--- /dev/null
+# 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
+# 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
+# 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.
+# - The Software is provided "as is", without warranty of any kind, express or
+# implied, including but not limited to the warranties of merchantability,
+# fitness for a particular purpose and noninfringement. In no event shall the
+# authors or copyright holders be liable for any claim, damages or other
+# 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
+
+
+#
+# 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'
+ # 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)]
+ # 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
+
+
+#
+# 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]
+
+# ---- 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'}")
+
+
+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}")
+
+
+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}")
+
+
+# ---- 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
+
+
+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
+
+
+# ---- 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
+
+
+def _random_vector(n):
+ return [complex(random.uniform(-1.0, 1.0),
+ random.uniform(-1.0, 1.0)) for _ in range(n)]
+
+
+if __name__ == "__main__":
+ main()
projects / libreriscv.git / commitdiff