# Relation between SER and the channel capacity

Is there a relation between Symbol Error Rate (SER) and the channel capacity?

What is {Modulation Order x (1-SER) / symbol duration [bps]} called?

• To give a contradictory answer to the one provided by Jim Clay, in some sense there is no direct relation between channel capacity and SER: only between channel capacity and SNR as described in Jim's answer. It is true that as a rule-of-thumb, SER decreases as SNR increases, but there are different modulation schemes and even different implementations that can be used, and if all that one is told is that the SER on a particular channel is $3.1415926\times 10^{-5}$, say, then the SNR is not known unless additional assumptions are made about modulation scheme and the implementation. Dec 30 '14 at 15:24
• @DilipSarwate For what it's worth, I agree with pretty much everything you said. I still think it's fair to say that there is a relationship since, if you hold all else equal, decreasing SER means increasing channel capacity, but that's a minor quibble. Dec 30 '14 at 18:03
• @JimClay I think the point I am trying to make is that by simply using a different modulation scheme or a better (less suboptimal) implementation or error-control coding, one could decrease SER without any change in SNR that would result in a change in the channel capacity. Thus, knowing just what the SER is (and nothing else) does not allow us to make anything other than a guess at what the SNR is and what the corresponding value of the channel capacity is. Dec 30 '14 at 19:16
• On the Relationships Between Average Channel Capacity, Average Bit Error Rate, Outage Probability, and Outage Capacity Over Additive White Gaussian Noise Channels ieeexplore.ieee.org/document/8986655 Jun 2 '20 at 19:38

Yes, there is a relationship between SER and channel capacity. The channel capacity equation is-

$$C = B\log_2(1 + \frac{S}{N})$$ where $C$ is the channel capacity in bits/s, $B$ is the bandwidth in Hz, $S$ is the average received signal power, and $N$ is the average noise power across the bandwidth $B$.

Looking at the equation, it is clear that increasing the SNR will increase the channel capacity. Increasing SNR does, of course, also decrease the symbol error rate (SER). Thus, there is an inverse relationship between SER and channel capacity.

What is {Modulation Order x (1-SER) / symbol duration [bps]} called?

As far as I am aware it does not have a name. Having a bit stream where some of the bits are good and some of the bits are bad is generally not very useful. Shannon's theorem assumes that error correction codes are used to reduce the number of errors to an arbitrarily low rate. This reduces the number of generated bits/s, but increases the reliability of those bits. The "channel capacity" of a channel is the number of bits it can produce with an arbitrarily low error rate.