signals and systems final 1 2 3 3.a 3.b 3.c 4 4.a

signals and systems final

SS_final_hw.pdf

1

real and odd →∫−∞∞x(t) dt=0\text{real and odd }\rightarrow \int_{-\infty}^{\infty}x(t)\ dt =0 12∫02∣asin⁡(πt)∣2dt=1a2∫02∣sin⁡(πt)∣2dt=2a2∫02cos⁡(πt)2dt=2a2=2x(t)=±2sin⁡(πt)\begin{aligned} \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{aligned}\\ x(t)=\pm \sqrt{2} \sin(\pi t)

2

ak=15∫05x(t)e−jk2π5t dta0=15∫05x(t) dt=52ak=15∫05x(t)e−jk2π5t dtak=15{x(t)e−jk2π5t−jk2π5t}∣05 dt\begin{aligned} 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} \} \Big|_0^5 \ dt\\ \end{aligned}

3

3.a

h(t)=if t≥1 then e−2(t−1) else 0h(t)=\text{if }t\geq 1 \text{ then } e^{-2(t-1)} \text{ else } 0

3.b

yes

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

3.c

yes

∫−∞∞∣h(t)∣ dt=∫1∞∣e−2(t−1)∣ dt<∞\int_{-\infty}^{\infty}|h(t)| \ dt=\int_{1}^{\infty}|e^{-2(t-1)}| \ dt < \infty

4

4.a