Problem $1[5$ points$]$: Consider the LTI system with Bode magnitude and phase response given below If the input to that system is:

${}_1=1ra\frac{d}{s}$

$x(t)=2cos(t)-10sin(10t+45)+cos(200t+120)$

${}_2=10ra\frac{d}{s}$

${}_3=200ra\frac{d}{s}$

Determine the output $y(t).$

$y(t)=A2cos(t+1B)$

$-10$Csin$(10t+45+D)$

$+$Ecos$(200t+120+f)$

$A=|H(1)|=-20,B\frac{?}{{?}_H}=(1)=-2$

$C=|H(10)|=-22,D\frac{?}{{?}_H}=(10)=-48$

$E=|H(200)|=-45,F\frac{?}{{?}_H}=(200)=-86$

plugging in:

$y(t)=-20(2)cos(t-2)-10(-22)sin(10t+45-48)+(-45)cos(200t+120-86)$

$y(t)=-40cos(t-2)+220sin(10t-3)-45cos(200t+34)$