Question

4. Let h(t), (t), and y(t), for -oo < oo, be the impulse response function, the input, and the output of a linear time-invariant system, respectively. Give the following spectra: Input magnitude spectrum: Input phase spectrum: ex(2) T/2 Output magnitude spectrum: tY() Output phase spectrum: ey (2) / 2 Find H() from the above spectra and from the fact that H() 0 for not belonging to the interval (-2,2). Find the impulse response function h(t) from H() found above. Is the system causal?

Answer 1

1XC2)1 = 2, if -4 4ns4 o, otheroise 2 +/2 2 1Yce)l 2) , -2 < 22 2 7 oftuerwise 2.0 T2 H2) 12) (711(0711 (ulx (v)x71()x HIA (A+2) -242 2 otueraoise 1

(u)H7 ()X ()k -TT Q 7 U (vlH = IH(R). (2) JLHA す(442)-eT RG (-2,0 (R+2).ejT R6 (0,2) 2E (-2,0 H) (2 12) -1) ne(0,2) HA) 2) (-2,9) oftuerwiee んt) = RT1 int CR+2).e da -2 jeも んも) = (2+2)-e 2 da -2 -9-

dn R+2) e eat T= (a42). e2t eiat 1 at2) limit (-2,2) 2 -2 j.2t Y242)- a3 I : 2 t 2t -j2t 1 (-24 2) - e 2 C-2 6) (-2 t) 2it e 2t 2jt I = 1 Rt 2t 2t

8T1 hle) J2t e 2t 2t 2t 8T1 j2t ej2t 4. e ,-2jt e 16 7Tt 2 t 2t Rt j2nt 니.e j2t e 16 7Tt 1 2 2 t j24 4-e 16T7 t sin (2t) t 2t dystem i ralue dloesn't depenol Arecent lausal as on future alue