I have Channel capacity analysis figures for two wireless channels show below:

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My question:

While ergodic capacity in the first subplots are very clear in conveying the channel capacity, How to make an assessment of channel capacity based on P(Capacity < x) in second subplots and P(Cond. Num < x) in the third subplots for each of the above channels, individually and relatively?


1 Answer 1


What the CDF of the capacity tells you is how it distributes, which allows you to say something about the quantiles. Ergodic capacity tells you: what capacity will we see on average? Quantiles are more relevant in practice: what capacity can we guarantee with 95% certainty? 99%? 99.9%? For this, you need a CDF and more often than not, you'll see a heavy-tailed distribution there.

Now, what you would like your system to do is a CDF that's very steep (yours are looking pretty good), which implies a well localized PDF, i.e., a low variance. It means that you'll have the 20-something bsp/Hz not just on average but with a pretty high probability. If the CDF were less steep you might still have 20 on average but with variations between 10 and 30, which is not that useful in practice if there is a certain data rate you want to guarantee.

For the condition number: this is an important aspect in MIMO channels if you want to maximize data rate by applying some form of spatial multiplexing, i.e., sending different data streams from your transmit antennas either towards different users (like in a multi-user MISO oder MIMO system) or towards your different RX antennas in a P2P system. For this to work, the MIMO system needs to offer enough spatial degrees of freedom. It needs to be full rank for it to work properly, and it will work even better of all the spatial eigenmodes are equally strong, which means the channel's singular values being almost equal (think of the singular values as the "strength" of individual spatial eigenmodes). This ideal case implies a condition number close to 1. The further it is from one, the larger the eigenvalue spread, which means some spatial eigenmodes will be weaker and thus be able to sustain only a lower data rate.

  • $\begingroup$ Thanks for the answer. So for MIMO systems which used beamforming or Tx diversity rather than spatial multiplexing, it looks like condition number would not be the right way to analyze the channel capacity. $\endgroup$
    – Naveen
    Aug 29, 2019 at 17:26
  • $\begingroup$ So let's say there are two scenarios, one in which the beam is aligned with the strongest cluster(path with maximum gain) of the channel and another in which the beam in not aligned with any strong cluster. How would you analyze the channel capacity for these two scenarios in terms of three plots? $\endgroup$
    – Naveen
    Aug 29, 2019 at 17:28
  • $\begingroup$ My understanding is aligned beam would give higher capacity for known Tx in first plot as well as higher cut-off point(where there is steep gradient) in the second Capacity CDF plot compared to un-aligned beam. Is that correct? $\endgroup$
    – Naveen
    Aug 29, 2019 at 17:30
  • $\begingroup$ Yes, you're right: if you apply TX beamforming, you're only using the dominant eigenmode of the channel, so the condition number wouldn't be indicative of your performance. And yes, the aligned beam should give higher ergodic capacity as well as higher outage capacities compared to the unaligned one. $\endgroup$
    – Florian
    Aug 30, 2019 at 7:27
  • $\begingroup$ One more question. As you can see in the first subplot in each plot, the SNR is offset by channel STD. Does the STD of channel matrix influence the SNR in any way? $\endgroup$
    – Naveen
    Sep 4, 2019 at 21:21

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