Let N(t) be a Poisson point process with parameter À > 0. Define {X(t)}t>o to be the random process that takes ±1 values such that P(X(0) = 1) = P(X(0) = − 1) = ½, and X(t) changes the polarity when N(t) increases, i.e., X(t) = (−1)N(¹) X (0). (a) Show that such a process is WSS by showing that E[X(t)] = 0 for all t and E[X(t₁)X(t₂)] = e-2x|t1-t₂l hint 1: X(t₁)X(t₂) = 1 or −1 depending on N(t₂) —– N(t₁) being even or odd, respectively. hint 2: for any (positive or negative) real number a, eª = (b) Find the PSD of X(t). (c) Suppose that we use an ideal low-pass filter with the frequency response H(f) = filter X(t) and obtain the random process Y(t). Find the average power of Y(t), i.e., E[Y²(t)]. [₁ |f|<1 |f|≥1 to
Let N(t) be a Poisson point process with parameter À > 0. Define {X(t)}t>o to be the random process that takes ±1 values such that P(X(0) = 1) = P(X(0) = − 1) = ½, and X(t) changes the polarity when N(t) increases, i.e., X(t) = (−1)N(¹) X (0). (a) Show that such a process is WSS by showing that E[X(t)] = 0 for all t and E[X(t₁)X(t₂)] = e-2x|t1-t₂l hint 1: X(t₁)X(t₂) = 1 or −1 depending on N(t₂) —– N(t₁) being even or odd, respectively. hint 2: for any (positive or negative) real number a, eª = (b) Find the PSD of X(t). (c) Suppose that we use an ideal low-pass filter with the frequency response H(f) = filter X(t) and obtain the random process Y(t). Find the average power of Y(t), i.e., E[Y²(t)]. [₁ |f|<1 |f|≥1 to
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![. Let N(t) be a Poisson point process with parameter d > 0. Define {X(t)}t>o to be the random process
that takes +1 values such that P(X(0) = 1) = P(X(0) = –1) = }, and X(t) changes the polarity
when N(t) increases, i.e., X(t) = (-1)N()X(0).
%3D
(a) Show that such a process is WSS by showing that E[X(t)] = 0 for all t and E[X(t1)X(t2)] =
hint 1: X(t1)X(t2) = 1 or –1 depending on N(t2) – N(t1) being even or odd, respectively.
hint 2: for any (positive or negative) real number a, eª = Eo
(b) Find the PSD of X(t).
1 [f|< 1
(c) Suppose that we use an ideal low-pass filter with the frequency response H(f) =
to
|0 ]f| >1
filter X(t) and obtain the random process Y(t). Find the average power of Y (t), i.e., E[Y²(t)].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2F09d0e91e-a97d-47ab-82db-2c816b08ba5b%2Fkmppbtd_processed.png&w=3840&q=75)
Transcribed Image Text:. Let N(t) be a Poisson point process with parameter d > 0. Define {X(t)}t>o to be the random process
that takes +1 values such that P(X(0) = 1) = P(X(0) = –1) = }, and X(t) changes the polarity
when N(t) increases, i.e., X(t) = (-1)N()X(0).
%3D
(a) Show that such a process is WSS by showing that E[X(t)] = 0 for all t and E[X(t1)X(t2)] =
hint 1: X(t1)X(t2) = 1 or –1 depending on N(t2) – N(t1) being even or odd, respectively.
hint 2: for any (positive or negative) real number a, eª = Eo
(b) Find the PSD of X(t).
1 [f|< 1
(c) Suppose that we use an ideal low-pass filter with the frequency response H(f) =
to
|0 ]f| >1
filter X(t) and obtain the random process Y(t). Find the average power of Y (t), i.e., E[Y²(t)].
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 33 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman