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

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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 $h (n) = W n 2 / 2 " role="presentation" style="position:relative;" tabindex="0">$

As developed here, the chirp $\mathit{z}$-transform evaluates the $\mathit{z}$-transform at equally spaced points on the unit circle. A slight modification allows evaluation on a spiral and in segments and allows savings with only some input values are nonzero or when only some output values are needed.

Two Matlab programs to calculate an arbitrary length DFT using the chirp $\mathit{z}$-transform is shown in Pre.

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