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Communication Theory/2
I. Frigyes
2009-10/II.
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Frigyes: Hírkelm 2
http://docs.mht.bme.hu/~frigyes/hirkelmhirkelm01bEnglish
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2. Transmission of digital signals over analog channels: effect of
noise
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Frigyes: Hírkelm 4
Introductory comments
• Theory of digital transmission is (at least partly) application of decision theory
• Definition of digital signals/transmission:• Finite number of signal shapes (M)• Each has the same finite duration (T)• The receiver knows (a priori) the
signal shapes (they are stored)• So the task of the receiver is hypothesis
testing.
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Frigyes: Hírkelm 5
Introductory comments
• Quality parameter: error probability• (I.e. the costs are:• )• Erroneous decision may be caused by:• additíve noise• linear distortion• nonlinear distortion• additive interference (CCI, ACI)• false knowledge of a
parameter• e.g. synchronizing error
MkiCC kiii ,...,2,1, ;1 ;0
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Frigyes: Hírkelm 6
Introductory comments – degrading effects causing false decision
DECISIONMAKER
BANDPASSFILTER
BANDPASSFILTER
FADINGCHANNEL +
n(t)
NONLINEARAMPLIFIER
s(t)
INTER-FERENCE
INTER-FERENCE
INTER-FERENCE
ωc
ωcz0(t)
z1(t)
z2(t)
ω1
ω2
CCI
ACI
ACI
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Frigyes: Hírkelm 7
Introductory comments
• Often it is not one signal of which the error probability is of interest but of a group of signals – e.g. of a frame.
• (A second quality parameter: erroneous recognition of T : the jitter.:
• )
T
Tˆ
pp
TTtJ ˆ
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Frigyes: Hírkelm 8
Introductory comments – improvement of performance parameters
Both performance parameters can be improved
Error probability:
ÁTVITELI CSATORNAFORRÁS KÓDOLÓ
DE-KÓDOLÓ NYELŐ
PE
PE,dec
Transmission ChannalSOURCE ENCODER
DE-CODER SINK
JitterTransmission
Channal
data
clock
ELASTICSTORE
JITTERFREE
CLOCK
data
clock
DIGITALSOURCE
Transmission Channal
SINK
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Frigyes: Hírkelm 9
Introductory comments – degrading effects causing false decision
• Comments:• 1. These effects canot be described by the
two performance parameters. The channel, at this level is an analog channel producing the effects seen in slide #6
• 2. Behavior of radio and optical channels are rather different. First we deal with the first and then show differences in the second
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Frigyes: Hírkelm 10
0.Transmission of single signals in additive Gaussian noise
• Among the many sources of error now we regard only this one
• Model to be investigated:
SOURCESIGNAL
GENERATOR +DECISION
MAKERSINK
TIMING (T)
n(t)
mi
{mi}, Pi
si(t) r(t)= si(t)+n(t)
ˆm
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Frigyes: Hírkelm 11
0.Transmission of single signals in additive Gaussian noise
• Specifications:• Pi a-priori probabilities are known• Support of the real time functions • is (0,T)• their energy is finite (E: square
integral of the time functions)
• mutual unique relationship (i.e.: their is no error in the transmitter)
tsi
tsm ii
T
ii dttsE0
2ˆ
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Frigyes: Hírkelm 12
0.Transmission of single signals in additive Gaussian noise
• Noise: Gaussian
• 0-mean
• stationary
• additive it’s drown so
• white
• Comment.: in white noise: σn=
2
2
1
2
1
n
n
n
n ep
const;0 nn
2
ˆ2
00 NTFkS B
n
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Frigyes: Hírkelm 13
0.Transmission of single signals in additive Gaussian noise
• Com.: white is an approximation More exact: Planck-formule:
• If hf/kBT0<<1:
• If hf/kBT0>>1:
• f =300 GHz,T0=30K:FkBT0 -0,1dB
• f=200 THz,T0=270K:FkBT0-127 dB
10
TBkhfn
e
hfFS
00 11
TFkTkhf
hfFS B
Bn
00 Tkhfn
BehfS
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Frigyes: Hírkelm 14
0.Transmission of single signals in additive Gaussian noise
• Decision: based on r(t)=si(t)+n(t).• Application of the general method:
independent samples would result in too high noise; correlated samples yield less information;
• they don’t specify bandwidth.• Contionouos investigation and appropriate
processing of signals yield most information. This is the subject of our next investigations.
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Frigyes: Hírkelm 15
0.Transmission of single signals in additive Gaussian noise
• Questions to be looked for:• 0. Vectorial representation of digital
signals• 1. The optimal receiver• 2. Error probability• 3. Coherent – non-coherent• 4. Optimal signal set• 5. Bandwidth occupation
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Frigyes: Hírkelm 16
0.Transmission of single signals in additive Gaussian noise
• Given the – somehow chosen – signal set
• We chose an orthonormal base:
• (ortonormal:• So that
• Of course
Mitsi ...1;
MDDjtj ;...1);(
D
lllii Mitats
1, ...1;
T
jiij dtttsa0
T
jllj lj
ljdttt
0;0
;1.
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Frigyes: Hírkelm 17
0.Transmission of single signals in additive Gaussian noise
• Thus: time functions are uniquely representet by D numbers (ai,1, ai,2 … ai,D)
• But: any structer represented by D numbers can be regarded as a vector of D dimensions
• I.e.• So we defined a vector space: signal
space• D is the dimensionality of the signal space
iDiiT
i aaats ...,, 21 is
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Frigyes: Hírkelm 18
0. Comment
• As said: DM • Earlier we saw: in the general case
dimensionality of the decision space is D=M-1.In the case of concretly defined signal waveforms (like now) decision can be made in the signal space; then D<M-1 is possible.
• (We had also the observation space, with D=N. In the case of continouos obser-vation D=∞, is not too important
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Frigyes: Hírkelm 19
0. How to chose the base?
1.1
21
212
1
1
11
dts
dtsdt
E
tst
0.
ˆ.2
122
1121
21122
22
121212122
dtsdtdtsE
dtEdth
tht
dtttsttstatsth
h
h
0
0
;ˆ.3
232
213212332
2321132
11331
23
3323213133
dtsdtdtsdtdtsdt
dtsdtdtsdtdtsdt
dtth
thttatatsth
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Frigyes: Hírkelm 20
0. How to chose the base?
• This can be done as long as we have signals (Gram-Schmidt ortogonalization)
• We see: there areM base functions at most. • But if some signal waveforms are linear
combinations of others these don’t introduce new dimensions
• E.g. dimensionality of M-ary PAM signal set is 1, of QAM signal set it is 2.
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Frigyes: Hírkelm 21
0. Scalar product
• Schalar product of two signal-space-vectors is the integral of their product:
• By the way from that: |si|2 = Ei
D
jkjij
D
j
D
l
T
ljklij
T T D
j
D
llkljijki
aadtttaa
dttatadttsts
11 1 0
0 0 1 1
kiss
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Frigyes: Hírkelm 22
0. Single signals: vectorial form of noise
• After the signal: noise should also be given in vectorial form.
• Of course: compo-nents of the noise vector can be written
• And by that: the noise vector
• But it is not true for the noise process that
T
jj dtttnn0
DT nnn ..., 21n
D
jjj tntn
1
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Frigyes: Hírkelm 23
0. Single signals: vectorial form of noise
• (A Gaussian process can not be linear combination of a finite number of functions.)
• So• We know that is orthogonal to the signal
space; as the signal is in the signal space, an efficient receiver can filter out this part of the noise – it is thus irrelevant from the point of view of reception. I.e. n contains that part of the noise what is relevant. (We’ll briefly come back to that.)
D
jjj tntntn
1
`
tn
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Frigyes: Hírkelm 24
0. Single signals: vectorial form of the link
• Thus we can investigate the vectorial model of this connection
SOURCESIGNALVECTOR
GEN.+ DECISION SINK
n
mi
{mi}, Pi
si r= si+n
ˆm
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Frigyes: Hírkelm 25
0. Single signals: vectorial representation
• Pdf of the noise in the signal space: σ2 was doubtful: of white noise is infinite.
• Without details: in the interval [0,T] Gs noise can be expanded according to any complete orthogonal series.
• Individual terms are independent and have equal σ2.
• Base of the signal space is part of such a complete base (we are interested only in that part).
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Frigyes: Hírkelm 26
0. Single signals: vectorial representation
• Thus pdf-s can be written:
0
2
20
1 ND
eN
p
n
n n
0
2
20
|
1| N
i
Diis eN
p
sr
r sr
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Frigyes: Hírkelm 27
0.The signal space
D
jjjii Mitats
1, ...1;
T
jiij dtttsa0
iDiiT
i aaats ...,, 21 is
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Frigyes: Hírkelm 28
0.The signal space - examples
• 1. (Antipodal) baseband NRZ signals:
s1(t) A=(E/T)1/2T
s2(t)=-s1(T)
φ(t)1/T1/2T
M=2D=1
s2 s1
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Frigyes: Hírkelm 29
0.The signal space - examples
• 2. BPSK signals
tAts
tAts
c
c
cos2
cos2
2
1
M=2D=2
Φ s1
s2
If Φ=π: antipodalD=1
tT
t
tT
t
c
c
sin2
cos2
2
1
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Frigyes: Hírkelm 30
0.The signal space - examples
• 3. QPSK signals tststAts
tststAts
c
c
242
131
;sin2
;cos2
M=4D=2
tT
t
tT
t
c
c
sin2
cos2
2
1
s1
s2
s3
s4
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Frigyes: Hírkelm 31
0.The signal space - examples
• 4. Ortogonal QFSK signals
3,2,1,0;2
;cos2
iT
titAts ci
M=4D=4
ttT
t ci cos2
s1
s2
s3
s4
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Frigyes: Hírkelm 32
0.The signal space - examples
• 5 Biortogonal signals
tststAts
tststAts
c
c
242
131
;cos2
;cos2
M=4D=2
tT
t
tT
t
c
c
cos2
cos2
2
1
s1
s2
s3
s4Note: just like QPSK
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Frigyes: Hírkelm 33
0.The signal space - examples
• 6. MQAM jelek
MiM
iqa
tAqtAats
ii
cjcii
....2,1;1
12,
sincos
MD=2
tT
t
tT
t
c
c
sin2
cos2
1
1
Example: M=16
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Frigyes: Hírkelm 34
1. Single signals: the optimal decision rule
• Decision rule now: a jeltér optimal partitioning of the signal space (resulting in minimal error probability)
• Pl: D=2s1
s2
s3 s4
s5
n
r
Before we had to partition the decision space
nr
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Frigyes: Hírkelm 35
1. Single signals: the optimal decision rule
• We’ve seen: risk is minimal if the a-posteriori probability is maximl. We the decide on what is the most likely, i.e.
• To proceed apply Bayes theorem:
Miii ,...2,1;max|Pr rs
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Frigyes: Hírkelm 36
1. Single signals: the optimal decision rule
• Thus the decision rule:
• Or: as denominator does not depend explicitly on i
r
ss|rrs
r
iii|sri p
p Pr.|Pr
Mi
p
pH i ,...2,1max;
Pr|:ˆ .
r
ssr
|r
iii|sr
MipH i ,...2,1max;Pr|:ˆ. iii|sr
ssr
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Frigyes: Hírkelm 37
1. Single signals: the optimal decision rule
• Logarithm: of the a-posteriory pdf:
• Finally
MiPN
NN
DP
i
i
,...2,1max;2ln
thus;ln2
ln
0
0
2
0
2ii
2
i
srsr
sr
MiEPNH iii ,...2,1max;2
1ln
2
1:ˆ
0 irs
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Frigyes: Hírkelm 38
1. Single signals: the optimal decision rule
• For an instant come back to the noise vector• We’ve seen:• • Details of the decision noise, taking the whole noise into
considerationl
• I.e. ‘n(t) in the optimal receiver is really irrelevant
tntntntnD
jjj `""`
1
n
0
0
0 10 1
`
`
T
i
T D
jjji
T
i
D
jjji
dttnts
dttntsdttntstnts irs
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Frigyes: Hírkelm 39
1. Optimal decider – vectorial form
× +
s1
½(N0lnP1-E1)
× +
s2
½(N0lnP2-E2)
× +
sM
½(N0lnPM-EM)
rCOMPARATOR
max
If E-s are equal itcan be omittedfrom the bias. Ifin addition Pi=1/M,the whole bias can be omitted.
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Frigyes: Hírkelm 40
1. Optimal decider (correlation)
T
dt0
×
s1(t)
+
½(N0lnP1-E1)
r(t)
COMPARATOR
max×
s2(t)
+
½(N0lnP2-E2)
T
dt0
×
sM(t)
+
½(N0lnPM-EM)
T
dt0
Sense of scalar productis known
Question: are all elements of the model
needed?
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Frigyes: Hírkelm 41
SOURCESIGNAL
GENERTOR + DECISION SINK
Timing (T)
n(t)
mi
{mi}, Pi
M
si(t) r(t)= si(t)+n(t)
ˆm
s(t) known
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Frigyes: Hírkelm 42
1. Single signals: the optimal decision rule
• Comment: if Pi≡1/M (equal a-priori prob.)
• I.e. we have to decide on which is closest
MiN
H
MiN
H
i
i
,...2,1min;:ˆ
,...2,1max;:ˆ
0
2
0
2
i
i
sr
sr
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Frigyes: Hírkelm 43
1. Optimal decider (matched filter)
• Correlation is a linear operation (multiplication by a signal independent of r(t)and integration).
• But: a linear operation can also be done with a linear filter thus an equivalent filter can also be found – its impulse response is h(t).
Tti
Tt
i
i
T
i
tTstrdtTsr
dsrdttstr
0
h(t)=si(T-t)↓
It is causal!
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Frigyes: Hírkelm 44
1. Optimal decider (matched filter)
r(t)
COMPARATOR
max
+
½(N0lnP1-E1)
s1(T-t)
+
½(N0lnP2-E2)
s2(T-t)
+
½(N0lnPM-EM)
sM(T-t)
t=T
![Page 45: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/45.jpg)
Frigyes: Hírkelm 45
1.Some of the previous examples together with decision boundaries
• 1. (Antipodal) NRZ baseband signals
s2(t)=-s1(T)
s1(t) AT
φ(t)1/T1/2T
M=2D=1
s1 s2
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Frigyes: Hírkelm 46
1.Some of the previous examples together with decision boundaries
• MQAM signals
MjiM
iqa
tAqtAats
ji
cjcii
....2,1,;1
12,
sincos
MD=2
tT
t
tT
t
c
c
sin2
cos2
1
1
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Frigyes: Hírkelm 47
1.Some of the previous examples together with decision boundaries
• 3. QPSK tststAts
tststAts
c
c
242
131
;sin2
;cos2
M=4D=2
tT
t
tT
t
c
c
sin2
cos2
2
1
s1
s2
s3
s4
![Page 48: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/48.jpg)
Frigyes: Hírkelm 48
1. Optimal reception in the optical band
• We’ve seen that there is no termal noise in the optical band.
• On the other hand there is shot noise. (We’ve seen – without refering to optics – the effect of Poisson noise.)
• To some detail later.
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Frigyes: Hírkelm 49
2. Error probability in the optimal detector
• Based on the precedings: conditional probability of correct decision (condition: si is transmitted):
• Total probability of correct decision:
• And the error probability
iV
iC dvpP iisrsrs ||
iV
M
iiC dvpPP iisr
sr ||1
CE PP 1
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Frigyes: Hírkelm 50
2. Error probability in the optimal detector
• If the a-priori probabilities are equal
• Or, if the constellation is in addition symmetric
M
i iVC dvp
MP
1| |
1iisrsr
1
1| |V
C dvpP srisr
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Frigyes: Hírkelm 51
2. Error probability in the optimal detector
• Important comment: we see: error probability depends only on the vectorial constellation and it does not depend on the signal wave forms.
• Thus e.g. antipodal NRZ and antipodal PSK, QPSK and biortogonal FSK yield identical error probability
• Thus choice of wave forms can be based on different criteria.
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Frigyes: Hírkelm 52
2. Example: antipodal signal set
• PE can be computed directly (PC is not needed):
• And: it is symmetricEd
0
0
20
0
erfc2
11
N
Edre
NP N
Er
E
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Frigyes: Hírkelm 53
2. Example: QPSK (and biortogonal) signal set
2Ed
0
2
0 2erfc
4
1
2erfc
N
E
N
EPE
2
EPP antipod
EQPSK
E
You see: the signal space is rotatedby 450; this doesn’t change the situation
How did we get it?
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Frigyes: Hírkelm 54
2. Example: QAM signal set
There are 3 different signalvectors, also if M>16).After much calculation(nearly exact formula):
12
;erfc/1120
M
Ed
N
dMP
peak
E
d
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Frigyes: Hírkelm 55
2. Example for preparing a bound : general binary signal set
2212
211222
1
021
211
111
;
;;
aE
attstts
E
dttsts
aE
tstts
T
General binary signal set:
d1+d2
s1
s2
φ1
φ2
ψ2
ψ1
s2-s1
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Frigyes: Hírkelm 56
2. Example for preparing a bound : general binary signal set
• The original (φ1,φ2) coordinate system can be transformed into (ψ1,ψ2)-be. We see that, decision result depends only on co-ordinate ψ2 of the transformed system. (r.s) – what is a (scalar) Gauss random variable. So it is a 1D situation
• This is a slightly modified version of (H0: noise, H1: signal+noise) case: H0:r=s1+n; H1:r=s2+n
• Thus the optimal decision:
![Page 57: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/57.jpg)
Frigyes: Hírkelm 57
2. Példa, korlát előkészítésére: általános bináris jelkészlet
In the new co-ordinate system
2:ˆ
2:ˆ
1222
1221
ssrss
ssrss
Comments: we assumed Pi=0,5; if notη must also be taken into account.Vector notation was not used as now s is 1D
d1+d2
s1
s2
φ1
φ2
ψ2
s2-s1
ψ1
Ed
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Frigyes: Hírkelm 58
2. Example for preparing a bound : general binary signal set
• But so: the role of di2 is the same as of E in the
antipodal signal set. Thus error probability:
• Application to M-ary signal set:
0
erfc2
1
N
dPE
dmin
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Frigyes: Hírkelm 59
2. Error probability in the optimal detector: a bound
• From that we get a bound for the M-ary signal set:
• As Gaussian pdf changes very fast false decision between closest is of decisive proportion. I.e. error rate is nearly equal to an antipodal case with energy :
• (Miért <?)
0
minerfc2
1
N
dPE
2dE
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Frigyes: Hírkelm 60
3. Special case: carrier wave transmission
tAtv ccos2• We know: a sinusoid
has 3 parameters:• It is fully known if we
know all:• A: AGC
• ωc: cristall control
• Φ: ?
Φ can be transmitted on aseparate route: coherent. Or we accept that it is a(uniformly distributed) random variable: non-coherent
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Frigyes: Hírkelm 61
3. Special case: carrier transmission, coherent case
SOURCESIGNAL
GEN. + DECIDER SINK
Timing (T)
n(t)
mi
{mi}, Pi
si(t) r(t)= si(t)+n(t)
mˆ
Phase (Φ)
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Frigyes: Hírkelm 62
3. Special case: carrier wave transmission – coherent system• In this case everything is valid. Optimal detector
can be simplified
T
dt0
×
cos ωct
×
1
cos ωct0 KOMP
BPSK
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Frigyes: Hírkelm 63
3. Special case: carrier wave transmission – coherent system
QPSK
T
dt0
×
cos ωct
×
1
cos ωct0 COMP
T
dt0
×
sin ωct
×
1
sin ωct0 COMP
+ LOGICS
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Frigyes: Hírkelm 64
3. Special case: carrier wave transmission – coherent system
64QAM
T
dt0
×
cos ωct
×
1/7, 3/7, 5/7, 1
cos ωctQUANTIZER
T
dt0
×
sin ωct
×
1/7, 3/7, 5/7, 1
sin ωctQUANTIZER
+ LOGICS
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Frigyes: Hírkelm 65
3. Special case: carrier wave transmission – coherent system
BFSK – orthogonal
T
dt0
×
cos ωc1t0 COMP
T
dt0
×
cos ωc2t
1
VCO +
+
-
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Frigyes: Hírkelm 66
3/a A few words on noncoherent detection
• First: in this case phase cannot contain information. Further
• Phase is unknown. The criterion that the receiver knows the signals must be eased: it knows it, except one parameter.
• According to precedings, PE is minimal if the a-posteriori probability is maximal. Again it will bw assumed that Pi =1/M.
• Real signal-waveform #i is
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Frigyes: Hírkelm 67
3/a A few words on noncoherent detection
sc
iciici
iciir
icii
ss
tttattta
tttats
tttats
sinsincoscos
cos
then waveformreceived the;cos
• Pdf of r (to be maximized) first in conditional form:
sincos21
exp1
sincos1
exp1
,|,
0
0
0
22
00
sci
2sr
2cr
isicisc
rrs
srsrsrr
Ne
N
NNp
NiE
D
D
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Frigyes: Hírkelm 68
3/a A few words on noncoherent detection
• We assume that Φ is of uniform distribution.• The total pdf is then
• Omitting terms not depending on si we have
• This integral is known:
dpp
2
0
,|,2
1|, iscisc srrsrr
0
22
0
2
0 0
0
22I
sincos21
exp2
1|,
N
dN
ep NiE
sici
sciisc
rsrs
rrssrr
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Frigyes: Hírkelm 69
3/a A few words on noncoherent detection
• Simplification possibilities: if signals have equal energy multiplier e-Ei/N0 can be omitted.
• Further in this case: as I0 is monotonously increasing növekszik, we have to maximize its argument; thus for that case the optimal noncoherent decision rule is:
Mi
Mi
s
s
...,2,1max; or
...,2,1max;
22
22
iic
iic
srsr
srsr
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Frigyes: Hírkelm 70
3/a A few words on noncoherent detection: optimal detector
• This is formed by an envelope detector. Thus an optimal noncoherent system
r(t)
COMPARATOR
max
s1(T-t)
s2(T-t)
sM(T-t)
t=T
ENVELOPEDETECTOR
ENVELOPEDETECTOR
ENVELOPEDETECTOR
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Frigyes: Hírkelm 71
3/a A few words on noncoherent detection: error probability
• Only only for binary orthogonal signal set :
• Note: it does not differ much from orthogonal:
02
2
1 NEE eP
000
2exp2
1~
2erfc
2
1NE
NEN
EPE
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Frigyes: Hírkelm 72
3/a A few words on noncoherent detection; example: FSK
BFSK – orthogonal
0 COMP
1
VCO +
+
-
ENVELOPEDETECTOR
ENVELOPEDETECTOR
BANDPASSω1
BANDPASSω2
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Frigyes: Hírkelm 73
3/a A few words on noncoherent detection
• Then (find out): • Why for coherent?• (In practice it is nearly exclusively used.)
000
2exp2
1~
2erfc
2
1NE
NEN
EPE
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Frigyes: Hírkelm 74
4. Optimizing the signal set
• Up to now signal set was assumed to be known. The receiver was optimized, analyzed based on that knowledge, etc.
• Of course: the transmitter is also designed by ourselves so the signal set must also be optimized
• As we know that performance depends only on the vector constellation, that must be optimized
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Frigyes: Hírkelm 75
4. Optimizing the signal set
• We’ve seen that error performance improves if the signals are more fare away. Thus signal vectors must be chosen according to that: signals should be far away from each other.
• One way is to increase energy – but this is no true optimizing .
• Thus optimization criterion: • signal vectors should be as far as possible• but none of them should be of energy
higher than a specified Emax.
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Frigyes: Hírkelm 76
4. Optimizing the signal set
• A more precise drafting in the signal space:
• Put M points on the surface or in the interior of a sphere of dimensionality DM and radius r so, that their distance should be as large as possible; (r2 is the prescribed maximal energ). (Sphere packaging problem.)
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Frigyes: Hírkelm 77
4. Optimizing the signal set
• Solution of this problem of variation calcule is the regular simplex signal set:
• Characteristics are:
• In practice only the case M=2, s2=-s1 is of significance
1
1
max
max2
M
E
E
MD
ki
i
ss
s
M = 2
M = 3
M = 4
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Frigyes: Hírkelm 78
4. Optimizing the signal set
• A practical problem: the case D≤2
• I.e.: PSK is the best if M6; if M>6, joint modulation of amplitude and phase is better (of course: a surface rather than a line is available)
M=2 M=4 M=6 M=7
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Frigyes: Hírkelm 79
4. Optimizing the signal setWith larger M : along a regular triangular grid.(Honeycomb modulation)
Not much applied, for practical reasons; nearly as good: a rectangular grid(i.e. QAM)
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Frigyes: Hírkelm 80
5. Frequency band occupation, etc
• 3 questions arise• occupied band?• physical meaning of dimensionality?• why to talk about M>2)?• Frequency band: if a function is of finite support,
that of its Fourier transform is infinit (meaning?)• Theoretically optimized band is thus infinite• but practically: band where most of the
energy is (e.g. 90%) is of course finite. (Of the order of 1/T)
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Frigyes: Hírkelm 81
5. Frequency band occupation, etc
• Dimensionality: dimension-theorem: practically occupied bans (whatever is definition) is proportional to dimensionality.
• To verify examples: PPM and orthogonal FSK)
τ
T M=4, D=4
s1(t)
s2(t)
s3(t)
s4(t)
W=1/τ=4/T=D/T
M=4, D=4
1/T
W=4/T=D/T
fHz
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Frigyes: Hírkelm 82
5. Frequency band occupation, etc
• Decrease of W: increase ofT
• Only possibility: not bit-by-bit, rather several bits together (united into 1 symbol)
• Then: M=2n; and: Ts=nT
• If D remains constant:Wn=W1/n
• Cost of that: we have to distinguish M rather than 2 signals (i.e. points)
• For what higher power is needed
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Frigyes: Hírkelm 83
5. Frequency band occupation, etc
2
11
11
12 if
12
1;
ME
Edd
MEdEd
nn
nn
En
dn
d1= E1
•Detailed: QAM vs. antipodError prob (nearly) equal ifd1=dn
or powers
n
122
TE
nTE
P
P22n
1
n
1
n
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Frigyes: Hírkelm 84
5. Frequency band occupation, etc
• Conclusion: occupied band can arbitrarily be increased if
• n bits are unied into one symbol;• And this applied to a 2D modulation• But its cost: exponental increase of signal
energy and nearly exponental increase of signal power
• Locsl optimum: 4QAM (QPSK): band decreases to its half while power remains unchanged [2(22/2-1)/2=1]
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Frigyes: Hírkelm 85
5. Frequency band occupation, etc
• n=2; M=4 (QPSK) Pn/P1= 0 dB
• n=4 M=16 (16QAM) 7 dB
• n=6 M=64 (64QAM) 12 dB
• n=8 M=256 (256QAM) 17 dB
• n=10 M=1024 (1024QAM) 23dB
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Frigyes: Hírkelm 86
Some special characteristics of the optical frequency band
• We’ve seen that tere is no thermal noise in the optical band.
• On the other hand ther is shot noise. (Without refering to optics we saw the effect of Poisson-distributed noise.)
• Some more detailed in what follows.
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Frigyes: Hírkelm 87
Some special characteristics of the optical frequency band
Somewhat modified model:
SOURCEOPTICAL
GEN+MOD +PHOTO
DETECTORDECIDER
Időzítés (T)
b(t)
NYESINKLŐ+
N(t)
b(t): optical background noiseN(t): noise of the amplifier(being electrical amp.)
M=2
Modulation: nearly alwaysOn-Off intensitymodulation
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Frigyes: Hírkelm 88
Optical frequency band: some elements of the model
• b(t): has at least three elements:
• noise really of the background (free space optical systems,
FSO)
• dark current of the photodetectos
• non-ideal extinction of the modulator
• Photodetector: electrons are generated by the entering photons
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Frigyes: Hírkelm 89
Optical frequency band: some elements of the model
• Two different types:
• PIN dióda: 1 photon1 electron
but not sure
• APD: 1 photon1 electron but not sure however secondery electrons to each electron
(APD: Avalanche Photo Diode)
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Frigyes: Hírkelm 90
Optical frequency band: some elements of the model
PMhf
eI
Phf
eI
hf
TP
e
TI
hf
E
e
Qnn
c
q
c
q
c
q
c
qphotqel
det
detdet
:APD
:PIN
Detected current is proportional to optical power(!)
![Page 91: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/91.jpg)
Frigyes: Hírkelm 91
Some specialities of the optical frequency band
0
: sintensitie
0 ;cos2
0
112
21
P
PPA
tstAts
s
c
• These are averageintensities. The lightsource emits photons in Poisson-distribution (?). Some of these arrives to the detector
• The fotodetector is an electron- counter
• m: average number ofelectrons during T seconds
Signal waveforms:
!
Prk
emk
mk
![Page 92: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/92.jpg)
Frigyes: Hírkelm 92
Optimal optical detection (we’ve seen in an example)
• First assume that there is no thermal noise• The two hypotheses:
• H0: there is only b(t) van – power: Pb (i.e. s0 is transmited)
• H1: (s1) is transmiited – power: Ps1+Pb
c
bq
c
bsq hf
TPm
hf
TPPm
0
11 ;
A két hipotézis: !
Pr: ;!
Pr:0
00
11
1 k
emkH
k
emkH
mkmk
![Page 93: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/93.jpg)
Frigyes: Hírkelm 93
Optimal optical detection (we’ve seen in an example)
• The likelihood-ratio:
• Decision rule:
• For being precise:
01
0
1 mm
k
em
mk
001
01
101
01
:ln
ln if
;:ln
ln if
Hmm
mmk
Hmm
mmk
5,0)1Pr()0Pr(
;1:ln
ln if 01
01
01
QHQHmm
mmk
![Page 94: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/94.jpg)
Frigyes: Hírkelm 94
Optimal optical detection (we’ve seen in an example)
b
s
s
c
qk
P
P
P
hf
Tk
1
1
1ln
1:H;0:H :then
0:0
10
kk
kP kb
•Like before, we assume thatln η=0. Threshold is then
1. special case: Pb=0(No background noise)
1|0Pr2
10|1Pr
2
1EP
=0
=
!! exp 1
hf
TP qs
Rather unpleasant: it depends both on Ps1/Pb and on Ps1
![Page 95: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/95.jpg)
Frigyes: Hírkelm 95
Optimal optical detection (we’ve seen in an example)
0|Pr4
11|Pr
4
1
0|Pr2
11|Pr
2
1
0|1Pr2
11|0Pr
2
1
kk
kk
E
kkkk
kkkk
P
2. Pb0
1 2
00
110
01
0
11
!4
1
!2!2 kkk
mkkmkkkmkk
k
km
E k
emem
k
me
k
meP
![Page 96: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/96.jpg)
Frigyes: Hírkelm 96
Some (further) specialities of the optical band
• The case of shot+thermal noise
• Noise then: Poisson + Gaussian – i.e. discrete +continuous . (Discrete RV will be called x)
PD
RL
+
N(t)
DECISION
Udiscrete
x(t)
r(t)=N(t)+x(t)
![Page 97: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/97.jpg)
Frigyes: Hírkelm 97
Some (further) specialities of the optical band
0
110
1,0;Pr;!
Pr
;;1,0;;
k
ik
ikix
imkiLi
k
sbbc
qii
L
ixxxxxpk
em
T
eRxx
PPPPPihf
TPm
T
keRtx
0
1,0;Pr;ˆix
ikiix ixxxmxpxp
PDF of x(discrete!)
H0 and H1 are distinguished in mi only; thus:
T
RNBRkFTNNp L
LBN0
022 2
4;,0; N is Gaussian:
![Page 98: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/98.jpg)
Frigyes: Hírkelm 98
Some (further) specialities of the optical band
2
0
2
0
2
0
,;Pr;
i.e.,,0;Pr; so And
,0;Pr;ˆ
i
x
iir
i
x
iir
i
x
iirr
Nxr
xrxmrp
xrxmrp
rxrxmrprp
rprprpNxr
i
i
i
Decision rule: threshold uk, where pr(r;m1)=pr(r;m0)and the error prob.:
ku rku
rE drmrpdrmrpP 10 ;2
1;
2
1
![Page 99: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/99.jpg)
Frigyes: Hírkelm 99
Some (further) specialities of the optical band
Relevant pdf: sum of Gaussians (of course: different for Pb és Pb+ Ps1).
Consequently PE: sum of erfc-s
![Page 100: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/100.jpg)
Frigyes: Hírkelm 100
Some (further) specialities of the optical band
• This is rather complex.
• But: if mi is rather large, Poisson distribution is approximately Gaussian
• And then E/N0 yields PE-t (erfc) – i.e. only SNR has to be known
EDDIG!!
![Page 101: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/101.jpg)
Frigyes: Hírkelm 101
Conditional Poisson distribution
• We know: if time between two events is independent,
• but the expected value of the events/sec is constant,
• the number of events is of Poisson- distribution.
• But: if there is noise too: then the expected value is of random variation
• Distribution then is conditional Poisson (CP)
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Frigyes: Hírkelm 102
Conditional Poisson distribution
• CP distribution:• Not too much
can be said in general Although characteristic function can be given.
• With Poisson:
dmmpmkk
k
emmk
m
mk
0|PrPr
!|Pr
k
0
Pr RV discretefor or
E
juk
juxx
juxx
ekuU
dxexpeuU
10
0
Finally
!ˆ
!
juemk
k
myk
mk
ju
k
jukmk
k
euU
ek
yeuUmey
ek
emuU
NB: Note that Ux(u)=[F(px(x)]
![Page 103: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/103.jpg)
Frigyes: Hírkelm 103
Feltételes Poisson eloszlás
• If CP: conditional characteristic function
• Thus• On the other hand
(definition)
• Comparing these
1
|juem
k emuU
dmmpeuU m
juem
k
0
1
0
duempuU jmum
1juejumk uUuU
![Page 104: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/104.jpg)
Frigyes: Hírkelm 104
Is bacground noise really Poisson?
• Some complicated formulas:• Példa az előzőkre: amiből kiindultunk, vagyis jel
(modulálatlan szinusz) + háttérzaj (tekintsük most Gaussi-fehér-nek)
• Szinusz+zaj: komplex burkolója: komplex Gs-folyamat, aminek a várható értéke >0. Ennek az absz. négyzet-integrálja ami a (val.vált) m (várható érték).
• Ennek az (ismert) karakt. függv-e.:
00 1
exp1
1
juN
juE
juNuU
BT
E
Itt B az optikai sávszélességT a megfigyelési idő – ez alatti elektron-számra vagyunk kíváncsiak
![Page 105: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/105.jpg)
Frigyes: Hírkelm 105
Jogos volt-e Poisson-eloszlásúnak tekinteni a háttérzajt?
• Alkalmazzuk az helyettesítést;
közvetlenül ez a detektált opt. energiának volna a kar. függv.-e; ha az energiákat (E és N0) még megszorozzuk ηq/hfc-vel, az elektronszám-ét kapjuk:
ju
c
q
ju
c
q
BT
ju
c
qk
eNhf
eEhf
eNhf
uU
11
1
exp
11
1
00
1jueju
Emlékszünk: uUxpxpuU xxxx1 FF
![Page 106: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/106.jpg)
Frigyes: Hírkelm 106
Jogos volt-e Poisson-eloszlásúnak tekinteni a háttérzajt?
• Így a karakterisztikus függvényt vissza kell transzformálni
• Ez a szép Laguerre- eloszláshoz vezet:
00
1
00
0
11
exp
1
Pr
Nhf
Nhf
Ehf
Nhf
Ehf
Nhf
Nhf
k
c
q
c
q
sc
q
BT
k
c
q
sc
q
BTk
c
q
k
c
q
L
![Page 107: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/107.jpg)
Frigyes: Hírkelm 107
Jogos volt-e Poisson-eloszlásúnak tekinteni a háttérzajt?
• Definíciók és alkalmazás a spec. esetre:
• Laguerre-polinóm: M-ed rendű, k-ad fokú)
• Továbbá: • Akkor:
!0 j
x
jk
kMx
jk
j
M
kL
1
1
0
c
q
elektromos
optikai
hf
N
B
BBT
!
;
1
0
0
0
k
BT
k
kBT
hf
N
hf
Nhf
Nkk
c
q
k
c
q
c
q
chf
NqBT
BT
c
q
e
hf
N
0
01
1
![Page 108: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/108.jpg)
Frigyes: Hírkelm 108
Jogos volt-e Poisson-eloszlásúnak tekinteni a háttérzajt?
• Ezekkel a Laguerre polinóm
• És a Laguerre eloszás közelítőleg
• Ez bizony Poisson:
• Ez éppen az, amit szerettünk volna:
!Pr
0
0 k
eBTNE
hfk
BTNsEchf
qk
sc
q
k
s
c
q
c
q
sc
q
BT
kBT
N
E
kN
hfN
hf
Ehf
L
000
1
!
1
1
00 ;; BNPTPEBTNEhf
m bsssc
q
bsc
q PPhf
Tm
![Page 109: Communication Theory/2 I. Frigyes 2009-10/II.. Frigyes: Hírkelm2 frigyes/hirkelm frigyes/hirkelm hirkelm01bEnglish](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649e3c5503460f94b2e50b/html5/thumbnails/109.jpg)
Frigyes: Hírkelm 109
Jogos volt-e Poisson-eloszlásúnak tekinteni a háttérzajt?
• Még egy kérdés: igaz-e, hogy ?
• Adatok: PE=10-9 (ideálisan 20 foton/(s1 bit); gyak.: 15 dB-lel
nagyobb) 2,5 Gbit/sec (T=400 psec)
fc=200 THzPs/Pb: 10 dBB(opt) =1 THz: 1 psec alatt 0,15 zaj-foton keletkezik
10 cq hfN