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I have the decision problem for 4 hypotheses as follows: $$H_j: Y_k=N_k-s_{jk},\ k=1,2,\ldots,n;\ j=0,1,2,3.$$ where signals are $s_{jk}=E_0\sin(w_cT(k-1)+(j+\frac{1}{2})\frac{\pi}{2}).$ $$$$ In vector form: $$\equiv H_{j}: \underline{Y}=\underline{N}+\underline{s}_j;\ j=0,1,2,3.$$ $$$$ How can I find the minimum error probability for equally likely signals in i.i.d. $N(0,\sigma^2)$ noise. (Thess signals are not orthogonal). how can I cobtain orthonormal signals for solving this problem? Thank you in advance.

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