FFT

def dft(x):
    """
    离散傅立叶变换
    https://zh.wikipedia.org/wiki/%E7%A6%BB%E6%95%A3%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2
    https://zh.wikipedia.org/wiki/%E5%BF%AB%E9%80%9F%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2
    https://pythonnumericalmethods.berkeley.edu/notebooks/chapter24.02-Discrete-Fourier-Transform.html
    """
    N = len(x)
    # print("dft data size: ", N)
    # 循环方式
    y = np.zeros(N, dtype="complex_")
    for k in range(N):
        t0 = x * np.exp(-2j * np.pi * k * np.arange(N)/N)
        # t1 = x * np.power(-1, -2 * k * np.arange(n)/n)  e^(i * pi) = -1 的替换，没有虚数
        y[k] = np.sum(t0)

    # 矩阵方式 虽然优雅，内存消耗极大O(n*n)
    # n = np.arange(N)
    # k = n.reshape((N, 1))
    # e = np.exp(-2j * np.pi * k * n / N)
    # y = e @ x
    return y

def fft(x):
	"""
	https://pythonnumericalmethods.berkeley.edu/notebooks/chapter24.03-Fast-Fourier-Transform.html
	https://jakevdp.github.io/blog/2013/08/28/understanding-the-fft/
	
	A recursive implementation of 
	the 1D Cooley-Tukey FFT, the 
	input should have a length of 
	power of 2. 
	"""
	N = len(x)
	if N == 1:
		return x
	else:
		X_even = fft(x[::2])
		X_odd = fft(x[1::2])
		factor = \
			np.exp(-2j*np.pi*np.arange(N) / N)

		X = np.concatenate(
			[X_even+factor[:int(N/2)]*X_odd,
				X_even+factor[int(N/2):]*X_odd])
		return X

# 
# 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
		return transform_bluestein(vec, inverse)


# 
# 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)]
	vec = [vec[reverse_bits(i, levels)] for i in range(n)]  # Copy with bit-reversed permutation
	
	# 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 discrete Fourier transform (DFT) of the given complex vector, returning the result as a new vector.
# The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
# Uses Bluestein's chirp z-transform algorithm.
# 
def transform_bluestein(vec, inverse):
	# Find a power-of-2 convolution length m such that m >= n * 2 + 1
	n = len(vec)
	if n == 0:
		return []
	m = 2**((n * 2).bit_length())
	
	coef = (1 if inverse else -1) * cmath.pi / n
	exptable = [cmath.rect(1, (i * i % (n * 2)) * coef) for i in range(n)]  # Trigonometric table
	avec = [(x * y) for (x, y) in zip(vec, exptable)] + [0] * (m - n)  # Temporary vectors and preprocessing
	bvec = exptable[ : n] + [0] * (m - (n * 2 - 1)) + exptable[ : 0 : -1]
	bvec = [x.conjugate() for x in bvec]
	cvec = convolve(avec, bvec, False)[ : n]  # Convolution
	return [(x * y) for (x, y) in zip(cvec, exptable)]  # Postprocessing


# 
# 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]

## 以上所有完整代码见fft.py