Question

For each of the following signals, compute the complex exponential Fourier series by using trigonometric identities, and then sketch the amplitude and phase spectra for all values of k. (a) x() = cos(51 - 7/4) (b) X(t) = sin 1 + cost

Answer 1

a) X(t) = cos (57 - inwet Σ C, e Y = -0 Exponential Fourier Series X(t) = W. = Fundamental Angular frequency (6) List-6) . & To X(t) = COS = 2 ist ist + 2 Gal - c, = 1 / 4 (- ç 2-1/4 ; Cu = ² <^/4 alol Amplitude spectaum: 2 > nork & Lan 지니 →n ork -794 CS Scanned with CamScanner

b) X(t) = sint + cost + sin g., sînt) EVO ( cos(?) cost (4-1) - x(t) = cos( 1 (ty) --- X(1) 1 + e ta T ↑ Ial VE VE 1 >k Lck c) X(t) cos (t-1) + sin(+-) (cost. Cosi + sînt sînl) + ( <fnt cos e cosi – ein +-) cost + (Sini + cos + ) sint cost sinş) 0.06 Cost + 1.2 sint 1721 1.2 E 1.2.1 { cose cost + 0106 COS eragesint) = 10721 Cos (t-6) = 1,721 cos(t-88 1.821 (4-160) + 1.121 i(t-280) 0:06 1.521 X(t) ging 112 1.121 2 oz. tan torin CS

k Ck >k -880 d ) X(t) = COS at sinat LA cose sin (A-1) = sinACOSB COSA Sing = SACOSB + COSA sºn B sin (A+B) = [ (sinst + sînt desteisty + X(t) = sin(A-B ) + sin(A+B) = 2 sinacos B jst -ist { u6= 1 rod/sec. } x(t) = 1134 ju 1 1 1 GE Cs Cas N Tal 0 5 90° istan k CSS