# Difference: Ergodic channel capacity and spectral efficiency

Can someone explain what is the difference between the capacity and ergodic capacity? Ergodic capacity and spectral efficiency?

• "spectral efficiency" is a property of a system. "capacity", of every kind, is a property of a channel. – Marcus Müller Jul 27 '20 at 11:42

Loosely speaking:

• Capacity is the supremum of data rate that one can send data with arbitrarily small error probability over a given channel;
• ergodic capacity is also the supremum of rate with arbitrarily small error probability, but for fading channels, in assuming the fading process is ergodic, as the term "ergodic" suggests;
• spectral efficiency is data rate divided by occupied bandwidth.

Therefore,

• Ergodic capacity is not capacity. Capacity is not ergodic capacity. They are channel-specific and we do not necessarily know how to achieve them.
• Spectral efficiency is system-specific, i.e. we have already had a system, as said MarcusMuller in his comment. We ignore error probability in using this. Nonetheless, a spectral efficiency is usually implicitly linked with an "acceptable" error probability.

In short, they are very different performance indicators with very different assumptions. In academy, you are free to choose any indicator that supports your research.

• What is the difference between "outage " and "ergodic"? – Jang Lee Aug 25 '20 at 11:43
• @JangLee you should add some context. They are different by their very definitions. – AlexTP Aug 25 '20 at 11:56
• Capcity compute as $log_2(det( I+ SNR/Nt \cdot H \cdot H^H))$, is it ergodic capcity? – Jang Lee Aug 26 '20 at 6:05
• @JangLee This is not outage capacity. This is not ergodic capacity by its definition either. Indeed, this can be interpreted as the capacity computed for a static MIMO channel characterized by matrix $H$ by assuming $H$ is known at receiver. Note that many channels with different assumptions can come up with the same formula. Read this chapter, Section 8.2.1, you may find something similar web.stanford.edu/~dntse/Chapters_PDF/… – AlexTP Aug 26 '20 at 7:50