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( 2ft)dt
0
S( f )
2(I )
4 (2f ) 2
2
( lower
4(I ) 2
1
1 2
upper )((
) (2f ) 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)2Probability 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(-2t)
(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( 2t ) 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.