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.

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 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