I am trying to find a value from a given mathematical function whose input is supposed to be derived from FFT components. Essentially, I follow the procedure from [1] which assumes that a given sine wave (regular wave) is composed of a cosine and sine wave of different amplitude but the same frequency.

How can I find the amplitudes of each decomposed wave if the FFT shows just one peak at the wave frequency?

The signal is a regular sine wave superimposed with 2 waves traveling in the opposite direction. We have two singals (i.e. two measurment devices) whose components are required for the mathemtical equation. So,

Signal1 = Wave1 + Wave2 = a1cos(kx-ft+e1) + a2cos(kx+ft+e2) = A1cos(ft) + B1sin(ft)

Signal2 = Wave1 + Wave2 = a1cos(kx-ft+e1) + a2cos(kx+ft+e2) = A2cos(ft) + B2sin(ft) where,

  • a = ampltiude
  • k = wavenumber
  • f = frequency
  • e = phase
  • t = time

The paper states A1,A2 ,B1 and A2 can be estimated through Fourier Analysis to be plugged into a equation.

[1] Yoshimi Goda, Yasumasa Suzuki, "Estimation of Incident and Reflected Waves in Random Wave Experiments", Coastal Engineering 1976 -1977

  • 1
    $\begingroup$ Hint: $e^{jx}=?$ You know a formula for this! $\endgroup$
    – mmmm
    Jul 27 at 21:41
  • $\begingroup$ Sorry, I'm a completely new to this. A direct answer would be most helpful. $\endgroup$
    – user244717
    Jul 27 at 21:52
  • $\begingroup$ Euler's equation ? $\endgroup$
    – user244717
    Jul 27 at 22:08
  • $\begingroup$ Unfortunately, the cited article is not accessible (behind a paywall). Is it really true that it describes a "sine wave [...] [to be] composed of a cosine and a sine wave"? Or is it a complex wave that's composed of a cosine and a sine? $\endgroup$
    – applesoup
    Jul 28 at 22:37
  • $\begingroup$ @applesoup Essentially, a given signal which is a regular sin/cos wave is composed of two superposed waves: Wave = Wave1 + Wave2 = a1*cos(x-ft+e1) + a2*cos(x+ft+e2) = Acos(ft) + Bsin(ft). The paper states that A and B can be derived from FFT to be plugged in another equation. $\endgroup$
    – user244717
    Jul 29 at 8:42

The generalized form $Ae^{j\theta}$ with $A$ and $\theta$ as real numbers is a phasor with magnitude $K$ and angle $\theta$. A "positive" frequency $\Omega_o$ is such a phasor with constant magnitude $A$ and rotating counter-clockwise with angle that increases linearly with time as $\Omega_o t$. (And similarly a negative frequency would be rotating clock-wise). The real and imaginary components of such a single frequency tone would be given as $\cos(\Omega_o t)$ and $\sin(\Omega_o t)$ respectively which is Euler's formula. Therefore any such "single tone" $Ae^{j\Omega_o t}$ would have cosine and sine components. The Discrete Fourier Transform is composed of such "single tones" with each bin represented in the time domain as $Ae^{j k \omega_o n}$, where $\omega_o = 2\pi/N$.

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  • $\begingroup$ Okay, I'm beginning to understand. So in my case, The amplitude signal from my FFT is single tone and I can split it into the sine and cosine wave components as 'A(cos(Ωt) + j sin(Ωt) )'? $\endgroup$
    – user244717
    Jul 28 at 10:11
  • $\begingroup$ Yes - in Signal Processing I find it easier to conceptualize a single frequency as a single phasor rotating on the complex plane rather than sines and cosines which each phaser consists of. A single sinusoidal waveform has two such rotating phasors so in my abstraction consists of two single frequencies: a positive frequency and a negative frequency. The is given by $2\cos(x)=e^{jx}+e^{-jx}$ $\endgroup$ Jul 28 at 12:01
  • $\begingroup$ In that case, wouldn't the amplitudes of these two phasor components be equal, since the only difference is the phase? Also going back to the original question, the goal is to derive Ycos(xt) +Zsin(xt) from a sine wave, from what I understood Y= A (FFT signal) and B = Ai ? $\endgroup$
    – user244717
    Jul 28 at 15:53
  • $\begingroup$ In which case? If you are referring to a cosine then yes they have equal amplitude and opposite phase. To derive Ycos(xt) + Zsin(xt) from Asin(xt+ phi) I assume you mean which is really nothing specific to an FFT but a simple trig identitiy, as listed here: en.wikipedia.org/wiki/…. $\endgroup$ Jul 28 at 22:28
  • $\begingroup$ maybe if I show the equation like how I did for the other commenter. The signal is a wave which can be , according to the literature, defined as : Wave = Wave1 + Wave2 = a1*cos(x-ft+e1) + a2*cos(x+ft+e2) = Acos(ft) + Bsin(ft). I have the Wave signal and need to find A and B. From your answer, I assumed A = amp(Wave) and B = j * amp(Wave). $\endgroup$
    – user244717
    Jul 29 at 8:41

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