Question

I’m reading how the cooley tukey method works, but I have a few problems with the following python script:

def fft_CT_twiddles(x, inverse = False, verbose = False, twiddles = None) :
    """
    Computes the DFT of x using Cooley-Tukey's FFT algorithm. Twiddle factors
    are precalculated in the first function call, then passed down recursively.
    """
    t = time.clock()
    N = len(x)
    inv = -1 if not inverse else 1
    if N % 2 :
        return dft(x, inverse, False)
    M = N // 2
    if not twiddles :
        twiddles = [math.e**(inv*2j*math.pi*k/N) for k in xrange(M)]+[N]
    x_e = x[::2]
    x_o  = x[1::2]
    X_e = fft_CT_twiddles(x_e, inverse, False, twiddles)
    X_o  = fft_CT_twiddles(x_o, inverse, False, twiddles)
    X = []
    for k in range(M) :
        X += [X_e[k] + X_o[k] * twiddles[k * twiddles[-1] // N]]
    for k in range(M,N) :
        X += [X_e[k-M] - X_o[k-M] * twiddles[(k - M) * twiddles[-1] // N]]
    if inverse :
        X = [j/2 for j in X]
    t = time.clock() - t
    if verbose :
        print "Computed","an inverse" if inverse else "a","CT FFT of size",N,
        print "in",t,"sec."
    return X

What does the
twiddles = [math.e**(inv*2j*math.pi*k/N) for k in xrange(M)]+[N]
line does? Seems like an array but why the +[N]?

And why accessing the twiddles[-1] value then?

I can’t figure this out

Answer 1

Trying to explain code when the level of expertise of the person asking the question is unknown is difficult. That said here it goes:

1. python has a concatenation operator for sequences nl. + so the twiddles = line creates a sequence of some sort and appends the number N to it.
2. twiddles[-1] accesses the last element in the sequence here the number N as the comments suggest.
3. the twiddles sequence expression uses complex numbers to generate a sequence consisting of the N points on the unit circle by dividing it in N equal slices.