FFT unexpected frequency shift after window function application

Question

I got this python code for FFT calculation of a sound signal:

from math import *
from cmath import exp, pi

def fft(x):
    N = len(x)
    if N <= 1: return x
    even = fft(x[0::2])
    odd =  fft(x[1::2])
    return ([even[k] + exp(-2j * pi * k / N) * odd[k] for k in xrange(N / 2)] + 
            [even[k] - exp(-2j * pi * k / N) * odd[k] for k in xrange(N / 2)])

N = 64

res = [sin(k) for k in xrange(N)]

# Window function
a = 2*pi/(N-1)
for k in xrange(N):
    z = a * res[k]
    res[k] = 0.42659 - 0.49656*cos(z) + 0.076849*cos(2*z)


res = fft(res)

for k in xrange(N/2):
    # get the amplitude...
    sqr = sqrt(res[k].real * res[k].real + res[k].imag * res[k].imag)
    if sqr > 0:
        print 20 * log10(sqr)  # ...in decibels
    else:
        print "-INF"

I got the following results:

WITHOUT window function (commented-out):

-20.3017238269
-16.9192604108
-12.5089302395
-8.97999530657
-5.96033201086
-3.12975820108
-0.242090896634
2.97021879504
6.95134203457
12.8752188937
29.5096108632 <-- PEAK
17.1668404562
10.6485650284
7.24321329787
4.98448122464
3.3242079033
2.03154022635
0.987966110459
0.124898554197
-0.600705355004
-1.21748302238
-1.74534177237
-2.1985940834
-2.5878009699
-2.92091067118
-3.20399051424
-3.44171254421
-3.63768393032
-3.79467588076
-3.91478386211
-3.99953964778
-4.04998822971

WITH window function:

-6.55943077129
-65.8567720414
-65.7987645827
-65.7012678903
-65.5625673034
-65.380788761
-65.1529344157
-64.8750852394
-64.5420675211
-64.1470597764
-63.6810080181
-63.131731575
-62.4825087571
-61.7097419947
-60.7788888801
-59.6368610687
-58.1964601495
-56.3001921054
-53.6185951634
-49.2554491173
-38.3322646561 <-- PEAK
-43.3318138698
-52.0838904305
-56.7277347745
-60.2038755771
-62.9772322874
-65.442363488
-67.7550361967
-70.0212827894
-72.3056579688
-74.5822818952
-76.5522909937

The peak appears shifted for some reason. It is a 2x frequency shift! To check the results, I tried this Java applet:
http://www.random-science-tools.com/maths/FFT.htm
And it appears that the results WITHOUT any window function are the correct ones (peak at 1 third of the spectrum). Instead, if I apply the window function in my python script the peak shows at 2/3 of the spectrum.

Is this supposed to happen? How do I fix it?

Answer 1

Ok, In the meanwhile I realized what was wrong. The window function as I wrote it in the question was totally meaningless.

This is the correct one:

a = 2*pi/(N-1)

for k in xrange(N):
    z = a * k
    res[k] *= 0.42659 - 0.49656*cos(z) + 0.076849*cos(2*z) # Blackman

Results:

-63.8888312044
-62.1859660802
-59.4560808775
-57.5235455007
-57.0010514385
-59.4284419437
-66.6535724743
-46.1441434426
-2.31562840406
16.0873761957
22.4136439765 <-- PEAK
19.5784749467
6.43274013629
-28.3842042716
-55.5273291654
-68.8982705127
-53.3843989911
-49.731974213
-48.3131204305
-47.6953570892
-47.4386151256
-47.361972079
-47.3787962267
-47.4434419084
-47.530228024
-47.6240076874
-47.7155325706
-47.799012933
-47.870764286
-47.9284264139
-47.9705003855
-47.9960714351

The peak is now exactly where it is supposed to be.

---

Some other windows you may want to try:

res[k] *= 0.355768 - 0.487396*cos(z) + 0.144232*cos(2*z) - 0.012604*cos(3*z)
res[k] *= 1 - 1.93*cos(z) + 1.29*cos(2*z) - 0.388*cos(3*z) + 0.028*cos(4*z)
res[k] *= 1 - 1.985844164102*cos(z) + 1.791176438506*cos(2*z) - 1.282075284005*cos(3*z) + 0.667777530266*cos(4*z) - 0.240160796576*cos(5*z) + 0.056656381764*cos(6*z) - 0.008134974479*cos(7*z) + 0.000624544650*cos(8*z) - 0.000019808998*cos(9*z) + 0.000000132974*cos(10*z)

In order: Nuttall, FTSRS, HFT248D.

https://holometer.fnal.gov/GH_FFT.pdf