Communication Theory Homework 3

Communication Theory Homework 3

1. A source generates 0s and 1s randomly according to a Bernoulli distribution, where the
probability of a 0 is 0.3 and the probability of a 1 is 0.7. The value is then sent across a long
wire, and corrupted by thermal noise. A system at the other end receives the corrupted signal
and guesses if it received a 1 or a 0. It has an error probability (either guessing a 1 was a 0
or guessing a 0 was a 1) of 0.2. If the receiver guesses it received a 1, what is the probability
a 1 was transmitted?
2. Let Θ be a random variable uniformly distributed from 0 to π. Let X = cos Θ and Y = sin Θ.
Are X and Y uncorrelated? Are they independent? Are they orthogonal?
3. Let X(t) be a random process defined by X(t) = A + Bt where A and B are independent
random variables uniformly distributed from -1 to 1. Find mX(t) and RX(t1, t2). Is the
process WSS? If not, is it cyclostationary? If the answer to either question is yes, find the
PSD of X.
4. Let X(t) = Y cos(ω0t)−Z sin(ω0t), where Y and Z are zero-mean independent Gaussians with
variance σ
2
. Find mX(t) and RX(t1, t2). Is the process WSS? If not, is it cyclostationary? If
the answer to either question is yes, find the PSD of X.
5. If X(t) has PSD SX(ω), what is the PSD of 4X0
(t) + X(t − T)?
6. In the discrete-time case, suppose we have a random process defined by {Xn} for n ∈ Z. If
we observe N samples of the random process, {xn} for n = 1, 2, . . . , N, we define the sample
autocorrelation as
RX(m) = ( 1
N−m
PN−m
n=1 xnxn+m, m = 0, 1, . . .
1
N−|m|
PN
n=|m|
xnxn+m, m = −1, −2, . . .
This approximates the autocorrelation. In practice, we don’t let m take infinitely many values,
but instead look at it over a finite range of values from −M to M for some integer M. The
Power Spectral Density is computed through the Wiener Khinchin Theorem, using the DFT
rather than the CTFT:
SX(ω) = X
M
m=−M
RX(m)e
−jωm
2M+1
Using MATLAB, write a function that takes an input vector of N samples and an integer M
as inputs, and returns the autocorrelation and PSD.
7. Use your function from above to plot the power spectral density white noise with some variance
you choose so as to validate your function’s operation. Make sure to use sufficiently large N
and M.
THERE IS A PROBLEM 8 DON’T STOP HERE
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8. Use your function to plot the PSD of the of X(t) from problem 4 with ω = 10000 rad/s, as
well as the PSD of the integral of X(t). Please don’t perform an integral numerically (except
to check your work if you want to) – instead, use the relationship of PSDs at the input and
output of an LTI system.
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