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4.3: The Chirp Z-Transform or Bluestein's Algorithm

  • Page ID
    1981
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    The DFT of \(x(n)\) evaluates the \(\mathit{Z}\)-transform of \(x(n)\) on \(N\) equally spaced points on the unit circle in the \(\mathit{z}\) plane. Using a nonlinear change of variables, one can create a structure which is equivalent to modulation and filtering \(x(n)\) by a “chirp" signal.

    The mathematical identity

    \[(k-n)^{2}=k^{2}-2kn+n^{2} \nonumber \]

    gives

    \[nk=\frac{\left ( n^{2}-(k-n)^{2}+k^{2}\right )}{2} \nonumber \]

    which substituted into the definition of the DFT in Multidimensional Index Mapping Equation gives

    \[C(k)=\left \{ \sum_{n=0}^{N-1} [x(n)W^{n^{2}/2}]W^{-(k-n)^{2}/2}\right \}W^{k^{2}/2} \nonumber \]

    This equation can be interpreted as first multiplying (modulating) the data \(x(n)\) by a chirp sequence \(W^{n^{2}/2}\) then convolving (filtering) it, then finally multiplying the filter output by the chirp sequence to give the DFT.

    Define the chirp sequence or signal as h(n)=Wn2/2h(n)=Wn2/2" role="presentation" style="position:relative;" tabindex="0">


    This page titled 4.3: The Chirp Z-Transform or Bluestein's Algorithm is shared under a CC BY license and was authored, remixed, and/or curated by C. Sidney Burrus.

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