Does a UMP test exist for the following system where a known signal y(t) gets corrupted by zero-mean gaussian noise v(t) of spectral height 1.
Channel A:: z(t) = y(t) + v(t) :: Hypothesis 0
Channel B:: z(t) = a y(t) + v(t) :: Hypothesis 1
where 'a' is a constant in the range 0 < a < 1.
My simplification of the problem by finding the Likelihood ratio is as follows:
$$ f_{Z|H_{0}}(z|H_0) = \frac{1}{\sqrt {2\pi} (\sigma_y^2+\sigma_v^2)^{\frac{1}{2}}} e^{\frac{-(z-\mu_y)^2}{2(\sigma_y^2+\sigma_v^2)}} $$
$$ f_{Z|H_{1}}(z|H_1) = \frac{1}{\sqrt {2\pi} (\sigma_y^2+\sigma_v^2)^{\frac{1}{2}}} e^{\frac{-(z-a\mu_y)^2}{2(\sigma_y^2+\sigma_v^2)}} $$
$$ L(y) = \frac{f_{Z|H_{1}}(z|H_1)}{f_{Z|H_{0}}(z|H_0)} \space \space \space \frac{\frac{H_1}{>}}{\frac{<}{H_0}} \space \space \space \eta $$
$$ L(y) = -2\mu_y(1-a).z \space \space \space \frac{>}{<} \space \space \space 2(\sigma_y^2+\sigma_v^2).\ln(\eta) - \mu_y^2(1-a^2) $$
Now then the test depends on z. But Why or why not is it the UMP test?