3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker...
Transcript of 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker...
![Page 1: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/1.jpg)
Introduction Random Variables Random Processes Noise Characterization
3: Random Variables, Processes, and Noise
Y. Yoganandam, Runa Kumari, and S. R. Zinka
Department of Electrical & Electronics EngineeringBITS Pilani, Hyderbad Campus
August 10 & 12, 2015
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 2: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/2.jpg)
Introduction Random Variables Random Processes Noise Characterization
Outline
1 Introduction
2 Random Variables
3 Random Processes
4 Noise Characterization
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 3: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/3.jpg)
Introduction Random Variables Random Processes Noise Characterization
Outline
1 Introduction
2 Random Variables
3 Random Processes
4 Noise Characterization
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 4: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/4.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Inputsignal
Inputtransducer Transmitter
Transmittedsignal
Channel
Recivedsignal
Reciver
Outputsignal
Outputtransducer
Outputmessage
Inputmessage
Distortionand
noise
Noise can be internal too ...
Noise is the undesired signal that gets added to the desired signal andreaches the destination.
Noise ultimately determines the threshold for the minimum signal that canbe reliably detected by a receiver.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 5: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/5.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Inputsignal
Inputtransducer Transmitter
Transmittedsignal
Channel
Recivedsignal
Reciver
Outputsignal
Outputtransducer
Outputmessage
Inputmessage
Distortionand
noise
Noise can be internal too ...
Noise is the undesired signal that gets added to the desired signal andreaches the destination.
Noise ultimately determines the threshold for the minimum signal that canbe reliably detected by a receiver.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 6: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/6.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Inputsignal
Inputtransducer Transmitter
Transmittedsignal
Channel
Recivedsignal
Reciver
Outputsignal
Outputtransducer
Outputmessage
Inputmessage
Distortionand
noise
Noise can be internal too ...
Noise is the undesired signal that gets added to the desired signal andreaches the destination.
Noise ultimately determines the threshold for the minimum signal that canbe reliably detected by a receiver.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 7: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/7.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Inputsignal
Inputtransducer Transmitter
Transmittedsignal
Channel
Recivedsignal
Reciver
Outputsignal
Outputtransducer
Outputmessage
Inputmessage
Distortionand
noise
Noise can be internal too ...
Noise is the undesired signal that gets added to the desired signal andreaches the destination.
Noise ultimately determines the threshold for the minimum signal that canbe reliably detected by a receiver.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 8: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/8.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Noise lives with the desired signal. Neither amplification nor the filtering canalleviate the effect of noise on the desired signal.
The only way to keep away from the effects of noise is to see that less amountof noise, relative to the desired signal, is present at the destination.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 9: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/9.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Noise lives with the desired signal. Neither amplification nor the filtering canalleviate the effect of noise on the desired signal.
The only way to keep away from the effects of noise is to see that less amountof noise, relative to the desired signal, is present at the destination.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 10: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/10.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise
Noise lives with the desired signal. Neither amplification nor the filtering canalleviate the effect of noise on the desired signal.
The only way to keep away from the effects of noise is to see that less amountof noise, relative to the desired signal, is present at the destination.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 11: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/11.jpg)
Introduction Random Variables Random Processes Noise Characterization
External Noise - Sources
• Noise from stars including the sun (20MHz – 1.5GHz)• Lightning (2MHz – 10MHz)• Thermal noise from the ground• Cosmic background noise from the sky
• Man made noise, e.g., spark plugs, engine noise, etc (1MHz – 500MHz)• Radio, TV, and cellular stations• Wireless devices• Microwave ovens
• Deliberate jamming devices
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 12: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/12.jpg)
Introduction Random Variables Random Processes Noise Characterization
External Noise - Sources
• Noise from stars including the sun (20MHz – 1.5GHz)• Lightning (2MHz – 10MHz)• Thermal noise from the ground• Cosmic background noise from the sky
• Man made noise, e.g., spark plugs, engine noise, etc (1MHz – 500MHz)• Radio, TV, and cellular stations• Wireless devices• Microwave ovens• Deliberate jamming devices
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 13: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/13.jpg)
Introduction Random Variables Random Processes Noise Characterization
External Noise - Sources
• Noise from stars including the sun (20MHz – 1.5GHz)• Lightning (2MHz – 10MHz)• Thermal noise from the ground• Cosmic background noise from the sky• Man made noise, e.g., spark plugs, engine noise, etc (1MHz – 500MHz)• Radio, TV, and cellular stations• Wireless devices• Microwave ovens• Deliberate jamming devices
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 14: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/14.jpg)
Introduction Random Variables Random Processes Noise Characterization
Internal Noise - Sources
• Thermal noise is the most basic type of noise, being caused by thermalvibration of bound charges. It is also known as Johnson or Nyquistnoise.
• Shot noise is due to random fluctuations of charge carriers in an electrontube or solid-state device.
• Flicker noise occurs in solid-state components and vacuum tubes.Flicker noise power varies inversely with frequency, and so is oftencalled 1/f noise.
• Plasma noise is caused by random motion of charges in an ionized gas,such as a plasma, the ionosphere, or sparking electrical contacts.
• Quantum noise results from the quantized nature of charge carriers andphotons; it is often insignificant relative to other noise sources.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 15: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/15.jpg)
Introduction Random Variables Random Processes Noise Characterization
Internal Noise - Sources
• Thermal noise is the most basic type of noise, being caused by thermalvibration of bound charges. It is also known as Johnson or Nyquistnoise.
• Shot noise is due to random fluctuations of charge carriers in an electrontube or solid-state device.
• Flicker noise occurs in solid-state components and vacuum tubes.Flicker noise power varies inversely with frequency, and so is oftencalled 1/f noise.
• Plasma noise is caused by random motion of charges in an ionized gas,such as a plasma, the ionosphere, or sparking electrical contacts.
• Quantum noise results from the quantized nature of charge carriers andphotons; it is often insignificant relative to other noise sources.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 16: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/16.jpg)
Introduction Random Variables Random Processes Noise Characterization
Internal Noise - Sources
• Thermal noise is the most basic type of noise, being caused by thermalvibration of bound charges. It is also known as Johnson or Nyquistnoise.
• Shot noise is due to random fluctuations of charge carriers in an electrontube or solid-state device.
• Flicker noise occurs in solid-state components and vacuum tubes.Flicker noise power varies inversely with frequency, and so is oftencalled 1/f noise.
• Plasma noise is caused by random motion of charges in an ionized gas,such as a plasma, the ionosphere, or sparking electrical contacts.
• Quantum noise results from the quantized nature of charge carriers andphotons; it is often insignificant relative to other noise sources.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 17: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/17.jpg)
Introduction Random Variables Random Processes Noise Characterization
Internal Noise - Sources
• Thermal noise is the most basic type of noise, being caused by thermalvibration of bound charges. It is also known as Johnson or Nyquistnoise.
• Shot noise is due to random fluctuations of charge carriers in an electrontube or solid-state device.
• Flicker noise occurs in solid-state components and vacuum tubes.Flicker noise power varies inversely with frequency, and so is oftencalled 1/f noise.
• Plasma noise is caused by random motion of charges in an ionized gas,such as a plasma, the ionosphere, or sparking electrical contacts.
• Quantum noise results from the quantized nature of charge carriers andphotons; it is often insignificant relative to other noise sources.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 18: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/18.jpg)
Introduction Random Variables Random Processes Noise Characterization
Internal Noise - Sources
• Thermal noise is the most basic type of noise, being caused by thermalvibration of bound charges. It is also known as Johnson or Nyquistnoise.
• Shot noise is due to random fluctuations of charge carriers in an electrontube or solid-state device.
• Flicker noise occurs in solid-state components and vacuum tubes.Flicker noise power varies inversely with frequency, and so is oftencalled 1/f noise.
• Plasma noise is caused by random motion of charges in an ionized gas,such as a plasma, the ionosphere, or sparking electrical contacts.
• Quantum noise results from the quantized nature of charge carriers andphotons; it is often insignificant relative to other noise sources.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 19: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/19.jpg)
Introduction Random Variables Random Processes Noise Characterization
Internal Noise - Sources
• Thermal noise is the most basic type of noise, being caused by thermalvibration of bound charges. It is also known as Johnson or Nyquistnoise.
• Shot noise is due to random fluctuations of charge carriers in an electrontube or solid-state device.
• Flicker noise occurs in solid-state components and vacuum tubes.Flicker noise power varies inversely with frequency, and so is oftencalled 1/f noise.
• Plasma noise is caused by random motion of charges in an ionized gas,such as a plasma, the ionosphere, or sparking electrical contacts.
• Quantum noise results from the quantized nature of charge carriers andphotons; it is often insignificant relative to other noise sources.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 20: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/20.jpg)
Introduction Random Variables Random Processes Noise Characterization
Desired vs Undesired Noise
In some cases, such as radiometers or radio astronomy systems, the desiredsignal is actually the noise power received by an antenna, and it is necessary
to distinguish between the received noise power and the undesired noisegenerated by the receiver system itself.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 21: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/21.jpg)
Introduction Random Variables Random Processes Noise Characterization
As you have seen, there are many types of noise sources. However, we willconcentrate more on thermal noise. Can you guess the reason?
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 22: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/22.jpg)
Introduction Random Variables Random Processes Noise Characterization
Before proceeding further, let’s recap the theory of random variables andprocesses ...
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 23: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/23.jpg)
Introduction Random Variables Random Processes Noise Characterization
Outline
1 Introduction
2 Random Variables
3 Random Processes
4 Noise Characterization
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 24: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/24.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 25: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/25.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 26: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/26.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 27: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/27.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 28: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/28.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 29: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/29.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 30: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/30.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 31: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/31.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 32: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/32.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Variable
Outcome of a random experi-ment could be numerical or non-numerical.
Non-numerical example: coin toss-ing experiment – Head or Tail.
For non numerical outcomes, onecan assign certain numbers.
We may assign 1 for Head and -1for Tail.
When such numerical values areassigned to a variable, the variableis called a random variable.
For the coin tossing experiment,variable x can take either 1 or -1 de-pending on the outcome.
Probability of a random variable xtaking values xi is Px(xi).
Discrete random variable: If {xi}are distinct , then x is a discrete ran-dom variable, such that
∑i
Px (xi) = 1.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 33: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/33.jpg)
Introduction Random Variables Random Processes Noise Characterization
Continuous Random Variables
Continuous RV can assume any value in a given interval.
Probability density function (PDF) px (x) is the appropriate definition for con-tinuous random variables.
Probability for a continuous RV is defined in terms of PDF as
P (x1 ≤ x ≤ x2) =
ˆ x2
x1
px (x) dx. (1)
Of course, ˆ ∞
−∞px (x) dx = 1. (2)
And cumulative distribution function (CDF) is defined as
Fx (x) = P (x ≤ x) =ˆ x
−∞px (x) dx. (3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 34: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/34.jpg)
Introduction Random Variables Random Processes Noise Characterization
Continuous Random Variables
Continuous RV can assume any value in a given interval.
Probability density function (PDF) px (x) is the appropriate definition for con-tinuous random variables.
Probability for a continuous RV is defined in terms of PDF as
P (x1 ≤ x ≤ x2) =
ˆ x2
x1
px (x) dx. (1)
Of course, ˆ ∞
−∞px (x) dx = 1. (2)
And cumulative distribution function (CDF) is defined as
Fx (x) = P (x ≤ x) =ˆ x
−∞px (x) dx. (3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 35: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/35.jpg)
Introduction Random Variables Random Processes Noise Characterization
Continuous Random Variables
Continuous RV can assume any value in a given interval.
Probability density function (PDF) px (x) is the appropriate definition for con-tinuous random variables.
Probability for a continuous RV is defined in terms of PDF as
P (x1 ≤ x ≤ x2) =
ˆ x2
x1
px (x) dx. (1)
Of course, ˆ ∞
−∞px (x) dx = 1. (2)
And cumulative distribution function (CDF) is defined as
Fx (x) = P (x ≤ x) =ˆ x
−∞px (x) dx. (3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 36: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/36.jpg)
Introduction Random Variables Random Processes Noise Characterization
Continuous Random Variables
Continuous RV can assume any value in a given interval.
Probability density function (PDF) px (x) is the appropriate definition for con-tinuous random variables.
Probability for a continuous RV is defined in terms of PDF as
P (x1 ≤ x ≤ x2) =
ˆ x2
x1
px (x) dx. (1)
Of course, ˆ ∞
−∞px (x) dx = 1. (2)
And cumulative distribution function (CDF) is defined as
Fx (x) = P (x ≤ x) =ˆ x
−∞px (x) dx. (3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 37: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/37.jpg)
Introduction Random Variables Random Processes Noise Characterization
Continuous Random Variables
Continuous RV can assume any value in a given interval.
Probability density function (PDF) px (x) is the appropriate definition for con-tinuous random variables.
Probability for a continuous RV is defined in terms of PDF as
P (x1 ≤ x ≤ x2) =
ˆ x2
x1
px (x) dx. (1)
Of course, ˆ ∞
−∞px (x) dx = 1. (2)
And cumulative distribution function (CDF) is defined as
Fx (x) = P (x ≤ x) =ˆ x
−∞px (x) dx. (3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 38: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/38.jpg)
Introduction Random Variables Random Processes Noise Characterization
Continuous Random Variables
Continuous RV can assume any value in a given interval.
Probability density function (PDF) px (x) is the appropriate definition for con-tinuous random variables.
Probability for a continuous RV is defined in terms of PDF as
P (x1 ≤ x ≤ x2) =
ˆ x2
x1
px (x) dx. (1)
Of course, ˆ ∞
−∞px (x) dx = 1. (2)
And cumulative distribution function (CDF) is defined as
Fx (x) = P (x ≤ x) =ˆ x
−∞px (x) dx. (3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 39: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/39.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Means
For a continuous RV case, the mean is given by
x̄ = E [x] =ˆ ∞
−∞xpx (x) dx. (4)
Mean of a function (y = g (x)) of a random variable is
g (x) = E [g (x)] =ˆ ∞
−∞g (x) px (x) dx. (5)
The above expression can be generalized for two random variables is of arandom variable as
g (x, y) = E [g (x, y)] =ˆ ∞
−∞
ˆ ∞
−∞g (x, y) pxy (x, y) dx. (6)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 40: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/40.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Means
For a continuous RV case, the mean is given by
x̄ = E [x] =ˆ ∞
−∞xpx (x) dx. (4)
Mean of a function (y = g (x)) of a random variable is
g (x) = E [g (x)] =ˆ ∞
−∞g (x) px (x) dx. (5)
The above expression can be generalized for two random variables is of arandom variable as
g (x, y) = E [g (x, y)] =ˆ ∞
−∞
ˆ ∞
−∞g (x, y) pxy (x, y) dx. (6)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 41: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/41.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Means
For a continuous RV case, the mean is given by
x̄ = E [x] =ˆ ∞
−∞xpx (x) dx. (4)
Mean of a function (y = g (x)) of a random variable is
g (x) = E [g (x)] =ˆ ∞
−∞g (x) px (x) dx. (5)
The above expression can be generalized for two random variables is of arandom variable as
g (x, y) = E [g (x, y)] =ˆ ∞
−∞
ˆ ∞
−∞g (x, y) pxy (x, y) dx. (6)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 42: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/42.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Means
For a continuous RV case, the mean is given by
x̄ = E [x] =ˆ ∞
−∞xpx (x) dx. (4)
Mean of a function (y = g (x)) of a random variable is
g (x) = E [g (x)] =ˆ ∞
−∞g (x) px (x) dx. (5)
The above expression can be generalized for two random variables is of arandom variable as
g (x, y) = E [g (x, y)] =ˆ ∞
−∞
ˆ ∞
−∞g (x, y) pxy (x, y) dx. (6)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 43: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/43.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Moments
The nth moment of a random variable x is defined as
xn =
ˆ ∞
−∞xnpx (x) dx. (7)
Similarly, the nth central moment of a random variable x is defined as
(x− x)n =
ˆ ∞
−∞(x− x)n px (x) dx. (8)
The second central moment of an RV x is called variance and denoted by σ2x ,
where σx is known as standard deviation.
σ2x = (x− x)2
= x2 − 2x2 + x2 = x2 − x2 (9)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 44: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/44.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Moments
The nth moment of a random variable x is defined as
xn =
ˆ ∞
−∞xnpx (x) dx. (7)
Similarly, the nth central moment of a random variable x is defined as
(x− x)n =
ˆ ∞
−∞(x− x)n px (x) dx. (8)
The second central moment of an RV x is called variance and denoted by σ2x ,
where σx is known as standard deviation.
σ2x = (x− x)2
= x2 − 2x2 + x2 = x2 − x2 (9)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 45: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/45.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Moments
The nth moment of a random variable x is defined as
xn =
ˆ ∞
−∞xnpx (x) dx. (7)
Similarly, the nth central moment of a random variable x is defined as
(x− x)n =
ˆ ∞
−∞(x− x)n px (x) dx. (8)
The second central moment of an RV x is called variance and denoted by σ2x ,
where σx is known as standard deviation.
σ2x = (x− x)2
= x2 − 2x2 + x2 = x2 − x2 (9)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 46: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/46.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Moments
The nth moment of a random variable x is defined as
xn =
ˆ ∞
−∞xnpx (x) dx. (7)
Similarly, the nth central moment of a random variable x is defined as
(x− x)n =
ˆ ∞
−∞(x− x)n px (x) dx. (8)
The second central moment of an RV x is called variance and denoted by σ2x ,
where σx is known as standard deviation.
σ2x = (x− x)2
= x2 − 2x2 + x2 = x2 − x2 (9)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 47: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/47.jpg)
Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Corollary
If x and x are independent RVs and
z = x + y, (10)
thenz = x + y (11)
σ2z = σ2
x + σ2y . (12)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Corollary
If x and x are independent RVs and
z = x + y, (10)
thenz = x + y (11)
σ2z = σ2
x + σ2y . (12)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Statistical Averages – Corollary
If x and x are independent RVs and
z = x + y, (10)
thenz = x + y (11)
σ2z = σ2
x + σ2y . (12)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 50: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/50.jpg)
Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem
If x and x are independent RVs and
z = x + y, (13)
then
pz (z) =ˆ ∞
−∞px (x) py (z− x) dx. (14)
From the above equation, it is clear that the PDF pz (z) is the convolution ofPDFs px (x) and py (y) . This result can be extended to n number of RVs.
Under certain conditions, sum of large number of independent random vari-ables tends to be a Gaussian random variable, independent of the PDFs of therandom variables involved. This is the central-limit theorem.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem
If x and x are independent RVs and
z = x + y, (13)
then
pz (z) =ˆ ∞
−∞px (x) py (z− x) dx. (14)
From the above equation, it is clear that the PDF pz (z) is the convolution ofPDFs px (x) and py (y) . This result can be extended to n number of RVs.
Under certain conditions, sum of large number of independent random vari-ables tends to be a Gaussian random variable, independent of the PDFs of therandom variables involved. This is the central-limit theorem.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 52: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/52.jpg)
Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem
If x and x are independent RVs and
z = x + y, (13)
then
pz (z) =ˆ ∞
−∞px (x) py (z− x) dx. (14)
From the above equation, it is clear that the PDF pz (z) is the convolution ofPDFs px (x) and py (y) . This result can be extended to n number of RVs.
Under certain conditions, sum of large number of independent random vari-ables tends to be a Gaussian random variable, independent of the PDFs of therandom variables involved. This is the central-limit theorem.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 53: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/53.jpg)
Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem
If x and x are independent RVs and
z = x + y, (13)
then
pz (z) =ˆ ∞
−∞px (x) py (z− x) dx. (14)
From the above equation, it is clear that the PDF pz (z) is the convolution ofPDFs px (x) and py (y) . This result can be extended to n number of RVs.
Under certain conditions, sum of large number of independent random vari-ables tends to be a Gaussian random variable, independent of the PDFs of therandom variables involved. This is the central-limit theorem.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 54: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/54.jpg)
Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem – Demonstration
px(x)
x1-1
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem – Demonstration
px(x)
x1-1
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem – Demonstration
px(x) px(x) * px(x)
x x1-1 2-2
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Central-Limit Theorem – Demonstration
px(x) px(x) * px(x) px(x) * px(x) * px(x)
x xx1-1 2-2 -3 3
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ= 2 1.0,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 59: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/59.jpg)
Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ= 2 1.0,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 60: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/60.jpg)
Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ=0,μ=
2 0.2,σ =2 1.0,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 61: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/61.jpg)
Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ=0,μ=0,μ=
2 0.2,σ =2 1.0,σ =2 5.0,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ=0,μ=0,μ=−2,μ=
2 0.2,σ =2 1.0,σ =2 5.0,σ =2 0.5,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
x
0.8
0.6
0.4
0.2
0.0
1.0
−5 −3 1 3 5−1 0 2 4−2−4
0,μ= 2 1.0,σ =
F x(x
)
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ=0,μ=0,μ=−2,μ=
2 0.2,σ =2 1.0,σ =2 5.0,σ =2 0.5,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF
- 1
1 2 3
- 3
- 2
- 1
x
0.8
0.6
0.4
0.2
0.0
1.0
−5 −3 1 3 5−1 0 2 4−2−4
0,μ=0,μ=0,μ=−2,μ=
2 0.2,σ =2 1.0,σ =2 5.0,σ =2 0.5,σ =
F x(x
)
p x(x
)
0.8
0.6
0.4
0.2
0.0
−5 −3 1 3 5
x
1.0
−1 0 2 4−2−4
0,μ=0,μ=0,μ=−2,μ=
2 0.2,σ =2 1.0,σ =2 5.0,σ =2 0.5,σ =
px (x) =1
σ√
2πe−(x−µ)2/2σ2
Fx (x) =1
σ√
2π
ˆ x
−∞e−(x−µ)2/2σ2
dx = 1−Q(
x− µ
σ
)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF – Interpretation
0.0
0.1
0.2
0.3
0.4
−2σ −1σ 1σ−3σ 3σ0 2σ
34.1% 34.1%
13.6%2.1%
13.6% 0.1%0.1% 2.1%
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The Gaussian (Normal) PDF – Interpretation
0.0
0.1
0.2
0.3
0.4
−2σ −1σ 1σ−3σ 3σ0 2σ
34.1% 34.1%
13.6%2.1%
13.6% 0.1%0.1% 2.1%
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Outline
1 Introduction
2 Random Variables
3 Random Processes
4 Noise Characterization
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
x(t, ζ1)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
x(t, ζ1)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
x(t, ζ1)
x(t, ζ2)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
Sample function
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
Ensemble
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
t
t
t1
x(t1) = x1
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
Randomvariable
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
t
t
t
t
t1 t2
x(t1) = x1 x(t2) = x2
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
Randomvariable
Randomvariable x isa function
of time
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process
A random variable that is a function of time is called a random process orstochastic process.
In other words, a random process is just a collection of an infinite number ofRVs, which are generally dependent.
So, a random process is completely described by the joint PDF
px1x2···xn (x1, x2, . . . , xn)
which can also be expressed as
px1x2···xn (x1, x2, . . . , xn; t1, t2, . . . , tn)
or simply
px (x; t)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process ... A Few More Things
We can always derive a lower order PDF from a higher order PDF by integra-tion. For instance,
px1 (x1) =
ˆ ∞
−∞px1x2 (x1, x2) dx2. (15)
The mean x (t) of a random process x (t) can be determined from the first-order PDF as
x (t) =ˆ ∞
−∞xpx (x; t) dx. (16)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Random Process ... A Few More Things
We can always derive a lower order PDF from a higher order PDF by integra-tion. For instance,
px1 (x1) =
ˆ ∞
−∞px1x2 (x1, x2) dx2. (15)
The mean x (t) of a random process x (t) can be determined from the first-order PDF as
x (t) =ˆ ∞
−∞xpx (x; t) dx. (16)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 86: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/86.jpg)
Introduction Random Variables Random Processes Noise Characterization
Random Process ... A Few More Things
We can always derive a lower order PDF from a higher order PDF by integra-tion. For instance,
px1 (x1) =
ˆ ∞
−∞px1x2 (x1, x2) dx2. (15)
The mean x (t) of a random process x (t) can be determined from the first-order PDF as
x (t) =ˆ ∞
−∞xpx (x; t) dx. (16)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 87: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/87.jpg)
Introduction Random Variables Random Processes Noise Characterization
A Few Other Types of Random Process
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
A Few Other Types of Random Process
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
A Few Other Types of Random Process
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
t
t
t
t
x(t, ζ1)
x(t, ζ2)
x(t, ζ3)
x(t, ζ4)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Auto-Correlation of a Random Process
x(t)
x1 x2
t1 t2
t
t
t
t
τ
Rx (t1, t2) = x (t1) x (t2) = x1x2 =
ˆ ∞
−∞
ˆ ∞
−∞x1x2px1x2 (x1, x2) dx1dx2
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Auto-Correlation of a Random Processx(t)
x1 x2
t1 t2
t
t
t
t
τ
Rx (t1, t2) = x (t1) x (t2) = x1x2 =
ˆ ∞
−∞
ˆ ∞
−∞x1x2px1x2 (x1, x2) dx1dx2
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Auto-Correlation of a Random Processx(t) y(t)
x1 x2
t1 t2 t1 t2
y1 y2
t
t
t
t
t
t
t
t
τ τ
Rx (t1, t2) = x (t1) x (t2) = x1x2 =
ˆ ∞
−∞
ˆ ∞
−∞x1x2px1x2 (x1, x2) dx1dx2
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Auto-Correlation of a Random Processx(t) y(t)
x1 x2
t1 t2 t1 t2
y1 y2
t
t
t
t
t
t
t
t
τ
Rx(τ)Ry(τ)
τ τ
Rx (t1, t2) = x (t1) x (t2) = x1x2 =
ˆ ∞
−∞
ˆ ∞
−∞x1x2px1x2 (x1, x2) dx1dx2
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 94: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/94.jpg)
Introduction Random Variables Random Processes Noise Characterization
Auto-Correlation of a Random Processx(t) y(t)
x1 x2
t1 t2 t1 t2
y1 y2
t
t
t
t
t
t
t
t
τ
Rx(τ)Ry(τ)
τ τ
Rx (t1, t2) = x (t1) x (t2) = x1x2 =
ˆ ∞
−∞
ˆ ∞
−∞x1x2px1x2 (x1, x2) dx1dx2
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 95: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/95.jpg)
Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
A random process whose statistical characteristics do not change with time isclassified as a stationary random process.
t
t
t
t
t
t
t
t
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
A random process whose statistical characteristics do not change with time isclassified as a stationary random process.
t
t
t
t
t
t
t
t
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
A random process whose statistical characteristics do not change with time isclassified as a stationary random process.
t
t
t
t
t
t
t
t
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
px (x; t) = px (x) (17)
Rx (t1, t2) = Rx (t2 − t1) (18)
Rx (τ) = x (t) x (t + τ) (19)
A process is called stationary process only when first-order as well as all thehigher order PDFs, such as px1x2 ...xn (x1, x2, . . . , xn) are all independent of thechoice of origin.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
px (x; t) = px (x) (17)
Rx (t1, t2) = Rx (t2 − t1) (18)
Rx (τ) = x (t) x (t + τ) (19)
A process is called stationary process only when first-order as well as all thehigher order PDFs, such as px1x2 ...xn (x1, x2, . . . , xn) are all independent of thechoice of origin.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 100: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/100.jpg)
Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
px (x; t) = px (x) (17)
Rx (t1, t2) = Rx (t2 − t1) (18)
Rx (τ) = x (t) x (t + τ) (19)
A process is called stationary process only when first-order as well as all thehigher order PDFs, such as px1x2 ...xn (x1, x2, . . . , xn) are all independent of thechoice of origin.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 101: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/101.jpg)
Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
px (x; t) = px (x) (17)
Rx (t1, t2) = Rx (t2 − t1) (18)
Rx (τ) = x (t) x (t + τ) (19)
A process is called stationary process only when first-order as well as all thehigher order PDFs, such as px1x2 ...xn (x1, x2, . . . , xn) are all independent of thechoice of origin.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 102: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/102.jpg)
Introduction Random Variables Random Processes Noise Characterization
Stationary Random Process
px (x; t) = px (x) (17)
Rx (t1, t2) = Rx (t2 − t1) (18)
Rx (τ) = x (t) x (t + τ) (19)
A process is called stationary process only when first-order as well as all thehigher order PDFs, such as px1x2 ...xn (x1, x2, . . . , xn) are all independent of thechoice of origin.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 103: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/103.jpg)
Introduction Random Variables Random Processes Noise Characterization
Wide-Sense Stationary Random Process
A process may not be stationary in strict sense, yet it may have a mean valueand an auto-correlation function that are independent of the shift of time ori-gin.
This meansx (t) = constant (20)
andRx (t1, t2) = Rx (τ) , τ = t2 − t1. (21)
Such a process is known as wide-sense stationary, or weakly stationary pro-cess.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Wide-Sense Stationary Random Process
A process may not be stationary in strict sense, yet it may have a mean valueand an auto-correlation function that are independent of the shift of time ori-gin.
This meansx (t) = constant (20)
andRx (t1, t2) = Rx (τ) , τ = t2 − t1. (21)
Such a process is known as wide-sense stationary, or weakly stationary pro-cess.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 105: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/105.jpg)
Introduction Random Variables Random Processes Noise Characterization
Wide-Sense Stationary Random Process
A process may not be stationary in strict sense, yet it may have a mean valueand an auto-correlation function that are independent of the shift of time ori-gin.
This meansx (t) = constant (20)
andRx (t1, t2) = Rx (τ) , τ = t2 − t1. (21)
Such a process is known as wide-sense stationary, or weakly stationary pro-cess.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 106: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/106.jpg)
Introduction Random Variables Random Processes Noise Characterization
Wide-Sense Stationary Random Process
A process may not be stationary in strict sense, yet it may have a mean valueand an auto-correlation function that are independent of the shift of time ori-gin.
This meansx (t) = constant (20)
andRx (t1, t2) = Rx (τ) , τ = t2 − t1. (21)
Such a process is known as wide-sense stationary, or weakly stationary pro-cess.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 107: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/107.jpg)
Introduction Random Variables Random Processes Noise Characterization
Ergodic Random Process
In ergodic process, ensemble averages are equal to the time averages of anysample function. Thus for an ergodic process x (t) ,
x (t) = x̃ (t)Rx (τ) = Rx (τ) ,
where
x̃ (t) = limT→∞
1T
ˆ T/2
−T/2x (t) dt
and
Rx (τ) = limT→∞
1T
ˆ T/2
−T/2x (t) x (t + τ) dt.
An Ergodic process is neccessarily a stationary process; but the converse is nottrue.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Ergodic Random Process
In ergodic process, ensemble averages are equal to the time averages of anysample function. Thus for an ergodic process x (t) ,
x (t) = x̃ (t)Rx (τ) = Rx (τ) ,
where
x̃ (t) = limT→∞
1T
ˆ T/2
−T/2x (t) dt
and
Rx (τ) = limT→∞
1T
ˆ T/2
−T/2x (t) x (t + τ) dt.
An Ergodic process is neccessarily a stationary process; but the converse is nottrue.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Ergodic Random Process
In ergodic process, ensemble averages are equal to the time averages of anysample function. Thus for an ergodic process x (t) ,
x (t) = x̃ (t)Rx (τ) = Rx (τ) ,
where
x̃ (t) = limT→∞
1T
ˆ T/2
−T/2x (t) dt
and
Rx (τ) = limT→∞
1T
ˆ T/2
−T/2x (t) x (t + τ) dt.
An Ergodic process is neccessarily a stationary process; but the converse is nottrue.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 110: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/110.jpg)
Introduction Random Variables Random Processes Noise Characterization
Ergodic Random Process
In ergodic process, ensemble averages are equal to the time averages of anysample function. Thus for an ergodic process x (t) ,
x (t) = x̃ (t)Rx (τ) = Rx (τ) ,
where
x̃ (t) = limT→∞
1T
ˆ T/2
−T/2x (t) dt
and
Rx (τ) = limT→∞
1T
ˆ T/2
−T/2x (t) x (t + τ) dt.
An Ergodic process is neccessarily a stationary process; but the converse is nottrue.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 111: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/111.jpg)
Introduction Random Variables Random Processes Noise Characterization
Why Ergodic Random Processes Notion is Important?
Because many of the stationary processes encountered in practice are ergodicwith respect to at least send-order averages, i.e., with respect to mean and
auto-correlation values.
And we need only the first- and second-order averages when we are dealingwith linear systems.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Why Ergodic Random Processes Notion is Important?
Because many of the stationary processes encountered in practice are ergodicwith respect to at least send-order averages, i.e., with respect to mean and
auto-correlation values.
And we need only the first- and second-order averages when we are dealingwith linear systems.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 113: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/113.jpg)
Introduction Random Variables Random Processes Noise Characterization
Why Ergodic Random Processes Notion is Important?
Because many of the stationary processes encountered in practice are ergodicwith respect to at least send-order averages, i.e., with respect to mean and
auto-correlation values.
And we need only the first- and second-order averages when we are dealingwith linear systems.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 114: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/114.jpg)
Introduction Random Variables Random Processes Noise Characterization
Classification of Random Processes
Random process
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Classification of Random Processes
Random process
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Classification of Random Processes
Random process
Wide-sense stationary
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Classification of Random Processes
Random process
Wide-sense stationary
Stationary
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Classification of Random Processes
Random process
Wide-sense stationary
Stationary
Ergodic
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
PSD of a Random Process
PSD of a random process x (t) is given as
Sx (ω) = limT→∞
[|XT (ω)|2
T
], (22)
where XT (ω) is the Fourier transform of the truncated random processx (t) rect (t/T) .
PSD is related to auto-correlation function as
Sx (ω) =
ˆ ∞
−∞Rx (τ) e−jωtdτ (23)
where Rx (τ) = x∗ (t) x (t + τ).
The average power of a wide-sense random process x (t) is its mean squarevalue
Px = x2 = Rx (0) =1
2π
ˆ ∞
−∞Sx (ω) dω. (24)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
PSD of a Random Process
PSD of a random process x (t) is given as
Sx (ω) = limT→∞
[|XT (ω)|2
T
], (22)
where XT (ω) is the Fourier transform of the truncated random processx (t) rect (t/T) .
PSD is related to auto-correlation function as
Sx (ω) =
ˆ ∞
−∞Rx (τ) e−jωtdτ (23)
where Rx (τ) = x∗ (t) x (t + τ).
The average power of a wide-sense random process x (t) is its mean squarevalue
Px = x2 = Rx (0) =1
2π
ˆ ∞
−∞Sx (ω) dω. (24)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 121: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/121.jpg)
Introduction Random Variables Random Processes Noise Characterization
PSD of a Random Process
PSD of a random process x (t) is given as
Sx (ω) = limT→∞
[|XT (ω)|2
T
], (22)
where XT (ω) is the Fourier transform of the truncated random processx (t) rect (t/T) .
PSD is related to auto-correlation function as
Sx (ω) =
ˆ ∞
−∞Rx (τ) e−jωtdτ (23)
where Rx (τ) = x∗ (t) x (t + τ).
The average power of a wide-sense random process x (t) is its mean squarevalue
Px = x2 = Rx (0) =1
2π
ˆ ∞
−∞Sx (ω) dω. (24)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 122: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/122.jpg)
Introduction Random Variables Random Processes Noise Characterization
PSD of a Random Process
PSD of a random process x (t) is given as
Sx (ω) = limT→∞
[|XT (ω)|2
T
], (22)
where XT (ω) is the Fourier transform of the truncated random processx (t) rect (t/T) .
PSD is related to auto-correlation function as
Sx (ω) =
ˆ ∞
−∞Rx (τ) e−jωtdτ (23)
where Rx (τ) = x∗ (t) x (t + τ).
The average power of a wide-sense random process x (t) is its mean squarevalue
Px = x2 = Rx (0) =1
2π
ˆ ∞
−∞Sx (ω) dω. (24)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 123: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/123.jpg)
Introduction Random Variables Random Processes Noise Characterization
Transmission of Random Processes Through LinearSystems
x(t) y(t)h(t)
H(ω)
Ry (τ) = h (τ) ∗ h (−τ) ∗ Rx (τ) (25)
Sy (ω) = |H (ω)|2 Sx (ω) (26)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 124: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/124.jpg)
Introduction Random Variables Random Processes Noise Characterization
Transmission of Random Processes Through LinearSystems
x(t) y(t)h(t)
H(ω)
Ry (τ) = h (τ) ∗ h (−τ) ∗ Rx (τ) (25)
Sy (ω) = |H (ω)|2 Sx (ω) (26)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 125: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/125.jpg)
Introduction Random Variables Random Processes Noise Characterization
Transmission of Random Processes Through LinearSystems
x(t) y(t)h(t)
H(ω)
Ry (τ) = h (τ) ∗ h (−τ) ∗ Rx (τ) (25)
Sy (ω) = |H (ω)|2 Sx (ω) (26)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 126: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/126.jpg)
Introduction Random Variables Random Processes Noise Characterization
Transmission of Random Processes Through LinearSystems
x(t) y(t)h(t)
H(ω)
Ry (τ) = h (τ) ∗ h (−τ) ∗ Rx (τ) (25)
Sy (ω) = |H (ω)|2 Sx (ω) (26)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 127: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/127.jpg)
Introduction Random Variables Random Processes Noise Characterization
Sum of Random Processes
If two stationary processes (at least in the wide sense) x (t) and y (t) are addedto form a process z (t), i.e.,
z (t) = x (t) + y (t) ,
thenRz (τ) = Rx (τ) + Ry (τ) + 2 x y. (27)
Most processes of interest in communication problems have zero means. So,if x (t) and y (t) are uncorrelated with either x = 0 or y = 0
Rz (τ) = Rx (τ) + Ry (τ) (28)
andSz (ω) = Sx (ω) + Sy (ω) . (29)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Sum of Random Processes
If two stationary processes (at least in the wide sense) x (t) and y (t) are addedto form a process z (t), i.e.,
z (t) = x (t) + y (t) ,
thenRz (τ) = Rx (τ) + Ry (τ) + 2 x y. (27)
Most processes of interest in communication problems have zero means. So,if x (t) and y (t) are uncorrelated with either x = 0 or y = 0
Rz (τ) = Rx (τ) + Ry (τ) (28)
andSz (ω) = Sx (ω) + Sy (ω) . (29)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Sum of Random Processes
If two stationary processes (at least in the wide sense) x (t) and y (t) are addedto form a process z (t), i.e.,
z (t) = x (t) + y (t) ,
thenRz (τ) = Rx (τ) + Ry (τ) + 2 x y. (27)
Most processes of interest in communication problems have zero means. So,if x (t) and y (t) are uncorrelated with either x = 0 or y = 0
Rz (τ) = Rx (τ) + Ry (τ) (28)
andSz (ω) = Sx (ω) + Sy (ω) . (29)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Outline
1 Introduction
2 Random Variables
3 Random Processes
4 Noise Characterization
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
As we have seen, there are many types of noise sources. However, we willconcentrate more on thermal noise for the reasons metioned ealier. So, let’s
study about thermal noise in this section.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal (Johnson-Nyquist) Noise
T°K
R v(t)
v(t)
t
Random motions of electrons produce small, random voltage fluctuations atthe resistor terminals, which have a zero average value but a nonzero rootmean square (rms) value given by Planck’s blackbody radiation law,
Vn =
√4hfBR
ehf /kT − 1, (30)
where h and k are Plank’s and Boltzmann’s constants, respectively.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal (Johnson-Nyquist) Noise
T°K
R v(t)
v(t)
t
Random motions of electrons produce small, random voltage fluctuations atthe resistor terminals, which have a zero average value but a nonzero rootmean square (rms) value given by Planck’s blackbody radiation law,
Vn =
√4hfBR
ehf /kT − 1, (30)
where h and k are Plank’s and Boltzmann’s constants, respectively.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal (Johnson-Nyquist) Noise
T°K
R v(t)
v(t)
t
Random motions of electrons produce small, random voltage fluctuations atthe resistor terminals, which have a zero average value but a nonzero rootmean square (rms) value given by Planck’s blackbody radiation law,
Vn =
√4hfBR
ehf /kT − 1, (30)
where h and k are Plank’s and Boltzmann’s constants, respectively.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Rayleigh–Jeans Approximation
At microwave frequencies, where hf � kT, (30) can be simplified to
Vn =√
4kTBR. (31)
For very high frequencies or very low temperatures, however, this approxi-mation may be invalid, in which case (30) should be used.�
�From the equation (31), it is evident that thermal noise is independent of
frequency. So, thermal noise is a white noise.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Rayleigh–Jeans Approximation
At microwave frequencies, where hf � kT, (30) can be simplified to
Vn =√
4kTBR. (31)
For very high frequencies or very low temperatures, however, this approxi-mation may be invalid, in which case (30) should be used.�
�From the equation (31), it is evident that thermal noise is independent of
frequency. So, thermal noise is a white noise.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Rayleigh–Jeans Approximation
At microwave frequencies, where hf � kT, (30) can be simplified to
Vn =√
4kTBR. (31)
For very high frequencies or very low temperatures, however, this approxi-mation may be invalid, in which case (30) should be used.
�
�From the equation (31), it is evident that thermal noise is independent of
frequency. So, thermal noise is a white noise.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Rayleigh–Jeans Approximation
At microwave frequencies, where hf � kT, (30) can be simplified to
Vn =√
4kTBR. (31)
For very high frequencies or very low temperatures, however, this approxi-mation may be invalid, in which case (30) should be used.�
�From the equation (31), it is evident that thermal noise is independent of
frequency. So, thermal noise is a white noise.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Maximum Available Noise Power
R
R
Vn
Idealbandpassfilter
B
Power delivered to the load shown in the above figure is
Pn =
(Vn
2R
)2R = kTB, (32)
since Vn is an rms voltage. This important result gives the maximum availablenoise power from the noisy resistor at temperature T .��
�
Independent white noise sources can be treated as Gaussian-distributedrandom variables, so the noise powers (variances) of independent noise
sources are additive.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Maximum Available Noise Power
R
R
Vn
Idealbandpassfilter
B
Power delivered to the load shown in the above figure is
Pn =
(Vn
2R
)2R = kTB, (32)
since Vn is an rms voltage. This important result gives the maximum availablenoise power from the noisy resistor at temperature T .��
�
Independent white noise sources can be treated as Gaussian-distributedrandom variables, so the noise powers (variances) of independent noise
sources are additive.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Maximum Available Noise Power
R
R
Vn
Idealbandpassfilter
B
Power delivered to the load shown in the above figure is
Pn =
(Vn
2R
)2R = kTB, (32)
since Vn is an rms voltage. This important result gives the maximum availablenoise power from the noisy resistor at temperature T .
��
�
Independent white noise sources can be treated as Gaussian-distributedrandom variables, so the noise powers (variances) of independent noise
sources are additive.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Maximum Available Noise Power
R
R
Vn
Idealbandpassfilter
B
Power delivered to the load shown in the above figure is
Pn =
(Vn
2R
)2R = kTB, (32)
since Vn is an rms voltage. This important result gives the maximum availablenoise power from the noisy resistor at temperature T .��
�
Independent white noise sources can be treated as Gaussian-distributedrandom variables, so the noise powers (variances) of independent noise
sources are additive.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal Noise - Some Observations
• As B→ 0, Pn → 0. This means that systems with smaller bandwidthscollect less noise power.
• As T → 0, Pn → 0. This means that cooler devices and componentsgenerate less noise power.
• As B→ ∞, Pn → ∞. This is the so-called ultraviolet catastrophe, whichdoes not occur in reality because Rayleigh-Jeans approximation is validonly when hf � kT.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal Noise - Some Observations
• As B→ 0, Pn → 0. This means that systems with smaller bandwidthscollect less noise power.
• As T → 0, Pn → 0. This means that cooler devices and componentsgenerate less noise power.
• As B→ ∞, Pn → ∞. This is the so-called ultraviolet catastrophe, whichdoes not occur in reality because Rayleigh-Jeans approximation is validonly when hf � kT.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal Noise - Some Observations
• As B→ 0, Pn → 0. This means that systems with smaller bandwidthscollect less noise power.
• As T → 0, Pn → 0. This means that cooler devices and componentsgenerate less noise power.
• As B→ ∞, Pn → ∞. This is the so-called ultraviolet catastrophe, whichdoes not occur in reality because Rayleigh-Jeans approximation is validonly when hf � kT.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Thermal Noise - Some Observations
• As B→ 0, Pn → 0. This means that systems with smaller bandwidthscollect less noise power.
• As T → 0, Pn → 0. This means that cooler devices and componentsgenerate less noise power.
• As B→ ∞, Pn → ∞. This is the so-called ultraviolet catastrophe, whichdoes not occur in reality because Rayleigh-Jeans approximation is validonly when hf � kT.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
The characterization of noise effects in communication systems in terms ofnoise temperature and noise figure will apply to all types of noise, regardless
of the source, as long as the spectrum of the noise is relatively flat over thebandwidth of the system.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature
RR
No No
Te
Te = kBRR
NoArbitrarywhitenoisesource
If an arbitrary source of noise (thermal or non-thermal) is white, it can bemodeled as an equivalent thermal noise source, and characterized with anequivalent noise temperature, Te.
Te =N0kB
(33)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature
RR
No No
Te
Te = kBRR
NoArbitrarywhitenoisesource
If an arbitrary source of noise (thermal or non-thermal) is white, it can bemodeled as an equivalent thermal noise source, and characterized with anequivalent noise temperature, Te.
Te =N0kB
(33)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature
RR
No No
Te
Te = kBRR
NoArbitrarywhitenoisesource
If an arbitrary source of noise (thermal or non-thermal) is white, it can bemodeled as an equivalent thermal noise source, and characterized with anequivalent noise temperature, Te.
Te =N0kB
(33)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Noisy Amplifier
R RNoiselessamplifier
Ni No = GkTeB
NoTe =GkB
G
R RNoisy
amplifier
Ts = 0 K
Ni = 0 No
eGT
Te =N0
GkB(34)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Noisy Amplifier
R RNoiselessamplifier
Ni No = GkTeB
NoTe =GkB
G
R RNoisy
amplifier
Ts = 0 K
Ni = 0 No
eGT
Te =N0
GkB(34)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Noise Figure
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
Noise figure, F, is a measure of this reduction in signal-to-noise ratio, and isdefined as
F =Si/NiSo/No
≥ 1. (35)
By definition, the input noise power is assumed to be the noise power result-ing from a matched resistor at T0 = 290K, i.e., Ni = kT0B.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Noise Figure
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
Noise figure, F, is a measure of this reduction in signal-to-noise ratio, and isdefined as
F =Si/NiSo/No
≥ 1. (35)
By definition, the input noise power is assumed to be the noise power result-ing from a matched resistor at T0 = 290K, i.e., Ni = kT0B.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Noise Figure
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
Noise figure, F, is a measure of this reduction in signal-to-noise ratio, and isdefined as
F =Si/NiSo/No
≥ 1. (35)
By definition, the input noise power is assumed to be the noise power result-ing from a matched resistor at T0 = 290K, i.e., Ni = kT0B.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Noise Figure
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
Noise figure, F, is a measure of this reduction in signal-to-noise ratio, and isdefined as
F =Si/NiSo/No
≥ 1. (35)
By definition, the input noise power is assumed to be the noise power result-ing from a matched resistor at T0 = 290K, i.e., Ni = kT0B.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Relation Between F and Te
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
F =SiSo× No
Ni
=1G× Gk (T0 + Te)B
kT0B
= 1 +Te
T0≥ 1. (36)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Relation Between F and Te
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
F =SiSo× No
Ni
=1G× Gk (T0 + Te)B
kT0B
= 1 +Te
T0≥ 1. (36)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Relation Between F and Te
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
F =SiSo× No
Ni
=1G× Gk (T0 + Te)B
kT0B
= 1 +Te
T0≥ 1. (36)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Relation Between F and Te
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
F =SiSo× No
Ni
=1G× Gk (T0 + Te)B
kT0B
= 1 +Te
T0≥ 1. (36)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Relation Between F and Te
R
R
Pi = Si +Ni Po = So +No
NoisynetworkG, B, Te
T0 = 290 K
F =SiSo× No
Ni
=1G× Gk (T0 + Te)B
kT0B
= 1 +Te
T0≥ 1. (36)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
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Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
The noise power at the output of the second stage is
No = G2N1 + G2kTe2B = G2G1k (T0 + Te1)B + G2kTe2B.
For the equivalent system we have
No = G1G2k (Te,cas + T0)B.
So, comparing the above equations gives
Te,cas = Te1 +Te2G1
. (37)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 163: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/163.jpg)
Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
The noise power at the output of the second stage is
No = G2N1 + G2kTe2B = G2G1k (T0 + Te1)B + G2kTe2B.
For the equivalent system we have
No = G1G2k (Te,cas + T0)B.
So, comparing the above equations gives
Te,cas = Te1 +Te2G1
. (37)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 164: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/164.jpg)
Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
The noise power at the output of the second stage is
No = G2N1 + G2kTe2B = G2G1k (T0 + Te1)B + G2kTe2B.
For the equivalent system we have
No = G1G2k (Te,cas + T0)B.
So, comparing the above equations gives
Te,cas = Te1 +Te2G1
. (37)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 165: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/165.jpg)
Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
The noise power at the output of the second stage is
No = G2N1 + G2kTe2B = G2G1k (T0 + Te1)B + G2kTe2B.
For the equivalent system we have
No = G1G2k (Te,cas + T0)B.
So, comparing the above equations gives
Te,cas = Te1 +Te2G1
. (37)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 166: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/166.jpg)
Introduction Random Variables Random Processes Noise Characterization
Equivalent Noise Temperature of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
The noise power at the output of the second stage is
No = G2N1 + G2kTe2B = G2G1k (T0 + Te1)B + G2kTe2B.
For the equivalent system we have
No = G1G2k (Te,cas + T0)B.
So, comparing the above equations gives
Te,cas = Te1 +Te2G1
. (37)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 167: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/167.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise Figure of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
Te,cas = Te1 +Te2G1
.
⇒ T0 (Fcas − 1) = T0 (F1 − 1) +T0 (F2 − 1)
G1
⇒ Fcas = F1 +F2 − 1
G1. (38)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 168: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/168.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise Figure of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
Te,cas = Te1 +Te2G1
.
⇒ T0 (Fcas − 1) = T0 (F1 − 1) +T0 (F2 − 1)
G1
⇒ Fcas = F1 +F2 − 1
G1. (38)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 169: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/169.jpg)
Introduction Random Variables Random Processes Noise Characterization
Noise Figure of a Cascaded System
G1F1Te1
G2F2Te2
G1G2FcasTecas
Ni No
No
T0
Ni
T0
N1
Te,cas = Te1 +Te2G1
.
⇒ T0 (Fcas − 1) = T0 (F1 − 1) +T0 (F2 − 1)
G1
⇒ Fcas = F1 +F2 − 1
G1. (38)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 170: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/170.jpg)
Introduction Random Variables Random Processes Noise Characterization
Generalization
For an arbitrary number of stages, Te,cas and Fcas are given as
Te,cas = Te1 +Te2G1
+Te3
G1G2+ · · · , and (39)
Fcas = F1 +F2 − 1
G1+
F3 − 1G1G2
+ · · · . (40)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 171: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/171.jpg)
Introduction Random Variables Random Processes Noise Characterization
Generalization
For an arbitrary number of stages, Te,cas and Fcas are given as
Te,cas = Te1 +Te2G1
+Te3
G1G2+ · · · , and (39)
Fcas = F1 +F2 − 1
G1+
F3 − 1G1G2
+ · · · . (40)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 172: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/172.jpg)
Introduction Random Variables Random Processes Noise Characterization
Generalization
For an arbitrary number of stages, Te,cas and Fcas are given as
Te,cas = Te1 +Te2G1
+Te3
G1G2+ · · · , and (39)
Fcas = F1 +F2 − 1
G1+
F3 − 1G1G2
+ · · · . (40)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 173: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/173.jpg)
Introduction Random Variables Random Processes Noise Characterization
Generalization
For an arbitrary number of stages, Te,cas and Fcas are given as
Te,cas = Te1 +Te2G1
+Te3
G1G2+ · · · , and (39)
Fcas = F1 +F2 − 1
G1+
F3 − 1G1G2
+ · · · . (40)
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 174: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/174.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
RT
No = kTBL, T, Zo = R
Ni = kTB
Consider a lossy line (lossy lines or networks containing passive elementscan generate only thermal noise) with a matched source resistor that is also at
temperature T as shown in the figure.
The power gain, G, of a lossy line is less than unity; the loss factor, L, can bedefined as L = 1/G > 1.
Input noise power is kTB. Because the entire system is in thermal equilibriumat the temperature T , and thermal noise power is independent of the
resistance value, the output noise power also must be No = kTB.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 175: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/175.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
RT
No = kTBL, T, Zo = R
Ni = kTB
Consider a lossy line (lossy lines or networks containing passive elementscan generate only thermal noise) with a matched source resistor that is also at
temperature T as shown in the figure.
The power gain, G, of a lossy line is less than unity; the loss factor, L, can bedefined as L = 1/G > 1.
Input noise power is kTB. Because the entire system is in thermal equilibriumat the temperature T , and thermal noise power is independent of the
resistance value, the output noise power also must be No = kTB.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 176: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/176.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
RT
No = kTBL, T, Zo = R
Ni = kTB
Consider a lossy line (lossy lines or networks containing passive elementscan generate only thermal noise) with a matched source resistor that is also at
temperature T as shown in the figure.
The power gain, G, of a lossy line is less than unity; the loss factor, L, can bedefined as L = 1/G > 1.
Input noise power is kTB. Because the entire system is in thermal equilibriumat the temperature T , and thermal noise power is independent of the
resistance value, the output noise power also must be No = kTB.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 177: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/177.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
RT
No = kTBL, T, Zo = R
Ni = kTB
Consider a lossy line (lossy lines or networks containing passive elementscan generate only thermal noise) with a matched source resistor that is also at
temperature T as shown in the figure.
The power gain, G, of a lossy line is less than unity; the loss factor, L, can bedefined as L = 1/G > 1.
Input noise power is kTB. Because the entire system is in thermal equilibriumat the temperature T , and thermal noise power is independent of the
resistance value, the output noise power also must be No = kTB.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 178: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/178.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
RT
No = kTBL, T, Zo = R
Ni = kTB
Consider a lossy line (lossy lines or networks containing passive elementscan generate only thermal noise) with a matched source resistor that is also at
temperature T as shown in the figure.
The power gain, G, of a lossy line is less than unity; the loss factor, L, can bedefined as L = 1/G > 1.
Input noise power is kTB. Because the entire system is in thermal equilibriumat the temperature T , and thermal noise power is independent of the
resistance value, the output noise power also must be No = kTB.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 179: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/179.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
Thus, we have
No = kTB = GkTB + GNadded
where Nadded is the noise generated by the line, as if it appeared at the inputterminals of the line. Rearranging the above equation gives
Nadded = kTB1−G
G= kTB (L− 1) .
From the above equation, it is clear that the equivalent noise temperature ofthe lossy line is
Te = (L− 1)T.
So, noise figure of the lossy line is
F = 1 +Te
T0= 1 + (L− 1)
TT0
.
When T = T0, for lossy networks, F = L.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 180: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/180.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
Thus, we have
No = kTB = GkTB + GNadded
where Nadded is the noise generated by the line, as if it appeared at the inputterminals of the line.
Rearranging the above equation gives
Nadded = kTB1−G
G= kTB (L− 1) .
From the above equation, it is clear that the equivalent noise temperature ofthe lossy line is
Te = (L− 1)T.
So, noise figure of the lossy line is
F = 1 +Te
T0= 1 + (L− 1)
TT0
.
When T = T0, for lossy networks, F = L.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 181: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/181.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
Thus, we have
No = kTB = GkTB + GNadded
where Nadded is the noise generated by the line, as if it appeared at the inputterminals of the line. Rearranging the above equation gives
Nadded = kTB1−G
G= kTB (L− 1) .
From the above equation, it is clear that the equivalent noise temperature ofthe lossy line is
Te = (L− 1)T.
So, noise figure of the lossy line is
F = 1 +Te
T0= 1 + (L− 1)
TT0
.
When T = T0, for lossy networks, F = L.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 182: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/182.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
Thus, we have
No = kTB = GkTB + GNadded
where Nadded is the noise generated by the line, as if it appeared at the inputterminals of the line. Rearranging the above equation gives
Nadded = kTB1−G
G= kTB (L− 1) .
From the above equation, it is clear that the equivalent noise temperature ofthe lossy line is
Te = (L− 1)T.
So, noise figure of the lossy line is
F = 1 +Te
T0= 1 + (L− 1)
TT0
.
When T = T0, for lossy networks, F = L.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 183: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/183.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
Thus, we have
No = kTB = GkTB + GNadded
where Nadded is the noise generated by the line, as if it appeared at the inputterminals of the line. Rearranging the above equation gives
Nadded = kTB1−G
G= kTB (L− 1) .
From the above equation, it is clear that the equivalent noise temperature ofthe lossy line is
Te = (L− 1)T.
So, noise figure of the lossy line is
F = 1 +Te
T0= 1 + (L− 1)
TT0
.
When T = T0, for lossy networks, F = L.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad
![Page 184: 3: Random Variables, Processes, and Noise2016/01/31 · tube or solid-state device. Flicker noiseoccurs in solid-state components and vacuum tubes. Flicker noise power varies inversely](https://reader033.fdocuments.us/reader033/viewer/2022060800/6083ef54f8ec900ce43c282a/html5/thumbnails/184.jpg)
Introduction Random Variables Random Processes Noise Characterization
Lossy Line held at Physical Temperature T
Thus, we have
No = kTB = GkTB + GNadded
where Nadded is the noise generated by the line, as if it appeared at the inputterminals of the line. Rearranging the above equation gives
Nadded = kTB1−G
G= kTB (L− 1) .
From the above equation, it is clear that the equivalent noise temperature ofthe lossy line is
Te = (L− 1)T.
So, noise figure of the lossy line is
F = 1 +Te
T0= 1 + (L− 1)
TT0
.
When T = T0, for lossy networks, F = L.
3: Random Variables, Processes, and Noise Communication Systems, Dept. of EEE, BITS Hyderabad