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I did an experiment of detecting an unknown DC level in AWGN with unknown variance according to Steven M Kay's Detection Theory. For one set of samples $x(i)=a+n(i)$, the GLRT is $T_x=N\log(\hat{\sigma}_{x0}^2/\hat{\sigma}_{x1}^2)$ where $\hat{\sigma}_{x0}^2$ and $\hat{\sigma}_{x1}^2$ is the estimate of variance under H0 and H1. For another set of samples $y(i)=b+n'(i)$, the GLRT is $T_y=N\log(\hat{\sigma}_{y0}^2/\hat{\sigma}_{y1}^2)$. I added the two statistics and compared to a threshold of a given false alarm rate $T_x+T_y>\gamma_1$. In another way, the two statistics could be computed as $T_x'=N\bar{x}^2/\hat{\sigma}_{x1}^2$ and $T_y'=N\bar{y}^2/\hat{\sigma}_{y1}^2$ where $\bar{x}$ and $\bar{y}$ is the mean of x and y. I also added the two statistics and compared to another threshold of the same false alarm rate $T_x'+T_y'>\gamma_2$. The two sums of statistics are different, but surprisingly the detection rates coincide with each other. How to explain this?

c1119
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    Asking for ahem a friend who's far from his uni library: what's GLRT? Generalized Linear something? – really not Constantine A. B. Jan 19 '24 at 22:54
  • GLRT is generalized likelihood ratio test. Please refer to Steven M Kay's book Fundamentals of Statistical Signal Processing Detection Theory. – c1119 Jan 20 '24 at 01:07
  • As said, I forgot my university library at home... – really not Constantine A. B. Jan 20 '24 at 08:05
  • It's not really clear what you are writing, could you update your post with more details such as what H0 and H1 are, what you mean by "coincide"? Even better if you can detail an example to show where you are having trouble. Thanks! – Dan Boschen Jan 21 '24 at 15:04
  • H0 is the hypothesis there is no DC. H1 is the hypothesis there is DC. By "coincide" I mean the two curves of detection rate for a set of SNRs are so close to each other that they look like one curve. I think I'm using the same notation as the book. I'm cofused that the two expressions of statistics are different but the two curves are almost the same. My trouble is how to explain this phenomenon. – c1119 Jan 22 '24 at 02:18

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