FFT shows (wierd) sine wave in spectrum

Question

I don't know if this is a programming or math question, but I have put together some short examples of FFT. I load in a 440hz wave and add some sine waves on top, but for some reason, the spectrum has a 'wave' that I don't understand. The spectrum, as I understand it, should have the same |Y(freq)| value for all frequencies.

from pylab import plot, show, xlabel, ylabel, subplot
from scipy import fft, arange
from numpy import linspace, array
# from scipy.io.wavfile import read,write
import scikits.audiolab as audio
import math

def plotSpectru(y,Fs):
    n = len(y) # lungime semnal
    k = arange(n)
    T = n/Fs
    frq = k/T # two sides frequency range
    frq = frq[range(n/2)] # one side frequency range

    Y = fft(y)/n # fft computing and normalization
    Y = Y[range(n/2)]

    plot(frq,abs(Y),'r') # plotting the spectrum
    xlabel('Freq (Hz)')
    ylabel('|Y(freq)|')

Fs = 44100;  # sampling rate

# (data, rate, bits) = audio.wavread('440Hz_44100Hz_16bit_05sec.wav')
(data, rate, bits) = audio.wavread('250Hz_44100Hz_16bit_05sec.wav')

for n in xrange(0,4*120, 4):
    n=n/40.
    data = array([x+math.sin(n*idx) for idx,x in enumerate(data)])

y=data[:]
lungime=len(y)
timp=len(y)/44100.
t=linspace(0,timp,len(y))

subplot(2,1,1)
plot(t,y, color="green")
xlabel('Time')
ylabel('Amplitude')
subplot(2,1,2)
plotSpectru(y,Fs)
show()

Wired sine wave through spectrum

Frank Zalkow · Accepted Answer · 2014-03-31T17:42:45.210

I do not fully understand how you want to calculate your sine wave? Here is a small example how to add a sine wave with numpy.

duration = len(data)/float(rate)
t = np.linspace(0, len(data), rate*duration)
sinewave = np.sin(2*np.pi*440*t)

data += sinewave

EDIT

Sorry, I rered your question and recognized my answer did't matched your question. Even of you really add all positive frequencies (that are analyzed by the fft), you get no uniform |Y(freq)|.

duration = len(data)/float(rate)
t = np.linspace(0, len(data), rate*duration)
allfreqs = np.fft.fftfreq(len(data), 1.0/rate)

for f in allfreqs[:len(allfreqs)/2]:
    data += np.sin(2*np.pi*f*t)

enter image description here

As far as I understand it's because of interference. If you add so many sine waves, it's likely, that some become weaker, some become stronger.

If you assign a random phase to each wave, things look different:

duration = len(data)/float(rate)
t = np.linspace(0, len(data), rate*duration)
allfreqs = np.fft.fftfreq(len(data), 1.0/rate)

for f in allfreqs[:len(allfreqs)/2]:
    data += np.sin(2*np.pi*f*t + np.random.rand()*np.pi)

enter image description here

Yeah, my thought was also something about destructive interference. Although, that can be it. Interference is only possible when the frequencies are identical (right?). Your first example, isn't it showing a non-uniform graph, because you input a square wave, instead of a sine wave? — Mads Y, Apr 01 '14 at 16:53
I didn't gave a square wave explicitly. I summed up all sine waves with the same phase and so they summed up to a quasi square wave. There is also interference between sine waves with different frequencies, like [beating](https://en.wikipedia.org/wiki/Beat_(acoustics)). — Frank Zalkow, Apr 02 '14 at 08:01

FFT shows (wierd) sine wave in spectrum

1 Answers1