Transcript Document
Modulation and Demodulation
EECS 233
Fall, 2002
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Modulation
Conversion of digital electrical data into optical format
On-off keying (OOK) is the most commonly used scheme
NRZ format most commonly used
RZ format (requiring 2X NRZ bandwidth)
Pulse format (not commercially deployed)
bit interval
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DC Balance
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
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
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
Suppose n independent trials are performed
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
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 lnp 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?
Normal distribution was introduced by French
mathematician A. De Moivre in 1733.
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
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
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
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
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
Shot noise (Poisson process ~ Gaussian)
is2 (t ) 2eI Be
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
RP2
2e( RP I d ) Be 4(k BT / RL ) Fn Be
Thermal-noise limit (practical cases)
SNR
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
Noise-equivalent power (NEP)
Minimum optical power per unit bandwidth required to
produce SNR = 1
Detectivity
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
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
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
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
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
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
Photocurrent at the input side of receiver optical power
I GRP
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Noise of Optical Preamp
Optical power is proportional to square of electric
field
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
RP2
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
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 )
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
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 1.6561010 B( A) 2
RL
For pin with front end amplifier, Gm=1, Prec Q th / R 26dBm
2
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
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
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
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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