \)

This rate represents the upper bound for hybrid systems.

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5. Final Compact Model Summary

Transmit: \( \mathbf{x} = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} \)

Receive: \( \mathbf{r} = \mathbf{W}_{\text{BB}}^H \mathbf{W}_{\text{RF}}^H \mathbf{H} \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s} + \mathbf{n}_{\text{eff}} \)

Effective Channel: \( \tilde{\mathbf{H}} = \mathbf{W}_{\text{RF}}^H \mathbf{H} \mathbf{F}_{\text{RF}} \)

Rate: \( R = \log_2 \det \left( \mathbf{I} + \frac{1}{\sigma_n^2 N_s} (\mathbf{W}_{\text{BB}}^H \tilde{\mathbf{H}} \mathbf{F}_{\text{BB}})(\mathbf{W}_{\text{BB}}^H \tilde{\mathbf{H}} \mathbf{F}_{\text{BB}})^H \right) \)

Further Reading

1.