Editor's Note: This translation completed by theunblock andjingliang. Again recommendation: 'If you do not understand the text read Fourier transform, then come to strangle me [full version],' Fast Fourier Transform (Fast Fourier Transform) signal processing and data analysis is the most important areas One algorithm. There is no formal background in computer science courses, I use this algorithm for many years, but this week I suddenly remembered that he had never thought about why the FFT can so quickly computing discrete Fourier transform. I opened an old algorithm 2015 Nike Free 5.0 book, enjoy the JW Cooley and Nike Lunareclipse John Tukey in 1965 article, seemingly simple calculation techniques to explain this stuff. Objective of this paper is deeply Cooley-Tukey FFT algorithm, explaining its roots as a 'symmetry', and with some intuitive python code theory into reality. I hope that this study enables data scientists (such as me) on the background principle of this algorithm is a more comprehensive understanding. FFT (Fast Fourier Transform) is itself a discrete Fourier transform (Discrete Fourier Transform) of fast algorithm, so that from the original algorithm complexity O (N ^ 2) becomes O (NlogN), Discrete Fourier Transform DFT , as more people are familiar with the continuous Fourier transform, there are positive and inverse form defined as follows: xn to Xk airspace is transformed to the frequency domain, Nike Lunareclipse this transformation will help power spectra of the signal, and make certain issues computing more efficient. In fact, you can also look at our upcoming books on astronomy and statistics X (here are some illustrations and python code). As an example, you can see my article, 'solving the Schrodinger equation with python,' it is how to use the FFT would simplify otherwise complex differential equations. Because FFT in so many areas so useful, python provides many standard tools and packaging to calculate it. NumPy and SciPy has been fully tested packaged FFT library, located in the sub-module numpy.fft and scipy.fftpack. The fastest FFT I know that in FFTW package, and you can use it in the python pyFFTW package. Although so far, but still temporarily put aside these libraries first, consider how the python from scratch using the original calculation of FFT. Computing discrete Fourier transform of simplicity, we only care about positive change, but also because of the inverse transform in a very similar manner to be able to do it. Look at the DFT expression above, it's just an intuitive linear operation: x vector matrix multiplication, matrix M can be expressed as thought, we can simply use matrix multiplication to calculate DFT: import numpy as npdef DFT_slow (x): \u0026 quot; \u0026 quot; \u0026 quot; Compute the discrete Fourier Transform of the 1D array x \u0026 quot; \u0026 quot; \u0026 quot; x = np.asarray (x, dtype = float) N = x.shape [0] n = np.arange (N) k = n.reshape ((N, 1)) M = np.exp (-2j * np.pi * k * n / N) return np.dot (M, x) comparison numpy built-in FFT function, we come to the results Careful examination, x = np.random.random (1024) np.allclose (DFT_slow (x), np.fft.fft (x)) Output: True To validate our algorithm now how slow, compared to under two Execution time% timeit DFT_slow (x)% timeit np.fft.fft (x) output: 10 loops, best of 3: 75.4 ms per loop10000 loops, best of 3: 25.5 us per loop using this simplified implementation, as expected, we 579827-300 Sport Turquoise Nike LeBron X PS Elite Online slow a thousand times. But the problem is not so simple. For the length of the input vector N, FFT is O (N logN) level, and our slow algorithm is O (N ^ 2) level. This means, FFT finished with 50 milliseconds capable of living for our algorithm is slow to almost 20 hours! So how speed FFT bin is it? The answer is that he takes advantage of symmetry. One of the most important means of discrete Fourier transform algorithm designer symmetry possession, it is to use the issue of symmetry. If you can show the problem clearly related to a part of another part, then Nike Air Max you only need to calculate the result of a child, thus saving computational cost. Cooley and Tukey It is using this method to export FFT. First we look at the value of. According to the above expression, Release: For all integers n, exp [2π in] = 1. The last line shows the DFT good symmetry: simply expand it: empathy for all integers i. As will soon see, this symmetry can be utilized in faster computing DFT. DFT to FFT: Using Symmetry Cooley and Tukey proved, DFT calculations can be divided into two parts. From the definition of the DFT was: We will look similar single DFT is divided into two smaller DFT. An odd one even. So far, we have not calculated the cost savings, each section contains the (N / 2) * to calculate the amount of N, in general, that is, N ^ 2. The trick is to use every part of the symmetry. Because the range of k is 0≤k \u0026 lt; N, and n ranges 0≤n \u0026 lt; M≡N / 2, from the symmetry of the above, we only make half the amount calculated for each sub-problem. O (N ^ 2) becomes O (M ^ 2). But we are not stopped to this step, as long as our little Fourier transform is an even multiple of, you can further divide and conquer, until the decomposition of the sub-problems small to divide and conquer by improving efficiency, close to the limit, this recursive It is O (n logn) levels. This recursive algorithm can be quickly implemented in python, the problem is when the sub exploded to the appropriate size, and then back to the original kind of 'slow method.' def FFT (x): \u0026 quot; \u0026 quot; \u0026 quot; A recursive implementation of the 1D Cooley-Tukey FFT \u0026 quot; \u0026 quot; \u0026 quot; x = np.asarray (x, dtype = float) N = x.shape [0] if N% 2 \u0026 gt; 0: raise ValueError (\u0026 quot; size of x must be a power of 2 \u0026 quot;) elif N \u0026 lt; = 32: # this cutoff should be optimized return DFT_slow (x) else: X_even = FFT (x [:: Mens Nike Free Run 3 Shoes Grey 3 2 ]) X_odd = FFT (x [1 :: 2]) factor = np.exp (-2j * np.pi * np.arange (N) / N) return np.concatenate ([X_even + factor [: N / 2 ] * X_odd, X_even + factor [N / 2:] * X_odd]) Now we make a quick check to see if the results are correct: x = np.random.random (1024) np.allclose (FFT (x), Air Jordan Outlet np .fft.fft (x)) True then compared with the 'slow approach' run time under:% timeit DFT_slow (x)% timeit FFT (x)% timeit np.fft.fft (x) 10 loops, best of 3: 77.6 ms per loop100 loops, best of 3: 4.07 ms per loop10000 loops, best of 3: 24.7 us per loop algorithm is now an order of magnitude faster than before. Moreover, our recursive algorithm asymptotic to Mens Nike Free 3.0 Wool Skin Shoes Red Grey O (n logn). We realized the FFT. It should be noted that we have not done numpy built FFT algorithm, which is expected. numpy of FFTPACK Fortran algorithms fft behind is implemented, after years of tuning. In addition, our solutions NumPy, while Python stack recursion involved and the distribution of many of the temporary array, which significantly increases the computation time. When would also like to speed up the case, a good method is to use Python / NumPy work, as far as possible double counting to quantify. We can do to eliminate the recursive calculation process, so that our python FFT more efficient. Note that the above NumPy to quantify recursive FFT implementation, the lowest level of recursion, we made N / 32 times of matrix-vector multiplication. Our algorithm will benefit from these into the matrix-vector multiplication Matrix-time calculation - matrix product. In each layer of recursion, repetitive calculations can be vectorized. Because NumPy very good at this kind of operation, we can use this to achieve vectorized FFT. def FFT_vectorized (x): \u0026 quot; \u0026 quot; \u0026 quot; A vectorized, non-recursive version of the Cooley-Tukey FFT \u0026 quot; \u0026 quot; \u0026 quot; x = np.asarray (x, dtype = float) N = x.shape [0] if np.log2 (N)% 1 \u0026 gt; 0: Nike Lunareclipse raise ValueError (\u0026 quot; size of x must be a power of 2 \u0026 quot;) # N_min here is equivalent to the stopping condition above, # and should be a power of 2 N_min = min (N, 32) # Perform an O [N ^ 2] DFT on all length-N_min sub-problems at once n = np.arange (N_min) k = n [:, None] M = np.exp (-2j * np.pi * n * k / N_min) X = np.dot (M, x.reshape ((N_min, -1))) # build-up each level of the recursive calculation all at once while X.shape [0 ] \u0026 lt; N: X_even = X [:,: X.shape [1] / 2] X_odd = X [:, X.shape [1] / 2:] factor = np.exp (-1j * np.pi * np.arange (X.shape [0]) / X.shape [0]) [:, None] X = np.vstack ([X_even + factor * X_odd, X_even - factor * X_odd]) return X.ravel () x = np.random.random (1024) np.allclose (FFT_vectorized (x), np.fft.fft (x)) True because our more dramatically enhance the efficiency of Air Max 2011 Womens Blue Mens Nike Free 3.0 Wool Skin Shoes Red Grey Black the algorithm, so to be a bigger test (not including DFT_slow) x = np.random.random (1024 * 16)% timeit FFT (x)% timeit FFT_vectorized (x)% timeit 554988 316 Nike Zoom KD V Jade Pink White Sale np.fft.fft (x) 10 loops, best of 3: 72.8 ms per loop100 loops, best of 3: 4.11 ms per loop1000 loops, best of 3: 505 us per loop has raised us to achieve a level. Here we are PFBASE FFTPACK within approximately 10, with only a few dozen lines Python + NumPy code. Although there is no corresponding calculations to prove, Python version is far superior to FFTPACK source, and that you can browse from here. So FFTPACK is how to get this last point accelerate it? Maybe it's just a detailed logbook, FFTPACK spent a lot of time Nike Lunarglide+ 2 to ensure that any sub-calculation can be reused. We here numpy version involves additional memory allocation and copying, for some low-level language such as Fortran can easily control and minimize memory usage. And the Cooley-Tukey algorithm also can make it into more than two portions (as we used here radix-2 Cooley-Tukey FFT algorithm), and Mens Nike Free Run 3 Shoes Blue 3 other more advanced FFT algorithm may also be able to be applied, including those based on convolution fundamentally different approaches (eg Bluestein algorithm and Rader's algorithm). Ideas and methods of combining these extension, you can make the array size does not satisfy even a power of 2, FFT can be quickly implemented. I hope they provide a lot of Mens Nike Free 3.0 Wool Skin Shoes Grey Yellow background on how the FFT data analysis carried out, although pure Python functions do not apply in practice. As a data scientist, we can defer the introduction of the basic application tools can contain black box, it is by our colleagues in the Senate more algorithms thought into the build, but I firmly believe: when the underlying algorithms are applied to the data we have a better We understand, we will also become better practitioners.understood Fast Fourier Transform (FFT) algorithm