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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  Hierarchy of probability density of random process

Markov Process  Markov 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.

Noise  Noise 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 noise  Shot noise  Impulse noise

Thermal Noise  Thermal 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.

Generalization  Nyquist’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 figure  The 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 3  Problem 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 Noise  We 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 Noise  When the noise spectrum is flat, we call it white noise.  The spectral density is given by

Filtered (Colored) Noise  When passed through a LTI filter with transfer function H(f), we have  Example: noise passed through RC network

Noise Equivalent Bandwidth  Average noise power:  Noise equivalent bandwidth:  The filtered noise is What about the RC circuit?

Illustration of Equivalent Bandwidth

Bandpass Noise  Bandpass noise results when white noise passes through a bandpass filter.

SNR  The predetection signal-to-noise ratio is given by  We also define a system parameter (W is the low pass filter bandwidth)

Quadrature Components  The bandpass noise can be written as  The power spectral densities are identical lowpass functions related to G_n(f):

Envelope and Phase  The envelope of bandpass noise is a Rayleigh random variable  The phase distribution is uniform over [0,2π]

Impulse Noise  The 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 Noise  Shot 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 Distribution  Consider 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

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