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Z-transform for a signal with an unknown sampling rate

  1. Mar 5, 2010 #1
    1. The problem statement, all variables and given/known data
    Given a ramp function x(t) = t*heaviside(t) with the known z-transform ( z / (z-1)^2 ).
    If the time-domain signal has been delayed by 2 seconds, then sampled with an unknown sampling rate (T). How could I get the z-transform (one-sided) for the discrete signal, for any values of T ?
    I guess my question is really how to express that delay in the Z-domain. I have searched a lot and couldn't find any good resources.


    2. Relevant equations
    original signal : x(t) = t*heaviside(t) - X(z) = z / (z-1)^2
    delayed signal: x(t) = (t-2)*heaviside(t-2) - X(z) = ?


    3. The attempt at a solution
    I have been trying a lot with this problem. The last (and seemingly to me) the closest I've got is :
    X(z) = T*( z / (z-1)^2 )*(z^-s)
    where :
    s = 2/T if T < 1
    s = 2*T if 1<= T < 2
    s = T if T >= 2
    Of course, if s turns out to be a non-integer value, the solution would be wrong.

    Any help on that is greatly appreciated.

    Regards,
    MT
     
    Last edited: Mar 5, 2010
  2. jcsd
  3. Mar 5, 2010 #2
    Another take :
    X(z) = T*( z / (z-1)^2 )*(z^-ceil(s))
    where :
    s = 2/T if T <= 1
    s = T if T > 1

    Seems right to me, but is the equation is in an acceptable/correct mathematical form ??
     
  4. Mar 5, 2010 #3
    Uh ! I'm sorry, the conditions I wrote are again wrong, here's my final approach :
    X(z) = T*( z / (z-1)^2 )*(z^-ceil(2/T))

    Again, it seems to me that it's functionally correct, although I'm not sure if it's syntactically correct too.
     
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