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```Chapter 3. Noise
Husheng Li
The University of Tennessee
Homework 2
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?
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
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)
 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.
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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