# Informations about Fourier Transform for a Python project (sound manipulation)

I have a project in Python with a friend where we want to manipulate music sound, and we'll need Fourier Transforms, so I made research online and wanted to know if I understand correctly the concepts.
Here is what I think I understand so far:

(the module numpy is noted np)

For a temporal signal x(t) made to an array x, sampled at a constant interval dt with N samples, sampled at a sampling rate sr:

• the Fourier transform decompose a signal x(t) in an infinite sum: $$x(t)= A_{1}e^{iw_1t} + A_{2}e^{iw_2t}+A_{3}e^{iw_3t}+...+ A_{n}e^{iw_nt} \\$$ where $$A_n = r_ne^{ip_n}$$, and $$w_n$$ is a frequency
• amplitude at the given frequency $$w_n$$: $$r_n = np.abs(A_n) = \sqrt{real(A_n)^2 + complex(A_n)^2}$$
• phase at the given frequency $$w_n$$: $$p_n = np.angle(A)$$
• np.fft.fftfreq(N, 1/sr): returns all the frequencies $$w$$ present in the wave as an array: $$[w_1, w_2, w_3, ..., w_n]$$
• np.fft.fft(array x): returns a 2D array with complex values which correspond to $$[A_1, A_2, ..., A_n]$$
• period of the signal: $$T = dt \times N$$
• fundamental frequency (Hz): $$df = 1/T$$
• fundamental frequency (rad/sec): $$dw = 2π/T$$
• fundamental frequency (adimensional): $$f = np.fft.fftfreq(N)\times N \times df$$ ?
• fundamental frequency (adimensional): $$w = np.fft.fftfreq(N) \times N \times dw$$ ?
• duration = number of frames (length of my array) $$\div$$ sample rate ?

• In order to plot/make a spectrogram, I need to have the duration of my wave, the frequencies ($$w_1, w_2, ..., w_n$$) in it, and the amplitudes for each frequencies ($$A_1, A_2, ..., A_n$$). For each value of t of my wave duration, I have the frequencies. I then calculate the amplitude of all the frequencies since its a function of time (e.g: $$A_1e^{iw_1t}$$) ?