For random process X = 6 and Rxx (t, t+t) = 36 + 25 exp(|t|). Consider following statements:
(i) X(t) is first order stationary.
(ii) X(t) has total average power of 36 W.
(iii) X(t) is a wide sense stationary.
(iv) X(t) has a periodic component.
Q. Which of the following is true?
X Constant and Rxx() is not a function of t, so X(t) is a wide sense stationary. So (i) is false & (iii) is true. Pxx = Rxx(0) 36+25 = 61. Thus (ii) is false if X(t) has a periodic component, then RXX(t) will have a periodic component with the same period. Thus (iv) is false.
White noise with power density No/2 = 6 microW/Hz is applied to an ideal filter of gain 1 and bandwidth W rad/s. If the output’s average noise power is 15 watts, the bandwidth W is
Pyy = 1/2p Integral(?xx(w) |H(w)|^2 dw ) from plus infinity to minus infinity. Hence solve for W.
(Q.3-Q.4) The two-level semi-random binary process is defined by X(t) A or -A where (n 1)T < t < nt and the levels A and -A occur with equal probability. T is a positive constant and n = 0, ±1, ±2.
The mean value E[X(t)] is
E[X(t)] = A P(A) – (-A)P(-A) which is zero.
The auto correlation Rxx(t1 = 0.5T, t2 = 0.7T) will be
Here Rxx is AxA if both t1 and t2 are different and zero if they are same. Hence the answer is AxA.
Air craft of Jet Airways at Ahmedabad airport arrive according to a Poisson process at a rate of 12 per hour. All aircraft are handled by one air traffic controller. If the controller takes a 2 – minute coffee break, what is the probability that he will miss one or more arriving aircraft?
P (miss/or more aircraft) = 1 – P(miss 0) = 1 – P(0 arrive).
A stationary random process X(t) is applied to the input of a system for which h(t) = u(t) t2e(-8t). If E[X(t)] = 2, the mean value of the system’s response Y(t) is
The mean value of Y(t) is integral of h(t)dt over negative infinity to positive infinity which gives the value equal to 3/128.
A random process is defined by X(t) + A where A is continuous random variable uniformly distributed on(0,1).
The auto correlation function and mean of the process is
E[X(t)X(t+t)] = 1/3 and E[X(t)] = 1/2 respectively.
(Q.8-Q.9) The auto correlation function of a stationary ergodic random process is shown below.
Q. The mean value E[X(t)] is
Lim |t| tends to infinity, Rxx(t) = 20 = X2. hence X is sqrt(20).
The E[X2(t)] is
Rxx(0) = X2 = 50.
The variance is
Here X = 0, y = 0, Rxx(0) = 5, Ryy(0) = 10. The only value that satisfies all the given conditions is 30.