Transcript Document

Modulation and Demodulation
EECS 233
Fall, 2002
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Modulation
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Conversion of digital electrical data into optical format
On-off keying (OOK) is the most commonly used scheme
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NRZ format  most commonly used
RZ format (requiring 2X NRZ bandwidth)
Pulse format (not commercially deployed)
bit interval
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DC Balance
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Typical problem for all OOK formats
Average transmitted power is not constant  decision
threshold varies with ratio of 1s to 0s
Use line coding or scrambling to solve the problem
Binary block line code  encodes a block of k bits into n
bits i.e. (k,n) code
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Scrambling
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Fiber Channel standard  (8, 10)
Fiber distributed data interface (FDDI)  (4,5)
Most data links
1-to-1 mapping of data stream and coded data streme
Minimize prob. of long 1s and 0s
Does not require more BW
Does not guarantee DC balance
Most telecom links
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Demodulation: Receiver
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Recovering the transmitted data
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Recovering the bit clock
Determining the bit value within each bit interval
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Binomial Random Variable
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Suppose n independent trials are performed
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each results in a success with probability p and in a
failure with probability 1-p.
X represents the number of successes that occur in n
trials, then X is said to be a binomial RV with parameter
(n,p).
The probability mass function (pmf)
n!
n i
p(i ) 
p i 1  p 
i!(n  i )!
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Poisson Random Variable
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First introduced by S.D. Poisson in 1837
regarding probability theory applied to lawsuits,
criminal trials and the like.
Approximates a binomial random variable with
parameters (n, p) when n is large and p is small
and lnp is moderate.
X is a Poisson random variable, taking the values
0, 1, 2, …, with parameter l
i
p(i)  P X  i   e
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l
l
i!
Prove to yourself that this comes from the Binomial pmf
l is in a sense the average success (occurrence rate)
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Normal or Gaussian?
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Normal distribution was introduced by French
mathematician A. De Moivre in 1733.
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1809, K.F. Gauss, a German mathematician, applied it to
predict astronomical entities… it became known as
Gaussian distribution.
Late 1800s, most believe majority data would follow the
distribution  called normal distribution
DeMoivre-Laplace limit theorem
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When n is large, a binomial RV will have approx. the same
distribution as a normal RV with the same mean and variance.
Central limit theorem
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Used to approximate probabilities of coin tossing
Called it exponential bell-shaped curve
The sum of a large number of independent RVs has a
distribution that is approx. normal.
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Bit Error Rate: Ideal Receiver
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Light signal with power P and B is bit rate
#Photons/sec=P/hn
Ave # Photons/bit interval = P/(hnB
p ( n)  P  X  n   e  l
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ln
n!
Where l  P / hnB
l
P[0|1]= p(0)  e
1  P hnB
BER=p(1)P[0|1]+p(0)P[1|0] = e
2
-12
For 10 BER, we need an average of 27
photons per bit
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Normal Distribution
f ( y) 
1
2 
e
( y   ) 2 2
2
 2= variance = (STD)2
 = mean



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
f ( y)dy 
1
2 
e
( y   ) 2 2
2
dy  1

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Receiver Design
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Front End Configuration
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High-impedance amplifier
Transimpedance front-end can achieve high
bandwidth (small RC constant), high sensitivity
and low thermal noise (high load resistance)
RL
Rin 
G
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PIN Basis
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Increase width of i-region
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Increase RC constant
Increase Responsivity
Gain-Bandwidth trade-off
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Materials for PIN diodes
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Noise in PIN Diodes
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Shot noise (Poisson process ~ Gaussian)
is2 (t )  2eI Be
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R is responsivity of detector
Be elec. BW of detector (typically between bit rate and
½ bit rate)
Thermal noise (Gaussian process)
it2 (t ) 
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I  RP
4 k BT
Fn Be
RL
Fn is the noise figure of front-end amplifier, typically 3-5
dB.
Total noise power
I 2
 is2 (t )  it2 (t )
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Signal to Noise Ratio
SNR 
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RP2
2e( RP  I d ) Be  4(k BT / RL ) Fn Be
Thermal-noise limit (practical cases)
SNR 
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4k BTFn Be
Shot-noise limit
SNR 
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RL R 2 P 2
RP
2eBe

P
2hnBe
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Some terminology
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Noise-equivalent power (NEP)
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Minimum optical power per unit bandwidth required to
produce SNR = 1
Detectivity
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The inverse of NEP
P
NEP 
~ 1  10 pW / Hz 2
Be
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Avalanche Photodiodes
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Electron multiplication was first observed in p-n junctions in Silicon and
Germanium, by McKay, K. G. and K. B. McAfee in 1953 (Phys Rev.
91:1079)
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Electron Multiplication Effect
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A high E-field in the depletion region causes carriers to have enough kinetic
energy to “kick” new electrons from valence band up to conduction band –
Avalanche Multiplication
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Positive Feedback
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When both electrons and holes can ionize, both will contribute to gain.
However, because they move in opposite directions, a positive
feedback loop arises.
Positive feedback degrades both response bandwidth and noise, by a
factor equal to loop-gain plus one. (from general feedback theory)
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Ionization Cofficient
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Ionization coefficient
is the number of new
e-h pairs generated
per unit length by one
photo-carrier.
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APD Structures
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Advanced APD structures have two regions: one for photon
absorption, and the other for carrier multiplication. This is so
that both absorption length and multiplicative gain can be
optimized separately.
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APD Gain
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One can derive the gain of an APD. Derivation is
similar to gain/loss in lasers, fibers etc., except now
carriers move in both directions, complicating things.
For the case where >>, the gain is as expected:
Gm=exp (L).

   e   L
For , gain is: Gm 
  e   L
1
For =, Gm 
1  L
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Noise in APD
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Shot noise
is2 (t )  2eRPB eGm FA Gm 


FA (Gm )   Gm  1   2  G1m
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
I  Gm RP
Where FA is the excess noise factor
InGaAs detectors typically have / =0.7
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Optical Preamp
Optical
Preamp
G
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Receiver
RX
Spontaneous noise power
PN  nsp hn (G 1)Bo  Pn (G 1)Bo
nsp  N 2  N1
N 2 = population inversion factor
Bo  Optical bandwdith
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Photocurrent at the input side of receiver  optical power
I  GRP
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Noise of Optical Preamp
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Optical power is proportional to square of electric
field
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Includes beating noise of signal-spontaneous and
spontaneous-spontaneous
2
is2 (t )  2eR GP  Pn (G  1) Bo Be   shot
4 k BT
Fn Be
RL
2
2
isignal
(
t
)

4
R
GPPn (G  1) Be
 spon
it2 (t ) 
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When G is large, signal-spontaneous beat noise
dominates over shot and thermal noise
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Noise Figure
Optical
Preamp
SNRi
SNRi 
G
Receiver
SNRo
RX
RP2
2e( RP  I d ) Be  4(k BT / RL ) Fn Be
No thermal noise, since the signal has not hit the detection circuit.
Ignore dark current.
2
SNRo 
Noise Figure
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GRP
4R 2GPPn (G  1) Be
SNRi ( RP) 2  4R 2GPPn (G  1) Be
Fn 

 2nsp
2
SNRo
2RePBe  GRP
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Bit Error Rates
erfc( x) 
2

e


 y2
dy
x
 Q 
1
erfc

2
 2
I1  I 0
Q
= Optical SNR
1   0
BER 
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Typically we like to know what is the minimum receiver
power for a given BER
Receiver sensitivity definition= minimum average optical
power for a given BER (typical 1e-12)
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Receiver Sensitivity
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Prec : Min. average power to achieve 1e-12 BER
Prec  ( P0  P1 ) / 2  P1 / 2
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For BER=1e-12, Q=7
Q  ( I1  I 0 ) /( 0  1 )  I1 /( 0  1 )
P1  Q  ( 0   1 ) / Gm R
Assuming I0=0
Prec  Q  ( 0  1 ) / 2Gm R
 0 2   th2 1   th   shot
2
 shot
 2eR  2Prec  BeGm FA Gm 
2
2
2
Q
Prec   eB e FA (Gm )Q   th / Gm 
R
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Sensitivity Improvement in APD
Q
Prec   eB e FA (Gm )Q   th / Gm 
R
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APD gain reduces the effective thermal noise, but increases
effective shot noise.
Assume l=1.55 m, R=1.25 A/W, =1, T=300 K, Be=B/2 and
B=109 bits/sec, Fn=3dB (front end amplifier), RL=100W
4k BT
 th 
Fn Be  1.6561010 B( A) 2
RL
For pin with front end amplifier, Gm=1, Prec  Q   th / R  26dBm
2
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For APD with Gm=10 and FA=1.3 (from kA=0.7)
Prec  Q  th /(RGm )  36dBm
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Receiver Sensitivity: Optical Preamp
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Prec : Min. average power to achieve 1e-12 BER
Prec  ( P0  P1 ) / 2  P1 / 2
For BER=1e-12, Q=7
Q  ( I1  I 0 ) /( 0  1 )  I1 /( 0  1 )  I1 / 1
I1  RGP1
12  4R2GP1Pn (G 1)Be
Q
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GP1
2 Pn (G  1) Be
For large G,
Prec  2Q2 Pn Be
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Improvement w. Optical Preamp
Prec  2Q2 Pn Be
Pn  nsp hn
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For nsp=1, l=1.55 m, R=1.25 A/W, =1, T=300 K, Be=B/2 and
B=109 bits/sec
Prec  98 hn  Be
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Or number of photons per bit = 2Q2 = 98
Prec  50dBm
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PIN and APD Sensitivity
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An APD typically has 10 dB better sensitivity than a PIN.
-36 dBm sensitivity at 2.5 GB/s is possible.
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Typical Spec-sheet of a 10Gb pin
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References
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B.E.A. Saleh and M.C. Teich, “Fundamentals of
Photonics”
G. P. Agrawal, “Fiber-Optic Communication Systems”
A. Yariv, “Optical Electronics in Modern
Communications”
Silvano Donati, “Photodetectors : devices, circuits,
and applications”
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