1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Solve derivative of least squares matrix equation

  1. Mar 10, 2016 #1
    1. The problem statement, all variables and given/known data

    I am designing a MIMO communication system, with input signal s, channel H and transform matrix T. The received signal is corrupted by noise.

    2. Relevant equations

    The received signal is r = Hs+n

    And then it is transformed (compressed) by:

    y = Tr

    And then its estimate s_hat is computed:

    s_hat = inv(TH)*y = inv(H)inv(T)THs + inv(H)inv(T)Tn

    Set C = inv(H)inv(T)Tn
    I want to find an optimum T based on the least squares solution:

    D = norm(s-s_hat)^2
    dD/dT = 0
    3. The attempt at a solution

    [itex]D= (s-s_{hat})^{H}(s-s_{hat})=0[/itex]
    [itex]D = (s-H^{-1}T^{-1}{THs})^{H}(s-H^{-1}T^{-1}THs)[/itex]
    [itex]D = ||s|| -s^{H}H^{-1}T^{-1}{THs}-s^{H}C-s^{H}H^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}s+(s^{H}H^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}s)(H^{-1}T^{-1}{THs})-sH^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}C-C^{H}s-C^{H}H^{-1}T^{-1}{THs}+C^{H}C[/itex]

    How do I find the derivative dD/dT? Suppose that I do find it, how then do I proceed to obtain T alone on one side of the equation?

    I would also like to get some ideas on which book covers this kind of matrix algebra.
  2. jcsd
  3. Mar 10, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The matrix ##T## disappears from your expression for ##D## as you have written it:
    ##H^{-1} T^{-1} T H = H^{-1} H = I## (the unit matrix), because ##T^{-1}T = I## and ##H^{-1}H = I##. So, what you have written is, basically, ##D = (s-s)^H (s-s)##, which is just 0 for all ##H, T##.
  4. Mar 10, 2016 #3
    But since s_hat is corrupted by noise then this will not be exactly true?
  5. Mar 10, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I am just going by what you wrote. Perhaps what you wrote is not appropriate.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted