What Are the Optimal Values of α and β for LTE SIMO SNR?

[madmike159]

Homework Statement

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?

Homework Equations

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

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.

 

[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?

 

[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.

 

[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

SNR_z = 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)?

 

[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.

 

[madmike159]

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

 

[marcusl]

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

 

[madmike159]

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

Since |k| = (k²)^1/2

 

[marcusl]

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

 

[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.

 

[marcusl]

Don't give up. It does simplify.

 

[madmike159]

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

 

[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.

 

[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.

 

[marcusl]

You do use the quotient rule to differentiate?

Yes.

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.

I worked it last night and got the expected result.

 

[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

 

[marcusl]

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

 

[madmike159]

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

 

[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)

 

[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?)

 

[madmike159]

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)

 

[marcusl]

Good, you had me really worried!

 

[madmike159]

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

 

[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.

 

[madmike159]

Ok thanks. When you say the missing t term, should there be a t on the top of the left fraction?

 

[marcusl]

Probably, I can't remember your attachment exactly. You should be able to figure it out.