# FFT and convolution test (Python)
-#
+#
# Copyright (c) 2020 Project Nayuki. (MIT License)
# https://www.nayuki.io/page/free-small-fft-in-multiple-languages
-#
+#
# Permission is hereby granted, free of charge, to any person obtaining a copy
-# of this software and associated documentation files (the "Software"), to deal
+# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
-# copies of the Software, and to permit persons to whom the Software is
+# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# - The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
# liability, whether in an action of contract, tort or otherwise, arising
# from, out of or in connection with the Software or the use or other
# dealings in the Software.
-#
+#
import cmath, math, random
-#
+#
# Computes the discrete Fourier transform (DFT) or inverse transform of the
# given complex vector, returning the result as a new vector.
# The vector can have any length. This is a wrapper function. The inverse
# transform does not perform scaling, so it is not a true inverse.
#
-def transform(vec, inverse):
- n = len(vec)
- if n == 0:
- return []
- elif n & (n - 1) == 0: # Is power of 2
- return transform_radix2(vec, inverse)
- else: # More complicated algorithm for arbitrary sizes
- assert False
+def transform(vec, inverse, generators=False):
+ n = len(vec)
+ if n == 0:
+ return []
+ elif n & (n - 1) == 0: # Is power of 2
+ return transform_radix2(vec, inverse, generators)
+ else: # More complicated algorithm for arbitrary sizes
+ assert False
-#
+#
# Computes the discrete Fourier transform (DFT) of the given complex vector,
# returning the result as a new vector.
# The vector's length must be a power of 2. Uses the Cooley-Tukey
# decimation-in-time radix-2 algorithm.
-#
-def transform_radix2(vec, inverse):
- # Returns the integer whose value is the reverse of the lowest 'width'
+#
+def transform_radix2(vec, inverse, generators_mode):
+ # Returns the integer whose value is the reverse of the lowest 'width'
# bits of the integer 'val'.
- def reverse_bits(val, width):
- result = 0
- for _ in range(width):
- result = (result << 1) | (val & 1)
- val >>= 1
- return result
-
- # Initialization
- n = len(vec)
- levels = n.bit_length() - 1
- if 2**levels != n:
- raise ValueError("Length is not a power of 2")
- # Now, levels = log2(n)
- coef = (2 if inverse else -2) * cmath.pi / n
- exptable = [cmath.rect(1, i * coef) for i in range(n // 2)]
+ def reverse_bits(val, width):
+ result = 0
+ for _ in range(width):
+ result = (result << 1) | (val & 1)
+ val >>= 1
+ return result
+
+ # Initialization
+ n = len(vec)
+ levels = n.bit_length() - 1
+ if 2**levels != n:
+ raise ValueError("Length is not a power of 2")
+ # Now, levels = log2(n)
+ coef = (2 if inverse else -2) * cmath.pi / n
+ exptable = [cmath.rect(1, i * coef) for i in range(n // 2)]
# Copy with bit-reversed permutation
- vec = [vec[reverse_bits(i, levels)] for i in range(n)]
-
- # Radix-2 decimation-in-time FFT
- size = 2
- while size <= n:
- halfsize = size // 2
- tablestep = n // size
- for i in range(0, n, size):
- k = 0
- for j in range(i, i + halfsize):
- temp = vec[j + halfsize] * exptable[k]
- vec[j + halfsize] = vec[j] - temp
- vec[j] += temp
- k += tablestep
- size *= 2
- return vec
-
-
-#
+ vec = [vec[reverse_bits(i, levels)] for i in range(n)]
+
+ # Radix-2 decimation-in-time FFT
+ size = 2
+ while size <= n:
+ halfsize = size // 2
+ tablestep = n // size
+ for i in range(0, n, size):
+ k = 0
+ for j in range(i, i + halfsize):
+ temp = vec[j + halfsize] * exptable[k]
+ vec[j + halfsize] = vec[j] - temp
+ vec[j] += temp
+ k += tablestep
+ size *= 2
+ return vec
+
+
+#
# Computes the circular convolution of the given real or complex vectors,
# returning the result as a new vector. Each vector's length must be the same.
# realoutput=True: Extract the real part of the convolution, so that the
# output is a list of floats. This is useful if both inputs are real.
# realoutput=False: The output is always a list of complex numbers
# (even if both inputs are real).
-#
+#
def convolve(xvec, yvec, realoutput=True):
- assert len(xvec) == len(yvec)
- n = len(xvec)
- xvec = transform(xvec, False)
- yvec = transform(yvec, False)
- for i in range(n):
- xvec[i] *= yvec[i]
- xvec = transform(xvec, True)
-
- # Scaling (because this FFT implementation omits it) and postprocessing
- if realoutput:
- return [(val.real / n) for val in xvec]
- else:
- return [(val / n) for val in xvec]
+ assert len(xvec) == len(yvec)
+ n = len(xvec)
+ xvec = transform(xvec, False)
+ yvec = transform(yvec, False)
+ for i in range(n):
+ xvec[i] *= yvec[i]
+ xvec = transform(xvec, True)
+
+ # Scaling (because this FFT implementation omits it) and postprocessing
+ if realoutput:
+ return [(val.real / n) for val in xvec]
+ else:
+ return [(val / n) for val in xvec]
# ---- Main and test functions ----
def main():
- global _maxlogerr
-
- # Test power-of-2 size FFTs
- for i in range(0, 12 + 1):
- _test_fft(1 << i)
-
- # Test power-of-2 size convolutions
- for i in range(0, 12 + 1):
- _test_convolution(1 << i)
-
- print()
- print(f"Max log err = {_maxlogerr:.1f}")
- print(f"Test {'passed' if _maxlogerr < -10 else 'failed'}")
+ global _maxlogerr
+
+ # Test power-of-2 size FFTs
+ for i in range(0, 12 + 1):
+ _test_fft(1 << i)
+
+ # Test power-of-2 size convolutions
+ for i in range(0, 12 + 1):
+ _test_convolution(1 << i)
+
+ print()
+ print(f"Max log err = {_maxlogerr:.1f}")
+ print(f"Test {'passed' if _maxlogerr < -10 else 'failed'}")
def _test_fft(size):
- input = _random_vector(size)
- expect = _naive_dft(input, False)
- actual = transform(input, False)
- err = _log10_rms_err(expect, actual)
-
- actual = [(x / size) for x in expect]
- actual = transform(actual, True)
- err = max(_log10_rms_err(input, actual), err)
- print(f"fftsize={size:4d} logerr={err:5.1f}")
+ input = _random_vector(size)
+ expect = _naive_dft(input, False)
+ actual = transform(input, False)
+ err = _log10_rms_err(expect, actual)
+
+ actual = [(x / size) for x in expect]
+ actual = transform(actual, True)
+ err = max(_log10_rms_err(input, actual), err)
+ print(f"fftsize={size:4d} logerr={err:5.1f}")
def _test_convolution(size):
- input0 = _random_vector(size)
- input1 = _random_vector(size)
- expect = _naive_convolution(input0, input1)
- actual = convolve(input0, input1, False)
- print(f"convsize={size:4d} logerr={_log10_rms_err(expect, actual):5.1f}")
+ input0 = _random_vector(size)
+ input1 = _random_vector(size)
+ expect = _naive_convolution(input0, input1)
+ actual = convolve(input0, input1, False)
+ print(f"convsize={size:4d} logerr={_log10_rms_err(expect, actual):5.1f}")
# ---- Naive reference computation functions ----
def _naive_dft(input, inverse):
- n = len(input)
- output = []
- if n == 0:
- return output
- coef = (2 if inverse else -2) * math.pi / n
- for k in range(n): # For each output element
- s = 0
- for t in range(n): # For each input element
- s += input[t] * cmath.rect(1, (t * k % n) * coef)
- output.append(s)
- return output
+ n = len(input)
+ output = []
+ if n == 0:
+ return output
+ coef = (2 if inverse else -2) * math.pi / n
+ for k in range(n): # For each output element
+ s = 0
+ for t in range(n): # For each input element
+ s += input[t] * cmath.rect(1, (t * k % n) * coef)
+ output.append(s)
+ return output
def _naive_convolution(xvec, yvec):
- assert len(xvec) == len(yvec)
- n = len(xvec)
- result = [0] * n
- for i in range(n):
- for j in range(n):
- result[(i + j) % n] += xvec[i] * yvec[j]
- return result
+ assert len(xvec) == len(yvec)
+ n = len(xvec)
+ result = [0] * n
+ for i in range(n):
+ for j in range(n):
+ result[(i + j) % n] += xvec[i] * yvec[j]
+ return result
# ---- Utility functions ----
_maxlogerr = -math.inf
def _log10_rms_err(xvec, yvec):
- global _maxlogerr
- assert len(xvec) == len(yvec)
- err = 10.0**(-99 * 2)
- for (x, y) in zip(xvec, yvec):
- err += abs(x - y) ** 2
- err = math.sqrt(err / max(len(xvec), 1)) # a root mean square (RMS) error
- err = math.log10(err)
- _maxlogerr = max(err, _maxlogerr)
- return err
+ global _maxlogerr
+ assert len(xvec) == len(yvec)
+ err = 10.0**(-99 * 2)
+ for (x, y) in zip(xvec, yvec):
+ err += abs(x - y) ** 2
+ err = math.sqrt(err / max(len(xvec), 1)) # a root mean square (RMS) error
+ err = math.log10(err)
+ _maxlogerr = max(err, _maxlogerr)
+ return err
def _random_vector(n):
- return [complex(random.uniform(-1.0, 1.0),
+ return [complex(random.uniform(-1.0, 1.0),
random.uniform(-1.0, 1.0)) for _ in range(n)]
if __name__ == "__main__":
- main()
+ main()
projects / libreriscv.git / commitdiff