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Discrete Time Convolution of Sums

  1. Mar 27, 2013 #1
    Evaluate the following discrete-time convolution:

    y[n] = cos([itex]\frac{1}{2}[/itex][itex]\pi[/itex]n)*2[itex]^{n}[/itex]u[-n+2]

    Here is my sloppy attempt:

    y[n] = [itex]\sum[/itex]cos([itex]\frac{1}{2}[/itex][itex]\pi[/itex]k)2[itex]^{n-k}[/itex]u[-n-k+2] from k = -∞ to ∞

    = [itex]\sum[/itex]cos([itex]\frac{1}{2}[/itex][itex]\pi[/itex]k)2[itex]^{n-k}[/itex] from k = -∞ to 2

    We can re-write the cos as [e[itex]^{0.5j\pi}[/itex]-e[itex]^{-0.5j\pi}[/itex]]0.5

    using the property of summation of geometric series:

    0.5[itex]\sum[/itex]2[itex]^{n-k}[/itex](e[itex]^{0.5j\pi k}[/itex]-e[itex]^{-0.5j \pi k}[/itex])

    from k = -∞ to 2

    so more or less am I on the right track?
  2. jcsd
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