In this module, we will derive an expansion for continuous-time, periodic functions, and in doing so, derive the Continuous Time Fourier Series (CTFS).

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems, calculating the output of an LTI system $\mathscr{H}$ given ${e}^{st}$ as an input amounts to simple multiplication, where $H\left(s\right)\in \mathbb{C}$ is the eigenvalue corresponding to s. As shown in the figure, a simple exponential input would yield the output

$$y\left(t\right)=H\left(s\right){e}^{st}$$

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Using this and the fact that $\mathscr{H}$ is linear, calculating $y\left(t\right)$ for combinations of complex exponentials is also straightforward.

$${c}_{1}{e}^{{s}_{1}t}+{c}_{2}{e}^{{s}_{2}t}\to {c}_{1}H\left({s}_{1}\right){e}^{{s}_{1}t}+{c}_{2}H\left({s}_{2}\right){e}^{{s}_{2}t}$$ $$\sum _{n}^{}{c}_{n}{e}^{{s}_{n}t}\to \sum _{n}^{}{c}_{n}H\left({s}_{n}\right){e}^{{s}_{n}t}$$

The action of $H$ on an input such
as those in the two equations above is easy to explain.
** $\mathscr{H}$ independently
scales** each exponential component
${e}^{{s}_{n}t}$
by a different complex number
$H\left({s}_{n}\right)\in \mathbb{C}$. As such, if we can write a function
$f\left(t\right)$
as a combination of complex exponentials it allows us to easily calculate the output of a system.

Joseph Fourier demonstrated that an arbitrary $f\left(t\right)$ can be written as a linear combination of harmonic complex sinusoids

$$f\left(t\right)=\sum _{n=-\infty}^{\infty}{c}_{n}{e}^{j{\omega}_{0}nt}$$

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The
${c}_{n}$ - called the Fourier coefficients -
tell us "how much" of the sinusoid
${e}^{j{\omega}_{0}nt}$ is in
$f\left(t\right)$.
The formula shows
$f\left(t\right)$ as a sum of complex exponentials, each of which is easily processed by an
LTI system (since it is an eigenfunction of
**every** LTI system). Mathematically,
it tells us that the set of
complex exponentials
$\left\{\forall n,n\in \mathbb{Z}:\left({e}^{j{\omega}_{0}nt}\right)\right\}$ form a basis for the space of T-periodic continuous
time functions.

Example 1

We know from Euler's formula that $cos\left(\omega t\right)+sin\left(\omega t\right)=\frac{1-j}{2}{e}^{j\omega t}+\frac{1+j}{2}{e}^{-j\omega t}.$

Guitar Oscillations on an iPhone

Finding the coefficients of the Fourier series expansion involves some algebraic manipulation of the synthesis formula. First of all we will multiply both sides of the equation by ${e}^{-(j{\omega}_{0}kt)}$, where $k\in \mathbb{Z}$.

$$f\left(t\right){e}^{-(j{\omega}_{0}kt)}=\sum _{n=-\infty}^{\infty}{c}_{n}{e}^{j{\omega}_{0}nt}{e}^{-(j{\omega}_{0}kt)}$$

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$${\int}_{0}^{T}f\left(t\right){e}^{-(j{\omega}_{0}kt)}dt={\int}_{0}^{T}\sum _{n=-\infty}^{\infty}{c}_{n}{e}^{j{\omega}_{0}nt}{e}^{-(j{\omega}_{0}kt)}dt$$

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$${\int}_{0}^{T}f\left(t\right){e}^{-(j{\omega}_{0}kt)}dt=\sum _{n=-\infty}^{\infty}{c}_{n}{\int}_{0}^{T}{e}^{j{\omega}_{0}(n-k)t}dt$$

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$$\forall n,n=k:\left({\int}_{0}^{T}{e}^{j{\omega}_{0}(n-k)t}dt=T\right)$$

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$$\forall n,n\ne k:\left({\int}_{0}^{T}{e}^{j{\omega}_{0}(n-k)t}dt={\int}_{0}^{T}\mathrm{cos}\left({\omega}_{0}(n-k)t\right)dt+j{\int}_{0}^{T}\mathrm{sin}\left({\omega}_{0}(n-k)t\right)dt\right)$$

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$${\int}_{0}^{T}\mathrm{cos}\left({\omega}_{0}(n-k)t\right)dt=0$$

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$${\int}_{0}^{T}{e}^{j{\omega}_{0}(n-k)t}dt=\{\begin{array}{l}T\text{if}n=k\\ 0\text{otherwise}\end{array}$$

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$$\forall n,n=k:\left({\int}_{0}^{T}f\left(t\right){e}^{-(j{\omega}_{0}nt)}dt=T{c}_{n}\right)$$

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$${c}_{n}=\frac{1}{T}{\int}_{0}^{T}f\left(t\right){e}^{-(j{\omega}_{0}nt)}dt$$

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Example 2

Consider the square wave function given by

$$x\left(t\right)=\left\{\begin{array}{cc}1/2& t\le 1/2\\ -1/2& t>1/2\end{array}\right.$$

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on the unit interval $t\in \mathbb{Z}[0,1)$.

$$\begin{array}{cc}\hfill {c}_{k}& ={\int}_{0}^{1}x\left(t\right){e}^{-j2\pi kt}dt\hfill \\ & ={\int}_{0}^{1/2}\frac{1}{2}{e}^{-j2\pi kt}dt-{\int}_{1/2}^{1}\frac{1}{2}{e}^{-j2\pi kt}dt\hfill \\ & =\frac{j\left(-1+{e}^{j\pi k}\right)}{2\pi k}\hfill \end{array}$$

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Thus, the Fourier coefficients of this function found using the Fourier series analysis formula are

$${c}_{k}=\left\{\begin{array}{cc}-j/\pi k& \mathrm{k}\mathrm{odd}\\ 0& \mathrm{k}\mathrm{even}\end{array}\right..$$

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Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.

$$f\left(t\right)=\sum _{n=-\infty}^{\infty}{c}_{n}{e}^{j{\omega}_{0}nt}$$

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$${c}_{n}=\frac{1}{T}{\int}_{0}^{T}f\left(t\right){e}^{-(j{\omega}_{0}nt)}dt$$

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