Question

Question 3: a) Find the power spectral density for the cosine signal and also compute power in the signal. b) Find The autocorrelation of X(t)=5cos[2(3)+1/3]

Digital communications

Answer 1

X(t) = cos (antet) be the cosine @. Let signal. we know that Auto Power Spectral correlation function density forms as and fourrier Transtorm pair. Rx(T) = _ x() X(t-) dt. 2 T Rx (T) = 1 cos (2x6ct) cos ante(t-) dt. 2T * S* cos extet [cos arbet cos arket cos extet cos antct cos anfet + sin axbct sins cosa Abct cos 2x507 + cos ar bet sin 2xhet sina Act ] dt -T + cos y abet) cosa fcł t sin yatt 2 2T sin othet). dt. Scanned with CamScanner

cos 275cm i dt ta c os 2xb, I cos utct dba I sin that sin antc I dt el cos atte. ат - dt = = cos axbet x 25 + cos2x+ct cos 25ct I n cos abct dt -T not in the range = = cos antes [ ut si un fet) .F 50.0 (1) cosaatet of sin untet L o Unte -T J sinuttet sinaabot = 0 (since sine is oda signal). cRxlhda vuit CamScanned cos (zabet) toto

Rxlt) = ţ cos (axbet) pso and Auto correlation forms as fourier transtorm pair Fit - Fourier Transform pso = FM [ Rx CT »] = of Fot [ cos2abet) = ef- = [ 8(t-te) + 8(r+bc) PSD = (SCH-te) + 6(++bc)] power → Value of Huto correlation function at (T=0 Rx(0) = cos (2x15cco) Rx(0) = power = I watts Scanned with CS CamScarler

© 2(t) = 5 cos(273 + + 7/3). Auto-correlation bunction PxLtda A+ S 26+) X(+ +). T 5 cos (2x 3 + + 7/3) .5 cos (ax 3 (t+1) +*}3) tt o I S cos (ort + /3) cos (6* (t+t) + /3). rt. 25 r'a cos (oxt+/3) Cos (6* (+ +t) + 2/3) 2.2T -T e cos A COSB = cos( A-B) + Cos CA+B) scos [ort 5. cat - c** - wa] CS Scanned with CamScanner t cos [676 + +676 +677 +*/3] 3

t $606 [-6mt) + cos [ 2 Cox++ *1946rt). © S 106(61) = 603 or ). a s 2T . cosſo (** +7]p) +6*c] = 0 . of St (not in range 25 x cos 6*1 = Rmxlt) RX(t)= 12.5 cos (671) Check ? of the 1. Rx (0) = 12.5 Roco) S gives the power signal act) (t)= 5 cos [ 27 (3). + */3) power = ) V2 512.50 -Both are same · Hence the obtained Scanned with CamScanner is correct Auto - correlation

Answer 2

a) cosine Signai act) = A cos (215518) Auto correlation function of XC) Rx CC) = 52ct) x[t-t) dt Rx (T) = A2 cos (218ct) power spectral density (PSD) J sect) is Rx Cam FT + PSD : PSP and auto correlation are Fowgies transbom ραία. .: Four transform of coslett fct) -> $($cf-9c]+(ft fe P50 = 4* [864-fe) +86+to] APSD - yof -fc powers in the signal = = Area under PSD jpsd df

To b) x(t) = 500$ ( 27 (3)[+]) Auto corelation of xct) RxGt) = Sx(+) x(+-6) dt - SocosCoucou thu ) scos(2m02764-) vyjdt 5. cos(a) cos(b) = } [cos (a+h) 4 cos (a+b)] - 525 cos (21667L+ 9) + COS( 217163)2) de То = 5 cos(21 (3)T) Rx (T) = 12.5 cos (21637)