- #1

TheSodesa

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## Homework Statement

Let

\begin{equation*}

f(t) = 2 + \cos\left( 3t - \frac{\pi}{6} \right) + \frac{1}{4}\cos\left( \frac{1}{2}t + \frac{\pi}{3} \right) + \sin^2(t)

\end{equation*}

Determine the period ##T## and fundamental frequency ##\omega_0## of ##f## and draw images of its amplitude and phase spectra.

## Homework Equations

If

\begin{align}

\hat{f}

&= \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n \omega t) + b_n \sin(n \omega t)\\

&= c_0 + \sum_{n = 1}^{\infty} \bigg( c_n e^{jn \omega t} + c_n^* e^{-j n \omega t}\bigg)=c_0 + \sum_{-\infty}^{\infty} c_n e^{jn \omega t}\\

&= c_0 + \sum_{n = 1}^{\infty} 2|c_n| \cos(n \omega t + \theta_n)

\end{align}

is the Fourier-series of the function ##f##, then the amplitude spectrum is defined as the sequence

\begin{equation}

\ldots ,|c_3^*|, |c_2^*|, |c_1^*|,|c_1|,|c_2|,|c_3|,|c_4|,\ldots

\end{equation}

and the phase spectrum as the corresponding sequence

\begin{equation}

\ldots, -\theta_3,-\theta_2, -\theta_1, \theta_1,\theta_2,\theta_3,\theta_4,\ldots

\end{equation}

## The Attempt at a Solution

I had no trouble answering the first two questions: By drawing the following picture I was able to quess that the smallest period ##T## was ##4\pi## and test for it.

The test came out positive. This gave me the fundamental frequency of ##\omega_0 = \frac{2\pi}{T} = \frac{1}{2}##, and therefore the upper harmonics as its integer multiples.

Now, however, I'm at a standstill. I'm a bit reluctant to start deriving the Fourier series for this function, since my professor has been in the habit of giving excercises, where taking shorcuts is possible using fancy threoms mentioned in the course handout, where I've still gone and wasted time deriving the series because I didn't notice a certain footnote, or something was not adequately well explained for my needs. Not to mention, that the integrals are going to be rather nasty.

My question therefore is: "Is there a way to extract the amplitude and phase spectra from just knowing the fundamental frequency of the given function?"

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