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
Wn (...y-2 y-1 y0 | y1 ) = Wn (y0 | y1 )
 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
SNRin
F=
SNRout
 The noise figure is defined as
NF =10 log(F)
Te
 The noise factor is given by 1+ , where T_e and
T0
T_0 are the noise and physical temperatures. For
a cascaded system, the noise factor is given by
F = F1 +
F2 -1 F3 -1
+
+...
G1
G1G2
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:
n2
- 2
1
p(n) =
e 2s n
2ps n2
 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