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Math Help - Difference equation: (1) step response, (2) homogeneous solution, (3) ...

  1. #1
    Member courteous's Avatar
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    Difference equation: (1) step response, (2) homogeneous solution, (3) ...

    1. The problem
    Consider the difference equation y[n] - \frac{5}{6}y[n-1] + \frac{1}{6}y[n-2] = \frac{1}{3}x[n-1].

    What is the
    (a) step response for the causal system satisfying this difference equation,
    (b) general form of the homogeneous solution of the difference equation?
    (c) Consider a different system satisfying the difference equation that is neither causal nor LTI, but that has y[0] = y[1] = 1. Find the response of this system to x[n] = \delta[n].

    2. Relevant equations and definitions
    Oppenheim's definition of (b), the general form of the homogeneous solution:
    y_h[n] = \sum_{m=1}^{N} A_m z_m^{n} \text{    ................... (2.97)}

    3. The attempt at a solution
    (a) step response, s[n]
    I have calculated the impulse response h[n] = 2( \frac{1}{2^n} - \frac{1}{3^n}) u[n], and put it to use in calculating s[n] (where I get stuck):

    s[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] = 2 \sum_{k=-\infty}^{\infty} u[n-k] ( \frac{1}{2^n} - \frac{1}{3^n}) u[n] = ?

    (b) general form of the homogeneous solution
    I have no idea (similar problem as (a) in this question on MHF) ... the solution ought to be y_h[n] = A_1 \frac{1}{2^n} + A_2 \frac{1}{3^n}.

    (c) response of a particular system to x[n] = \delta[n]
    Confused as well. I do get particular values for y[n]:
    y[-1] = 41/5,
    y[0] = 1,
    y[1] = 1,
    y[2] = 2/3,
    ...
    continuing in this manner (and hoping for a pattern to emerge) surely isn't the way to go?

    The solution given is:
    y[n] = 4 \frac{1}{2^n} - 3 \frac{1}{3^n} - 2 \frac{1}{2^n} u[-n-1] + 2 \frac{1}{3^n} u[-n-1]. Other answers are possible.
    Last edited by courteous; September 30th 2012 at 06:02 AM.
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  2. #2
    Member courteous's Avatar
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    Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...

    Bumping in hope of an answer.
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  3. #3
    Member courteous's Avatar
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    Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...

    Found the method for solving (b); still need help with (a) and (c).
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  4. #4
    Member courteous's Avatar
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    Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...

    Got (a), still plead for help with (c).
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