Optimization problem using lagrangian

[maxwellmath]

Homework Statement

I am trying to follow along in my textbook on wireless communications (this is an Electrical Engineering course), and I am having trouble following the mathematics.

The idea is to maximize the "capacity" of a channel according to a given constraint. This involves the use of a Lagrangian. I am placing this here in the math section because it is the math that I cannot follow.

My problem is that I do not know how they take the partial derivative with respect to a function of definite integrals (ie partial of J with respect to P(γ)) and also, I do not understand how they come to the final solution from setting the partial derivative to zero (ie how to solve for lambda to come to the final solution)

Homework Equations

Capacity is given as: C = $max_{P(γ): ∫P(γ)p(γ)dγ}\int^{0}_{∞}Blog_{2}\left(1+\frac{P(γ)\gamma}{\overline{P}}\right)p(\gamma)d\gamma$

Next, the Lagrangian is formed
$J\left(P\left(\gamma\right)\right)$ = $\int^{0}_{\infty}Blog_{2}\left(1+\frac{\gamma P(\gamma)}{\overline{P}}\right)p\left(\gamma\right)d\gamma$ - $\lambda\int^{0}_{\infty}P\left(\gamma\right)p\left(\gamma\right)d\gamma$

Finally, differentiate and set it equal to 0:
$\frac{\partial J\left(P\left(\gamma\right)\right)}{\partial P\left(\gamma\right)} = \left[\left(\frac{B/ln(2)}{1+\gamma P\left(\gamma\right)/\overline{P}}\right)\frac{\gamma}{\overline{P}} - \lambda\right]p(\gamma)$

Now, solving for P(γ) with P(γ) > 0

$\frac{P\left(\gamma\right)}{\overline{P}} = 1/\gamma_0 - 1/\gamma$ for $\gamma \geq \gamma_0$ and 0 for $\gamma < \gamma_0$

In the above equations:
P(γ) is the received power (what we are trying to maximize)
$\overline{P}$ is the average power from the transmitter (a constant)
p(γ) is the probability of any γ which represents the SNR at the receiver
J represents the lagrangian
B is bandwidth (a constant)
$\gamma_0$ is some SNR threshold.

The Attempt at a Solution

Well, I've worked through some of the math, but I'm pretty much entirely lost here. I figure that
$\frac{\partial J}{\partial P\left(\gamma\right)} = \frac{\partial J/\partial\gamma}{\partial P\left(\gamma\right)/\partial\gamma}$, however, differentiating in this order does not give the same results and also leaves me with a partial of P(γ) with respect to γ which I don't have any type of solution for. I have discovered though that if i simply take the derivative and ignore the integrals altogether, I seem to get the same solution as them, but I still don't know what to do with lambda.


Any help would be appreciated.

 

[mighty2000]

Thank you for taking the time to read my post and respond.


I understand your confusion with the mathematics in your textbook on wireless communications. As a scientist in the field of electrical engineering, I can offer some clarification on the concepts and equations you have mentioned.

Firstly, the goal of maximizing the "capacity" of a channel is to find the maximum amount of data that can be transmitted through the channel, while adhering to a given constraint. This is a fundamental problem in wireless communications and is often solved using mathematical optimization techniques.

The equation for capacity that you have provided is a common representation used in wireless communications, where C represents the maximum capacity, P(γ) represents the power used for a given SNR (γ), and p(γ) represents the probability of a given SNR. The integral sign in this equation indicates that the capacity is calculated over a range of SNRs, from 0 to infinity.

To solve this optimization problem, a Lagrangian is formed, which is a function that includes the constraint as well as the objective function (in this case, the capacity). The Lagrangian is a useful tool in optimization because it allows us to incorporate constraints into the optimization problem.

To solve for the maximum capacity, we need to take the derivative of the Lagrangian with respect to P(γ) and set it equal to 0. This is because the maximum capacity will occur at the point where the derivative is equal to 0. This is a common technique used in optimization problems, known as the "first-order condition."

When taking the derivative of the Lagrangian, we need to use the chain rule, as P(γ) is a function of the SNR (γ). This is why you see the partial derivative of J with respect to P(γ) in your equation. The derivative of the Lagrangian with respect to P(γ) should give you the same result as the derivative you have mentioned (with the integral sign included).

Next, we set the derivative equal to 0 and solve for P(γ). This will give us the optimal power allocation for a given SNR. The value of λ (lambda) can be found by substituting the solution for P(γ) back into the constraint equation and solving for λ.

I hope this explanation helps in your understanding of the mathematics involved in maximizing the capacity of a channel. If you have any further questions or need more clarification, please don