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Homework 2 Fall 2016 AERE432 Due 9/16(F)
Name ______________________________
Problem 1. (20pts) Often times, the noise that corrupts a given signal is assumed to be white noise. Consider a
bandlimited white noise random process N (t ) that has mean  N  0 and standard deviation  N  0.1volts . Its bandwidth
(BW) extends to 10 MHz.
(a)(10pts) Plot its power spectral density (psd), including numbers related to the above given information. Moreover,
explain how you arrived at those numbers. [Also, plot the 2-sided psd, and be sure to include units.]
Solution: The noise power is  N2  0.01volts2 . This must equal the area associated with S N ( f ) . Hence,.
Figure 1.1 Plot of the noise psd.

(b)(10pts) Recall that R N ( ) 
S
N
( f ) e i 2
f
df . Use this formula to compute an explicit expression for RN ( ) .

Solution:
(c)(5pts) Extra Credit Compute a plot of your expression for RN ( ) . [Include your code in the Appendix.] Then validate
your plot by noting the value of RN (0) .
Answer:
Figure 1.2 Plot of RN ( ) .
2
RY ( )
Problem 2. (15pts) It is suggested that a certain real
process, Y (t ) , has the autocorrelation function shown at the right. Is this
possible? Justify your answer by calculating and then plotting the psd.
 Y2
1

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Figure 2.1 Plot of the process autocorrelation.
Solution:
Figure 2.1 Plot of the process psd.
Problem 3. A wss process, X (t ) , has RX ( )   X2 e  | | . For Y (t )  aX (t )  b , obtain the expression for RY ( ) .
Solution:
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Problem 4. (30pts) When using an atomic force microscope, it is essential that the scope base be as stable as possible.
This problem addresses two wss discrete-time random process models for the vibration, X (k ) , of the base.
(a)(5pts) Assume that X (k ) is zero mean white noise with variance  X2  9 . Compute (i) R X (m) , and from your
expression compute (ii) S X () . [This is a typical model choice of researchers not familiar with random processes.]
Solution:
(i)
(ii)
(b)(10pts) Assume that X (k ) is zero mean non-white noise with variance  X2  9 . Specifically, assume that:
X (k )  0.8 X (k  1)  W (k )
(1)
 W2
where W (k ) is a white noise process. Find the numerical value for
in the following manner:
First, multiply (1) by X (k ) , and then take the expected value of this equation. Second, multiply (1) by X (k  m) for m  1 ,
and then take the expected value of this equation. For m  1 you should end up with two equations in the unknowns
RX (1) and  W2 , from which you can easily arrive at a numerical value for  W2 .
Solution:
(c)(10pts) Overlay plots a sample realization of {X (k )}100
k 1 for each model. Then comment on how they visually differ.
[Note: Include your code in the Appendix. Also, choose the initial condition so that the process is, indeed, wss.]
Solution:
Figure 4.1 Plots of an n=1000 partial realization for X1 (model 1) and X2 (model 2).
COMMENT:
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Problem 5. (20pts) This problem addresses the data generated in Problem 4 in greater detail. Recall that the

autocorrelation function for a wss process is defined as: R X (m)  E ( X k X k  m ) . The process is said to be ergodic if:
1
n  n
lim
n


X k X k m

k 1

lim R X (m)
n 
pr

(5.1)
R X ( m)

where the equality in in relation probability. The quantity RX (m) is called the lagged-product estimator of R X (m) .
(a)(15pts) Write your own Matlab code for computing R X (m ) . Then use it to obtain plots of RX (m)m20 for each data


20
set in Problem 4. Then overlay plots of R X ( m )20
m  20 .
Solution:
Figure 5.1 Plots of the lagged-product autocorrelations (solid lines) and true autocorrelations (dashed lines) for the
models in Problem 3.
(b)(5pts) Discuss how the plots give a more rigorous basis to your comment in Problem 4(c).
Discussion:
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Appendix