Rank & Condition Number FAQ
What is the "Rank" of a wireless channel matrix mathematically?
Mathematically, the Rank of a matrix $H$ is the number of linearly independent rows or columns. In your simulator, it is found by performing Singular Value Decomposition (SVD) and counting the number of non-zero singular values ($\sigma$). If a $4 \times 4$ matrix has a Rank of 2, it means only two "paths" are truly independent.
How does Channel Rank apply to MIMO systems?
In MIMO (Multiple-Input Multiple-Output), the Rank defines the maximum number of independent data streams (spatial multiplexing) you can send simultaneously. A Full Rank channel is ideal because it allows for the highest possible data rate by using all available antennas for different data layers.
What is the "Condition Number" and how is it calculated?
The Condition Number ($\kappa$) is the ratio of the largest singular value to the smallest singular value: $\kappa = \sigma_{max} / \sigma_{min}$. A condition number of 1 (0dB) is considered "perfectly conditioned," whereas a high condition number (e.g., 100 or more) indicates an "ill-conditioned" matrix.
Why is a high Condition Number bad for wireless receivers?
A high condition number means the channel is nearly singular. When the receiver tries to "invert" the channel to recover the data (using Zero-Forcing or MMSE), it will accidentally amplify the background noise. This leads to a high Bit Error Rate (BER) even if the signal power is high.