Question

can someone please explain why they only considered 0,2 and 5 as their frequencies and didn't include -2 and -5? Also, how did they get the angles for the changes column? please explain with steps. thank you

1. (i) (8 pts) The input signal z(t) to a continuous time (CT) linear time-invariant (LTI) system is given by x(t) 12 cos 2t +sin 5t The output y(t) is found to be given by y(t) 3-4 sin 2t 0.5 sin 5t At what values of the frequency w can H(jw), the frequency response of the LTI system, be determined? Determine H(jw) at these frequencies. Display your results in a table showing the frequencies and the corresponding values of H(ju). Justify your answers. "Tat maps加esok, out) tChnge UJ 椭2 | 2co, 2t |-Y.sm ㄳ | Cos IC at a LT ST 2 2016 Q ar speche olo ns at certan

Answer 1

For practical LTI system it is generally considered that they exist for t>0 and hence while considering frequency response only positive frequencies are taken in account and generally laplace transform is used for such cases and for analysis s= jw is substituted. However if we want complete frequency spectrum of system i e also having consideration of input for negative time fourier transform is used that will give values for both positive and negative frequencies. So for complete spectrum negative frequency amplitude is also required. In practical scenario there is no meaning of negative frequencies. So here it is not considered . Further due to symmetry phase shift of +ve and -ve frequency and amplitude will be same always.

For dc component angle will be zero as no phase shift.

For input of 2 cos2t output is -4 sin2t = 4 cos(2t+$\pi$/2) so phase shift with respect to input is $\pi$/2 and amplitude gain = 4/2 = 2

For input of sin5t output is -0.5 sin 5t = 0.5 sin(5t+$\pi$) so phase shift with respect to input is $\pi$ and amplitude gain = 0.5/1 = 1/2