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Find Impulse Response of LTI system given transfer function

  1. Dec 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the impulse response of a system with transfer function H(S) = (s+3)/(s^2+2s+1)

    or H(S)=(s+3)/[(s+1)^2]

    2. Relevant equations

    Poles are s1=s2=-1

    y = Ae^st + Be^st

    3. The attempt at a solution

    In my notes I do not have an answer for the case when there is only one pole (root) to the denominator of the transfer function.

    I know the cover-up method and it doesn't look like it will work here.

    Also tried solving for the step response and differentiating to get the impulse response as shown here: http://tinyurl.com/c5uhzcv

    But I cannot find the step response that way either. Thanks!
     
  2. jcsd
  3. Dec 14, 2012 #2
    I'm sure you do -- look for mention of repeated roots.

    A partial fraction expansion of a root repeated to the nth power involves sums of all fractions with the root raised to powers 1 through n. This is because the common denominator of these terms is (s+1)n and the numerator must be able to achieve a power in s of (n-1) to be completely general.

    In this case,

    [itex]H(s)=\frac{s+3}{(s+1)^2}=\frac{A}{s+1}+\frac{B}{(s+1)^2}[/itex]
     
  4. Dec 14, 2012 #3
    If you add the fractions back together, the A will add an As term and the B will supply a constant term in the numerator.

    You already know the inverse transform of the first fraction. Whenever factors in the denominator are taken to a power, in the time domain they are multiplied by t to that power-1. So the second fraction will invert as t^1 times the inverse of 1/(s+1)
     
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