zadoff chu sequence cyclic shift

Discussion in 'Math Research' started by dgse, Jun 4, 2009.

  1. dgse

    dgse Guest

    How to prove the different cyclic shift of the same root zadoff chu
    sequence is orthogonal to each other?
    dgse, Jun 4, 2009
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  2. "dgse"
    Let the length of the sequence be L, a positive integer. Write s = exp(- 2
    pi i / L). The Z-C sequence is
    Z(n) = s^{t(n)} where t(n) = n(n+1)/2.
    To say Z is orthogonal to each of its L-1 shifts means
    sum_{n = 0 to L-1} s^( t(n) - t(n+k) ) = 0
    for k = 1, 2, ... L-1.
    Observe that t(n) - t(n+k) is linear in n. Therefore (omitting some details)
    the summands are evenly distributed around the unit circle, and the sum is
    therefore zero. The details aren't hard.

    Larry Hammick, Aug 23, 2009
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