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Degrees of Freedom in MIMO Systems

In MIMO (Multiple-Input Multiple-Output) wireless systems, the degrees of freedom (DoF) refer to the maximum number of independent data streams that can be transmitted simultaneously over the channel without interfering with each other.

Basic Formula

For a point-to-point MIMO system with:

N_t transmit antennas
N_r receive antennas

DoF = min( N_t, N_r )

Meaning

This number represents the maximum number of parallel data streams the channel can support.

4×2 MIMO → DoF = 2
8×8 MIMO → DoF = 8
2×1 MISO → DoF = 1

Why is this the DoF?

Each independent stream needs:

a unique transmit spatial direction, and
a unique receive spatial dimension

So the limiting factor is the side with fewer antennas.

DoF & Rank of MIMO Channel Matrix

If the channel matrix H has rank r, then:

DoF = rank(H) ≤ min(N_t, N_r)

In rich-scattering environments, the channel usually has full rank, so the simple formula holds.

Multiuser Systems

MIMO MAC (uplink): DoF = sum of users’ transmit antennas, limited by BS antennas
MIMO BC (downlink): DoF = min(total BS antennas, total user antennas)
Massive MIMO: DoF scales with the number of base-station antennas

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