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I really appreciate an answer or any hint how to approach this problem:

Problem:

A receiver has two antennas. The first antenna receives $Y_1=hx+n_1$, where $x\in\{-A,+A\}$, $h$ estimated channel, and $n_1$~$N(0,\sigma^2)$. The second antenna receives $Y_2=n_1+n_2$, where $n_1$ is identical to the noise at antenna 1, $n_2$~$N(0,\sigma^2)$ and two noise are iid.

The question is whether $Y_2$ can be used to improve the performance of system? If yes, how?

I think yes, because $I(Y_1;Y_2)\neq 0$, but I don't know how to combine them linearly.

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  • $\begingroup$ please define your symbols and terminology in more detail. I assume that x is a signal, h an impulse response, hx may mean convolution. Is it time domain for frequency domain? What is $I(Y_1;Y_2)$. What is iid ? $\endgroup$
    – Hilmar
    Jun 18 at 4:55
  • $\begingroup$ @Hilmar Thanks for your comment. h is channel coefficient (scalar), which is simply multiplied with message x in time domain. iid means independent and identically distributed. Also, I( ; ) denotes the mutual information between its arguments. $\endgroup$
    – zstr
    Jun 18 at 5:49
  • $\begingroup$ As a hint consider how variances add when we sum independent noise, so the variance would double if there were two independent identically distributed noise sources added (so the standard deviation which is a magnitude quantity would go up by the square root of 2). The correlated signal itself however would double in magnitude. The added detail is what do you do when the two signals are not equal in signal to noise ratio, as would be the case here. $\endgroup$ Jun 18 at 13:23
  • $\begingroup$ Hint: What does $Y_2$ tell you about $x$ that you don't already know by looking at $Y_1$? Alternatively, what is $I(Y_2;x)$? $\endgroup$
    – MBaz
    Jun 18 at 14:39
  • $\begingroup$ Dear @MBaz , I really appreciate your hint. During the interview, I have been asked to linearly combine $Y_1$ and $Y_2$ such that the SNR is improved ( Assume you have estimated $h$ beforehand). I appreciate your time. $\endgroup$
    – zstr
    Jun 19 at 3:05

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