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ECE 459 Fall 2000
Handout # 2 August 29, 2000
HOMEWORK ASSIGNMENT 1
Reading: ECE 434 notes, lecture notes (lectures 1-3), Proakis (Chapter 4), papers referenced in lecture notes.
Note: The first 4 questions are meant to review your background in probability and random processes (ECE 434 material). The last 4 questions are on complex baseband representations.
Due Date: Tuesday, September 5, 2000 (in class)
- (17 pts total) Random vectors and covariance matrices.
(a) (4 pts) Let X be a random n-vector with mean m and covariance matrix Σ. Give expressions for the mean and variance of
∑n i=1 aiYi^ in terms of^ m^ and^ Σ. (b) (4 pts) Using the above result (or by other means), show that for mutually uncorrelated random variables, the variance of the sum is the sum of the variances. (c) (2 pts) Is the vector [1 − 1 − 1]>^ a right eigenvector of the following matrix?
(d) (7 pts) Consider a random vector X = [X 1 X 2 ]>^ with mean [1 2]>^ and covariance matrix
[
]
Form the vector Y = [Y 1 Y 2 ]>^ as Y = AX + b. Find A and b such that Y is zero mean with covariance matrix I. (Hint: Diagonalize Σ.)
- (15 pts total) Bounds on the Q function.
Q(x) =
x
e−t (^2) / 2 √ 2 π
dt
(a) (8 pts) For x > 0 show that the following upper and lower bounds hold for the Q function: ( 1 −
x^2
e−x (^2) / 2
x
2 π
≤ Q(x) ≤ e−x (^2) / 2
x
2 π
Hint: For the upper bound, write the integrand as a product of 1/t and te−t (^2) / 2 , use in- tegration by parts, and bound. For the lower bound, integrate by parts once more and bound.
(b) (7 pts) As you know from your undergraduate communications course, the bit error proba- bility for BPSK signaling in additive white Gaussian noise (AWGN) with PSD N 0 /2 is given by:
Pe = Q
2 Eb N 0
where Eb is the bit energy. Plot the error probabililty Pe (on a log scale) versus signal-to-noise ratio Eb/N 0 (in dB) using Matlab or Mathematica. (You may need to use an appropriately modified version of the error function in these packages.) Consider Eb/N 0 ranging from −5 dB to 15 dB. Also plot the bounds and compare.
- (20 pts total) Rayleigh and Ricean Random Variables. Let X and Y be independent N (0, σ^2 ) random variables, and define R and Θ by:
R =
X^2 + Y 2 , and Θ = tan−^1
Y
X
(Assume Θ ∈ [−π, π].)
(a) (4 pts) Find the joint pdf of R and Θ. Are R and Θ independent? (b) (4 pts) Find the marginal pdf’s of R and Θ. (The pdf of R is called a Rayleigh pdf). (c) (4 pts) Find the pdf of R^2. (You should see that it is an exponential.) (d) (8 pts) Now assume that X ∼ N (a, σ^2 ) and X ∼ N (b, σ^2 ) independent, where a and b are deterministic and possibly nonzero. Find the joint pdf of R and Θ and from this the marginal pdf of R. Express the latter in terms of the modified Bessel function of the first kind:
I 0 (x) =
2 π
∫ (^) π
−π
exp(x cos φ)dφ
The pdf of R in this case is called Ricean.
- (18 pts total) Random Processes
(a) (4 pts) Let {X(t)} be a WSS process with ACF RX (τ ). Find E
[
(X(1) + X(2))^2
]
(b) (6 pts) Suppose V is a zero-mean Gaussian random variable and define the processes X(t) = V t and Y (t) = V 2 t, for −∞ < t < ∞. (a) Find the crosscorrelation function RX,Y (t + τ, t). (b) Are the two random processes jointly WSS? (c) Are the two random processes uncorrelated? (c) (4 pts) A random process {X(t)} is given by
X(t) = cos(2πf 0 t + Θ)
where Θ is uniformly distributed on [−π, π]. Find the power spectral density SX (f ). (d) (4 pts) An ideal white noise process {n(t)} with PSD Sn(f ) = N 20 is input to linear system with impulse response h(t) = e−tu(t). Let {Y (t)} denote the output process. Find SY (f ) and RY (τ ).
