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Chapter 3 . Noise. Husheng Li The University of Tennessee. Homework 2. Deadline: Sept. 16, 2013. Random Process. For a random process in the discrete time domain, we use to represent the probability distribution of n samples. If the random process is stationary, we have - PowerPoint PPT Presentation
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Chapter 3. NoiseHusheng LiThe University of Tennessee
Homework 2Deadline: Sept. 16, 2013
Random ProcessFor a random process in the discrete time
domain, we use to represent the probability distribution of n samples.
If the random process is stationary, we have
Hierarchy of probability density of random process
Markov ProcessMarkov process is a special type of random
process.For each Markov process, we have
Intuitively, in a Markov process, given the current system state, the future system state is independent of the previous history.
NoiseNoise is the negative factor impairing
communication qualities. Without noise, we may transmit as much as we want without errors.
In this chapter, we study the mechanisms, properties and descriptions of various types of noise.
We follow the classical book:D. Middleton, An Introduction to Statistical Communication Theory, Peninsula Publishing, 1987
Three Types of Noise In this chapter, we consider three types of
noises:Thermal noiseShot noiseImpulse noise
Thermal NoiseThermal noise is the result of the random
motion of the free electrons in a conductor with temperature T.
The random movement results in a random current I(t).
Two equivalent representations of a resistance at temperature T:
Spectrum of Thermal Current (detailed model)Using the theory of electrons (such as free path),
we obtain the spectrum of thermal current
When the wave length is 10^-6cm and T0=300K, the spectrum begins to depart from the uniform response when f is more than 10^13 rad/s.In the range of wireless signal, we can consider the thermal noise as ‘white’.
The voltage spectrum is given by
An Alternative Derivation
We can have another approach to derive the Nyquist equation:
Quiz Problem 1. Given the following band pass signal:
write down the equivalent baseband signal in both time and frequency domains. Problem 2. Consider a two-path wireless channel
with the following output:
write down the frequency domain transfer function.
GeneralizationNyquist’s result is mot limited to purely
resistive elements in an equilibrium state, but can also be directly extended to general (passive) linear systems.
Noise Factor and figureThe noise factor of a system is defined as
The noise figure is defined as
The noise factor is given by , where T_e and
T_0 are the noise and physical temperatures. For a cascaded system, the noise factor is given by
Homework 3Problem 1. If the temperature is 300K and the
signal bandwidth is 1MHz, what is the value of noise power?
Problem 2. Consider a series of devices with gains G1, G2, …, Gn and noise temperature T1, T2, …, Tn. What is the expression of the noise temperature of these concatenated devices?
Problem 3. What is the expectation and variance of Poisson distribution?
Deadline: Sept. 23, 2013.
Distribution of Thermal NoiseWe can assume that the thermal noise is
Gaussian distributed:
Usually we also assume that the thermal noise is white, i.e., the noise is independent for different time slots.
In this case, we say that the communication channel is additive white Gaussian noise (AWGN).
White NoiseWhen the noise spectrum is flat, we call it
white noise.The spectral density is given by
Filtered (Colored) NoiseWhen passed through a LTI filter with transfer
function H(f), we have
Example: noise passed through RC network
Noise Equivalent BandwidthAverage noise power:
Noise equivalent bandwidth:
The filtered noise is
What about the RC circuit?
Illustration of Equivalent Bandwidth
Bandpass NoiseBandpass noise results when white noise
passes through a bandpass filter.
SNRThe predetection signal-to-noise ratio is given
by
We also define a system parameter (W is the low pass filter bandwidth)
Quadrature ComponentsThe bandpass noise can be
written as
The power spectral densities are identical lowpass functions related to G_n(f):
Envelope and PhaseThe envelope of bandpass noise is a Rayleigh
random variable
The phase distribution is uniform over [0,2π]
Impulse NoiseThe noise inherent in transmitting and
receiving systems is for the most part due to thermal effects in both the passive and active elements of the system.
Additional noise may enter a communication link through the medium of propagation. One common source is interference, which has a noticeable different statistical character.
A General Model We assume that the noise process X(t;a) is the
resultant of multiple events in the time interval (t,t+T).
We have
Poisson Noise In this model, the process X(t,a) is assumed to
be the result of the linear superposition of independent impulses.
Typical Impulsive Noises
Temperature-limited Shot NoiseShot noise is the name given to the noise that
arises in vacuum tubes and crystals because of the random emission and motion of electrons in these active elements.
Noise of this type appears as a randomly fluctuating component of the output current and along with thermal noise is an important factor inhibiting the performance of transmitting and receiving systems.
Expression of DistributionConsider the current of a temperature limited
diode. The current waves can be written as
The first order approximation is given by
Spectrum of Shot Noise