# LTE SIMO signal to noise ratio

1. Mar 24, 2013

### madmike159

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
2. Mar 24, 2013

### marcusl

The SNR of x is $$SNR_x=\frac{E[|h_1 t|^2]}{E[|n_1|^2]}.$$ 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?

3. Mar 24, 2013

### madmike159

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.

4. Mar 25, 2013

### madmike159

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)?

5. Mar 25, 2013

### marcusl

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 $$SNR_z=\frac{|(\alpha h_1+\beta h_2)t|^2}{E[|\alpha n_1+\beta n_2|^2]}=\frac{|(k h_1+h_2)t|^2}{|k|^2 \sigma_1^2+\sigma_2^2}.$$ 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
6. Mar 26, 2013

### madmike159

So to find the optimal value, I would differentiate with respect to k?

7. Mar 26, 2013

### marcusl

Yes, paying attention to the complex nature of your quantities (i.e., |k|^2=k k*).

8. Mar 26, 2013

### madmike159

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

Since |k| = (k2)1/2

9. Mar 26, 2013

### marcusl

Ok, if you aren't familiar with complex variables just treat everything as real.

10. Mar 26, 2013

### madmike159

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.

11. Mar 26, 2013

### marcusl

Don't give up. It does simplify.

12. Mar 26, 2013

### madmike159

Is with our without complex values, or does it not matter?

13. Mar 26, 2013

### marcusl

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.

14. Mar 26, 2013

### madmike159

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.

15. Mar 26, 2013

### marcusl

Yes.
I worked it last night and got the expected result.

16. Mar 26, 2013

### madmike159

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

17. Mar 26, 2013

### marcusl

Ok. At that time you can post your work and we find your error.

18. Mar 26, 2013

### madmike159

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

19. Mar 26, 2013

### madmike159

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
20. Mar 26, 2013

### marcusl

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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