ELEC5508 Chap.7 MIMO Channels and SVD
MIMO Channels and SVD
This chapter treats the MIMO matrix as the resource, not the antenna count. Singular-value decomposition rotates antenna coordinates into orthogonal channel modes. Their number gives rank and spatial degrees of freedom; their magnitudes determine conditioning and useful mode strength.
The chapter separates low-SNR beamforming and diversity benefits from high-SNR spatial multiplexing and links both to array geometry and correlation. Dimension and power checks are central. The channel has receive antennas as rows and transmit antennas as columns, and the number of streams cannot exceed rank or the smaller antenna count.
Comparing configurations requires the same total power unless a different constraint is stated. Array spacing and angular spread determine whether physical elements create distinct modes, so a rich hardware diagram does not by itself prove multiplexing capability.
What this chapter covers
- 01
Narrowband matrix model y equals Hx plus n
- 02
Singular-value decomposition
- 03
Rank and spatial degrees of freedom
- 04
Beamforming versus multiplexing
- 05
Channel conditioning and correlation
- 06
Angular resolution and array aperture
Worked example · free
Compute capacity from two useful modes
- +1Two non-zero singular values give rank 2.
- +1Equal power gives 6 SNR units to each active mode.
- +1Mode SNRs are 54 and 6.
- +1Capacity is log2(55) plus log2(7) = 8.588 bit/s/Hz.
- +1The zero mode cannot carry an independent stream.
Key terms
- MIMO
- Use of multiple transmit and receive antennas with a matrix-valued wireless channel.
- SVD
- Factorisation of a channel into transmit directions, singular gains and receive directions.
- Rank
- The number of linearly independent channel modes available for spatial processing.
- Condition number
- A measure of disparity between strongest and weakest useful singular directions.
- Spatial multiplexing
- Transmission of different data streams over independent spatial channel modes.
MIMO Channels and SVD FAQ
Does a four-antenna link support four streams?
Only if the channel matrix has four useful independent modes and the receiver has enough dimensions. Correlation or limited angular spread can reduce rank, leaving fewer streams even though the hardware count remains four.
When should MIMO use beamforming instead of multiplexing?
At low SNR or with one dominant singular mode, concentrating power into the strongest direction can be efficient. At high SNR with several strong modes, parallel streams exploit spatial degrees of freedom. The objective and CSI boundary decide the choice.
How does antenna correlation affect MIMO?
Correlation makes channel columns or rows more similar, reduces conditioning and can collapse useful rank. This weakens spatial multiplexing and diversity. Inspect covariance or singular values rather than inferring performance from antenna count alone.
How should I turn singular values into a transmission decision?
Write the dimensions of the channel matrix, count its non-zero singular values and compare their magnitudes. The count gives the number of useful spatial modes, while a large spread reveals poor conditioning and warns that weak modes will be noise-sensitive. At low total SNR, concentrating power on a strong mode may be preferable; at higher SNR, several well-supported modes can raise the multiplexing slope.
Keep total transmitted power fixed when comparing antenna configurations. State whether the objective is array gain, reliability through diversity or rate through multiplexing, because these goals lead to different signalling choices. Confirm the decision with the singular-value distribution rather than nominal antenna count or total channel energy alone.
Exam move
For every MIMO problem, write dimensions of H, count non-zero singular values and compare their magnitudes. Label the desired gain as power, diversity or multiplexing before selecting an architecture. Start with the matrix equation and annotate the dimension of every vector and matrix. This exposes impossible products before arithmetic begins. Next, practise reading singular-value lists.
Count non-zero values for rank, square them for mode power gains when required, and compare their spread to describe conditioning. Solve equal-power capacity examples before water-filling so mode contributions remain visible. Repeat at low and high total SNR and explain why beamforming dominates when power is scarce while multiple active modes determine the high-SNR slope.
Keep total transmitted power fixed when changing antenna count. Build a separate comparison between diversity and multiplexing: repeated or coded information across independent fades improves reliability, whereas different symbols across singular modes improve rate. For array questions, sketch physical directions and express aperture in wavelengths.
Explain how narrow beams and resolvable angles produce distinct angular bins, while close spacing or limited angular spread produces correlated columns. In simulation review, inspect the singular-value or condition-number distribution rather than only average channel energy. Two channels can have the same Frobenius norm but very different multiplexing performance.
Finish every answer by choosing a transmission strategy from the observed rank, SNR, CSI and reliability objective. The strategy should follow the channel evidence rather than the nominal number of antennas.
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