CODEWITHSUNDEEP
pythonsignal-processingfftnon-linear-regressiondft
Alejandro
Alejandro

Reputation: 949

Strange result from Fast Fourier transform signal reconstruction

I have some data which is shown in the below figure and am interested in finding some of its Fourier series coefficients.

r = np.array([119.80601628, 119.84629291, 119.85290735, 119.45778804,
   115.64497439, 105.58519852, 100.72765819, 100.04327702,
   100.08590518, 100.35824977, 101.58424993, 105.47976376,
   112.27556007, 117.07679226, 118.99998888, 119.60458086,
   119.78624424, 119.83022022, 119.36116943, 115.72323767,
   106.58946834, 101.19479124, 100.11537349, 100.13313755,
   100.41846106, 101.42255377, 104.33650237, 109.73625492,
   115.14763728, 118.24665037, 119.35359999, 119.68061835])

z = np.array([-411.42980545, -384.98596279, -358.13032372, -330.89578468,
   -303.39129113, -275.76248957, -248.24478443, -221.07069838,
   -194.33260984, -168.05271807, -142.19357982, -116.62090103,
    -91.15354178,  -65.56745626,  -39.65284757,  -13.29632162,
     13.54374939,   40.84929432,   68.50496394,   96.33720787,
    124.08525182,  151.36802193,  177.98791952,  204.0805317 ,
    229.85399128,  255.44727674,  281.02166554,  306.75399703,
    332.74638285,  359.05528646,  385.74336711,  412.8189858 ])

plt.plot(z, r, label='data')
plt.legend()

enter image description here Then I calculate the average sampling period, since it is not constant as seen in the Z variable:

l = []
for i in range(32-1):
    l.append(z[i]-z[i+1])
Ts = np.mean(l)

Then I calculate the fft: from scipy.fftpack import fft

rf = scipy.fftpack.fft(r)

For reconstruction of the signal then:

fs = 1/Ts

amp = np.abs(rf)/r.shape[0]
n = r.shape[0]
s = 0
for i in range(n//2):

    phi = np.angle(rf[i], deg=False)
    a   = amp[i]
    k   = i*fs/n

    s += a*np.cos(2*np.pi*k  *(z)  +phi)

plt.plot(z, s, label='fft result')
plt.plot(z, r, label='data')
plt.legend()

enter image description here

The result is strange however both in terms of amplitude and frequency.

Upvotes: 0

Views: 883

Answers (1)

Heinz M.
Heinz M.

Reputation: 684

The complex spectrum is a symmetric spectrum with the range of (-fMax/2, ..., +fMax/2). You only used the right hand positive part of the spectrum. This means, your reconstructed signal contains only half of the spectrums frequencies. Because the spectrum is symmetric, all you have to do is to double the calculated absolute values. However, there is an important exception. The DC value amplitude[0] must not be doubled.

Upvotes: 2

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