# Signals and systems problem

1. Jun 21, 2017

### crom1

1. The problem statement, all variables and given/known data
continuous time LTI system is given with differential equation y'(t) + 5y(t) = 10u(t)
a) Find transfer function of system and determine is it stable or not.
b)Find frequency characteristics (amplitude and phase angle) of given system
c) Find impulse response using Laplace transformation
d) Find zero-input response(I think that's the right word) for u(t) = 6cos(5t)μ(t)

3. The attempt at a solution
Ok, I got
a) Y(s) = 10/(s+5) * U(s) , pole is s=-5 , and Re(s)<0 so the system is stable
b) Plug s=jω in H(s)=10/(s+5) , then |H(jω)| = 10/(√s^2+ω^2) and ∠H(jω) = - arctan(ω/5)
c) for impulse response U(s) = 1, Y(s) = 10/(s+5) ⇒ y(t) = 10*e^(-5t) , t>0
d) Using frequency characteristics, we have ω=5, and |H(j5)| = √2 and ∠H(j5) = -π/4 so the response is
y(t) = 6√2 *cos(5t-pi/4)

I need someone to say if these are correct or not, if not i will post my full attempt and reasoning. Thanks

2. Jun 24, 2017

### rude man

10 U(s) is not part of the transfer function. It's a specific input; the transfer function gives output for any input.
OK but leave out the "10". Also, typo used "s" for "5".
U is a step function.The symbol for impulse input is δ(t). Your actual input is kδ(t), k = 1 volt-sec. Careful with δ(t), its units are time-1. Again, ditch the "10".
never heard of "zero input response". You mean mean input is 6cos(5t)U(t) or ??? Never saw μ(t) either.
Strange they never asked for the time response to the 10u(t) input ...