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If I'm correct, Fourier transform lets me know the magnitude and phase of any given frequency in a signal.

I have a sampled signal and I want to compute its Fourier transform (any frequency) without comparing it to the sine/cosine wave of that frequency. So I could create a function checking the wavelength in a signal (also containing other frequencies) to get magnitude of any frequency.

Is it possible? And why?

(I hope my question is not too confused this time)

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    $\begingroup$ If I read your question literally, meaning you are asking about the Fourier Transform specifically (not discrete Fourier Transform) and you are asking about sine specifically then yes of course as you can compute the FT using $F(\omega)= \int {x(t)e^{-j\omega t}}dt$ But of course $e^{-j\omega t}$ is composed of sine waves (and cosine waves) according to Euler's formula.. Can you detail your question further including what the purpose of this is? That may help get a better answer. $\endgroup$ – Dan Boschen Jun 17 '17 at 9:48
  • $\begingroup$ I've edited the question. $\endgroup$ – Juju17ification Jun 17 '17 at 10:16
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    $\begingroup$ If it is a sampled signal, then use the FFT (Fast Fourier Transform) which has the minimum processing to compute the full discrete Fourier Transform. If you are looking for only specific frequencies, then the Goertzel Algortihm could be faster depending on how many frequencies you need. $\endgroup$ – Dan Boschen Jun 17 '17 at 10:18
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    $\begingroup$ I just showed you that it was $\endgroup$ – Dan Boschen Jun 17 '17 at 12:32
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    $\begingroup$ To derive it but the implementation does not multiply the signal by sine and cosines directly---- in the end of course it is the same result but much simpler--- $\endgroup$ – Dan Boschen Jun 17 '17 at 15:01
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You are confusing the terms "frequency" and what a Fourier Transform lets you know. A frequency does not have a magnitude or phase, only a reciprocal of time value.

You don't get magnitudes decomposed by units of time unless you specify the basis vector set for which you are measuring the magnitude and phase. A Fourier Transform only does that "measurement" or decomposition for basis vectors consisting of sinusoids or the equivalent complex exponential set.

If you want some other basis vector set, it's not a Fourier Transform.

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