Skip to content

Waterfilling Power Allocation in MIMO and OFDM

The waterfilling algorithm is used in wireless communications to optimally allocate power across multiple channels (or subcarriers) to maximize data rate under a total power constraint.

Intuition

Imagine pouring water into vessels with different floor heights: - Each vessel = a channel - Floor height = inverse of channel gain - More power (water) goes to channels with better gains (deeper vessels)

Mathematical Formulation

Objective:

Maximize total data rate:

\[ \max_{\{P_i\}} \sum_{i=1}^{N} \log_2\left(1 + \frac{P_i |h_i|^2}{N_0}\right) \]

Subject to:

\[ \sum_{i=1}^{N} P_i = P_{\text{total}}, \quad P_i \geq 0 \]

Optimal Power Allocation:

\[ P_i = \left( \mu - \frac{N_0}{|h_i|^2} \right)^+ \]

Where: - \( \mu \): Water level (chosen to meet total power constraint) - \( (\cdot)^+ \): Zero if negative

Application in Wireless Systems

OFDM Frequency-Domain Waterfilling

  • Apply across subcarriers with different gains \( |H_k| \)
  • More power to strong subcarriers, none to weak ones

MIMO Spatial Waterfilling

  • Use SVD: \( \mathbf{H} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^H \)
  • Power allocated to eigenchannels \( \sigma_i^2 \):
\[ P_i = \left( \mu - \frac{N_0}{\sigma_i^2} \right)^+ \]

Summary Table

Use Case Domain Channel Basis
OFDM Frequency Subcarriers
MIMO Spatial Eigenmodes (SVD)
General Any Parallel AWGN channels

Benefits

  • Maximizes capacity
  • Skips bad (deep fade) channels
  • Dynamic and adaptive power use

Does the Transmitter Need CSI?

? Yes ? Channel State Information at the Transmitter (CSIT) is required for waterfilling.

Why?

Waterfilling allocates more power to better channels: - In OFDM, this means knowing each subcarrier's gain \( |H_k|^2 \) - In MIMO, this means knowing the SVD of the channel matrix \( \mathbf{H} \) to extract \( \sigma_i^2 \)

Without CSIT, the transmitter cannot: - Differentiate between good and bad channels - Apply waterfilling (no way to adjust \( P_i \))

Result:

  • Use uniform power allocation if no CSIT is available
  • Waterfilling is not possible without CSIT

_Last updated: June 06, 2025