Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are
independent random variables,...
Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are
independent random variables, each with
mean 0 and variance 1.
(a) Is Zt strictly stationary?
(b) Is Zt weakly stationary?
If u(t) = < sin(8t), cos(4t), t > and v(t) = < t,
cos(4t), sin(8t) >, use the formula below to find the given
derivative.
d/(dt)[u(t)* v(t)] =
u'(t)* v(t) +
u(t)* v'(t)
d/(dt)[u(t) x v(t)] =
<.______ , _________ , _______>
If u(t) = < sin(5t),
cos(5t), t > and
v(t) = < t, cos(5t),
sin(5t) >, use the formula below to find the given
derivative.
d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t)
d/dt [ u(t) x v(t)] = ?
2. Let {Zt , t = 0, ±1, ±2, ...} be a sequence of independent
random variables, each with mean EZt = 0 and variance Var(Zt) = σ 2
. Define Xt = ZtZt−1 + Zt−2.
• Compute the mean and the covariance function for Xt .
• Is {Xt} weakly stationary? Explain why.
Let {Zt} be independent normal random variables with mean 0 and
variance σ2. Let a, b, c be constants. Which of the
following processes are stationary? Evaluate mean and
autocovariance function.
(a) Xt = Ztcos(at) +
Zt−1sin(bt)
(b) Xt =a+bZt + cZt−2
(c) Xt = ZtZt−1
Define a random process by X(t) = Asin(50*pi*t)+B, where A and B
are independent random variables, E(A) = 0, SD(A) = 4, E(B) = -3,
SD(B) = 5.
A. Determine the mean function of X(t)
B. Determine the autocovariance function of X(t)
C. determine the variance function of X(t)
D. is X(t) wide-sense stationary? how can you tell?
Y(t) = 4sin(-120 pi t) cos(-200 pi t) sin(-200 pi
t)
Find all frequency responses of Y(f) and F{y(t)}
Find frequencies if y(t) is sampled at fs=40 and
fs=100
Pllt magnitude and phase response of F{y[n]} where the
range of n is -2pi to 2pi
Let (Un, U, n>1) be asequence of random variables such that
Un and U are independent, Un is N(0, 1+1/n), and U is N(0,1), for
each n≥1.
Calculate p(n)=P(|Un-U|<e), for all e>0.
Please give details as much as possible
Let U, V be iid Unif(0, 1) random variables, and set
M = max(U,V) and N = min (U,V)
(a) Find the conditional density of N given M = a for any value
of a ∈ (0, 1).
(b) Find Cov(M, N).
use matlab
y(t)=10*(cos(2*pi*500*t)+cos(2*pi*1000*t)+
cos(2*pi*1500*t)).
e) Down sample y(t) by a factor of 6. Sketch the Fourier
transform with appropriate frequency axis. Check if all frequency
components are correct?
Up-sample the time-domain signal obtained in e) by a factor of
6. Use appropriate filter for interpolation. Sketch the Fourier
transform of the up-sampled and filtered signal. Does the resulting
signal show all frequency components of the original signal
y(t)?