2.a) Using the result for P(I) in Question 1, prove that the n-th moment of complex Gaussian noise is

{I^n} = n!(2sigma^2)^n

You may use the following result for the Gamma integral

Gamma(n-1) = n! = integral x^n exp(-x)dx.

b) Hence, obtain an expression for the normalized moments {I^n}/{I}^n

c) Deduce that the variance of the intensity is sigma^2 = {I^2} - {I}^2 = {I}^2

p(I) = 1/{I}exp(-1/{I})

d) Calculate the probability p(Ic) that the intensity exceeds a threshold intensity Ic. Note tthat in radar systems a threshold level is set so that one can decide upon the presence or absence of a target.

e) Hence deduce that the probability that the intensity I exceeds the mean intensity {I} by a factor n is
p(I) = exp(-n)

Note: Question number 2 is predicated upon Question number 1 and as such Question number 1 is ONLY shown for 'background' purposes.

Probability Density Function - Complex Gaussian Noise

1. a) I think you have done some error in integrating; I am getting a better solution.

I will use "t" for phase angle theta, and "~" for infinity.

p(t) = (1/4*pi*sigma^2)*
Integral(from 0 to ~)[exp-(I/2Sigma^2)] dI

put k = -(I/2Sigma^2)

take derivatives on both sides

dk = -dI/(2Sigma^2)
dI = -(2Sigma^2)dk
and use the fact that integral (e^k dk) = e^k

Hence, p(t) = (-2Sigma^2/4*pi*sigma^2)*
(from 0 to ~)[exp-(I/2Sigma^2)]

p(t) = -1/(2*pi)*[0-1] = 1/(2*pi)
------------------------------------------
b) p(I) = ...

Solution Summary

This solution solves for (a) the nth moment [In] of the complex Gaussian noise, (b) normalized moments, (c) variance of the intensity, (d) probability that the intensity exceeds a threshold intensity Ic and (e) the probability that the intensity I exceeds the mean intensity. This solution also provides revised figures for question 1.

A white Gaussiannoise process X(t) with power spectral density (PSD) N0/2=0.1 is input to an LTI filter with a transfer function H(f) given by
H(f) = 2, |f| <= W and 0, otherwise
The output is denoted Y(t).
(1) Find the autocorrelation function RX(t) of X(t).
(2) Find E[Y(t)].
(3) Find the PSD of Y(t).
(4) De

Suppose that an analog communications channel has an intended capacity of 10 Mbps. The bandwidth is 4 MHz. According to Shannon's law, what is the maximum signal-to-noise ratio (in dB) that can be tolerated on this channel?

Use Gaussian elimination; calculate the current, i1, i2, i3, in each branch of an electrical network that produced the following set of simultaneous equations:
2i_2 - i_2 +i_3 = -2
i_1 + 2i_2 + 3i_3 = -1
2i_1 + 2i_2 - i_3 = 8

X+Y+2z=6
3X+2Y+Z=9
X-Y=4
Use the system in above. Without interchanging any of the rows in the augmented matrix, what is the first value, which will be replaced with zero when using the Gaussian Elimination method?

I need some help completing the following:
Find the nodes x_i and the weights w_i so that the Gaussian quadrature of the sum from i=1 to 2 of (w_i) f(x_i) approximating the integral from -1 to 1 of f(x)dx is exact when f(x) is a polynomial of as high degree as possible.

Let X and Y be two jointly distributed Gaussian random variables with means ηX and ηY and variance σ2X and σ2Y respectively. The correlation coefficient between X and Y is ρ. Let Z be a new random variable defined by Z = X-Y. Is Z Gaussian? What is the variance of Z?

1. Use Gaussian elimination to complete this system of equations.
2x+3y+z-w=4
X+ y -z+w=7
3x-2y+3z-2w=-10
x-y-z+w=3.
2. Use gauss-jordan elimination to solve this system of equation.
a. x+2y+z=-2
x+y-2z+5
2x-y+z=3
b. X+ y+ z=1
2x+3y-4z=6

Approximate the following integrals using Gaussian quadrature with n = 3 and n = 4 then compare your results to the exact values of the integrals.
See attached file for full problem description.