(Solved): When determining the trigonometric Fourier series representation of a signal f(t), the analysis can ...

When determining the trigonometric Fourier series representation of a signal $f(t)$, the analysis can be simplified by exploiting the symmetry properties of $f(t)$. 1. (6 points) Show that if $f(t)$ is an even, real-valued periodic function of time with period $T_o$, then $\begin{array}{rl}{a}_0& =\frac{2}{T_o}{\int }_0^{T_o/2}f(t)dt\\ a_n& =\frac{4}{T_o}{\int }_0^{T_o/2}f(t)\cos \left(n{\omega }_{o}t\right)dt\\ b_n& =0\end{array}$ and $a_n$ is an even function of $n$. 2. (6 points) Show that if $f(t)$ is an odd, real-valued periodic function of time with period $T_o$, then $\begin{array}{l}{a}_0=0\\ a_n=0\\ b_n=\frac{4}{T_o}{\int }_0^{T_o/2}f(t)\sin \left(n{\omega }_{o}t\right)dt\end{array}$ and $b_n$ is an odd function of $n$. 3. (8 points) If a periodic signal $f(t)$ with fundamental period $T_o$ satisfies $f\left(t-\frac{T_o}{2}\right)=-f(t)$ the signal $f(t)$ is said to have a half-wave symmetry. In signals with half-wave symmetry, the two-halves of one period of the signal are of identical shape except that one half is the negative of the other. If a periodic signal $f(t)$ with fundamental period $T_o$ satisfies the the half-wave symmetry condition, then $f\left(t-\frac{T_o}{2}\right)=-f(t)$ In this case, show that all the even-numbered harmonic coefficients $\left(a_0,a_2,a_4,\dots ,b_2,b_4,\dots \right)$ vanish, and that the odd-numbered harmonic coefficients $\left(a_1,a_3,\dots ,b_1,b_3,\dots \right)$ are given by $\begin{array}{rl}{a}_n& =\frac{4}{T_o}{\int }_0^{T_o/2}f(t)\cos \left(n{\omega }_{o}t\right)dt\\ b_n& =\frac{4}{T_o}{\int }_0^{T_o/2}f(t)\sin \left(n{\omega }_{o}t\right)dt\end{array}$ where $a_n$ is an even function of $n$ while $b_n$ is an odd function of $n$.