Download Noise: An Introduction

Noise: An Introduction
Adapted from a presentation in:
Transmission Systems for Communications,
Bell Telephone Laboratories, 1970, Chapter 7
5/24/2017
Noise: An Introduction
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Noise: An Introduction
•
•
What is noise?
Waveforms with incomplete information
–
–
•
Analysis: how?
What can we determine?
Example: sine waves of unknown phase
–
–
–
–
•
Energy Spectral Density
Probability distribution function: P(v)
Probability density function: p(v)
Averages
Common probability density functions
–
–
•
•
•
Gaussian
Exponential
Noise in the real-world
Noise Measurement
Energy and Power Spectral densities
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Background Material
• Probability
– Discrete
– Continuous
• The Frequency Domain
– Fourier Series
– Fourier Transform
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Noise
• Definition
Any undesired signal that interferes with the
reproduction of a desired signal
• Categories
– Deterministic: predictable, often periodic, noise
often generated by machines
– Random: unpredictable noise, generated by a
“stochastic” process in nature or by machines
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Random Noise
• Unpredictable
– “Distribution” of values
– Frequency spectrum: distribution of energy
(as a function of frequency)
• We cannot know the details of the
waveform only its “average” behavior
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Noise analysis Introduction:
a sine wave of unknown phase
• Single-frequency interference
n(t) = A sin(nt + )
A and n are known, but  is not known
• We cannot know its value at time “t”
1.5
1
0.5
n1
0
-1.5
-1
-0.5
n2
0
0.5
1
1.5
2
2.5
n3
-0.5
-1
t = .17
-1.5
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Energy Spectral Density
Here the “Energy Spectral Density” is just the magnitude
squared of the Fourier transform of n(t)
N  
2
A2
    n       n 

4
since all of the energy is concentrated at n and each half
of the energy is at ±  since the Fourier transform is based
on the complex exponential not sine and cosine.
1.2
0.25(n)
0.25n
1
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0.2
0
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Probability Distribution
• The “distribution” of the ‘noise” values
– Consider the probability that at any time t the voltage is less than
or equal to a particular value “v” Pv   Pnt   v
•
n
The probabilities at some values are easy
– P(-A) = 0
– P(A) = 1
– P(0) = ½
• The actual equation is: P(vn) = ½ + (1/)arcsin(vn/A)
1.2
1
0.8
Shown for A=1
0.6
0.4
0.2
0
-1.5
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Probability Distribution
continued
• The actual equation is: P(vn) = ½ + (1/)arcsin(v/A)
1.2
1
0.8
Shown for A=1
0.6
0.4
0.2
0
-1.5
-1
-0.5
0
0.5
1
1.5
• Note that the noise spends more time near the extremes and less time
near zero. Think of a pendulum:
– It stops at the extremes and is moving slowly near them
– It move fastest at the bottom and therefore spends less time there.
• Another useful function is the derivative of P(vn): the
“Probability Density Function”, p(vn) (note the lower case p)
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Probability Density Function
• The area under a portion
of this curve is the
probability that the
voltage lies in that region.
• This PDF is zero for
|vn| > A
d
p(vn )  Pvn 
dv
1
p ( vn ) 
2
2
 A  vn
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2
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0
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Averages
• Time Average of signals
1 T2
n   T nt dt
T 2
• “Ensemble” Average
– Assemble a large number of examples of the noise signal.
(the set of all examples is the “ensemble”)
– At any particular time (t0) average the set of values of vn(t0)

1 K
or E(vn )   p(v)dv for the infinite set
vn  E (v)   vl

K l 1
to get the “Expected Value” of vn
• When the time and ensemble averages give the same value
(they usually do), the noise process is said to be “Ergodic”
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Averages (2)
• Now calculate the ensemble average of our
sinusoidal “noise”

E (vn )   v * p(v)dv

E (vn )  


v
 A  v
2

2 0.5
dv
• Which is obviously zero
(odd symmetry, balance point, etc.
as it should since this noise the has no DC component.)
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Averages (3)
• E[vn] is also known as the “First Moment” of p(vn)

E (vn )   v * p(v)dv

• We can also calculate other important moments of p(vn).
The “Second Central Moment” or “Variance” (2) is:

  E v  vn
2
    v  v  * p(v)dv
2

2
n

Which for our sinusoidal noise is:
2 


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v2
  A2  v

2 0.5
Noise: An Introduction
dv
13
Averages (4)
Integrating this requires “Integration by parts
 U * dV  U *V   VdU
2 


let
v2
  A2  v
U  v and dV 
v
 
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
A

v

A

1
0
2
dv
 A  v
2
Then dU  dv and V  
2

2 0.5
v

2 0.5
A
2

2 0.5
 v2


A
0.5
and

A

1
A
A
Noise: An Introduction
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v

2 0.5
dv
14
Averages (5)
Continuing
 
2
A


A

1
A
2
 v2

0.5
dv
A
A
v
sin 1  
2
 A  A
2
A2

2

2

  
   
 2 
A2

2
Which corresponds to the power of our sine wave noise
Note:  (without the “squared”) is called the “Standard Deviation” of
the noise and corresponds to the RMS value of the noise
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Common Probability Density Functions:
The Gaussian Distribution
1
pv  

 2
 
 v v 2



2 2 

Gaussian Disrtibution
= 0.5, Mean = 0.5
1
0.8
0.6
0.4
0.2
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
• Central Limit Theorem
The probability density function for a random variable that is the result
of adding the effects of many small contributors tends to be Gaussian
as the number of contributors gets large.
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Common Probability Density Functions:
The Exponential Distribution
pv    *  v for v  0 , 0 for v  0
Exponential Disrtibution
= 1
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
• Occurs naturally in discrete “Poison Processes”
– Time between occurrences
• Telephone calls
• Packets
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Common Noise Signals
•
•
•
•
Thermal Noise
Shot Noise
1/f Noise
Impulse Noise
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Thermal Noise
• From the Brownian motion of electrons in a
resistive material.
pn(f) = kT is the power spectrum where:
k = 1.3805 * 10-23 (Boltzmann’s constant) and
T is the absolute temperature (°Kelvin)
• This is a “white” noise (“flat” spectrum)
– From a color analogy
– White light has all colors at equal energy
• The probability distribution is Gaussian
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Thermal Noise (2)
• A more accurate model (Quantum Theory)
pn  f  
h* f

 h* f

kT


1
Which corrects for the high frequency roll off
(above 4000 GHz at room temperature)
• The power in the noise is simply
Pn = k*T*BW Watts or
Pn = -174 + 10*log10(BW) in dBm
(decibels relative to a milliwatt)
Note: dB = 10*log10 (P/Pref ) = 20*log10 (V/Vref )
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Shot Noise
• From the irregular flow of electrons
Irms = 2*q*I*f where:
q = 1.6 * 10-19 the charge on an electron
• This noise is proportional to the signal level
(not temperature)
• It is also white (flat spectrum) and Gaussian
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1/f Noise
• Generated by:
– irregularities in semiconductor doping
– contact noise
– Models many naturally occurring signals
• “speech”
• Textured silhouettes (Mountains, clouds,
rocky walls, forests, etc.)
• pn(f) =A / f  (0.8 <  < 1.5)
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Impulse Noise
• Random energy spikes, clicks and pops
– Common sources
• Lightning
• Vehicle ignition systems
– This is a white noise, but NOT Gaussian
• Adding multiple sources - more impulse noise
• An exception to the “Central Limit Theorem”
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Noise Measurement
• The Human Ear
– Average Performance
– The Cochlea
– Hearing Loss
• Noise Level
– A-Weighted
– C-Weighted
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Hearing Performance
(an average, good, ear)
• Frequency
response is a
function of
sound level
• 0 dB here is
the threshold
of hearing
• Higher
intensities
yield flatter
response
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The Cochlea
• A fluid-filled spiral
vibration sensor
– Spatial filter:
• Low frequencies travel the
full length
• High frequencies only
affect the near end
– Cillia: hairs put out signals
when moved
• Hearing damage occurs
when these are injured
• Those at the near end are
easily damaged (high
frequency hearing loss)
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Noise Intensity Levels:
The A- Weighted Filter
• Corresponds to the sensitivity of the ear at the threshold of
hearing; used to specify OSHA safety levels (dBA)
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An A-Weighting Filter
• Below is an active filter that will accurately
perform A-Weighting for sound measurements
Thanks to: Rod Elliott at http://sound.westhost.com/project17.htm
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Noise Intensity Levels:
The C- Weighted Filter
• Corresponds to the sensitivity of the ear at normal listening
levels; used to specify noise in telephone systems (dBC)
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Energy Spectral Density (ESD)
E

t  
f 2 t dt is the Energy in a time waveform,
1 
 j t


but f t  
F


d is the Inverse Fourier Transform




2

 1 

E
f t  
F    j t d  dt Substituti ng for one f t 
t  
 2   


1 
 j t





E
F

f
t

dt  d Interchang ing the order of integratio n


 t  

2   
but the inner integral is almost the Fourier Transform (except for the "-" jt )
1 
*






  the complex conjugate
E
F

F


d

but
F



F




2
1 
2
2
E
F   d so Fω  is the " Energy Spectral Density"




2
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Energy Spectral Density (ESD)
and Linear Systems
X()

H()
Y() = X() H()
1
Ey  
Y   df changing to f eliminates the
f  
2
Ey  

Ey  

f  
f  
2
H  * X   df so if H   and X   are uncorrelat ed
2
H   * X   df
2
2
Therefore the ESD of the output of a linear system
is obtained by multiplying the ESD of the input by
|H()|2
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Power Spectral Density (PSD)
• Functions that exist for all time have an
infinite energy so we define power as:
lim  1 T2 2

P
  T f t dt  this is energy/tim e
t   T 2

T
T
Define fT t   f t  for   t  and zero elsewhere
2
2

ET   FT   df which does exist and the Power is :
2

2
lim  1    lim  FT   
 df
P
ET    


t    T   t    T



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Power Spectral Density (PSD-2)
• As before, the function in the integral is a density. This
time it’s the PSD
2

lim  FT   
PSD 
t   T 


• Both the ESD and PSD functions are real and even
functions
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