Question

. Consider the continuous time signal, x(t) = e jω0t . Write an expression for the even and odd portions of this function, and show that the even and odd parts sum up to the original x(t). Do you recognize the even and odd functions? What are they called?

Answer #1

The continuous time signal x(t) = 2*sin
(2*π*100*t +
π/2) + sin (2*π*150*t) +
3*sin (2*π*300*t) is
sampled at 500 samples per second.
Write the mathematical expression x(n) of the sampled discrete
time signal. Show your work.
What are the discrete time frequencies obtained in
x(n)?

Given continuous analog
signal xa(t)=
e-100|t|. Write a
MATLAB program to display its frequency spectra using
FFT.
(Hints: To obtain the
sampling signals x(n) =
xa(nTs)
, the sampling interval
Tscan be less than
0.0025 sec)

Q No 1: Here R=18
A continuous time signal x(t) is
defined as
x(t) = {
-2R,
-0.65 < t < 0
R,
0
< t < 1
0.5R,
1 < t < 1.25
0,
otherwise
Then sketch and label following
signals
(i) x(-t)
[u(t+1.25)-u(t-0.75)] (ii) Ev{x(t)}
(iii) Od{x(t)} (iv) x(2.5t)
(v) x(0.25t)

1) Find the even and odd components of x(t)=e^(jt)
2) Determine whether x(t)=cost+sin√2 t is a periodic
signal?

(a) Write the expression for y as a function of
x and t in SI units for a sinusoidal wave
traveling along a rope in the negative x direction with
the following characteristics: A = 5.50 cm, λ =
90.0 cm, f = 6.00 Hz, and y(0, t) = 0 at
t = 0. (Use the following as necessary: x and
t.)
y =
(b) Write the expression for y as a function of x
and t for the wave...

Consider the time-dependent ground state wave function
Ψ(x,t ) for a quantum particle confined to an
impenetrable box.
(a) Show that the real and imaginary parts of Ψ(x,t) ,
separately, can be written as the sum of two travelling waves.
(b) Show that the decompositions in part (a) are consistent with
your understanding of the classical behavior of a particle in an
impenetrable box.

Write a Matlab program to find the autocorrelation of the
following signal :
g(t) =
e-atu(t) , a > 0
.
Then , use the Wiener-Khintchine theorem to determine the energy
spectral density of the signal .
Use this program to display the autocorrelation function and the
energy spectral density .

A continuous-time system with impulse response
h(t)=8(u(t-1)-u(t-9))
is excited by the signal
x(t)=3(u(t-3)-u(t-7))
and the system response is y(t).
Find the value of y(10).

(i) If a discrete random variable X has a moment generating
function
MX(t) = (1/2+(e^-t+e^t)/4)^2, all t
Find the probability mass function of X. (ii) Let X and Y be two
independent continuous random variables with moment generating
functions
MX(t)=1/sqrt(1-t) and MY(t)=1/(1-t)^3/2, t<1
Calculate E(X+Y)^2

(1) Recall on February 6 in class we discussed e 0 + e 2πi/n + e
4πi/n + · · · + e 2(n−1)πi/n = 0 and in order to explain why it was
true we needed to show that the sum of the real parts equals 0 and
the sum of the imaginary parts is equal to 0.
(a) In class I showed the following identity for n even using
the fact that sin(2π − x) = − sin(x):...

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