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Choosing the Best FFT Algorithm: Power-of-Two vs Prime-Factor vs Winograd, Study notes of Signals and Systems Theory

Saybrook UniversitySignals and Systems Theory

The choice of fft algorithms based on their computational complexity and specific use cases. Power-of-two algorithms like split-radix are generally best for arbitrary lengths due to their low operation counts and compatibility with modern processors. Prime-factor algorithms and winograd fourier transform algorithm (wfta) require fewer multiplications but have higher additions and are less commonly used due to their complexity. The chirp z-transform offers a universal method for computing ffts of any length but comes at a higher cost. The document also covers multi-dimensional ffts and applications with few time or frequency samples.

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Connexions module: m12060 1
Choosing the Best FFT Algorithm
∗
Douglas L. Jones
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
Power-of-two-length FFTs are usually best for general use when the DFT length is somewhat at
the user's discretion. The best power-of-two algorithm may depend on the type of computer to be
used. When FFTs must be of a specic length, a prime-factor algorithm when applicable or otherwise
a common-factor algorithm usually provide fastest performance, while the chirp z-transform is a single,
universal method for computing an FFT of any length.
1 Choosing an FFT length
The most commonly used FFT algorithms
by far
are the power-of-two-length FFT
1
algorithms. The
Prime Factor Algorithm (PFA)
2
and Winograd Fourier Transform Algorithm (WFTA)
3
require somewhat
fewer multiplies, but the overall dierence usually isn't sucient to warrant the extra diculty. This is
particularly true now that most processors have single-cycle pipelined hardware multipliers, so the total
operation count is more relevant. As can be seen from the following table, for similar lengths the split-
radix algorithm is comparable in total operations to the Prime Factor Algorithm, and is considerably better
than the WFTA, although the PFA and WTFA require fewer multiplications and more additions. Many
processors now support single cycle multiply-accumulate (MAC) operations; in the power-of-two algorithms
all multiplies can be combined with adds in MACs, so the number of additions is the most relevant indicator
of computational cost.
Representative FFT Operation Counts
FFT length Multiplies (real) Adds(real) Mults + Adds
Radix 2 1024 10248 30728 40976
Split Radix 1024 7172 27652 34824
Prime Factor Alg 1008 5804 29100 34904
Winograd FT Alg 1008 3548 34416 37964
Table 1
∗
Version 1.3: Feb 6, 2007 8:02 am US/Central
†
http://creativecommons.org/licenses/by/1.0
1
"Power-of-two FFTs" <http://cnx.org/content/m12059/latest/>
2
"The Prime Factor Algorithm" <http://cnx.org/content/m12033/latest/>
3
"FFTs of prime length and Rader's conversion": Section Winograd Fourier Transform Algorithm (WFTA)
<http://cnx.org/content/m12023/latest/#WFTA>
http://cnx.org/content/m12060/1.3/
pf3

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Choosing the Best FFT Algorithm

Douglas L. Jones

This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †

Abstract Power-of-two-length FFTs are usually best for general use when the DFT length is somewhat at the user's discretion. The best power-of-two algorithm may depend on the type of computer to be used. When FFTs must be of a specic length, a prime-factor algorithm when applicable or otherwise a common-factor algorithm usually provide fastest performance, while the chirp z-transform is a single, universal method for computing an FFT of any length.

1 Choosing an FFT length

The most commonly used FFT algorithms by far are the power-of-two-length FFT^1 algorithms. The Prime Factor Algorithm (PFA)^2 and Winograd Fourier Transform Algorithm (WFTA)^3 require somewhat fewer multiplies, but the overall dierence usually isn't sucient to warrant the extra diculty. This is particularly true now that most processors have single-cycle pipelined hardware multipliers, so the total operation count is more relevant. As can be seen from the following table, for similar lengths the split- radix algorithm is comparable in total operations to the Prime Factor Algorithm, and is considerably better than the WFTA, although the PFA and WTFA require fewer multiplications and more additions. Many processors now support single cycle multiply-accumulate (MAC) operations; in the power-of-two algorithms all multiplies can be combined with adds in MACs, so the number of additions is the most relevant indicator of computational cost.

Representative FFT Operation Counts

FFT length Multiplies (real) Adds(real) Mults + Adds Radix 2 1024 10248 30728 40976 Split Radix 1024 7172 27652 34824 Prime Factor Alg 1008 5804 29100 34904 Winograd FT Alg 1008 3548 34416 37964

Table 1 ∗Version 1.3: Feb 6, 2007 8:02 am US/Central †http://creativecommons.org/licenses/by/1. (^1) "Power-of-two FFTs" http://cnx.org/content/m12059/latest/ (^2) "The Prime Factor Algorithm" http://cnx.org/content/m12033/latest/ (^3) "FFTs of prime length and Rader's conversion": Section Winograd Fourier Transform Algorithm (WFTA) http://cnx.org/content/m12023/latest/#WFTA

The Winograd Fourier Transform Algorithm^4 is particularly dicult to program and is rarely used in practice. For applications in which the transform length is somewhat arbitrary (such as fast convolution or general spectrum analysis), the length is usually chosen to be a power of two. When a particular length is required (for example, in the USA each carrier has exactly 416 frequency channels in each band in the AMPS^5 cellular telephone standard), a Prime Factor Algorithm^6 for all the relatively prime terms is preferred, with a Common Factor Algorithm^7 for other non-prime lengths. Winograd's short-length modules^8 should be used for the prime-length factors that are not powers of two. The chirp z-transform^9 oers a universal way to compute any length DFT^10 (for example, Matlab^11 reportedly uses this method for lengths other than a power of two), at a few times higher cost than that of a CFA or PFA optimized for that specic length. The chirp z-transform^12 , along with Rader's conversion^13 , assure us that algorithms of O (N logN ) complexity exist for any DFT length N.

2 Selecting a power-of-two-length algorithm

The choice of a power-of-two algorithm may not just depend on computational complexity. The latest extensions of the split-radix algorithm^14 oer the lowest known power-of-two FFT operation counts, but the 10%-30% dierence may not make up for other factors such as regularity of structure or data ow, FFT programming tricks^15 , or special hardware features. For example, the decimation-in-time radix-2 FFT^16 is the fastest FFT on Texas Instruments'^17 TMS320C54x DSP microprocessors, because this processor family has special assembly-language instructions that accelerate this particular algorithm. On other hardware, radix- algorithms^18 may be more ecient. Some devices, such as AMI Semiconductor's^19 Toccata^20 ultra-low-power DSP microprocessor family, have on-chip FFT accelerators; it is always faster and more power-ecient to use these accelerators and whatever radix they prefer. For fast convolution^21 , the decimation-in-frequency^22 algorithms may be preferred because the bit-reversing can be bypassed; however, most DSP microprocessors provide zero-overhead bit-reversed indexing hardware and prefer decimation-in-time algorithms, so this may not be true for such machines. Good, compiler- or hardware-friendly programming always matters more than modest dierences in raw operation counts, so manufacturers' or good third-party FFT libraries are often the best choice. The module FFT programming tricks^23 references some good, free FFT software (including the FFTW^24 package) that is carefully coded to be compiler-friendly; such codes are likely to be considerably faster than codes written by the casual programmer.

(^4) "FFTs of prime length and Rader's conversion": Section Winograd Fourier Transform Algorithm (WFTA) http://cnx.org/content/m12023/latest/#WFTA (^5) http://en.wikipedia.org/wiki/AMPS (^6) "The Prime Factor Algorithm" http://cnx.org/content/m12033/latest/ (^7) "Multidimensional Index Maps" http://cnx.org/content/m12025/latest/ (^8) "FFTs of prime length and Rader's conversion" http://cnx.org/content/m12023/latest/ (^9) "Chirp-z Transform" http://cnx.org/content/m12013/latest/ (^10) "Spectrum Analysis Using the Discrete Fourier Transform" http://cnx.org/content/m12032/latest/ (^11) http://www.mathworks.com/products/matlab/ (^12) "Chirp-z Transform" http://cnx.org/content/m12013/latest/ (^13) "FFTs of prime length and Rader's conversion": Section Rader's Conversion http://cnx.org/content/m12023/latest/#Radersconv (^14) "Split-radix FFT Algorithms" http://cnx.org/content/m12031/latest/ (^15) "Ecient FFT Algorithm and Programming Tricks" http://cnx.org/content/m12021/latest/ (^16) "Decimation-in-time (DIT) Radix-2 FFT" http://cnx.org/content/m12016/latest/ (^17) http://www.ti.com/ (^18) "Radix-4 FFT Algorithms" http://cnx.org/content/m12027/latest/ (^19) http://www.amis.com (^20) http://www.amis.com/products/dsp/toccata_plus.html (^21) "Fast Convolution" http://cnx.org/content/m12022/latest/ (^22) "Decimation-in-Frequency (DIF) Radix-2 FFT" http://cnx.org/content/m12018/latest/ (^23) "Ecient FFT Algorithm and Programming Tricks" http://cnx.org/content/m12021/latest/ (^24) http://www.tw.org/