this is for holding javascript data
Joseph Nofech edited begin_flushleft_begin_align_S_2__1.tex
almost 8 years ago
Commit id: fb98f6ae1f9e8b36047f337a6b0ade564e1c9f45
deletions | additions
diff --git a/begin_flushleft_begin_align_S_2__1.tex b/begin_flushleft_begin_align_S_2__1.tex
index 8b6aa93..c41055f 100644
--- a/begin_flushleft_begin_align_S_2__1.tex
+++ b/begin_flushleft_begin_align_S_2__1.tex
...
&= \left< I_{signal}^2 + 2I_{signal}I_{noise} - 2I_{signal}I'_{signal} - 2I_{signal}I'_{noise} + I_{noise}^2 - 2I_{noise}I'_{signal} - 2I_{noise}I'_{noise} + (I')_{signal}^2+2I'_{signal}I'_{noise} + (I')_{noise}^2 \right>\\
&= \left< I_{signal}^2 - 2I_{signal}I'_{signal} + (I')_{signal}^2 + I_{noise}^2 - 2I_{noise}I'_{noise} + (I')_{noise}^2 \right> + \left< 2I_{signal}I_{noise} - 2I_{signal}I'_{noise} - 2I_{noise}I'_{signal} - 2I'_{signal}I'_{noise} \right> \\
&= \left< \left[ I_{signal} - I'_{signal} \right]^2 + \left[ I_{noise}^2 - I'_{noise} \right]^2 \right> + \left< 2I_{signal}I_{noise} - 2I_{signal}I'_{noise} - 2I_{noise}I'_{signal} - 2I'_{signal}I'_{noise} \right> \\
&= \left< \left[ I_{signal} - I'_{signal} \right]^2 \right> + \left< \left[ I_{noise}^2 - I'_{noise} \right]^2 \right> + \left<
2I_{signal}I_{noise} 2I_{noise}I_{signal} - 2I_{signal}I'_{noise} - 2I_{noise}I'_{signal} - 2I'_{signal}I'_{noise} \right> \\
&\approx \left< \left[ I_{signal} - I'_{signal} \right]^2 \right> + \left< \left[ I_{noise}^2 - I'_{noise} \right]^2 \right>\\
&\approx S_{2(signal)} + S_{2(noise)}\\
\end{align*}