ECE4024 Chap.9 MIMO and Spatial Multiplexing
MIMO and Spatial Multiplexing
MIMO represents the radio channel as a matrix connecting every transmit antenna to every receive antenna. This chapter uses rank and singular modes to separate diversity, beamforming and spatial multiplexing objectives, then develops a zero-forcing detector with explicit dimension and remultiplication checks.
The practical limit is not antenna count alone: correlation, weak singular values, noise and estimation accuracy determine how many streams remain useful.
What this chapter covers
- 01
Transmit and receive vectors
- 02
MIMO channel matrices
- 03
Rank and independent spatial modes
- 04
Diversity versus spatial multiplexing
- 05
Maximal-ratio transmission
- 06
Singular-value decomposition and beamforming
- 07
Zero-forcing MIMO detection
- 08
Conditioning, noise gain and practical stream count
Worked example · free
Recover two mixed spatial streams
- +2Recognise the upper-triangular inverse H inverse = [[1, -0.25], [0, 1]].
- +1First recovered component is 1 x 2.75 - 0.25 x 3 = 2.
- +1Second recovered component is 3.
- +1Therefore x-hat = [2, 3].
- +1Verify H[2,3] = [2 + 0.75, 3] = [2.75, 3].
- +1State that a nearly singular channel would amplify noise even though a formal inverse might exist.
Key terms
- Channel matrix
- A complex matrix whose entries describe the gain from each transmit antenna to each receive antenna.
- Matrix rank
- The number of independent channel dimensions, which bounds the number of algebraically separable spatial streams.
- Singular value
- A nonnegative mode strength produced by singular-value decomposition, indicating how strongly a spatial channel direction is supported.
- Beamforming
- The use of antenna weights to align transmitted or received energy coherently in a selected spatial direction.
- Spatial stream
- An independent information sequence transmitted through a separable MIMO channel mode at the same time and frequency as other streams.
- ZF detector
- A linear MIMO receiver that applies a channel left inverse to cancel inter-stream mixing in the ideal model.
MIMO and Spatial Multiplexing FAQ
What limits the number of spatial streams?
The count is bounded by transmit dimensions, receive dimensions and channel rank. Weak singular values, noise, correlation and imperfect channel estimates can reduce the number that is reliable in practice.
How does spatial multiplexing differ from diversity?
Spatial multiplexing sends different symbols on separable modes to raise rate. Diversity sends redundant evidence across independently fading dimensions to improve reliability. A design may trade between these objectives.
Why can zero forcing amplify MIMO noise?
When channel columns are nearly dependent, the left inverse needs large coefficients to separate them. Those coefficients also multiply receiver noise, so algebraic cancellation can produce a poor noisy estimate.
What does singular-value decomposition provide?
It gives transmit and receive directions that rotate the matrix channel into parallel modes. The singular values describe the strength of those modes and help explain beamforming, multiplexing and conditioning.
Does adding an antenna always add a stream?
No. The new channel direction must be independent and strong enough, and the other side must have enough dimensions to observe it. Antenna count alone does not guarantee additional rank or useful SNR.
Exam move
Write the dimensions of every vector and matrix before multiplying. Practise rank tests and simple two-by-two inverses, then verify detector outputs by remultiplying through the channel. Compare diversity, beamforming and multiplexing by their objectives rather than by antenna count. Use singular values to discuss both stream strength and conditioning.
When a question asks for a capacity advantage, state the assumptions about rank, channel knowledge, SNR and power allocation. Do not call a formal inverse a good receiver until you have considered its noise cost. Begin each matrix problem by drawing transmit antennas, receive antennas and the coefficient connecting every pair.
Translate that picture into y equals Hx plus n, label the dimensions and state whether rows index receivers or transmitters. Work a small multiplication forward before attempting detection; this catches transposed channel conventions. For rank, test whether columns carry distinguishable spatial signatures rather than merely counting nonzero entries.
Use determinant or row dependence for a two-by-two example, then use singular values to discuss weak and strong modes. Practise ZF detection by forming the appropriate inverse or pseudoinverse, multiplying it by the channel and verifying the identity relation. Inspect the same calculation when columns are nearly parallel: large detector coefficients signal noise amplification even though an algebraic inverse exists.
Contrast MMSE regularisation by explaining what prior signal and noise scales it balances. For beamforming, trace how transmit and receive weights create a scalar effective channel and state where channel knowledge is available. For spatial multiplexing, count streams using antenna limits and channel rank, then explain why practical stream adaptation also considers SNR and conditioning.
When discussing capacity, identify bandwidth, SNR convention, power allocation and channel knowledge rather than quoting a headline antenna multiplier. Finish with an objective table: diversity protects reliability, beamforming concentrates energy and multiplexing raises parallel rate. Real systems may combine them, but each claim must still be tied to the channel matrix and operating point.
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