代做XJEL3430 Digital Communications PROBLEM SET 3代做留学生Matlab程序

XJEL3430 Digital Communications

PROBLEM SET 3

Problem 1: Error Probability

Suppose in a digital communication system, you are transmitting bits 1 and 0 with equal probability. If bit 1 is sent, you get, at the receiver, a random number in the form  1 + X1 , where X1  ~ N(0,0.5) . If you send bit 0, you get, at the receiver, a random number in the form -  1 + X0 , where X0  ~ N(0,0.5) . We assume that X1  and X0  are independent. Atthe receiver, however, we do not know which bit has been sent. In order to detect which bit has been sent, we look at the output of the receiver (the random number); if it is greater than zero, we assume bit one has been sent, and if it is less than or equal to zero we assume bit zero has been sent. Based on the above detection scheme, find the probability of error in this communications system.

Hint: Probability of error is the probability that your detected bit at the receiver is different from what has actually been sent. You may need to remind yourselves of the concept of conditional probability and the total probability theorem.

Problem 2: Energy or Power Signal

For each of the following signals, specify whether they are energy signals (i.e., have finite total energy) or power  signals  (i.e.,  do  not  have  finite  total  energy,  but  has  finite  total  average  power) .  Then  find  the corresponding total energy or total average power in each case.

a)   x(t) = t100, for 0<t<1, and zero otherwise.

b)   y(t) = sin(2t), t ∈ ℝ.

c)    Z(t) = e一t2, t ∈ ℝ.

Problem 3: Correlation Function and Energy/Power Spectral Density

a)    Find the auto-correlation function for the following signal:

x(t) = 1, for t ∈ [0,1], and zero otherwise.

b)    Find the correlation (i.e., the inner product) between the following two signals:

y(t) = sin(2πt), for t ∈ [0,1], and zero otherwise,

Z(t) = cos(2πt), for t ∈ [0,1], and zero otherwise.

c)    The auto-correlation function for an energy signal f(t) has the following form.

Determine the total energy of this signal.

d)    Find the energy spectral density of the signal in part (c). How much is its effective bandwidth in Hz?

Problem 4: White Gaussian Noise

Suppose n(t) is a white Gaussian noise with two-sided spectral density of N0/2.

a)    Is n(0) a random number, or a deterministic one? How about n(1) and n(0) + 2n(0.5)+ n(1)?

b)    Based on your answer in (a), justify that R = ∫0 1 n(t)dt is a random variable.

c)    Find the mean and variance of R (Check out the lecture notes for random processes to find a general expression for the mean and variance). What probability distribution R would have? Write down the corresponding probability density function.

d) Repeat (c) for R = ∫0 1 n(t)cos(1000t)dt. You can use the type of approximations that we used in carrier modulation when WT ≫ 1.