Agenda for supervisor meeting the 22th of March 2011

Transcription

1 Agenda for supervisor meeting the 22th of March 2011 Group 11gr842 A3-219 at 14:00 1 Approval of the agenda 2 Approval of minutes from last meeting 3 Status from the group Since last time the two teams have looked into different methods for spectrum analysis and there have been a detailed study of different FFT algorithms. In context to the different methods and FFT algorithms, the group have completed a number of worksheets. According to the time schedule, the different methods and algorithms should be finished on thursday 24th. This will not happen, so the time schedule has been updated. You will find it on the webpage. 4 Supervising needs We would like feedback on the material provided on the webpage. For R&S only, you do not have to comment on the material unless you want to and feel that you have the time for it. If you will give us feedback, please send it via Further work We shall start investigate different algorithms for both LPC and filter banks. According to the time schedule they should be done before the next meeting takes place. 1

2 6 AOB Use of sources 7 ext meeting 2

3 Minutes from supervisor meeting March 15th 2011 Group 11gr842 March 15, Approval of the agenda Approved. 2 Approval of minutes from last meeting Approved. 3 Status from the group We have split into two groups to work on separate tasks. One group is finishing the application description and the other is looking into which FFT algorithm would be the most optimal with respect to computational complexity, memory usage etc. 4 Supervising needs o worksheets are done for this meeting so the found specifications for 3G were discussed as the standard can be confusing. The found bandwidth is 20 MHz... It is emphasized that sample rate and are important. If the requirement for sample rate is low enough it could be sufficient to calculate the DFT instead of the FFT. The dynamic range can be a problem with a large bandwidth so a solution could be to lower the resolution in the outer bands where the attenuation is high. 1

4 5 Further work The table with applications as well as their description will be done for the next meeting. The second group will continue analyzing the FFT algorithms. 6 ext meeting Tuesday the 22nd of March. 14:00. Peter, Jes and Kristian will not be attending this meeting. 7 AOB Jes and Kristian would like to comment on worksheets when they have the time and not necessarily before the meeting. Methods and methodologies should be determined soon so it can be used in the analysis of algorithms. Kristian mentioned again that consideration about how to verify the project is important. A set of test vectors should be selected. 8 ext meeting 2

5 Chapter 3 Spectrum analysis methods When doing spectrum analysis it is of interest to know whether a method uses a parametric or a non-parametric model and whether it is a linear, logarithmic or perhaps inverse logarithmic scale. In this introduction the difference in parametric and non-parametric models will be explained. In the following sections, different spectrum analysis methods will be introduced and the pros and cons will be clarified. This section will not contain any mathematics for the use of these methods and algorithms for each methods is not included. In the end of the chapter, a table will be presented having different frequency ranges and the different spectrum analysis methods listed as the leading column and row. All the case studies will be plotted in the table to underline which frequency and method is applicable to each case study. Based on this table the methods will be evaluated and the methods that will be used in for further work in project will be chosen. The difference between parametric and non-parametric models will now be explained. The starting point is in probability theory. When describing a population it is advantageous if the distribution function are known beforehand. If the distribution function is known beforehand, the population can be described by a set of parameters. As an example consider a population X having a normal distribution. X can then be described by its mean and variance. This implies that X is described by a set of parameters, thus it is called a parametric inference. The same have its effect on a spectrum analysis. As an example the LPC could be considered. In LPC a model for the vocal tract is set up and the parameters for the model is estimated, thus it is a parametric model. This means that you can describe speech by a set of parameters. The key factor of a parametric model is that the collection of distribution functions are of a finite length as in LPC. In contrary to the parametric model, non-parametric model has a infinite set of parameters to describe the signal. As an example the DFT/FFT can be considered. 9

6 3. Spectrum analysis methods 3.1 Fast Fourier Transform Fast Fourier Transform (FFT) is a functionally equivalent of DFT and is usually used in real time analysing because of its speed. The algorithm uses only log 2 operations to display the spectrum. The difference in speed between DFT and FFT can be substantial, especially for long data sets where may be in the thousands or millions in practice. The computation time can be reduced by several orders of magnitude in such cases, and the improvement is roughly proportional to / log. The most well known FFT algorithms depend upon the factorization of, but there are also FFTs with O( log 2 ) complexity for all, even for prime. Many FFT algorithms only depend on primitive roots of unity (twiddle factors), that is, on number of complex multiplications and additions. FFT is in wide use and works very well for stationary signals. But if the signal is dynamical, has pulses or any other kind of interesting behaviour, then FFT is not appropriate and other methods should be used. Thus, FFT is very popular when it comes to implementation into spectrum analysers, especially the Cooley-Tukey algorithm since it is easier to implement compared to other FFT algorithms. But speed comes at a price of FFTs accuracy, or rather resolution. amely, if an - length FFT has a certain resolution, i.e. every 250 Hz, it will not be able to show a signal peak at 270 Hz. In other words, the user will not see that there are certain points in a signal. To have a better resolution a larger should be used. Although, the more samples that have to be processed, the more time is takes FFT to compute. 3.2 Wavelets Wavelets are functions that can come in many shapes. Different functions are chosen for different applications. The shape of the wavelet is decided by the shape and characteristics of the expected signal. For example, wavelets used for analysing oscillations in human brain are often based on the Morlet wavelet [17]. This wavelet is named after Jean Morlet one of the developers of the wavelet transform. Other functions used are for instance the Mexican hat or other functions that start and end at zero, and decay rapidly towards zero with time [3]. This can be seen in figure 3.1 where both the Morlet wavelet and the Mexican hat wavelet are shown. 10

7 3.2. Wavelets 1 Morlet wavelet 1 Mexican hat wavelet Amplitude Amplitude Time Time Figure 3.1: The Morlet wavelet and the Mexican hat wavelet. The basic wavelet can be real or complex, with the resulting wavelet transform being real or complex. The use of complex wavelets can come in handy for applications where the phase of the wavelet transform can contain important information. For use in a spectrum analyser the phase is generally not important, but it is needed to recreate the original signal. Wavelets gives more precision than the Fourier transform because of the windowing possibilities. When using wavelet analysis, the size of the windows depends on the frequency in the signal, as opposite to the windowed Fourier transform. Wavelets have a good time-frequency resolution and is therefore suitable for non-stationary signals [17]. Wavelets are used for many applications within signal processing, in everything from neuroscience and speech which are within the lower frequency range, and up to the high frequency ranges. It is also used in signal prcessing matters like pattern recognition, data compression and acoustics [13]. The length of a wavelet decreases as the frequency increases. This also means that there is a temporal and frequency resolution trade-off. For the higher frequencies the frequency resolution decreases while temporal resolution increases. S: Figure? The topic of wavelets are a rather comprehensive matter regarding the theory behind. The mathematical structure combines ideas from Fourier analysis with signal processing ideas about a two channel filter bank [11]. Although this section will not focus on the mathematics of wavelets, it is said that one disadvantage with wavelets is their complexity. There are several methods within wavelet transforms, and there are also several real-time methods. These methods introduces some problems due to the fact that it can require both a previously computed value and a not yet computed value at 11

8 3. Spectrum analysis methods the same time. This makes it complicated, but there are possible solutions to this and a good real-time functionality is possible to achieve. 3.3 Multitapers The multitaper method is based on the Fourier transform, as many other spectrum analysis methods are. In multitapers methods the signal is multiplied with special windows before frequency decomposition. These windows are design to prevent bleeding of power to nearby frequencies. This makes it very frequency-specific and makes it suitable for non-stationary signals with high dynamic range. It is also suitable for signal that changes rapidly [18]. Performing a Fourier transform is the next step after multiplying the window with the signal [17], so this method is not a unique methods, but is closely related to the original Fourier transform. Real-time?? Multitaper methods is commonly used in neuroscience and biomedical engineering, for instance with the purpose of measure brain oscillations in human or animal brains [17]. For these applications the frequency range is within the extremely low frequency (ELF) band. 3.4 Filter Bank ref to the sec. about case studies - JBK The filter bank method is a non-parametric method. It can be used in a large variety of spectrum analysis. Mapped onto the case studies made in, this method can be useful in the windmill-, speech-, concert-, and base station case. Generally seen, the filter banks method divide the frequency band into a number of smaller bands using a number of bandpass filters. The number of bandpass filters correspond linearly to the number of smaller frequency bands, thus the higher the number of filters used, the higher the resolution. The disadvantage of using a lot of bandpass filters is the increased number of calculations needed for every measurement. It should also be mentioned that the implementation of a bandpass filter can be done more or less complex, making it interesting for use in a spectrum analyser. In addition, the frequency range can be very high, but as a consequence the frequency resolution will decrease when the platform is utilised 100 %. When it comes to real-time implementation, the filter bank method is interesting because the bandpass filters can be calculated very fast and efficient on a DSP, thus real-time execution becomes more relevant. 3.5 Joint Time-Frequency Analysis The method of Joint Time-Frequency Analysis (JTFA) takes both time and frequency into account when analysing a signal. The Fourier transform is the most used method for spectral analysis, but this is not always the best way of analysing a signal. While 12

9 3.6. Super Resolution Spectral Analysis the Fourier transform is the connection between time and frequency domain, JTFA is a further development of the Fourier transform, where both time and frequency is analysed. While the power spectra from the Fourier transform is suitable for signals whose spectra does not change over time, the JTFA is usually a good alternative for signals varying over time. Signals changing over time, for instance speech signal, audio signals, biomedical signals, vibrations and seismic signals could be presented by a joint time-frequency representation [14]. For example, a spectrum of a speech signal will show the significant frequencies in the speech, but not how the frequencies evolve over time. Using a joint time-frequency representation, like a spectrogram, it will show graphically how the frequency in the speech signal change over time. There are several algorithms within the method of JTFA, both parametric and nonparametric, although the non-parametric methods are most used. As for the real-time perspective for this method, that depends on whether a non-parametric or parametric algorithm is used. Real-time functionality is easier for non-parametric model, than for parametric models. The most common method within JTFA is the short-time Fourier transform [8]. This is the simplest form of JTFA and also the easiest to compute and implement. It can be used when the frequency in the signal does not change dramatically. In case of rapid changes in frequency other methods within JTFA are recommended. These provides high performance at the expense of computational complexity. 3.6 Super Resolution Spectral Analysis The Super Resolution Spectral Analysis (SRSA) is a model-based alternative to the FFT [9]. That it is model-based means that a suitable model for the signal is conducted in advance, and that the spectrum analyser has to determine the coefficients of the model. With this model it is possible to predict the missing points from a dataset, and by this, achieve high resolution. This method is suitable for finding the spectrum for signals with only a limited amount of samples. For datasets with less than 100 samples, it could be used with great success. The SRSA method is computationally heavy, compared to the FFT. That is why it is not the preferred method for larger datasets. One other disadvantage of this methods is that some prior knowledge about the input signal is required. In order to use the SRSA method the analyser needs an estimate of the number of sinusoidal components in the input signal. That is the reason for some typical applications of the SRSA being speech analysis, biomedical research, noise and vibrations. Models for these applications are possible to develop for use in the spectrum analyser. It is crucial to use the most suitable model to get an appropriate spectrum analysis of the signal. Since this methods only is suitable for datasets with a limited amount of samples, there is not expected any problem due to real-time functionality. But this methods requires the models of the different signals to be developed, which can be an extremely time 13

10 3. Spectrum analysis methods consuming part of the development period. Within the field of speech signal analysis, this method can be somewhat similar to the LPC approach. 3.7 Linear Predictive Coding As stated in the introduction to this chapter, LPC is a parametric model that is well suited for analysis of the frequency content of speech. LPC uses a set of parameters to describe the speech signal and the parameters are found from the model which is set up to copy the speech. To understand the model, first speech will be described. can be divided into two different types of signals. The voiced and the unvoiced signals. The voiced signal is produced by pushing air up through the vocal chords which creates a signal with a fundamental tone. The signal then passes through the vocal tract which colors (filters) the signal. An example of a voiced sound is /a/ or /o/. The unvoiced sounds are merely produced by a noise-like signal. The model of the speech system can be imitated by the illustration on Figure 3.2. The voiced and unvoiced signals can be somewhat copied using either a filtered pulse train or filtered white noise, respectively. The vocal tract filter, through which the pulse train and white noise is lead, can be estimated as an all-pole filter. To sum up the required parameters that needs to be estimated to describe a speech signal is the voicing, gain, filter coefficients and pitch period. Due to the relatively slow physical changes in the vocal tract, the parameters can be considered stationary withing ms. Figure 3.2: Simplified model of the speech system. As the formants of the fundamental tone in a speech signal is attenuated in a logarithmic way (1/x 2 ) LPC is very well suited for analysing speech. For speech the frequency range is usually 20 20k Hz. From a real-time perspective, LPC is hard to do. This is due to large matrices that needs to be inverted and handled every ms. However optimisation for handling matrices are existing, thus making a real-time implementation more relevant. 14

11 3.8. Spectrum Analysis Method Summary 3.8 Spectrum Analysis Method Summary In this chapter different methods for analysing the spectrum of a signal have been introduced. The pros and cons for each method have been sketched and this summary will try to compare them to find the final methods that should be used in the spectrum analyser. The second method mentioned is to use wavelets on the signal in order to find the spectrum. This is a relatively new, but also a comprehensive method[3]. It is still a field of research, but has proven to be a good alternative to the FFT. Multitapers are based on the Fourier transform. This method is commonly used within the field of neuroscience, which is within the lower frequency range. Another method described is the Joint Time-Frequency Analysis. That is basically a spectrogram, showing how the frequencies evolves over time. This method is said to be useful for analysing speech signals and other signals varying over time. Super Resolution Spectral Analysis is parametric way of finding the spectrum. It is suitable for analysing a dataset of limited amount of samples and it is computationally heavy. The last method mentioned is the LPC. This has proven to be an excellent parametric method for spectrum analysis of a speech signals. The drawback is the large matrices that has to be inverted every ms, but there exists optimised algorithms to reduce the number of calculations. From what has been written in the earlier sections, three of the method are chosen. The DFT/FFT is very common and with a great variety of different algorithms. The method is preferable to use because because its use is very versatile as can be seen in Table 3.1. When considering wavelets, it too is also very versatile and with a lot of pros like the DFT/FFT. One critical point where wavelets are superior to the DFT/FFT is the resolution. However, wavelet algorithms often introduce higher complexity and are ELF SLF ULF VLF LF MF HF DFT/FFT Windmill Windmill Concert Concert Concert Basestation Basestation Basestation Wavelets eural Oscillations (ECG) Windmill Concert Windmill Concert Concert Multitapers eural Oscillations (ECG) Filter bank Windmill Windmill Concert Concert Concert JTFA SRSA LPC Biomedical Biomedical Biomedical Basestation Basestation Basestation Basestation Basestation Basestation Table 3.1: Division of frequency ranges. We need something on the DFT/FFT! - JBK Can it be used for anything else than speech?? - JBK Remember to use the table in this section! - JBK S: Do we want to choose three? 15

12 3. Spectrum analysis methods sometimes very inefficient. Another interesting method to use is the filter bank. This method has, like the DFT/FFT and wavelet methods, a large variety of different usages. In addition, the filter calculations can be optimised to run on a DSP, which makes it very fast and efficient. The last method chosen is the more specialised parametric method, namely LPC. This is chosen because of its very good approximation of speech and because speech is one of the case studies. ow four methods have been mentioned. One of them is wavelets, because it is be very desirable, but the complexity of this method combined with the time allocated for this project, means that this method is neglected. The final three method is therefore the DFT/FFT, filter bank, and the LPC. 16

13 Chapter 7 FFT algorithms The Discrete Fourier Transform (DFT) transforms data in the discrete time domain to the discrete frequency domain and is defined as (7.1) X[k] = where 1 n=0 x[n] W kn for k = 0,..., 1 (7.2) W = e j2π/ is also called the twiddle factor. When implementing the DFT the Fast Fourier Transform (FFT) is often used as it improves some issues with the DFT, namely computational efficiency (number of real multiplications and real additions) and quantization noise. Solving eq. (7.1) would for each k take complex multiplications and 1 complex additions. The complexity for the DFT in Big-O notation is O( 2 ) whereas the complexity of the FFT is reduced to O( log 2 ). The main strategy behind most FFT algorithms is to factor a length- DFT into a number of shorter-length DFTs, the outputs of which are reused multiple times (usually in additional short-length DFTs) to compute the final results. The lengths of the short DFTs correspond to integer factors of the DFT length,, leading to different algorithms for different lengths and factors. This section describes different algorithms for computing the FFT and evaluates them based on the given application, as described in section 3.8 on page JOC-I think we should just say that FFT is faster and then when explaining different algorithms say how much faster. C: describe req. for accuracy and freq. range here

14 7. FFT algorithms 7.1 The Cooley-Tukey algorithm The work done by Cooley and Tukey [5] makes it possible to calculate the -point DFT with = 2 m. To reduce the number of calculations the algorithm uses the principles of symmetry and periodicity of the twiddle factor, namely (7.3) (7.4) Periodicity: W kn Symmetry: W k( n) = W k(n+) = W kn = W (k+)n. = ( W kn ). better explanation From the unit circle in figure 7.1 it can be seen that periodicity of the signal enables twiddle factors to repeat themselves with period. Also, for every W kn, which is periodic with, there is an opposite W kn+/2 which equals the negative value of W kn, or in other words symmetry is used. This makes it possible to skip redundant multiplications as it is shown later. W /4 Im W /2 W 0 Re W 3/4 Figure 7.1: Unit circle with corresponding twiddle factors. The principle behind the Cooley-Tukey algorithm is to divide the DFT into two smaller DFTs of size /2 for each iteration and hence the requirement that = 2 m. In practice it is also possible to calculate the DFT when is not a power of two by zero-padding the sequence. This of course results in more operations but in many cases it is still more efficient than computing the DFT. The process of dividing the DFT into smaller DFTs can be done either by decimationin-time (DIT) or decimation-in-frequency (DIF). They both give the same result but the DIT reorganises the time samples before the DFT is computed, while the DIF rearranges the frequency samples. For DIT the DFT is seperated for n even and odd so that (7.5) (7.6) X[k] = 1 n=0 = n even x[n] W nk fork = 0,..., 1 x[n] W nk + n odd x[n] W nk. 28

15 7.2. The radix-2 algorithm By use of index mapping the index for n even is substituted with n = 2r and for n odd with n = 2r + 1 so that (7.7) (7.8) X[k] = = /2 1 r=0 /2 1 r=0 /2 1 x[2r] W (2r)k + x[2r] W (2r)k r=0 + W k x[2r + 1] W (2r+1)k /2 1 r=0 x[2r + 1] W (2r)k, and using the fact that W 2 = e j2π 2 = e j2π 1 /2 = W /2, the DFT can be rewritten as (7.9) (7.10) X[k] = /2 1 r=0 /2 1 x[2r] W/2 rk + W k = G[k] + W k H[k]. r=0 x[2r + 1] W rk /2 The two new DFTs, G[k] and H[k], can be computed independently as shown in figure? and then combined as in eq. (7.10) to produce the -point DFT. For large values of, G[k] and H[k] can in a similar way again be decomposed into smaller DFTs of size make figure /4, /8 and so on. For the DIF approach, index mapping is done for the index k so that the even-numbered frequency samples is given as (7.11) X[2r] = /2 1 n=0 and the odd-numbered frequency samples as (x[n] + x[n + (/2)]) W rn /2, (7.12) X[2r + 1] = /2 1 n=0 (x[n] + x[n + (/2)]) W n W rn /2 better explanation 7.2 The radix-2 algorithm The radix-2 algorithm is a special kind of Cooley-Tukey algorithm where = 2 m is used. It is the simplest kind of FFT algorithm, but very effective way of computing a DFT. For example, when DIT radix-2 has = 4 the equation (7.7) can be written as (7.13) X[k] = 1 r=0 x[2r] W 2rk r=0 x[2r + 1] W (2r+1)k 4, or in a matrix representation as X[0] W4 0 W4 0 W4 0 W 0 4 x[0] (7.14) X[1] X[2] = W4 0 W4 1 W4 2 W4 3 x[1] W4 0 W4 2 W4 4 W4 6 x[2]. X[3] W4 0 W4 3 W4 6 W4 9 x[3] 29

16 7. FFT algorithms As described in equation (7.3) the FFT uses the periodicity where for this example W4 4 = W 4 0, W 4 5 = W 4 1, W 4 6 = W 4 2 and W 4 9 = W 4 1, where now only W 4 0, W 4 1, W 4 2 and W 4 3 are used. Furthermore, equation (7.4) leads to W4 2 = W 4 0 and W 4 3 = W 4 1. Again, the number of twiddle factors that must be computed is reduced and only W4 0 and W 4 1 remain, where W4 0 = 1 and W 4 1 = j from equation (7.2). From the explanation above, the equation (7.14) can be rewritten as (7.15) X[0] x[0] X[1] X[2] = 1 j 1 j x[1] x[2]. X[3] 1 j 1 j x[3] Because of the periodicity and symmetry, FFT can use further simplifications that reduce the number of computations. amely, the equation (7.15) can be rewritten as (7.16) (7.17) stage 1 {}} { X[0] x[0] + x[2] X[1] X[2] = j x[0] x[2] x[1] + x[3] X[3] j x[1] x[3] stage 2 { }} { (x[0] + x[2]) + (x[1] + x[3]) = (x[0] + x[2]) (x[1] + x[3]) (x[0] x[2]) + j(x[1] x[3]) (x[0] x[2]) j(x[1] x[3]) The reason of such decomposition lays in the fact that a pair of even-numbered index and odd-numbered index spectral components have a common (complex) multiplication. For example, in equation (7.17) it can be seen that X[2] and X[3] have a common multiplication product j(x[1] x[3]) which can be computed only once. Generally, every pair of even-indexed and odd-indexed spectral component in FFT has a common product. This is the reason why the power-of-two FFTs are so effective, they use periodicity and symmetry which in the end lead to simplification because less data must be computed. Figure 7.2 displays an flow graph of the radix-2 example which has been made from stages in equations (7.16) and (7.17). Every node in a stage is composed of an addition (subtraction) of an upper node x[r] and lower node multiplied with the twiddle factor W rkx[r + /2]. An radix-2 FFT uses log 2 stages where every stage has additions and /2 multiplicatoins, although multiplication by 1 is treated as an addition. From figure 7.2 it can also be seen that computation savings are made by reusing the results from the previous stage. Basically, an -length DFT is divided into two /2-length DFTs, an /2-length into two /4-length DFTs and so on until only a 2-length DFT is left which has no complex multiplication. 30

17 7.3. The split-radix algorithm stage 0 stage 1 stage 2 x[0] X[0] x[1] 1 - X[1] x[2] 1 - X[2] x[3] j X[3] Figure 7.2: Signal flow graph (butterfly structure) for a = 4 radix-2 FFT. For larger there are more complex twiddle factors but as long as the = 2 m symmetry and periodicity rules apply. Also, since there is a known number of twiddle factors for a know they can be computed once and stored in a memory where they can be just used when needed. From figure 7.2 it can be seen that stage 1 uses 2 multiplications and 4 additions (subtractions), just like stage 2, which in total makes 8 additions and 4 multiplications. Thus, the radix-2 algorithm uses 5 log 2 real additions and multiplications for a given [5]. The decimation-in-frequency (DIF) radix-2 FFT partitions the DFT computation into even-indexed and odd-indexed outputs, which can each be computed by shorter-length DFTs of different combinations of input samples. Recursive application of this decomposition to the shorter-length DFTs results in the full radix-2 decimation-in-frequency FFT. explain how the O(5 log 2 ) is found 7.3 The split-radix algorithm The split-radix algorithm can be derived from the radix-2 and radix-4 algorithms. Where radix-4 is similar to radix-2 but it uses = 4 m. The decomposition of equation (7.1) of lenght for the split-radix is given as (7.18) X[k] = 2 1 n=0 x[2n] W 2nk n=0 x[4n + 1] W (4n+1)k n=0 x[4n + 3] W (4n+3)k. The first sum corresponds to the radix-2 with the size /2 and includes even-numbered indices of the DFT, and the last two sums corresponds to the radix-4 of size /4 which include the odd-numbered indices. The algorithm can be used only for = 4 m. The reason of a such decomposition is that radix-4 algorithm in some places of the butterfly stage has fewer non-trivial twiddle factors (those who are different from ±1 or ±j). Meanwhile, the radix-2 in some places lacks twiddle factors present in the radix-4 ref 31

18 7. FFT algorithms add figure How difficult is it to implement in comparison? structure, or has twiddle factors which require only additions since the multiplication is made by a j twiddle factor. The split-radix algorithm makes use of those facts and mixes those two algorithms, since all power-of-two algorithms have the same butterfly tree structure, in an order which reduces the total amount of complex multiplications. The split-radix algorithm was introduced by R. Yavne who proved that split radix has O(4 log ), which is 20% less computational operations then radix-2 algorithm. Steven G. Johnson and Matteo Frigo have improved the split-radix algorithm so it has O( 34 9 log log ( 1)log 2 log ( 1)log 2 + 8) (5.6% less operations then original split-radix algorithm) [12]. 7.4 The Prime-factor algorithm (PFA) The PFA belongs to a class of algorithms that do not use = 2 m. It re-expresses the DFT of a size = 1 2 as a two-dimensional 1 2 DFT, but only for the case where 1 and 2 are relatively prime. These smaller transforms of size 1 and 2 can then be evaluated by applying PFA recursively or by using some other FFT algorithm. The PFA gains efficiency by reuse of intermediate computations and by eliminating twiddle-factor multiplies, but require more operations than the power-of-two algorithms to compute the short DFTs of various prime lengths. In the end, the computational costs of the prime-factor and the power-of-two algorithms are comparable for similar lengths. PFA algorithm should not be confused with the Cooley-Tukey algorithm, which can use any factors (not necessarily relatively prime), but it has the disadvantage that it also requires extra multiplications by twiddle factors, in addition to the smaller transforms. PFA only works for relatively prime factors and that it requires a more complicated reindexing of the data based on the Chinese Remainder Theorem. However, PFA can be combined with a mixed-radix Cooley-Tukey, with the former factorising into relatively prime components and the latter handling repeated factors. ref PFA is also closely related to the nested Winograd FFT algorithm, where the latter performs the decomposed 1 by 2 transform via more sophisticated two-dimensional convolution techniques. Some older papers therefore also call Winograd s algorithm a PFA FFT. 7.5 The Winograd algorithm The development of this algorithm took place in a time where multiplications were expensive in computation time, area and power and therefore aims at reducing the number of multiplications. Reducing the number of multiplications comes at the expense of increasing the number of additions. This fact could be an issue in architectures where fast hardware multipliers are available since this algorithm does not make any advantage. 32

19 7.6. Rader s algorithm The Winograd algorithm is particularly difficult and complicated to program and it is different for every length of [4]. 7.6 Rader s algorithm This is a Cooley-Tukey-like factorisation but with purely imaginary twiddle factors, reducing multiplication at the cost of increasing additions and reducing numerical stability. It uses the circular convolution properties of prime number DFTs, much like the Winograd algorithm. Winograd extended Rader s algorithm to include prime-power DFT sizes pm (Winograd 1976; Winograd 1978), and today Rader s algorithm is sometimes described as a special case of Winograd s FFT algorithm. For composite sizes such as prime powers, the Cooley-Tukey FFT algorithm is much simpler and more practical to implement, so Rader s algorithm is typically only used for large-prime base cases of Cooley-Tukey s recursive decomposition of the DFT (Frigo and Johnson, 2005). Rader s FFT algorithm can be used to compute DFTs of length in log 2 operations when is a prime number. C: make proper references 7.7 The Sørensen algorithm Whereas most FFT algorithms assume that the length of the input and output sequence is equal, the Sørensen FFT has the possibility of computing the DFT with only a subset of points [15]. This is relevant when only a narrow spectrum of the sampled signal is needed. This can also be done with other types of FFTs by computing the whole sequence and discarding the unwanted results but the Sørensen FFT reduces the number of calculations by using a transform decomposition algorithm and only calculating the needed data points. 7.8 FFT algorithm summary The Prime-factor algorithm and Winograd algorithm require somewhat fewer multiplies compared to power-of-two algorithms, but the overall difference is usually not sufficient to warrant the extra difficulty. This is particularly true now that most processors have single-cycle pipelined hardware multipliers (MACs), so the total operation count is more relevant. FFT algorithms that use DFT lengths equal to a power of two are the simplest and by far the most commonly used. A drawback of mentioned algorithms is that they cannot be used for any DFT length, so for a specific length a prime-factor algorithm should be used. 33

20 7. FFT algorithms Algorithm Complexity Operations Memory usage Pros Cons Radix-2 Medium 5 log 2 B C D Split-radix Medium-high A B C D Winograd High A B C D Prime-factor High A B C D Sørensen?? A B C D Rader s?? A B C D Table 7.1: Algorithm comparison. 34