Chirp Z-transform

Bluestein's FFT algorithm (1968), commonly called the chirp-z algorithm (1969), is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a linear convolution. (The other algorithm for FFTs of prime sizes, Rader's algorithm, also works by rewriting the DFT as a convolution.)

Recall that the DFT is defined by the formula

$f_{j}=\sum _{k=0}^{n-1}x_{k}e^{-{\frac {2\pi i}{n}}jk}\qquad j=0,\dots ,n-1.$

If we replace the product jk in the exponent by the identity jk = -(j-k)²/2 + j²/2 + k²/2, we thus obtain:

$f_{j}=e^{-{\frac {\pi i}{n}}j^{2}}\sum _{k=0}^{n-1}\left(x_{k}e^{-{\frac {\pi i}{n}}k^{2}}\right)e^{{\frac {\pi i}{n}}(j-k)^{2}}\qquad j=0,\dots ,n-1.$

This summation is precisely a linear convolution of the two sequences a_k and b_k of length n (k = 0,...,n-1) defined by:

$a_{k}=x_{k}e^{-{\frac {\pi i}{n}}k^{2}}$

$b_{k}=e^{{\frac {\pi i}{n}}k^{2}},$

with the output of the convolution multiplied by n phase factors b_j^*. That is:

$f_{j}=b_{j}^{*}\sum _{k=0}^{n-1}a_{k}b_{j-k}\qquad j=0,\dots ,n-1.$

This convolution, in turn, can be performed with a pair of FFTs (plus the pre-computed FFT of b_k) via the convolution theorem. The key point is that these FFTs are not of the same length n: such a linear convolution can be computed exactly from FFTs only by zero-padding it to a length greater than or equal to 2n–1. In particular, one can pad to a power of two or some other highly composite size, for which the FFT can be efficiently performed by e.g. the Cooley-Tukey algorithm in O(n log n) time. Thus, Bluestein's algorithm provides an O(n log n) way to compute prime-size DFTs, albeit several times slower than the Cooley-Tukey algorithm for composite sizes.

The use of zero-padding for the convolution in Bluestein's algorithm deserves some additional comment. Suppose we zero-pad to a length N ≥ 2n–1. This means that a_k is extended to an array A_k of length N, where A_k = a_k for 0 ≤ k < n and A_k = 0 otherwise—the usual meaning of "zero-padding". However, because of the b_j–k term in the linear convolution, both positive and negative values of k are required for b_k (noting that b_–k = b_k). The periodic boundaries implied by the DFT of the zero-padded array mean that –k is equivalent to N–k. Thus, b_k is extended to an array B_k of length N, where B₀ = b₀, B_k = B_N–k = b_k for 0 < k < n, and B_k = 0 otherwise. A and B are then FFTed, multiplied pointwise, and inverse FFTed to obtain the linear convolution of a and b, according to the usual convolution theorem.

Chirp z-Transforms

In fact, Bluestein's algorithm can be used to compute more general transforms than the DFT, called chirp z-transforms (Rabiner, 1969); this is any transform of the form:

$f_{j}=\sum _{k=0}^{n-1}x_{k}z^{jk}\qquad j=0,\dots ,m-1,$

for an arbitrary complex number z;, and for differing numbers n and m of inputs and outputs. Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum (although the frequency resolution is still limited by the total sampling time), enhance arbitrary poles in transfer-function analyses, etcetera.

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References:

• Leo I. Bluestein, "A linear filtering approach to the computation of the discrete Fourier transform," Northeast Electronics Research and Engineering Meeting Record 10, 218-219 (1968).
• Lawrence R. Rabiner, Ronald W. Schafer, and Charles M. Rader, "The chirp z-transform algorithm and its application," Bell Syst. Tech. J. 48, 1249-1292 (1969).
• D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Review 33, 389-404 (1991). [Note that this terminology for the chirp z-transform is nonstandard: a fractional Fourier transform conventionally refers to an entirely different, continuous transform.]