# Homework Help: LTE SIMO signal to noise ratio

1. Mar 24, 2013

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

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

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.

Last edited: Mar 26, 2013
6. Mar 26, 2013

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

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

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

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

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

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

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

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?)

21. Mar 26, 2013

That was a typo, no idea why I put integrating, I did mean differentiating.
I believe that setting the differential to 0 would allow the minimum point to be found (that is the point where the noise is lowest/SNR is the highest)

22. Mar 26, 2013

### marcusl

Good, you had me really worried!

23. Mar 27, 2013

I take it what I did was still wrong though. Any pointers on what I am doing wrong?

24. Mar 27, 2013

### marcusl

Well your start is good but you have only just started. In fact, you don't even have an equation, just the right side of an expression. Keep going! Make an equation by setting your expression to zero. Don't forget to add the missing "t" to the first term.

Multiply through by the ()^2 denominator term, expand, collect terms and simplify to get a quadratic equation in k. Solve for k. Expand terms again, and simplify to get the answer.

25. Mar 27, 2013