Suppose a random process is X(t) = A[cos(2fot + T)] where A and Fo are constants

Question

Suppose a random process is X(t) = A[cos(2fot + T)] where A and Fo are constants and T is a random variable that is uniformly distributed over [0, 2p].

1. Is X(t) wide-sense stationary?
2. Find power spectral density of x(t).
3. Let Y(t) = X (t) - X (t - T) , find the power spectral density of Y(t)

PLEASE HELPS ME

Explanation / Answer

I am assuming T= 2pi to make question easy . X(t) is wide-sense stationary if: E{X(t)] = constant 8 E{X(t)} = ? x(t) f(x,t)dx -8 where, joint density function in which time is fixed and T, is a uniformly distributed random variable. Hence, f(x,t) = 1/2p in the interval 0=T=2p ..............2p E[X(t)] = ? Acos(2?ot + T) (1/2p) dT = -[sin(2?ot + T)] for T=0->2p ...............o = -[(sin2?ot)(cosT) + cos2?ot(sinT)] for T=0->2p = -[sin2?ot - sin2?ot] = 0 And, the other condition for a wide-sense stationary random process: Autocorrelation function: R(t) = E[X(t)X(t+t)] is only dependent on t R(t) = E[Acos(2?ot +T)Acos(2?ot + 2?ot + T)] = ½ A² E[cos2?ot] => t dependent only! Therefore, X(t) is a wide-sense stationary random process. 2. Spectral density of x(t): S(?) = lim E[|X(?)|²]/2T ..........T?8 X(?) = ? X(t) e^-j?t dt = ? Acos(2?ot + T) e^-j?t dt, for -T=t=T but since: cos(2?ot+T) = cos(2?ot)cosT - sin(2?ot)sinT and cosT = ½ (e^jT + e^-jT) X(?) = ½ A e^jT ? e^[j(?o - ?)t] dt + ½ A e^-jT ? e^[-j(?o + ?)t] dt X(?) = AT e^jT sin[?-?o)T]/(?-?o)T + AT e^-jt sin[?+?o)T]/(?+?o)T |(X(?)|² = |X(?)X*(?)| => E[|X(?)|²]/2T = ½ A²p {Tsin²[(?-?o)T]/[p(?-?o)T]² but lim (T/p)[sin(aT)/aT]² = d(a) S(?) = ½ A²p [d(?-?o) + d(?+?o)]

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.