Question

Let the signals x(t) and y(t) be the input and output signals to a differentiator, respectively. x(t) do y(t) (a) Let the autocorrelation of the signal x(t) be R (T) and the autocorrelation of the signal y(t) be R (T). If y(t)= X, express R, (T) in terms of R. (T) dt (b) Assume R (T) = 5e and find the power in the output signal y(t). JA, \f}<B (c) Assume that the power spectral density (PSD) of x(t) is S, (f) = { 10, otherwise. Find the power in the output signal y(t).

Answer 1

Gäven auto coroelation of the input XIH = Rall) = 5€ 247 alt dan sults ylt = da ces alf £ft XW) (اف لا د ے (lateى د الا = (xo بیر dt H(W) = ylw) = jw = jzif х( @). Qutocorreletion output Ry(t): – Rylt) <fot, St1w) Rx (T) ER:T, Sxlw) Solw) = Sxlw) HC) I Sylf) = Sx G) CHICHI Ry(t) ef:ty S(0) 5est snixe Wor et R.Tvalere Solw) = Sx(W) | H(W)") Solw) = 5 0 W/8 15w In = 50W/8 wr -678 →0

S8/w) = 5wv & W/S. dett) Els Qu? Alw) deh. - I.F.Taswe wy 7 TET BL 3,930 = D.AT VE CW TOWEJ from Comparing Od F(W) = ē W/ If.tfrw] - flH) If the Wif] = ezt De Tresowa ware] For Parent) e+2) = 20 cl (texty) ( :.37.-1) 220(eat ++l-ute atr] = 20 e 2tall- un] i auto correlation output Rylt) Ry (7) = great (1-44] & 4x562ty/1-46) = 4x Rx (T) (1–44 RYIT) = for IT) 461-44]

6 output power (PM) = solunda s = هاست r ما به = Vnt2f surowyrdo P = Ry (T) | I=0 Io = 20 watts O given Sxlf) = lo othewire P = I alt at = falso .: output powel = pr+Pg p (20+4) Wetts