Assume that data of the form Z = S + N are observed where S and N are independent, Gaussian random variables representing signal and noise, respectively, with zero means and variances σ²_s and σ²_n. Design a likelihood ratio tests for each of the following cases. Describe the decision regions in each case and explain your results.

(a) c₁₁ = c₂₂ = 0; c₂₁ = c₁₂; p₀ = q₀ = 1/2

(b) c₁₁ = c₂₂ = 0; c₂₁ = c₁₂; p₀ = 1/4; q₀ = 3/4

(c) c₁₁ = c₂₂ = 0; c₂₁ = 1/2 c₁₂; p₀ = q₀ = 1/2

(d) c₁₁ = c₂₂ = 0; c₂₁ = 2c₁₂; p₀ = q₀ = 1/2

Note that under either hypothesis, Z is a zero-mean Gaussian random variable. Consider what the variances are under hypothesis H₁ and H₂, respectively.