Informations about Fourier Transform for a Python project (sound manipulation)
0
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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 ?
I had some questions too:
- 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: $A_1e^{iw_1t}$) ?
- If I want to generate a more complex sound that just one frequency, I need to generate many frequencies which amplitudes vary over time ? I want to be able to generate samples (i.e: those kind of sound : link)
Thank you very much for your attention!
fourier-analysis fourier-transform python fast-fourier-transform music-theory
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
$endgroup$
add a comment |
0
$begingroup$
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 ?
I had some questions too:
- 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: $A_1e^{iw_1t}$) ?
- If I want to generate a more complex sound that just one frequency, I need to generate many frequencies which amplitudes vary over time ? I want to be able to generate samples (i.e: those kind of sound : link)
Thank you very much for your attention!
fourier-analysis fourier-transform python fast-fourier-transform music-theory
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
$endgroup$
add a comment |
0
0
0
$begingroup$
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 ?
I had some questions too:
- 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: $A_1e^{iw_1t}$) ?
- If I want to generate a more complex sound that just one frequency, I need to generate many frequencies which amplitudes vary over time ? I want to be able to generate samples (i.e: those kind of sound : link)
Thank you very much for your attention!
fourier-analysis fourier-transform python fast-fourier-transform music-theory
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
$endgroup$
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 ?
I had some questions too:
- 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: $A_1e^{iw_1t}$) ?
- If I want to generate a more complex sound that just one frequency, I need to generate many frequencies which amplitudes vary over time ? I want to be able to generate samples (i.e: those kind of sound : link)
Thank you very much for your attention!
fourier-analysis fourier-transform python fast-fourier-transform music-theory
fourier-analysis fourier-transform python fast-fourier-transform music-theory
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
asked Dec 6 '18 at 8:01
LutinRoseLutinRose
11
11
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