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I am just learning about spectrums and I am a little confused. I have given a simple example and where I get stuck.

Plot the spectrum of $x(t) = 5\cos\left(800\pi t + \frac{3 \pi}{7}\right)$.

So I use reverse-Euler's to write this as $$x(t) = \frac{5 e^{i \frac{3 \pi}{7}}}{2} e^{i 800\pi t} + \frac{5 e^{-i \frac{3 \pi}{7}}}{2} e^{-i 800\pi t}$$

  • What is the spectral amplitude in each case? I am unclear if the frequency of $800 \pi t$ has amplitude $5$ or $\frac{5 e^{-i \frac{3 \pi}{7}}}{2}$.

  • If it is the second do I make two seperate plots for the complex and real parts, or do I just plot $\bigg\lvert \frac{5 e^{-i \frac{3 \pi}{7}}}{2}\bigg\rvert $?

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Given your signal $x(t) = 5\cos(800\pi t + \frac{3 \pi}{7})$. And considering your correct decomposition of it into complex exponentials:

$$x(t) = \frac{5 e^{i \frac{3 \pi}{7}}}{2} e^{i 800\pi t} + \frac{5 e^{-i \frac{3 \pi}{7}}}{2} e^{-i 800\pi t}$$

You next apply the continuous-Fourier transform properties to get:

$$X(\Omega) = 5\pi \left( e^{i \frac{3 \pi}{7}} \delta(\Omega - 800\pi) + e^{-i \frac{3 \pi}{7}} \delta(\Omega + 800\pi) \right)$$

Now your Fourier transform has two impulses with weights of $5\pi e^{i \frac{3 \pi}{7}}$ and $5\pi e^{-i \frac{3 \pi}{7}}$ respectively.

The spectral amplitudes are infinite, by definition of the impulse function, but they have the weight respectively. And for the magnitude you would take the absolute value of the amplitudes, which is therefore infinite. Yet again you can represent the magnitudes of those impulses by their weights' absolute value.

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