# Why does a wave composed of two waves have an FFT with only one peak?

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.

• Hint: $e^{jx}=?$ You know a formula for this!
– mmmm
Jul 27 at 21:41
• Sorry, I'm a completely new to this. A direct answer would be most helpful. Jul 27 at 21:52
• Euler's equation ? Jul 27 at 22:08
• 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? Jul 28 at 22:37
• @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. 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 $$\omega_o = 2\pi/N$$.
• 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 $2\cos(x)=e^{jx}+e^{-jx}$ Jul 28 at 12:01