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Homework Help: Optimization problem using lagrangian

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data

    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)

    2. Relevant equations

    Capacity is given as: C = [itex]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[/itex]

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

    Finally, differentiate and set it equal to 0:
    [itex]\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)[/itex]

    Now, solving for P(γ) with P(γ) > 0

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

    In the above equations:
    P(γ) is the received power (what we are trying to maximize)
    [itex]\overline{P}[/itex] 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)
    [itex]\gamma_0[/itex] is some SNR threshold.

    3. 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
    [itex]\frac{\partial J}{\partial P\left(\gamma\right)} = \frac{\partial J/\partial\gamma}{\partial P\left(\gamma\right)/\partial\gamma}[/itex], 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.
  2. jcsd
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