Bluestein's FFT algorithm

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Bluestein's FFT algorithm (1968), commonly called the chirp z-transform 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.)

In fact, Bluestein's algorithm can be used to compute more general transforms than the DFT, based on the (unilateral) z-transform (Rabiner et al., 1969).

Algorithm

Recall that the DFT is defined by the formula

[ X_k = \sum_^ x_n e^ nk }\qquadk = 0,\dots,N-1. ]

If we replace the product nk in the exponent by the identity nk = –(k–n)²/2 + n²/2 + k²/2, we thus obtain:

[ X_k = e^ k^2 } \sum_^ \left( x_n e^ n^2 } \right) e^ (k-n)^2 }\qquadk = 0,\dots,N-1. ]

This summation is precisely a linear convolution of the two sequences a_n and b_n of length N (n = 0,...,N–1) defined by:

[a_n = x_n e^ n^2 }]

[b_n = e^ n^2 },]

with the output of the convolution multiplied by N phase factors b_k^*. That is:

[ X_k = b_k^* \sum_^ a_n b_ \qquad k = 0,\dots,N-1. ]

This convolution, in turn, can be performed with a pair of FFTs (plus the pre-computed FFT of b_n) 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 M ≥ 2N–1. This means that a_n is extended to an array A_n of length M, where A_n = a_n for 0 ≤ n < N and A_n = 0 otherwise—the usual meaning of "zero-padding". However, because of the b_k–n term in the linear convolution, both positive and negative values of n are required for b_n (noting that b_–n = b_n). The periodic boundaries implied by the DFT of the zero-padded array mean that –n is equivalent to M–n. Thus, b_n is extended to an array B_n of length M, where B₀ = b₀, B_n = B_M–n = b_n for 0 < n < N, and B_n = 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.

z-Transforms

Bluestein's algorithm can also be used to compute a more general transform based on the (unilateral) z-transform (Rabiner et al., 1969). In particular, it can compute any transform of the form:

[ X_k = \sum_^ x_n z^\qquadk = 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.

The algorithm was dubbed the chirp z-transform algorithm because, for the Fourier-transform case (|z| = 1), the sequence b_n from above is a complex sinusoid of linearly increasing frequency, which is called a (linear) chirp in radar systems.

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). Also published in: Rabiner, Shafer, and Rader, "The chirp z-transform algorithm," IEEE Trans. Audio Electroacoustics 17 (2), 86–92 (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 z-transform is nonstandard: a fractional Fourier transform conventionally refers to an entirely different, continuous transform.)
Lawrence Rabiner, "The chirp z-transform algorithm—a lesson in serendipity," IEEE Signal Processing Magazine 24, 118-119 (March 2004). (Historical commentary.)

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