np.fft.fft到底返回什么？

Question

I'm confused about understanding FFT's and how to apply them in python. From my understanding applying an fft to a 10-pixel 1D array should contain a list of 10 numbers (+2 for the "DC" component): (A_0, P_0, A_1, P_1, A_2, P_2, A_3, P_3, A_4, P_4, A_5, P_5)

A_0 = Amplitude of the average value, "DC" \lambda = \infty component
P_0 = Flat wave, so phase doesn't say anything meaningful
A_1 = Amplitude of the fundamental, \lambda = 1a
P_1 = Phase of the fundamental frequency component
A_2 = Amplitude of the first harmonic \lambda = 2a
P_2 = Phase of the first harmonic frequency component
A_3 = Amplitude of the second harmonic \lambda = 3a
P_3 = Phase of the second harmonic frequency component
A_4 = Amplitude of the third harmonic \lambda = 4a
P_4 = Phase of the third harmonic frequency component
A_5 = Amplitude of the Nyquist frequency
P_5 = Phase of the Nyquist

where a is the period. The Nyquist frequency is the highest frequency component that is present in the 1D array. The array below is of length 10 therefore 10 pixels. The fastest frequency possible is one in which the max value is in pixel i and the min value is in pixel i+1 . In other words the Nyquist frequency ( f_{Nyq} = pixel_{total}/2 ) for 10 pixels is 5.

Looking at the code below, when I use np.fft.fft(array) I get 10 numbers. What are these 10 numbers?

one_d_pixel_vals =  np.array([0.3, 0.4, 1.3, 3.4, 4.5, 4.2, 2.8, 2.4, 1.4, 0.1])
print(one_d_pixel_vals)

plt.plot(one_d_pixel_vals)
plt.yticks(np.arange(5))
plt.xticks(np.arange(9))
plt.grid(True)
plt.show()

one_d_fft = np.fft.fft(one_d_pixel_vals, norm='ortho')
print(one_d_fft)
print(len(one_d_fft))

>>> [ 6.57753753 +0.00000000e+00j -3.27588825 -6.42423464e-01j
0.01065311 +6.25512087e-01j  0.22429031 -4.34214340e-01j
0.25814049 -7.06019011e-02j -0.06324555 -1.12346671e-15j
0.25814049 +7.06019011e-02j  0.22429031 +4.34214340e-01j
0.01065311 -6.25512087e-01j -3.27588825 +6.42423464e-01j]
10

Answer 1

No, your interpretation is erroneous.

• a[0] is the mean.
• a[1] = np.conjugate(a[-1]) : represent the fundamental a[1]*exp(jwt)+a[-1]*exp(-jwt).
• a[k],a[-k] represent the k-th harmonic.

so the Fourier representation is sum(a[k] exp (kwt)) , k = -n//2 .. n//2 .