Transcript 2015.3.18

IEE5668 Noise and Fluctuations
02/25/2015 Lecture:
Pulse Train- Part I:
Two-Level Random Telegraph Signals (RTS)
Prof. Ming-Jer Chen
Dept. Electronics Engineering
National Chiao-Tung University
Self-Reading:
(Buckingham: Chapter 7 Burst Noise, p. 180; pp. 195-198 for RTS)
Pulse Train
Given the following pulse train:
It may come from one of the following signals:
1. Trapping-Detrapping in a device (due to defect)
(with no memory; well known RTS; time
exponential distribution)
2. Charging-Discharging (Trapping-Detrapping or
Capture-Emission) in a quantum dot
(with memory; time power-law distribution)
3. Shot noise in a quantum dot
4. Stochastic resonance in a nerve
5. Morse Code (telegraph signals)
6. Pulse from a pulse generator
Today, we will focus on RTS --- No memory --- Poisson event.
Other cases will be addressed later in the semester.
Three Key Parameters Characterizing a Two-Level RTS:
• RTS Magnitude I for given V
(you can use V for given I)
• Average Upper Time Constant upper
• Average Lower Time Constant lower
Note: If the two-level RTS you observed can be explained with the Poisson
process, then your RTS can be definitely said to be statistically stationary.
Review of Poisson Process
Three Properties of a Poisson Process
• No Memory to change the Future or Past Events;
• The Probability of Undergoing a Poisson Event in t being Proportional
to t;
• Ignorable Probability of finding more than one Poisson Event in t.
On the One Hand
The probability of finding exactly m transitions (that is, m Poisson events) ,
regardless of being in the upward or downward direction, in the time interval t is
Here
(t ) m e t
p(m;t ) 
m!
 is the mean number of the transitions per unit time
m, the Poisson variable, is equal to 0, 1, 2, 3,….
The mean and variance of the Poisson variable m both have the
same value of t.
On the Other Hand (two random variables: tupper and tlower)
A Poisson process also means the presence of an exponential distribution
for the upper time constant tupper and the lower time constant tlower:
p (t upper ) 
p (t lower ) 
1
 upper
1
 lower

e
e
tu p p er
 u p p er
Key: Time to Occurrence of Poisson Event;
Survival Time; Time to Survive;
Time to Capture; Time to Emission;
t
 lo w er
Lifetime; etc.
 lo w er
The mean and standard deviation of the exponential distribution for tupper
both have the same value of upper.
The mean and standard deviation of the exponential distribution for tlower
both have the same value of lower.
Here
1/upper is the probability per unit time of a downward transition in t or
between tupper and tupper+t
1/lower is the probability per unit time of an upward transition in t or
between tlower and tlower+t
exp(-tupper/upper) is the probability of no transition in tupper
exp(-tlower/lower) is the probability of no transition in tlower
Choice of the Origin of the Coordinate System – Case 1
At upper level, random variable x = I;
At lower level, x = 0.
Thus, we have, via the tree diagram, for the
autocorrelation function
<x(s)x(s+t)> = (I)2((upper/(upper+lower))P11(t) + 0 + 0 +0
s: time to start
<y>: the average of y
P11(t): probability of even number of transitions in time t, given starting in
high level - conditional probability
P10(t): probability of odd number of transitions in time t, given starting in
high level -> conditional probability
Thus, P11(t) + P10(t) = 1
and P11(t+dt) = P10(t)(dt/lower) + P11(t)(1-(dt/upper))
This leads to
dP11(t)/dt +P11(t)((1/lower)+(1/upper)) = 1/lower
The solution is
 upper
 lower
1
1
P11 (t ) 

exp( (

)t )
 lower   upper  lower   upper
 lower  upper
Note: P11(0) = 1.
The corresponding power spectral density can be
obtained via the Wiener-Khintchine theorem (The dc term
corresponds to a delta function at f = 0):

S ( f )  4  x( s) x( s  t )  cos( 2ft)dt 
0
S( f ) 
2(I ) 
4  (2f ) 2
2
( lower
4(I ) 2
1
1 2
  upper )((

)  (2f ) 2 )
 lower
 upper
for  = upper = lower
(I ) 2
P
The total power P
1
1
( lower   upper )(

)
In the RTS (or Lorentzian) spectrum;
 lower  upper
If upper = lower, then P is the maximum.
Choice of the Origin of the Coordinate System – Case 2
At upper level, random variable x = I/2;
At lower level, x = -I/2.
<x(s)x(s+t)> = (I/2)2Probability of an even number of transitions in t
- (I/2)2 probability of an odd number of transitions in t
Autocorrelation
function
= (I/2)2(p(0; t)+p(2; t)+p(4; t)+…..)
- (I/2)2(p(1; t)+p(3; t)+p(5; t)+….)
= (I/2)2 exp(-2t)
(Note: no dc term in this case)
Exponentially decaying with the
system’s memory time 1/2
1/ > 1/2  So, no memory.
The task to derive the following power spectral density is straightforward:
I 2
) /

I 2 
2
S ( f )  4  x( s) x( s  t )  cos tdt  4( )  exp( 2t ) cos tdt 
0
0
2
1   2 / 4 2
2(
Comparison with Case 1 confirms that what we see in noise power spectrum
stems from the time-domain fluctuations around the dc current, NOT the dc itself.
Keep in Mind:
If your measured random pulse train is due to RTS (no
memory), you must show the evidence:
it follows the exponential distribution and
it also comes from a Poisson event.
Otherwise, other mechanisms may be possible:
Power-law relationship (memory), Shot noise,
Stochastic resonance, Morse code, even from
a pulse generator, etc.