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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Conditioned on the $j$-th signal being transmitted, the likelihood function of the observation $(Y_1, \ldots, Y_n)$ is proportional to $$\exp\left(-\frac{1}{2}\sum_{k=1}^n (s_{kj}-Y_k)^2\right)$$ Since the signals are equally likely to be transmitted, the_ minimum-error-probability decision rul_e is the same as the maximum-likelihood decision rule, viz.

Choose the hypothesis that has the largest likelihood

which in this instance means deciding that the signal $s_j$ that is closest in Euclidean metric to the observation $Y$ is the one that is most likely to have been transmitted. In other words, compute the four sums $$Z_j = \sum_{k=1}^n (s_{kj}-Y_k)^2, ~~j = 0, 1, 2, 3$$ and decide that signal $s_j$ was transmitted if $Z_j < \min_{i: i\neq j} Z_i$

  • $\begingroup$ @ Dilip Sarwate: Thank you for your answer! However, my main problem is in the calculation minimum error probability($P_e$) where $P_c=1-P_e=P_0(Z_0>Z_1,Z_0>Z_2,Z_0>Z_3)$. Because the signals ($s_j$) are not ortogonal to have independecies between $Z_j$. $\endgroup$ – lewer Nov 23 '12 at 19:21
  • $\begingroup$ @lewer You need to state very carefully what you mean by minimum error probability. The minimum average error probability? The conditional minimum error probability given hat a specific $s_i$ was transmitted? The smallest probability that a given transmitted $s_i$ was demodulated as specific $s_j$? As a simple answer, read about the union bound or Bonfderroni inequalities. $\endgroup$ – Dilip Sarwate Nov 23 '12 at 23:09
  • $\begingroup$ @lewer See my lecture notes for some ideas on how to approach the problem. $\endgroup$ – Dilip Sarwate Nov 23 '12 at 23:18
  • $\begingroup$ @ Prof.Sarwate: Thank you for providing me your lecture notes in this field. My goal of minimum error probability is The conditional minimum error probability given that a specific $s_j$ was transmitted(on page.162 of your lecture notes)? $\endgroup$ – lewer Nov 24 '12 at 6:22

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