Frequency response Output y(t)

need_help

Joined Sep 7, 2009
23
f (t) = 5+4e^(j2t) + 4e^(-j2t)

H(ω) = (1+jω) / (2+jω)

Determine the steady-state output y(t) of the system H(ω) and express it as a real valuued signal

HINT: Use rule : e^(jωt) ---->LTI---> H(ω)e^(jωt)

any ideas???

t_n_k

Joined Mar 6, 2009
5,448
HINT: Use rule : e^(jωt) ---->LTI---> H(ω)e^(jωt)

any ideas???
What's LTI stand for?

need_help

Joined Sep 7, 2009
23
LTI = linear time invariant

t_n_k

Joined Mar 6, 2009
5,448
I'd use a rather basic approach

1. Realize that e^(j2t)+e^(-j2t) = 2 cos(2t)

2. The input f(t) is therefore a DC shifted sinusoidal function

3. H(0) = 0.5 for the DC component (i.e. you get half of the input DC value at the output)

4. H(2) = (1+j2)/(2+j2) for the AC component. Or H(2) = 0.75 + j0.25 - which will produce an attenuated version of the AC component with a phase lead.

and so on .....

Eventually the answer would be

v(t) = 2.5 + cos(2t+atan(1/3))

Hope this gives some insight anyway.