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EASWARI ENGINEERING COLLEGE (AUTONOMOUS),
CHENNAI – 89.
Team Members:
310622106148 Sri Durga R
310622106156 Srivatsan SK
310622106157 Stephanie A P
310622106168 Thanga Mari G
310622106178 Vasu V S
310622106179 Vijayakumar NK
Subject: Probability & Random Process
Subject Code: 191MAB404T
Group No.: 3
Date: 24-5-2024
AGENDA
Uniform & Exponential Distribution
• Introduction - Continuous Distribution
• Uniform Distribution
• Application based problem on Uniform Distribution
• Exponential Distribution
• Application based problem on Exponential Distribution
• Overview of Distribution
• Cross Power Spectral Density(CSD)
• Properties of CSD
• Association of CSD with PSD
• Applications of CSD
• Antenna Design
• Noise Source Identification
• Conclusion
2
Continuous Distribution
3
Uniform & Exponential Distribution
 A continuous distribution describes the
probabilities of the possible values of a
continuous random variable.
 Probabilities of continuous random
variables (X) are defined as the area under
the curve of its PDF.
Introduction
4
Uniform & Exponential Distribution
The uniform probability distribution
function is defined as
Uniform Distribution
5
Uniform & Exponential Distribution
The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between
zero and 15 minutes, inclusive.
Problem on Uniform Distribution
6
Uniform & Exponential Distribution
7
Uniform & Exponential Distribution
Exponential Distribution
The probability distribution function
of exponentially distributed random
variable is defined as
8
Uniform & Exponential Distribution
On the average, a certain computer part lasts
ten years. The length of time the computer part
lasts is exponentially distributed.
a) What is the probability that a computer part
lasts more than 7 years?
b) On the average, how long would five
computer parts last if they are used one after
another?
Problem on Exponential Distribution
9
Uniform & Exponential Distribution
c) Eighty percent of computer parts last at most how long?
d) What is the probability that a computer part lasts between nine and 11 years?
10
Uniform Distribution:
1. Equal Probability: All outcomes have an equal chance of occurring.
2. Continuous or Discrete: Can be used for both continuous and discrete data.
3. Applications: Used when there is no preference for any particular outcome.
Exponential Distribution:
1. Decreasing Probability: Small values are more likely than larger values.
2. Continuous: Used for continuous data.
3. Memoryless: The probability of an event occurring in the future is not affected
by the past.
4. Applications: Used to model the time between events in a Poisson process.
Uniform & Exponential Distribution
Overview of Distributions
11
 In signal processing, Cross
Spectral Density (CSD) is a
measure of the correlation
between two signals in the
frequency domain.
 It is used to determine how much
two signals are related to each
other in terms of their frequency
content.
Applications of Cross Power Spectral Density
Cross Power Spectral Density
 The cross power spectral density or
cross power spectrum 𝑆𝑋𝑌(𝜔) of two
continuous random process {X(t)}
and {Y(t)} is defined as the Fourier
transform of R𝑋𝑌(𝜏).
12
If 𝑆𝑋𝑌(𝜔) of two continuous random process {X(t)} and {Y(t)},
then the following properties hold:
Applications of Cross Power Spectral Density
Properties of CSD
13
 The PSD represents the distribution of a signal over a frequency spectrum. The
magnitude, or power, of the PSD is the mean-square value of the signal.
 The cross-spectral density provides similar information as the PSD but presents
it as a statistic for a pair of signals.
 In this case, cross-correlation is used to determine the power of the pair of
signals, hence the word cross. The CSD may also be called the cross power
spectral density.
Applications of Cross Power Spectral Density
Association of CSD with PSD
14
1.Vibration Analysis:
1.Machine Health
Monitoring
2.Structural Analysis
2.Acoustics:
1.Noise Source Identification
2.Room Acoustics
3.Oceanography:
1.Wave Prediction
2.Current Mapping
4.Meteorology:
1.Climate Analysis
2.Wind Speed Correlation
Applications of Cross Power Spectral Density
Applications of CSD
5.Electrical Engineering:
1.Power Quality Analysis
2.Signal Integrity
6.Biomedical Engineering:
1.Brain Signal Analysis
2.Heart Rate Variability
7.Geophysics:
1.Seismic Data Analysis
2.Resource Exploration
8.Telecommunications:
1.Network Analysis
2.Antenna Design
15
Applications of Cross Power Spectral Density
Antenna Design
16
Applications of Cross Power Spectral Density
Noise Source Identification
17
Overall, the Cross-Spectral Density is a powerful analytical tool
that enhances the understanding of the relationship between
signals in the frequency domain, providing critical insights for
designing and optimizing various signal processing and
communication systems.
Conclusion
18
 Continuous and discrete probability distributions – Minitab
 The Uniform Distribution - Statistics LibreTexts
 The Exponential Distribution - Statistics LibreTexts
 sixsigmastudyguide.com/exponential-distribution
 What is the Cross Spectral Density (CSD)? - Vibration Research
 Cross Power Spectral Density Spectrum for Noise Modelling and
Filter Design (cadence.com)
Reference
THANK YOU

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Uniform and exponential distribution ppt

  • 1. EASWARI ENGINEERING COLLEGE (AUTONOMOUS), CHENNAI – 89. Team Members: 310622106148 Sri Durga R 310622106156 Srivatsan SK 310622106157 Stephanie A P 310622106168 Thanga Mari G 310622106178 Vasu V S 310622106179 Vijayakumar NK Subject: Probability & Random Process Subject Code: 191MAB404T Group No.: 3 Date: 24-5-2024
  • 2. AGENDA Uniform & Exponential Distribution • Introduction - Continuous Distribution • Uniform Distribution • Application based problem on Uniform Distribution • Exponential Distribution • Application based problem on Exponential Distribution • Overview of Distribution • Cross Power Spectral Density(CSD) • Properties of CSD • Association of CSD with PSD • Applications of CSD • Antenna Design • Noise Source Identification • Conclusion 2
  • 3. Continuous Distribution 3 Uniform & Exponential Distribution  A continuous distribution describes the probabilities of the possible values of a continuous random variable.  Probabilities of continuous random variables (X) are defined as the area under the curve of its PDF. Introduction
  • 4. 4 Uniform & Exponential Distribution The uniform probability distribution function is defined as Uniform Distribution
  • 5. 5 Uniform & Exponential Distribution The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. Problem on Uniform Distribution
  • 6. 6 Uniform & Exponential Distribution
  • 7. 7 Uniform & Exponential Distribution Exponential Distribution The probability distribution function of exponentially distributed random variable is defined as
  • 8. 8 Uniform & Exponential Distribution On the average, a certain computer part lasts ten years. The length of time the computer part lasts is exponentially distributed. a) What is the probability that a computer part lasts more than 7 years? b) On the average, how long would five computer parts last if they are used one after another? Problem on Exponential Distribution
  • 9. 9 Uniform & Exponential Distribution c) Eighty percent of computer parts last at most how long? d) What is the probability that a computer part lasts between nine and 11 years?
  • 10. 10 Uniform Distribution: 1. Equal Probability: All outcomes have an equal chance of occurring. 2. Continuous or Discrete: Can be used for both continuous and discrete data. 3. Applications: Used when there is no preference for any particular outcome. Exponential Distribution: 1. Decreasing Probability: Small values are more likely than larger values. 2. Continuous: Used for continuous data. 3. Memoryless: The probability of an event occurring in the future is not affected by the past. 4. Applications: Used to model the time between events in a Poisson process. Uniform & Exponential Distribution Overview of Distributions
  • 11. 11  In signal processing, Cross Spectral Density (CSD) is a measure of the correlation between two signals in the frequency domain.  It is used to determine how much two signals are related to each other in terms of their frequency content. Applications of Cross Power Spectral Density Cross Power Spectral Density  The cross power spectral density or cross power spectrum 𝑆𝑋𝑌(𝜔) of two continuous random process {X(t)} and {Y(t)} is defined as the Fourier transform of R𝑋𝑌(𝜏).
  • 12. 12 If 𝑆𝑋𝑌(𝜔) of two continuous random process {X(t)} and {Y(t)}, then the following properties hold: Applications of Cross Power Spectral Density Properties of CSD
  • 13. 13  The PSD represents the distribution of a signal over a frequency spectrum. The magnitude, or power, of the PSD is the mean-square value of the signal.  The cross-spectral density provides similar information as the PSD but presents it as a statistic for a pair of signals.  In this case, cross-correlation is used to determine the power of the pair of signals, hence the word cross. The CSD may also be called the cross power spectral density. Applications of Cross Power Spectral Density Association of CSD with PSD
  • 14. 14 1.Vibration Analysis: 1.Machine Health Monitoring 2.Structural Analysis 2.Acoustics: 1.Noise Source Identification 2.Room Acoustics 3.Oceanography: 1.Wave Prediction 2.Current Mapping 4.Meteorology: 1.Climate Analysis 2.Wind Speed Correlation Applications of Cross Power Spectral Density Applications of CSD 5.Electrical Engineering: 1.Power Quality Analysis 2.Signal Integrity 6.Biomedical Engineering: 1.Brain Signal Analysis 2.Heart Rate Variability 7.Geophysics: 1.Seismic Data Analysis 2.Resource Exploration 8.Telecommunications: 1.Network Analysis 2.Antenna Design
  • 15. 15 Applications of Cross Power Spectral Density Antenna Design
  • 16. 16 Applications of Cross Power Spectral Density Noise Source Identification
  • 17. 17 Overall, the Cross-Spectral Density is a powerful analytical tool that enhances the understanding of the relationship between signals in the frequency domain, providing critical insights for designing and optimizing various signal processing and communication systems. Conclusion
  • 18. 18  Continuous and discrete probability distributions – Minitab  The Uniform Distribution - Statistics LibreTexts  The Exponential Distribution - Statistics LibreTexts  sixsigmastudyguide.com/exponential-distribution  What is the Cross Spectral Density (CSD)? - Vibration Research  Cross Power Spectral Density Spectrum for Noise Modelling and Filter Design (cadence.com) Reference