signals and systems final

SS_final_hw.pdf

1

$$\text{real and odd }\to {\int }_{-\infty }^{\infty }x(t)dt=0$$ $$\begin{gathered} \begin{array}{rl}\frac{1}{2}{\int }_0^2|a\sin (\pi t){|}^{2}dt& =1\\ a^2{\int }_0^2|\sin (\pi t){|}^{2}dt& =2\\ a^2{\int }_0^2\cos (\pi t{)}^{2}dt& =2\\ a^2& =2\end{array} \\ x(t)=\pm \sqrt{2}\sin (\pi t) \end{gathered}$$

2

$$\begin{array}{rl}{a}_k& =\frac{1}{5}{\int }_0^{5}x(t)e^{-jk\frac{2\pi }{5}t}dt\\ a_0& =\frac{1}{5}{\int }_0^{5}x(t)dt\\ & =\frac{5}{2}\\ a_k& =\frac{1}{5}{\int }_0^{5}x(t)e^{-jk\frac{2\pi }{5}t}dt\\ a_k& =\frac{1}{5}\{x(t)\frac{e^{-jk\frac{2\pi }{5}t}}{-jk\frac{2\pi }{5}t}\}{|}_0^{5}dt\end{array}$$

3

3.a

$$h(t)=\text{if }t\ge 1\text{ then }{e}^{-2(t-1)}\text{ else }0$$

3.b

yes $$h(t)=0\text{ when }t<0$$

3.c

yes $${\int }_{-\infty }^{\infty }|h(t)|dt={\int }_1^{\infty }|e^{-2(t-1)}|dt<\infty$$

4

4.a