Question

vebdav/pid-1888136-dt-content-rid-4139590.1/courses/Spring,2019 ELEC341 22498/Assignment1.pdf Problem 5 For the system showa in the figure below, the input signals are gil0-rece 0frect( 104t) gan-at), the low-pass filters have frequency responses M(아rect(0140,0007), H2(co-rect(m/20,000π), and the output of the low-pass filters yi(t) and y2(t) are multiplied to obtain the output signal y). (a) Are the low-pass filters distortion-less systems or not? Why? (b) Find and sketch Gi(a) and G() (c) Sketch Hi(a) and H2(m) (d) Find and sketch Yi(), Y2(o), and Y(o) (e) What is the bandwidth of Y'(a), ½(o) , and no)? what is the relation between the bandwidth of Y(a) and those of i(o) and Y(o)? g, (t) y, (0) H (f) y (1) = yi (t) ½(f) y, (t)

Answer 1

For distortion less -system, it's impulse response should be constant. i.e. magnitude should be constant. And magnitude of both filters H_1 (w) and H_2 (w) is constant, which is equal to '1'. So filters are distortion -less. We know that A rect (t/T) is given by so g_1 (t) = 10^4 rect (10^4 t) so by comparing A = 10^4, T = 10^-4 sec \r\n we know that G_1 (w) = integral^infinity_-infinity g_1 (t) middot e^-j w t dt G_1 (w) = integral^0.5 times 10^-4_-0.5 times 10^-4 10^4 e^-j w t dt = 10^4 (e^-j w t/-j w)^0.5 times 10^-4_-0.5 times 10^-4 = 10^4 (e^-j 0.5 times 10^-4 w/-j w + e^j 0.5 times 10^-4 w/jw) we know that e^I theta - e^j theta = j 2 sin theta so G_1 (w) = 10^4 (j 2 sin (0.5 times 10^-4 w)/jw G_1 (w) = 2 times 10^4 sin (5 times 10^-5 w)/w G_2 (w) = integral^infinity_-infinity g_2 (t) - e^-j w t dt