signals and systems mid 2 3 a b c 4 5 6

signals and systems mid

SS_mid_hw.pdf

2

odd signal of x[n]x[n] is {x[n],n>=0−x[−n],n<0 \begin{cases} x[n] &, n>=0\\ -x[-n] &, n<0\end{cases}

3

a

odd signal x[n]=−x[−n]x[n]=-x[-n]

b

odd signal x[n]=−x[−n]x[n]=-x[-n]
even signal x[n]=x[−n]x[n]=x[-n]
xo[n]={x[n],n>=0−x[−n],n<0x_o[n] = \begin{cases} x[n] &, n>=0\\ -x[-n] &, n<0\end{cases}

xe[n]{x[n],n>=0x[−n],n<0x_e[n] \begin{cases} x[n] &, n>=0\\ x[-n] &, n<0\end{cases}

product of xo[n]x_o[n] and xe[n]x_e[n] is {x[n]2,n>=0−x[−n]2,n<0 \begin{cases} x[n]^2 &, n>=0\\ -x[-n]^2 &, n<0\end{cases} still are odd signal since x[n]=−x[−n]x[n]=-x[-n]

c

x=xo+xex(t)2=xo(t)2+2xo(t)xe(t)+xe(t)2∫−∞∞x2(t) dt=∫−∞∞xo2(t) dt+∫−∞∞2xo(t)xe(t) dt+∫−∞∞xe2(t) dt∫−∞∞x2(t) dt=∫−∞∞xo2(t) dt+0+∫−∞∞xe2(t) dt∫−∞∞x2(t) dt=∫−∞∞xe2(t) dtx=x_o+x_e\\ x(t)^2=x_o(t)^2+2x_o(t) x_e(t) +x_e(t)^2\\ \begin{aligned}\\ \int_{-\infty}^{\infty}x^2(t) \ dt&=\int_{-\infty}^{\infty}x_o^2(t) \ dt+\int_{-\infty}^{\infty}2x_o(t) x_e(t) \ dt+\int_{-\infty}^{\infty}x_e^2(t) \ dt\\ \int_{-\infty}^{\infty}x^2(t) \ dt&=\int_{-\infty}^{\infty}x_o^2(t) \ dt+0+\int_{-\infty}^{\infty}x_e^2(t) \ dt\\ \int_{-\infty}^{\infty}x^2(t) \ dt&=\int_{-\infty}^{\infty}x_e^2(t) \ dt\\ \end{aligned}\\

4

lineartime invariant
y(t)=t2x(t−1)y(t)=t^2x(t-1)nono
y[n]=x[n+2]−x[n−3]y[n]=x[n+2]-x[n-3]yesyes
y[n]=Od{x[n]}y[n]=O_d\{x[n]\}yesno

5

period
x(t)=3cos⁡(4t+π3)x(t)=3 \cos (4t+\frac{\pi}{3})π2\frac{\pi}{2}
x(t)=Od{sin⁡(4πt)u(t)}x(t)=O_d\{\sin(4\pi t)u(t)\}0.50.5
x[n]=cos⁡(n8−π)x[n]=\cos( \frac{n}{8}-\pi)None\text{None}

6

memorylesstime invariantlinearcausalstable
y[n]=Ev{x[n−1]}y[n]=E_v\{x[n-1]\}❌✅✅✅✅
y[n]=x[n−2]−2x[n−3]y[n]=x[n-2]-2x[n-3]❌✅✅✅✅
y(t)=cos(3t+2)x(t)y(t)=cos(3t+2)x(t)✅✅❌✅✅
y[n]={0,if t<0x(t)+x(t−2),otherwisey[n]=\begin{cases} 0,& \text{if } t<0 \\ x(t)+x(t-2), &\text{otherwise}\end{cases}❌✅❌✅✅