# How can we measure the accuracy of channel estimation?

Suppose I have a channel matrix $$\mathbf{H}$$ and I estimate it as $$\mathbf{\hat{H}}$$. How can I measure the accuracy of the estimate? Is using the $$\mathtt{MSE}=\mathbb{E}\left\{\|\mathbf{H}-\mathbf{\hat{H}}\|^2\right\}$$ a good measure metric?

• whether something is a good metric or not depends on what the metric is going to be used for! So, who or what takes an interest in the metric, and why? Jan 30 at 12:47
• We just want to evaluate who good the channel estimation is, i.e., how far it is from the actual channel. We use Least Square channel estimation. Jan 30 at 13:50
• Concur with Marcus-- otherwise you could use a metric that would be meaningless for a given application. Clarify what application the channel will be used for and then under that application what is the metric that is used for "good enough" then show how your channel estimation works under the corner cases of the given application. For many applications this comes down to the bandwidth of the channel actually used and and SNR performance metric under that condition given the ideal receiver. One such approach is the corr coeff to a ref wfm under low SNR conditions with static offsets removed Jan 30 at 14:19
• yes, but "how far" depends on the metric you chose. For example, "how far" can mean "looking at all frequencies individually, what's the worst absolute error we make?", so just a classical maximum norm? But maybe to your statement, the average quadratic error is more important. But: that's something that we can't really know. So, repeating my question: So, who or what takes an interest in the metric, and why? Jan 30 at 14:38
• (this isn't just academic nitpicking – different channel estimate methods have different metrics under which they're optimal, and which one you pick depends on the type of equalizer you want to use and the kind of communications problem you're solving. See, for example, the difference between a time-domain MMSE equalizer and the frequency-domain ZF equalizer you could do in OFDM) Jan 30 at 14:51