# scipy/scipy

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 from __future__ import division, print_function, absolute_import from numpy import arange from numpy.fft.helper import fftshift, ifftshift, fftfreq from bisect import bisect_left __all__ = ['fftshift', 'ifftshift', 'fftfreq', 'rfftfreq', 'next_fast_len'] def rfftfreq(n, d=1.0): """DFT sample frequencies (for usage with rfft, irfft). The returned float array contains the frequency bins in cycles/unit (with zero at the start) given a window length `n` and a sample spacing `d`:: f = [0,1,1,2,2,...,n/2-1,n/2-1,n/2]/(d*n) if n is even f = [0,1,1,2,2,...,n/2-1,n/2-1,n/2,n/2]/(d*n) if n is odd Parameters ---------- n : int Window length. d : scalar, optional Sample spacing. Default is 1. Returns ------- out : ndarray The array of length `n`, containing the sample frequencies. Examples -------- >>> from scipy import fftpack >>> sig = np.array([-2, 8, 6, 4, 1, 0, 3, 5], dtype=float) >>> sig_fft = fftpack.rfft(sig) >>> n = sig_fft.size >>> timestep = 0.1 >>> freq = fftpack.rfftfreq(n, d=timestep) >>> freq array([ 0. , 1.25, 1.25, 2.5 , 2.5 , 3.75, 3.75, 5. ]) """ if not isinstance(n, int) or n < 0: raise ValueError("n = %s is not valid. " "n must be a nonnegative integer." % n) return (arange(1, n + 1, dtype=int) // 2) / float(n * d) def next_fast_len(target): """ Find the next fast size of input data to `fft`, for zero-padding, etc. SciPy's FFTPACK has efficient functions for radix {2, 3, 4, 5}, so this returns the next composite of the prime factors 2, 3, and 5 which is greater than or equal to `target`. (These are also known as 5-smooth numbers, regular numbers, or Hamming numbers.) Parameters ---------- target : int Length to start searching from. Must be a positive integer. Returns ------- out : int The first 5-smooth number greater than or equal to `target`. Notes ----- .. versionadded:: 0.18.0 Examples -------- On a particular machine, an FFT of prime length takes 133 ms: >>> from scipy import fftpack >>> min_len = 10007 # prime length is worst case for speed >>> a = np.random.randn(min_len) >>> b = fftpack.fft(a) Zero-padding to the next 5-smooth length reduces computation time to 211 us, a speedup of 630 times: >>> fftpack.helper.next_fast_len(min_len) 10125 >>> b = fftpack.fft(a, 10125) Rounding up to the next power of 2 is not optimal, taking 367 us to compute, 1.7 times as long as the 5-smooth size: >>> b = fftpack.fft(a, 16384) """ hams = (8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405, 432, 450, 480, 486, 500, 512, 540, 576, 600, 625, 640, 648, 675, 720, 729, 750, 768, 800, 810, 864, 900, 960, 972, 1000, 1024, 1080, 1125, 1152, 1200, 1215, 1250, 1280, 1296, 1350, 1440, 1458, 1500, 1536, 1600, 1620, 1728, 1800, 1875, 1920, 1944, 2000, 2025, 2048, 2160, 2187, 2250, 2304, 2400, 2430, 2500, 2560, 2592, 2700, 2880, 2916, 3000, 3072, 3125, 3200, 3240, 3375, 3456, 3600, 3645, 3750, 3840, 3888, 4000, 4050, 4096, 4320, 4374, 4500, 4608, 4800, 4860, 5000, 5120, 5184, 5400, 5625, 5760, 5832, 6000, 6075, 6144, 6250, 6400, 6480, 6561, 6750, 6912, 7200, 7290, 7500, 7680, 7776, 8000, 8100, 8192, 8640, 8748, 9000, 9216, 9375, 9600, 9720, 10000) if target <= 6: return target # Quickly check if it's already a power of 2 if not (target & (target-1)): return target # Get result quickly for small sizes, since FFT itself is similarly fast. if target <= hams[-1]: return hams[bisect_left(hams, target)] match = float('inf') # Anything found will be smaller p5 = 1 while p5 < target: p35 = p5 while p35 < target: # Ceiling integer division, avoiding conversion to float # (quotient = ceil(target / p35)) quotient = -(-target // p35) # Quickly find next power of 2 >= quotient p2 = 2**((quotient - 1).bit_length()) N = p2 * p35 if N == target: return N elif N < match: match = N p35 *= 3 if p35 == target: return p35 if p35 < match: match = p35 p5 *= 5 if p5 == target: return p5 if p5 < match: match = p5 return match