This document discusses different types of noise in communication systems. It defines random variables and random processes that are used to model noise. There are two main types of random variables: discrete and continuous. Noise can be modeled as random processes. Thermal noise arises from the random motion of electrons and is well modeled by a Gaussian process. Other types of noise discussed include shot noise and transit time noise. External noise sources include atmospheric noise, extraterrestrial noise from space, and man-made noise. Internal noise is generated within devices and circuits. White noise is defined as having a constant power spectral density across all frequencies.
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Impedance matching is a procedure for obtaining the maximum power transfer to a load. What is a goal for microwave design? If we can give maximum power to a load, we succeed in design. Impedance matching allows us to make that happen.
Impedance matching is a procedure for obtaining the maximum power transfer to a load. What is a goal for microwave design? If we can give maximum power to a load, we succeed in design. Impedance matching allows us to make that happen.
This presentation is based on the noise inherent in communication. This presentation includes the types of noise during the communication between one person to another.
This Presentation also includes the examples with the pictures.
Types of noise Environmental noise, cultural noise, semantic noise, psychological noise, physiological noise etc and barriers to communication.
This presentation covers types of noise in communication system, noise modelling, thermal noise, shot noise, experimental determination of noise figure, noise figure, friss formula with numerical.
Accurate Evaluation of Interharmonics of a Six Pulse, Full Wave - Three Phase...idescitation
Interharmonics are the non-integral multiples of
the system’s fundamental frequency. The interharmonic
components can be apprehended as the intermodulation of
the fundamental and harmonic components of the system with
any other frequency components introduced by the load. These
loads include static frequency converters, cyclo-converters,
induction motors, arc furnaces and all the loads not pulsating
synchronously with the fundamental frequency of the system.
The harmonic and interharmonic components inflict common
damage to the system and apart from these damages the
interharmonics also cause light flickering, sideband torques
on motor/generator and adverse effects on transformer and
motor components. To
filter/compensate the interharmonic
components, their accurate evaluation is essential and to
achieve the same the Iterative algorithm has been proposed.
The main cause of spectral leakage errors is the truncation of
the time-domain signal. The proposed adaptive approach
calculates the immaculate window width, eliminating the
spectral leakage errors in the frequency domain and thereby
the interharmonics/harmonics can be calculated accurately.
The algorithm does not require any inputs regarding the
system frequency and interharmonic constituents of the
system. The only parameter required is the signal sequence
obtained by sampling the analog signal at equidistant sampling
interval.
Performance Comparison of Power Quality Evaluation Using Advanced High Resolu...IOSRJEEE
Most of the conventional methods of power quality assesment in power systems are almost exclusively based on Fourier Transform that suffer from various inherent limitations. First limitation of an FFT based method is that of frequency resolution, whereas the second limitation is due to no coherent signal sampling of the data which proves itself as a leakage in spectrum domain. These two performance limitations of FFT or similar methods are particularly troublesome when analyzing short data records. To overcome from this problem, high regulation spectrum estimation methods can be used where resolution problem is not found. In this thesis, high resolution methods, such as MUSIC, root MUSIC and ESPRIT are discussed that use a different approach to spectral estimation; instead of trying to estimate the power spectral density (PSD) directly from the data, they model the data as the output of a linear system driven by white noise, and then attempt to estimate the parameters of that linear system. Detail Matlab simulations are carried out in order to investigate the performance of MUSIC, Root MUSIC and ESPRIT methods in estimating amplitude, power (squared amplitude) and frequency estimation of synthetic power signal both in clean and noisy conditions. Using mean square error (MSE) as the evaluation criterion, the variation of amplitude, power (squared amplitude) and frequency estimation are shown with respect to data sequence length and SNR and their influences on MSE are compared for the different methods as mentioned above.
Identification of the Memory Process in the Irregularly Sampled Discrete Time...idescitation
In the present work, we have considered the daily
signal of Solar Radio flux of 2800 Hz cited by National
Geographic Data Center, USA during the period from 29 th
October, 1972 to 28th February, 2013. We have applied Savitzky-
Golay nonlinear phase filter on the present discrete signal to
denoise it and after denoising Finite Variance Scaling Method
has been applied to investigate memory pattern in this discrete
time variant signal. Our result indicates that the present signal
of solar radio flux is of short memory which may in turn
suggest the multi-periodic and/or pseudo-periodic behaviour
of the present signal.
An Improved Detection and Classification Technique of Harmonic Signals in Pow...Yayah Zakaria
This paper introduces an improved detection and classification technique of harmonic signals in power distribution using time-frequency distribution (TFD) analysis which is spectrogram. The spectrogram is an appropriate approach to signify signals in jointly time-frequency domain and known as time frequency representation (TFR). The spectral information of signals can
be observed and estimated plainly from TFR due to identify the
characteristics of the signals. Based on rule-based classifier and the threshold settings that referred to IEEE Standard 1159 2009, the detection and classification of harmonic signals for 100 unique signals consist of various characteristic of harmonics are carried out successfully. The accuracy of proposed method is examined by using MAPE and the result show that the technique provides high accuracy. In addition, spectrogram also gives 100 percent correct classification of harmonic signals. It is proven that the proposed method is accurate, fast and cost efficient for detecting and classifying harmonic signals in distribution system.
An Improved Detection and Classification Technique of Harmonic Signals in Pow...IJECEIAES
This paper introduces an improved detection and classification technique of harmonic signals in power distribution using time-frequency distribution (TFD) analysis which is spectrogram. The spectrogram is an appropriate approach to signify signals in jointly time-frequency domain and known as time frequency representation (TFR). The spectral information of signals can be observed and estimated plainly from TFR due to identify the characteristics of the signals. Based on rule-based classifier and the threshold settings that referred to IEEE Standard 1159 2009, the detection and classification of harmonic signals for 100 unique signals consist of various characteristic of harmonics are carried out successfully. The accuracy of proposed method is examined by using MAPE and the result show that the technique provides high accuracy. In addition, spectrogram also gives 100 percent correct classification of harmonic signals. It is proven that the proposed method is accurate, fast and cost efficient for detecting and classifying harmonic signals in distribution system.
Shunt Faults Detection on Transmission Line by Waveletpaperpublications3
Abstract: Transmission line fault detection is a very important task because major portion of power system fault occurring in transmission system. This paper represents a fast and reliable method of transmission line shunt fault detection. MATLAB Simulink use for modeled an IEEE 9-bus test power system for case study of various faults. In proposed work Daubechies wavelet is applied for decomposition of fault transients. The application of wavelet analysis helps in accurate classification of the various fault patterns. Wavelet entropy measure based on wavelet analysis is able to observe the unsteady signals and complexity of the system at time-frequency plane.
The result shows that the proposed method is capable to detect all the shunt faults.
In order to improve sensing performance when the noise variance is not known, this paper considers a so-called
blind spectrum sensing technique that is based on eigenvalue models. In this paper, we employed the spiked population
models in order to identify the miss detection probability. At first, we try to estimate the unknown noise variance
based on the blind measurements at a secondary location. We then investigate the performance of detection, in terms
of both theoretical and empirical aspects, after applying this estimated noise variance result. In addition, we study the
effects of the number of SUs and the number of samples on the spectrum sensing performance.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Comparative detection and fault location in underground cables using Fourier...IJECEIAES
In this research, we create a single-phase to ground synthetic fault by the simulation of a three-phase cable system and identify the location using mathematical techniques of Fourier and modal transforms. Current and voltage signals are measured and analyzed for fault location by the reflection of the waves between the measured point and the fault location. By simulating the network and line modeling using alternative transient programs (ATP) and MATLAB software, two single-phase to ground faults are generated at different points of the line at times of 0.3 and 0.305 s. First, the fault waveforms are displayed in the ATP software, and then this waveform is transmitted to MATLAB and presented along with its phasor view over time. In addition to the waveforms, the detection and fault location indicators are presented in different states of fault. Fault resistances of 1, 100, and 1,000 ohms are considered for fault creation and modeling with low arch strength. The results show that the proposed method has an average fault of less than 0.25% to determine the fault location, which is perfectly correct. It is varied due to changing the conditions of time, resistance, location, and type of error but does not exceed the above value.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
1. CHAPTER 4: NOISE
Prepared by:
DR NOORSALIZA BINTI ABDULLAH
DEPARTMENT OF COMMUNICATION ENGINEERING
FACULTY OF ELECTRICAL AND ELECTRONIC ENGINEERING
2. DEFINATION OF RANDOM VARIABLES
A real random is mapping from the sample space Ω (or S) to the
set of real numbers.
A schematic diagram representing a random variable is given
below
Ω
ω1 ω2
ω3
X (ω1 )
X (ω2 )
ω4
X (ω3 ) X (ω4 )
Figure 4.1 : Random variables as a mapping from Ω to R
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
3. A random variable, usually written X, is a variable whose
possible values are numerical outcomes of a random
phenomenon, etc.; individuals values of the random variable X
are X(ω).
There are two types of random variables, which is Discrete
Random Variables and Continuous Random Variables.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
4. Discrete Random Variables
A sample space is discrete if the number of its elements are
finite or countable infinite, i.e., a discrete random variable is
one which may take on only a countable number of distinct
values such as 0,1,2,3,4,........
Examples of discrete random variables include the number of
children in a family, the Friday night attendance at a cinema,
the number of patients in a doctor's surgery, the number of
defective light bulbs in a box of ten.
A non-discrete sample space happens when the sample space
of the random experiment is infinite and uncountable.
Example of non-discrete sample space is randomly chosen
number from 0 to 1 (continuous random variables).
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
5. Continuous Random Variables
A continuous random variable is one which takes an infinite
number of possible values. Continuous random variables are
usually measurements.
Examples include height, weight, the amount of sugar in an
orange, the time required to run a mile.
A continuous random variable is not defined at specific values.
Instead, it is defined over an interval of values, and is
represented by the area under a curve (in advanced
mathematics, this is known as an integral).
The probability of observing any single value is equal to 0,
since the number of values which may be assumed by the
random variable is infinite.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
6. Figure 4.2 : Random variables (a) continuous (b) discrete.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
7. Example 4.1
Which of the following random variables are discrete and which are
continuous?
c)
X = Number of houses sold by real estate developer per week?
X = Number of heads in ten tosses of a coin?
X = Weight of a child at birth?
d)
X = Time required to run 100 yards?
a)
b)
Answer:
(a) Discrete (b) Discrete (c) Continuous (d) Continuous
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
8. SIGNALS: DETERMINISTIC VS. STOCHASTIC
DETERMINISTIC SIGNALS
Most introductions to signals and systems deal strictly with
deterministic signals as shown in Figure 4.3. Each value of
these signals are fixed and can be determined by a
mathematical expression, rule, or table.
Because of this, future values of any deterministic signal can
be calculated from past values. For this reason, these signals
are relatively easy to analyze as they do not change, and we
can make accurate assumptions about their past and future
behavior.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
9. RANDOM SIGNALS
Random signals cannot be characterized by a simple, welldefined mathematical equation and their future values cannot
be predicted.
Rather, we must use probability and statistics to analyze their
behavior.
Also, because of their randomness as shown in Figure 4.4,
average values from a collection of signals are usually studied
rather than analyzing one individual signal.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
10. Deterministic Signal
Figure 4.3: An example of a deterministic signal, the sine wave.
Random Signal
Figure 4.4: We have taken the above sine wave and added random noise to it to come up with a
noisy, or random, signal. These are the types of signals that we wish to learn how to deal with so
that we can recover the original sine wave.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
11. RANDOM PROCESSES
As mentioned before, in order to study random signals, we
want to look at a collection of these signals rather than just
one instance of that signal. This collection of signals is
called a random process.
Is an extension of random variables
Also known as Stochastic Process
Model Random Signal and Random Noise
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
12. Outcome of a random experiment is a function
An indexed set of random variables
Typically the index is time in communications
The difference between random variable and random process
is that for a random variable, an outcome is the sample space
mapped into a number, whereas for a random process it is
mapped into a function of time.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
13. Figure 4.5: Example of random process represent the temperature of a city at 20
hours.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
14. POWER SPECTRAL DENSITY
Random process is a collection of signals, and the spectral
characteristics of these signals determine the spectral
characteristic of the random process.
Slow varying signals (of a random process) have power concentrated at
low frequencies.
Fast changing signals (of a random process) have power concentrated
at high frequencies.
Power spectral density determines the power distribution (or
power spectrum) of the random process.
PSD of a random process X(t) is denoted by SX(f), denotes the
strength of power in the random process as a function of
frequency.
Units for PSD is Watts/Hz.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
15. RELATIONSHIP OF RANDOM PROCESS
AND NOISE
Unwanted electric signals come from variety of sources,
generally classified as human interference or naturally
occurring noise.
Human interference comes from other communication systems
and the effects of many unwanted signals can be reduced or
eliminated completely.
However there always remain inescapable random signals, that
present a fundamental limit to systems performance.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
16. THERMAL NOISE
Thermal noise is the noise
resulting from the random motion
of electrons in a conducting
medium.
Thermal noise arises from both the
photodetector and the load resistor.
Amplifier noise also contributes to
thermal noise.
A reduction in thermal noise is
possible by increasing the value of
the load resistor.
However, increasing the value of
the load resistor to reduce thermal
noise reduces the receiver
bandwidth.
Figure 4.6 Fluctuating voltage
produced by random movements of
mobile electrons.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
17. GAUSSIAN PROCESS
Gaussian process is important in
communication systems.
The main reason is that thermal
noise in electrical devices produced
by movement of electrons due to
thermal agitation is closely modeled
by a Gaussian process.
Due to the movements of electrons,
sum of small currents of a very large
number of sources was introduced.
Since majority sources are
independent, hence the total current
is sum of large number of random
variables.
Therefore the total currents has
Gaussian distribution.
Figure 4.7 Histogram of some noise voltage
measurements
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
18. Definition
A random process X(t) is a Gaussion process if for all n and all
(t1,t2,…,tn) the random variable {X(ti)}ni=1 have a jointly Gaussian
density function.
Gaussian or Normal Random Variables
( x − m )2
−
where m = mean
1
2σ 2
(4.1)
f X ( x) =
e
σ = standard deviation
2πσ
σ2 = variance
A Gaussian random variable with mean m and variance σ2 is denoted
by Ν(m, σ2)
Assuming X is a standard normal random variable, we defined the function
Q(x) as P(X > x). The Q function is given by relation
2
Q( x) = P( X > x) = ∫
∞
x
1 − t2
e dt
2π
(4.2)
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
19. The Q function represent the area under the tail of a standard random
variable.
It is well tabulated and used in analyzing the performance of
communication system.
Q(x) satisfy the following relations:
(4.3a)
Q(-x) = 1 – Q(x)
(4.3b)
Q(0) = ½
Q(∞) = 0
(4.3c)
Table 3.1 gives the value of this function for various value of x.
For Ν(m, σ2) random variable, a simple change of variable in the integral
that computes P(X > x) results in P(X > x) = Q[(x – m)/σ].
tail probability in Gaussian random variable.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
20. Figure 4.8: The Q-function as the area under the tail of a standard normal random variable.
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
21. Table 4.1 Table of the Q function
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
22. Example 4.2
X is a Gaussian random variable with mean 1 and variance 4. Find the
probability X between 5 and 7.
Ans.
We have m = 1 and σ = √4 = 2. Thus,
P( 5 < X < 7)
= P (X > 5) – P(X > 7)
= Q ((5 – 1)/2) – Q((7 – 1)/2)
= Q(2) – Q(3)
≈ 0.0214
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia
23. WHITE NOISE
There are many ways to characterize different noise sources, one is to
consider the spectral density, that is, the mean square fluctuation at any
particular frequency and how that varies with frequency.
In what follows, noise will be generated that has spectral densities that vary
as powers of inverse frequency, more precisely, the power spectra P(f) is
proportional to 1 / fβ for β ≥ 0.
When β = 0 the noise is referred to white noise, when β = 2, it is referred
to as Brownian noise, and when it is 1 it normally referred to simply as 1/f
noise which occurs very often in processes found in nature.
White process is a process in which all frequency component appear with
equal power, i.e. power spectral density is constant for all frequencies.
A process X(t) is called a white process if it has a flat spectral
density,i.e., if SX(f) is constant for all f.
Dept. Of Communication Engineering,
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24. White Noise, β = 0
β =0
β =2
Brownian noise
white noise
β =1
β =3
1/f noise
Dept. Of Communication Engineering,
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25. White Noise
Spectral density of white
noise is a constant, N0/2
N0
SX ( f ) =
2
SX (f)
N0
2
f
0
(3.4)
White noise power spectrum
-
Where N0 = kT
RXX ( )
Autocorrelation function:
N0
2
⎛N ⎞
RXX (τ ) = F −1 ⎜ 0 ⎟
⎝ 2 ⎠
∞
=
N 0 j 2π ft
∫ 2 e df
−∞
N
= 0 δ (τ )
2
k = Boltzmann’s constant = 1.38 x
0
(3.5)
10-23
White noise autocorrelation
-
Figure 4.9: White noise (a) power spectrum
(b) autocorrelation
Dept. Of Communication Engineering,
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26. Properties of Thermal Noise
Thermal noise is a stationary process
Thermal noise is a zero-mean process
Thermal noise is a Gaussian process
Thermal noise is a white noise with power spectral density
SX(f)=kT/2=Sn(f)=N0/2.
It is clear that power spectral density of thermal noise increase
with increasing the ambient temperature, therefore, keeping
electric circuit cool makes their noise level low.
Dept. Of Communication Engineering,
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27. TYPE OF NOISE
Noise can be divided into :
2 general categories
Correlated noise – implies relationship between the signal and the noise,
exist only when signal is present
Uncorrelated noise – present at all time, whether there is signal or not.
Under this category there are two broad categories which are:i) Internal noise
ii) External noise
Dept. Of Communication Engineering,
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28. UNCORRELATED NOISE
Can be divided into 2 categories
1.
External noise
Generated outside the device or circuit
Three primary sources are atmospheric, extraterrestrial and man made
(a) Atmospheric Noise
Naturally occurring electrical disturbance originate within Earth’s
atmosphere
Commonly called static electricity
Most static electricity is naturally occurring electrical conditions,
such as lighting
In the form of impulse, spread energy through wide range of
frequency
Insignificant at frequency above 30 MHz
Dept. Of Communication Engineering,
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29. (b) Extraterrestrial Noise
Consists of electrical signals that originate from outside earth
atmosphere, deep-space noise
Divide further into two
(i) Solar noise – generated directly from sun’s heat. There are 2
parts to solar noise:Quite condition when constant radiation intensity exist and
high intensity
Sporadic disturbance caused by sun spot activities and solar
flare-ups which occur every 11 years
(ii) Cosmic noise – continuously distributed throughout the
galaxies, small noise intensity because the sources of galactic
noise are located much further away from sun. It's also often
called as black-body noise.
Dept. Of Communication Engineering,
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Universiti Tun Hussein Onn Malaysia
30. (c) Man-made noise
Source – spark-producing mechanism such as from commutators in
electric motors, automobile ignition etc
Impulsive in nature, contains wide range of frequency that
propagate through space the same manner as radio waves
Most intense in populated metropolitan and industrial areas and is
therefore sometimes called industrial noise.
Dept. Of Communication Engineering,
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31. (d) Impulse noise
High amplitude peaks of short duration in the total noise spectrum.
Consists of sudden burst of irregularly shaped pulses.
More devastating on digital data,
Produce from electromechanical switches, electric motor etc.
(e) Interference
External noise
Signal from one source interfere with another signal.
It occurs when harmonics or cross product frequencies from one
source fall into the passband of the neighboring channel.
Usually occurs in radio-frequency spectrum
Dept. Of Communication Engineering,
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32. 2. Internal noise
Generated within a device or circuit.
3 primary kinds, shot noise, transit-time noise and thermal noise
(a) Shot noise
Caused by random arrival of carriers (hole and electron) at the
output element of an electronic device such as diode, field effect
transistor or bipolar transistor.
The currents carriers (ac and dc) are not moving in a continuous,
steady flow, as the distance they travel varies because of their
random paths of motion.
Shot noise randomly varying and is superimposed onto any signal
present.
When amplified, shot noise sounds similar to metal pellets falling
on a tin roof.
Sometimes called transistor noise
Dept. Of Communication Engineering,
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33. (b) Transit-time noise (Ttn)
Any modification to a stream of carriers as they pass from the input
to the output of a device produce irregular, random variation
(emitter to the collector in transistor).
Time it takes for a carrier to propagate through a device is an
appreciable part of the time of one cycle of the signal , the noise
become noticeable.
Ttn is transistors is determined by carrier mobility, bias voltage, and
transistor construction.
Carriers traveling from emitter to collector suffer from emitter
delay, base Ttn,and collector recombination-time and propagation
time delays.
If transmit delays are excessive at high frequencies, the device may
add more noise than amplification of the signal.
Dept. Of Communication Engineering,
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34. (c) Thermal noise
Due to rapid and random movement of electrons within a conductor
due to thermal agitation
Present in all electronic components and communication system.
Uniformly distributed across the entire electromagnetic frequency
spectrum, often referred as white noise.
Form of additive noise, meaning that it cannot be eliminated , and it
increases in intensity with the number of devices and circuit length.
Set as upper bound on the performance of communication system.
Temperature dependent, random and continuous and occurs at all
frequencies.
Dept. Of Communication Engineering,
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35. Noise Spectral Density
In communications, noise spectral density No is the noise
power per unit of bandwidth; that is, it is the power spectral
density of the noise.
It has units of watts/hertz, which is equivalent to watt-seconds
or joules.
If the noise is white, i.e., constant with frequency, then the
total noise power N in a bandwidth B is BNo.
This is utilized in Signal-to-noise ratio calculations.
The thermal noise density is given by No = kT, where k is
Boltzmann's constant in joules per kelvin, and T is the receiver
system noise temperature in kelvin.
No is commonly used in link budgets as the denominator of the
important figure-of-merit ratios Eb/No and Es/No.
Dept. Of Communication Engineering,
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36. NOISE POWER
Noise power is given as
N0
df
−B 2
= N0 B
PN = ∫
B
and can be written as
PN = kTB [W]
(3.6)
(3.7)
where
PN = noise power,
k = Boltzmann’s constant (1.38x10-23 J/K)
B = bandwidth,
T = absolute temperature (Kelvin)(17oC or 290K)
Dept. Of Communication Engineering,
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37. NOISE VOLTAGE
Figure 4.10 shows the equivalent
circuit for a thermal noise source.
Internal resistance RI in series
with the rms noise voltage VN.
For the worst condition, the load
resistance R = RI , noise voltage
dropped across R = half the noise
source (VR=VN/2) and
From equation 4.5 the noise
power PN , developed across the
load resistor = kTB
Figure 4.10 : Noise source equivalent
circuit
The mathematical expression :
PN
(V
= kTB = N
/ 2)
V N2
=
R
4R
2
(4.8a)
V N2 = 4 RkTB
VN =
4 RkTB
(4.8b)
Dept. Of Communication Engineering,
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Universiti Tun Hussein Onn Malaysia
38. OTHER NOISE SOURCES
1.
2.
3.
There are 3 other noise mechanisms that contribute to internally generated
noise in electronic devices.
Generation-Recombination Noise - The result of free carriers being
generated and recombining in semiconductor material. Can consider these
generation and recombination events to be random. This noise process can
be treated as shot noise process.
Temperature-Fluctuation Noise – The result of the fluctuating heat
exchange between a small body, such as transistor, and it’s environment
due to the fluctuations in the radiation and heat-conduction processes. If a
liquid or gas is flowing past the small body, fluctuation in heat convection
also occurs.
Flicker Noise – It is characterized by a spectral density that increases with
decreasing frequency. The dependence on spectral density on frequency is
often found to be proportional to the inverse first power of the frequency.
Sometimes referred as one-over-f noise.
Dept. Of Communication Engineering,
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39. Example 4.3
Calculate the thermal noise power available from any resistor at room
temperature (290 K) for a bandwidth of 1 MHz. Calculate also the
corresponding noise voltage, given that R = 50 Ω.
Ans
a) Thermal noise power
b) Noise voltage
N = kTB
VN =
= 1.38 × 10 − 23 × 290 × 1×10 6
= 4 × 50 × 4 × 10
= 0 . 895 μ V
= 4 × 10 −15W
4 RkTB
− 15
Dept. Of Communication Engineering,
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40. Example 4.4
For an electronic device operating at a temperature of 17 oC
with a bandwidth of 10 kHz, determine
a) Thermal noise power in watts and dBm
b) rms noise voltage for a 100 Ω internal resistance and 100
Ω load resistance.
Ans.
N = 1.38 ×10 −23 × 290 × 10 ×103 b) V = 4 RkTB
a)
N
= 4.002 × 10 −17 W
⎛ 4 × 10−17 ⎞
N (dBm ) = 10 log⎜
⎟
⎜ 1 × 10− 3 ⎟
⎠
⎝
= −134dBm
= 4 × 100 × 4 ×10 −17
= 0.127 μV (rms )
Dept. Of Communication Engineering,
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41. Example 4.5
Two resistor of 20 kΩ and 50 kΩ are at room temperature (290
K). For a bandwidth of 100 kHz, calculate the thermal noise
voltage generated by
1. each resistor
2. the two resistor in series
3. the two resistor in parallel
Dept. Of Communication Engineering,
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43. CORRELATED NOISE
1.
Mutually related to the signal, not present if there is no signal
Produced by nonlinear amplification, and include nonlinear
distortion such as harmonic and intermodulation distortion
Harmonic Distortion (HD)
Harmonic distortion – unwanted harmonics of a signal produced
through nonlinear amplification (nonlinear mixing). Harmonics are
integer multiples of the original signal.
There are various degrees of harmonic distortion.
2nd order HT, ratio of the rms amplitude of the second harmonic to the
rms amplitude of the fundamental.
3rd oder HT, ratio of the rms amplitude of the third harmonic to the rms
amplitude of the fundamental.
Total harmonic distortion (THD), ratio of the quadratic sum of the rms
values of all the higher harmonics to the rms value of the fundamental.
Dept. Of Communication Engineering,
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44. Figure 4.11(a) show the input and
output frequency spectrums for a
nonlinear device with a single input
frequency f1.
Mathematically, THD is
%THD =
vhigher
vfundamental
x100
Where,
%THD = percent total harmonic
distortion
vhigher = quadratic sum of the rms
2
2
2
voltages,
v2 + v3 + vn
vfundamental = rms voltage of the
fundamental frequency
(4.9)
Figure 4.11: Correlated noise:
(4.10)
(a) Harmonic distortion
(b) Intermodulation distortion
Dept. Of Communication Engineering,
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45. 2. Intermodulatin Distortion (ID)
Intermodulation distortion is the generation of unwanted sum and
difference frequency when two or more signal are amplified in a
nonlinear device such as large signal amplifier.
The sum and difference frequencies are called cross products.
Figure 4.11(b) show the input and output frequency spectrums for a
nonlinear device with two input frequencies (f1 and f2).
Mathematically, the sum and difference frequencies are
(4.11)
Cross products =mf1 ± nf2
Where f1 and f2 = fundamental frequencies, f1 > f2
m and n = positive integers between one and infinity
Dept. Of Communication Engineering,
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46. Example 4.6
Determine
2nd, 3rd and 12th harmonics for a 1 kHz repetitive wave.
b) Percent 2nd order, 3rd order and total harmonic distortion for a
fundamental frequency with an amplitude of 8 Vrms, a 2nd harmonic
amplitude of 0.2 Vrms and a 3rd harmonic amplitude of 0.1 Vrms.
a)
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47. Ans
a)
b)
2nd harmonic = 2×fundamental freq. = 2×1 kHz =2 kHz
3rd harmonic = 3×fundamental freq. = 3×1 kHz =3 kHz
12th harmonic = 12×fundamental freq. = 12×1 kHz =12 kHz
%
2nd
V2
0.2
× 100 = 2.5%
order = × 100 =
V1
8
% 3rd order =
V3
0.1
× 100 =
×100 = 1.25%
V1
8
0.2 2 + 0.12
× 100% = 2.795%
% THD =
8
Dept. Of Communication Engineering,
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48. Example 4.7
For a nonlinear amplifier with two input frequencies, 3 kHz and 8 kHz,
determine,
a) First three harmonics present in the output for each input frequency.
b) Cross product frequencies for values of m and n of 1 and 2.
Dept. Of Communication Engineering,
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49. Ans f1 = 8 kHz, f2 = 3 kHz
a)
For freqin =3kHz
1st harmonic = original signal freq. = 3 kHz
2nd harmonic = 2× original signal freq. = 2×3 kHz =6 kHz
3rd harmonic = 3× original signal freq. = 3×3 kHz =9 kHz
For freqin =8kHz
1st harmonic = original signal freq. = 8 kHz
2nd harmonic = 2× original signal freq. = 2×8 kHz =16 kHz
3rd harmonic = 3× original signal freq. = 3×8 kHz =24 kHz
b)
m
1
n
1
8±3
Cross Product
5kHz and 11kHz
1
2
2
2
1
2
8±6
16±3
16±6
2kHz and 14kHz
13kHz and 19kHz
10kHz and 22kHz
Dept. Of Communication Engineering,
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50. Table 4.2 Electrical Noise Source Summary
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51. SIGNAL-TO-NOISE RATIO (SNR)
Signal-to-noise power ratio (S/N) is the ratio of the signal power level to
the noise power
Mathematically,
S
P
= S
N PN
where,
(4.12)
PS = signal power (watts)
PN = noise power (watts)
In dB
S
PS
(dB) = 10log
N
PN
(4.13)
Dept. Of Communication Engineering,
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52. If the input and output resistances of the amplifier, receiver, or
network being evaluated are equal
⎛V ⎞
⎛V ⎞
S
( dB ) = 10 log ⎜ s 2 ⎟ = 10 log ⎜ s ⎟
N
⎝ Vn ⎠
⎝ Vn ⎠
2
⎛ Vs ⎞
= 20 log ⎜ ⎟
⎝ Vn ⎠
where
2
(4.14)
Vs = signal voltage (volts)
Vn = noise voltage (volts)
Dept. Of Communication Engineering,
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53. Example 4.8
For an amplifier with an output signal power of 10 W and an output noise
power of 0.01W, determine the S/N.
Ans
S/N =
10
= 1000
[unitless ]
0.01
S / N ( dB ) = 10 log 1000 = 30 [ dB ]
Example 4.9
For an amplifier with an output signal voltage of 4 V, an output noise voltage
of 0.005 V and an input and output resistance of 50 Ω, determine the S/N.
Ans
Vs
S/N =
VN
2
R =
2
42
= 640000
[unitless]
0.0052
S / N ( dB ) = 10 log 640000 = 58 [ dB ]
R
Dept. Of Communication Engineering,
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54. NOISE FACTOR (F) & NOISE FIGURE (NF)
Noise factor and noise figure are figures of merit to indicate how much a
signal deteriorate when it pass through a circuit or a series of circuits
Noise factor
F=
Noise figure
input signal-to-noise ratio
[unitless]
output signal-to-noise ratio
input signal-to-noise ratio [dB]
NF = 10log
output signal-to-noise ratio
= 10log F
(4.15)
(4.16)
For perfect noiseless circuit, F = 1, NF = 0 dB
Dept. Of Communication Engineering,
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55. For ideal noiseless amplifier with a power gain (AP), an input signal power
level (Si) and an input noise power level (Ni) as shows in Figure 4.12 (a).
The output signal level is simply APSi, and the output noise level is APNi.
Ap Si
Sout
Si
=
=
N out Ap N i N i
[unitless]
(4.17)
Figure 4.12 (b) shows a nonideal amplifier that generates an internal noise
Nd
Ap Si
Sout
Si
=
=
N out Ap N i + N d N i + N d Ap
[unitless]
(4.18)
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56. Figure 4.12 Noise Figure: (a) ideal, noiseless device (b) amplifier with
internally generated noise
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57. When two or more amplifiers are cascaded as shown in Figure
4.13, the total noise factor is the accumulation of the
individual noise factors. Friiss’ formula is used to calculate the
total noise factor of several cascaded amplifiers.
Mathematically, Friiss formula is
Fn − 1
F2 − 1 F3 − 1
[unitless] (4.19)
FT = F1 +
A1
+
A1 A2
+
A1 A2 ..... An −1
Figure 4.13 Noise figure of cascaded amplifiers
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58. Where
FT = total noise factor for n cascaded amplifiers
F1, F2, F3…n = noise factor, amplifier 1,2,3…n
A1, A2…. An = power gain, amplifier 1,2,…..n
Notification remarks
Change unit of all noise factors F and power gains A from [dB]
to [unitless] before insert its into Friss formula equation
Dept. Of Communication Engineering,
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59. Example 4.10
The input signal to a telecommunications receiver consists of 100 μW of
signal power and 1 μW of noise power. The receiver contributes an
additional 80 μW of noise, ND, and has a power gain of 20 dB. Compute
the input SNR, the output SNR and the receiver’s noise figure.
Ans.
a) Input SNR =
Si
100 × 10 -6
=
= 100 [ unitless ]
-6
Ni
1 × 10
Input SNR(dB) = 10 log 100 = 20 [ dB ]
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60. b) The output noise power = internal noise + amplified input noise
N out = N D + Ap N i = 80 μW + (100 × 1× 10 −6 W )
1.8 × 10 − 4 [W ]
=
The output signal power = amplified input signal
S out = Ap Si = 100 × 100 × 10 −6
× 10 − 2 [W ]
=1
S out
1× 10 -2
Output SNR=
=
= 55.56[unitless ]
-4
N out 1.8 × 10
Output SNR(dB) =
10 log 55 .56 = 17 .45[ dB ]
Dept. Of Communication Engineering,
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61. c) Noise Figure NF = 10 log
input SNR[unitless ]
100
= 10 log
output
SNR[unitless ]
55.56
= 2 . 55 [ dB ]
Dept. Of Communication Engineering,
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62. Example 4.11
For a non-ideal amplifier and the following parameters, determine
Input signal power = 2 x 10-10 W
Input noise power = 2 x 10-18 W
Power Gain = 1,000,000
Internal Noise (Nd) = 6 x 10-12 W
a.
b.
c.
Input S/N ratio (dB)
Output S/N ratio (dB)
Noise factor and noise figure
Dept. Of Communication Engineering,
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63. Ans
a) Input SNR
S i 2 × 10 -10
=
= 1 × 10 8 [unitless ]
N i 2 × 10 -18
10
Input SNR(dB) = log 100000000 = 80 [ dB ]
b) The output noise power Nout = ND + Ap Ni = 6×10−12 + (1×106 × 2×10−18)
×10−12[W ]
=8
The output signal power S out = Ap Si = 1×106 × 2 ×10 −10
2 ×10 − 4 [W ]
=
2 × 10 -4
= 74 [ dB ]
Output SNR(dB) = 10 log
- 12
8 × 10
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64. c)
Noise factor F =
input SNR [ unitless ] 100000000
=
= 4[ unitless ]
output SNR [unitless ]
25000000
Noise figure NF = 10 log 4 = 6.02[ dB ]
Dept. Of Communication Engineering,
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65. Example 4.12
For three cascaded amplifier stages, each with noise figures of 3 dB and power
gains of 10 dB, determine the total noise figure.
Ans.
Change all noise figure and power gain from [dB] unit to [unitless]
10
Power gain A = A = A = 10 10 = 10[unitless ]
1
2
3
3
10
Noise Factor F1 = F2 = F3 = 10 = 2[unitless ]
Using Friss formula ,
F − 1 F3 − 1
+
[unitless ]
Total noise factor FT = F1 + 2
A1
A1 A2
2 −1 2 −1
= 2+
+
10 10 × 10
[unitless ]
= 2.11
Total noise figure NFT = 10 log 2 . 11 = 3 . 24 [ dB ]
Dept. Of Communication Engineering,
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66. EQUIVALENT NOISE TEMPERATURE (Te)
The noise produced from thermal agitation is directly proportional to
temperature, thermal noise can be expressed in degrees as well as watts or
dBm.
Mathematically,
N
T=
KB
(4.20)
where T = environmental temperature (kelvin)
N = noise power (watts)
K = Boltzmann’s constant (1.38 x 10-23 J/K)
B = bandwidth (hertz)
Dept. Of Communication Engineering,
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67. Te is a hypothetical value that cannot be directly measured
Convenient parameter often used . It’s also indicates reduction in the
signal-to-noise ratio a signal undergoes as it propagates through a receiver.
The lower the Te , the better the quality of a receiver.
Typically values for Te , range from (20 K – 1000 K) for noisy receivers.
Mathematically,
(4.21)
Te = T (F − 1)
Where
Te =equivalent noise temperature (kelvin)
T = environmental temperature (290 K)
F = noise factor (unitless)
Conversely, F can be represented as a function of Te :
Te
F =1+
T
(4.22)
Dept. Of Communication Engineering,
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68. Example 4.13
Determine,
a) Noise figure for an equivalent noise temperature of 75 K.
b) Equivalent noise temperature for noise figure of 6 dB.
Ans.
a) Noise factor F = 1 + Te = 1 + 75 = 1 .258 [unitless ]
290
T
Noise figure NF = 10 log 1 . 258 = 1[ dB ]
b) Noise factor F = anti log(
NF
6
) = anti log( ) = 4[unitless]
10
10
Equivalent noise temperature Te = T ( F − 1) = 290 ( 4 − 1)
870[ K ]
=
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