# 3.3 Choosing the best fft algorithm

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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 specific 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.

## Choosing an fft length

The most commonly used FFT algorithms by far are the power-of-two-length FFT algorithms. The Prime Factor Algorithm (PFA) and Winograd Fourier Transform Algorithm (WFTA) require somewhat fewer multiplies, but the overall difference usually isn't sufficient to warrant the extra difficulty.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 thePFA 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 mostrelevant 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
The Winograd Fourier Transform Algorithm is particularly difficult to program and is rarely used in practice. For applications in which the transform length is somewhat arbitrary (suchas 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 cellular telephone standard), a Prime Factor Algorithm for all the relatively prime terms is preferred, with a Common Factor Algorithm for other non-prime lengths. Winograd's short-length modules should be used for the prime-length factors that are not powers of two.The chirp z-transform offers a universal way to compute any length DFT (for example, Matlab 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 specific length. The chirp z-transform , along with Rader's conversion , assure us that algorithms of $O(N\lg N)$ complexity exist for any DFT length $N$ .

## 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 offer the lowest known power-of-two FFT operation counts, but the 10%-30% difference may not make up for other factors such as regularity of structure or data flow, FFT programming tricks , or special hardware features. For example, the decimation-in-time radix-2 FFT is the fastest FFT on Texas Instruments' TMS320C54x DSP microprocessors, because this processor family has special assembly-language instructions that accelerate this particular algorithm. On other hardware, radix-4 algorithms may be more efficient. Some devices, such as AMI Semiconductor's Toccata ultra-low-power DSP microprocessor family, have on-chip FFT accelerators; it is always faster and more power-efficient to use these accelerators and whatever radix they prefer. For fast convolution , the decimation-in-frequency 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 differences in raw operation counts, so manufacturers' or good third-party FFT libraries are often the best choice.The module FFT programming tricks references some good, free FFT software (including the FFTW package) that is carefully coded to be compiler-friendly; such codes are likely to be considerably faster than codes written by the casual programmer.

## Multi-dimensional ffts

Multi-dimensional FFTs pose additional possibilities and problems. The orthogonality and separability of multi-dimensional DFTs allows them to be efficiently computed by a series of one-dimensional FFTs along each dimension.(For example, a two-dimensional DFT can quickly be computed by performing FFTs of each row of the data matrix followed by FFTs of all columns, or vice-versa.) Vector-radix FFTs have been developed with higher efficiency per sample than row-column algorithms. Multi-dimensional datasets, however, are often large and frequently exceed the cache size of the processor, and excessive cache misses may increase the computational time greatly, thus overwhelming any minor complexity reduction from a vector-radix algorithm.Either vector-radix FFTs must be carefully programmed to match the cache limitations of a specific processor, or a row-column approach should be used with matrix transposition in between to ensure data locality for high cache utilization throughout the computation.

## Few time or frequency samples

FFT algorithms gain their efficiency through intermediate computations that can be reused to compute many DFT frequency samples at once. Some applications require only a handful of frequency samples to be computed; when that number is of order less than $O(\lg N)$ , direct computation of those values via Goertzel's algorithm is faster. This has the additional advantage that any frequency, not just the equally-spaced DFT frequency samples,can be selected. Sorensen and Burrus developed algorithms for when most input samples are zero or only a block of DFT frequencies are needed, but the computational cost is of the same order.

Some applications, such as time-frequency analysis via the short-time Fourier transform or spectrogram , require DFTs of overlapped blocks of discrete-time samples. When the step-size between blocks is less than $O(\lg N)$ , the running FFT will be most efficient. (Note that any window must be applied via frequency-domain convolution, which is quite efficient for sinusoidal windows such as the Hamming window .) For step-sizes of $O(\lg N)$ or greater, computation of the DFT of each successive block via an FFT is faster.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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