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LTE SIMO signal to noise ratio

  1. Mar 24, 2013 #1

    madmike159

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    1. The problem statement, all variables and given/known data

    Base station transmits signal t which is received by two antennas such that:

    x = h1t + n1
    y = h2t + n2

    where n1 and n2 are independent additive white Gaussian noise with variances σ21 and σ22 respectively.
    h1 is the channel gain between the base station and receive antenna 1 and h2 is the gain between the base station and receive antenna 2.

    The receive terminal combines the signals x and y as follows:
    z = αx + βy

    Find the optimal values of α and β.
    What is the resulting SNR of z? How does this compare to the SNR of x and y?

    2. Relevant equations

    The only hint given was that k = α/β and that
    SNRz = f(k)

    3. The attempt at a solution

    I have tried finding the power of each signal since the variance is the power of the noise, and then trying to find a ratio of α/β to give a constant which should allow α and β to be adjusted to find the optimal values.

    However I haven't come to a solution that actually expresses SNR of z, x or y. Some of the calculations I have made are lengthy and don't arrive at a solution, but I could upload them if you wish.

    Any help or pointers would be greatly appreciated.
     
    Last edited: Mar 24, 2013
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  3. Mar 24, 2013 #2

    marcusl

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    The SNR of x is [tex]SNR_x=\frac{E[|h_1 t|^2]}{E[|n_1|^2]}.[/tex] With that hint, you can find the SNR of y and z. Divide SNR_z through by alpha to express it in terms of k.
    Note that the noise cross term in the denominator is zero (do you see why?). Differentiate SNR_z with respect to partial k and set to zero. That's the approach to solving the problem.

    The hard part is evaluating the partial derivatives, especially if the signals, noise and k are complex. Do you know if you are dealing with real or complex quantities?
     
  4. Mar 24, 2013 #3

    madmike159

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    I am not sure if it will be complex or not. My original post contains all the information we were given. I will have a go at finding the SNR of y and z and post my results here.
     
  5. Mar 25, 2013 #4

    madmike159

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    I have spoken to my lecturer and he said that the power of the received can be assumed to be constant, so the expectation can be omitted.

    So I assume we would get

    SNRz = α ( h1t / σ21) + β ( h2t / σ22)

    Then would it simply be a case of changing values of α and β to ensure the signal with the best SNR is chosen ( i.e z = 2x + 0.1y)?
     
  6. Mar 25, 2013 #5

    marcusl

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    No, not quite. First you have written the voltage SNR, but usually SNR is a power quantity (and using it is a must if your variables are complex). Second, the signal part of z is (αh_1 + βh_2)t, while the noise is αn_1 + βn_2. The power SNR is therefore [tex]SNR_z=\frac{|(\alpha h_1+\beta h_2)t|^2}{E[|\alpha n_1+\beta n_2|^2]}=\frac{|(k h_1+h_2)t|^2}{|k|^2 \sigma_1^2+\sigma_2^2}.[/tex] SNR_z varies from SNR_x to SNR_y as |k| varies between infinity and 0.

    Edit: Added expectation to denominator.
     
    Last edited: Mar 26, 2013
  7. Mar 26, 2013 #6

    madmike159

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    So to find the optimal value, I would differentiate with respect to k?
     
  8. Mar 26, 2013 #7

    marcusl

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    Yes, paying attention to the complex nature of your quantities (i.e., |k|^2=k k*).
     
  9. Mar 26, 2013 #8

    madmike159

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    I would have thought that |k|2 would = k2

    Since |k| = (k2)1/2
     
  10. Mar 26, 2013 #9

    marcusl

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    Ok, if you aren't familiar with complex variables just treat everything as real.
     
  11. Mar 26, 2013 #10

    madmike159

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    I understand that |k|^2=k k* if k is complex. I just tried differentiating SNR_z but ended up with a very long function (from the quotient rule), which didn't appear to simplify. I will try again with complex values.
     
  12. Mar 26, 2013 #11

    marcusl

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    Don't give up. It does simplify.
     
  13. Mar 26, 2013 #12

    madmike159

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    Is with our without complex values, or does it not matter?
     
  14. Mar 26, 2013 #13

    marcusl

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    That sentence doesn't make sense! Do it with reals, it's hard enough that way. If you are careful with your terms, when you simplify you'll have a famous result that has been used widely in wireless comms.
     
  15. Mar 26, 2013 #14

    madmike159

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    You do use the quotient rule to differentiate? I just tried again and got a very long answer where only one small part cancels out, certainly not something I recognize as a famous result.
     
  16. Mar 26, 2013 #15

    marcusl

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    Yes.
    I worked it last night and got the expected result.
     
  17. Mar 26, 2013 #16

    madmike159

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    Ok, well clearly I am going wrong somewhere or missing something important. I have a load of other questions to do for this assignment so once I have those done I will have another go at it.

    Thanks
     
  18. Mar 26, 2013 #17

    marcusl

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    Ok. At that time you can post your work and we find your error.
     
  19. Mar 26, 2013 #18

    madmike159

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    Yea sure, I will try right it out on here, its pretty long and messy though (a clear sign I have gone wrong).
     
  20. Mar 26, 2013 #19

    madmike159

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    I have just directly differentiated the top line and bottom line with respect to k, using the quotient rule (assuming that |k|2 = k2).

    This is what I got:

    http://gyazo.com/c5301f7e5c8f23d85622e1d2fbc9bb82

    (I can never get the math input working on here)
     
    Last edited: Mar 26, 2013
  21. Mar 26, 2013 #20

    marcusl

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    I can't access sites like that through the firewall at work so I can't see your work. But why are you integrating???? Differentiate w.r.t. k and set to zero! (Do you understand why you should be doing that?)
     
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