Feb 20

## SNR with PHA vs. number of bins

In this post, I will discuss the signal to noise ratio (SNR) of a photon counting detector with pulse height analysis (PHA). I will show that it approaches the ideal full-spectrum SNR as the number of bins gets large. I will also show that we can get quite close to the ideal value even with a small number of bins, which was one of the main points in my paper.

I will begin by re-visiting a result from my last post that the CramÃ¨r-Rao lower bound (CRLB) with multivariate normal log of photon count data assuming a constant covariance is nearly equal to the accurate matrix that includes the variation of the covariance. The derivation becomes problematic with a large number of bins since the mean value in each bin may be small enough so the normal approximation to the counts is not valid. I will show an alternate derivation that uses the Poisson model of the count data, which is valid for small counts. The result is formally the same as with the normal approximation so the CRLB formula can be applied for any number of bins. This result is new and was not included in my paper.

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