Bluestein's FFT algorithm

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Bluestein's FFT algorithm

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 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 $X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} nk } \qquad k = 0,\dots,N-1.$

If we replace the product nk in the exponent by the identity nk = –(kn)2/2 + n2/2 + k2/2, we thus obtain: $X_k = e^{-\frac{\pi i}{N} k^2 } \sum_{n=0}^{N-1} \left( x_n e^{-\frac{\pi i}{N} n^2 } \right) e^{\frac{\pi i}{N} (k-n)^2 } \qquad k = 0,\dots,N-1.$

This summation is precisely a convolution of the two sequences an and bn of length N (n = 0,...,N–1) defined by: $a_n = x_n e^{-\frac{\pi i}{N} n^2 }$ $b_n = e^{\frac{\pi i}{N} n^2 },$

with the output of the convolution multiplied by N phase factors bk*. That is: $X_k = b_k^* \sum_{n=0}^{N-1} a_n b_{k-n} \qquad k = 0,\dots,N-1.$

This convolution, in turn, can be performed with a pair of FFTs (plus the pre-computed FFT of bn) via the convolution theorem. The key point is that these FFTs are not of the same length N: such a 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 an is extended to an array An of length M, where An = an for 0 ≤ n < N and An = 0 otherwise—the usual meaning of "zero-padding". However, because of the bkn term in the convolution, both positive and negative values of n are required for bn (noting that bn = bn). The periodic boundaries implied by the DFT of the zero-padded array mean that –n is equivalent to Mn. Thus, bn is extended to an array Bn of length M, where B0 = b0, Bn = BMn = bn for 0 < n < N, and Bn = 0 otherwise. A and B are then FFTed, multiplied pointwise, and inverse FFTed to obtain the convolution of a and b, according to the usual convolution theorem.

Let us also be more precise about what type of convolution is required in Bluestein's algorithm for the DFT. If the sequence bn were periodic in n with period N, then it would be a cyclic convolution of length N, and the zero-padding would be for computational convenience only. However, this is not generally the case: $b_{n+N} = e^{\frac{\pi i}{N} (n+N)^2 } = b_n e^{\frac{\pi i}{N} (2Nn+N^2) } = (-1)^N b_n .$

Therefore, for N even the convolution is cyclic, but in this case N is composite and one would normally use a more efficient FFT algorithm such as Cooley–Tukey. For N odd, however, then bn is antiperiodic and we technically have a negacyclic convolution of length N. Such distinctions disappear when one zero-pads an to a length of at least 2N−1 as described above, however. It is perhaps easiest, therefore, to think of it as a subset of the outputs of a simple linear convolution (i.e. no conceptual "extensions" of the data, periodic or otherwise).

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: $X_k = \sum_{n=0}^{N-1} x_n z^{nk} \qquad k = 0,\dots,M-1,$

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 bn from above is a complex sinusoid of linearly increasing frequency, which is called a (linear) chirp in radar systems.

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