4.1 $U[x]=-1+\theta[x+1]+\theta[x-1]$

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The Fourier series representation of $U[x]$ is illustrated below. The three plots below illustrate the evaluation frequencies $f=1$, $f=2$, and $f=3$ respectively and include the left-half plane as well as the right-half plane. Note$U[0]=0$ and $U[x]$ is an odd function of $x$ (i.e. $U[-x]=-U[x]$).


Fourier Series for u[x] Evaluated at f=1.
Fourier series representation of $U[x]$ from $x=-5$ to $x=5$ evaluated at $f=1$.
Fourier Series for u[x] Evaluated at f=2.
Fourier series representation of $U[x]$ from $x=-5$ to $x=5$ evaluated at $f=2$.
Fourier Series for u[x] Evaluated at f=3.
Fourier series representation of $U[x]$ from $x=-5$ to $x=5$ evaluated at $f=3$.