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Milcom 05 paper ID=1226
DATA FORMAT CLASSIFICATION FOR AUTONOMOUS SOFTWARE DEFINED RADIOS
Marvin Simon, Dariush DivsalarJet Propulsion Laboratory, California Institute of Technology
4800 Oak Grove DrivePasadena, California 91109
[email protected], [email protected]
ABSTRACT
We present maximum-likelihood (ML) coherent andnoncoherent classifiers for discriminating between NRZand Manchester coded (biphase-L) data formats for binaryphase-shift-keying (BPSK) modulation. Such classificationof the data format is an essential element of so-calledautonomous software defined radio (SDR) receivers(similar to so-called cognitive SDR receivers in themilitary application) where it is desired that the receiverperform each of its functions by extracting the appropriateknowledge from the received signal and, if possible, withas little information of the other signal parameters aspossible. Small and large SNR approximations to the MLclassifiers are also proposed that lead to simplerimplementation with comparable performance in theirrespective SNR regions. Numerical performance resultsobtained by a combination of computer simulation and,wherever possible, theoretical analyses, are presented andcomparisons are made among the various configurationsbased on the probability of misclassification as aperformance criterion. Extensions to other modulationssuch as QPSK are readily accomplished using the samemethods described in the paper.
INTRODUCTION
In autonomous radio operation, aside from classifying themodulation type, e.g, deciding between BPSK and QPSK,it is also desirable to have an algorithm for choosing thedata format, e.g., NRZ versus Manchester encoding. In theabsence of subcarriers, when NRZ is employed the carriermust be fully suppressed whereas Manchester codingallows for the possibility of a residual carrier, if desired.With this consideration in mind, there exist two differentscenarios. In one case, independent of the data format, themodulations are assumed to be fully suppressed carrier. Inthe other case, which due to space limitations will be notbe considered here, an NRZ data format is always used ona fully suppressed carrier modulation whereas a residual
carrier modulation always employs Manchester codeddata. In this case, the data format classification algorithmand its performance will clearly be a function of themodulation index, i.e., the allocation of the power to thediscrete and data-modulated signal components. In thispaper, we derive the maximum-likelihood (ML)-baseddata format classification algorithms as well as reducedcomplexity versions of them obtained by applying suitableapproximations of the nonlinearities resulting from the MLformulation. As in previous classification problems of thistype, we shall first assume that all other system parametersare known. Following this, we relax the assumption ofknown carrier phase and, as was done for the modulationclassification investigation, we shall consider thenoncoherent version of the ML classifiers. Numericalperformance evaluation will be obtained by computersimulations and, wherever possible, by theoretical analysesto verify the simulation results.
MAXIMUM-LIKELIHOOD COHERENTCLASSIFIER OF DATA FORMAT FOR BPSK
We begin by considering suppressed carrier BPSKmodulation and a choice between NRZ and Manchesterencoding. Thus, the received signal is given by
r t P a p t nT t n tn bn
c( ) = -( )ÊËÁ
ˆ¯
+ ( )=-•
•
Â2 cosw (1)
where P is the signal power, an{ } is the sequence ofbinary independent, identically distributed (i.i.d.) datataking on values ±1 with equal probability, p t( ) is thepulse shape (the item to be classified), w c is the radiancarrier frequency and n t( ) is a bandpass additive whiteGaussian noise (AWGN) source with single-sided powerspectral density N0 W/Hz. Based on the above AWGNmodel, then for an observation of Kb bit intervals, theconditional likelihood function (CLF) is given by
2
p r t a p t
N Nr t P a p t nT t dt
CP
Na r t p t kT tdt
n
n bn
c
K T
k b ckT
k T
b b
b
b
( ) { } ( )( )= - ( ) - -( )Ê
ËÁˆ¯
È
ÎÍ
˘
˚˙
Ê
ËÁ
ˆ
¯˜
= ( ) -( )ÊËÁ
ˆ
= -•
•
+( )
ÂÚ
Ú
,
exp cos
exp cos
1 12
2 2
0 0
2
0
0
1
pw
w¯=
-
’k
Kb
0
1
(2)
where C is a constant that has no bearing on theclassification. Averaging over the i.i.d. data sequencegives
p r t p t CP
Nr t p t kT tdtb c
kT
k T
k
K
b
bb
( ) ( )( ) = ( ) -( )ÊËÁ
ˆ¯
+( )
=
-
Ú’ cosh cos2 2
0
1
0
1
w (3)
Finally, taking the logarithm of (3), we obtain the log-likelihood function (LLF)
L D= ( ) ( )( ) = ( ) -( )ÊËÁ
ˆ¯=
- +( )Â Úln ln cosh cosp r t p t
P
Nr t p t kT tdt
k
K
ckT
k Tb
b
b
0
1
0
12 2 w (4)
where we have ignored the additive constant lnC .For NRZ data, p t( ) is a unit rectangular pulse of
duration Tb , i.e,
p t
t Tb1
1 0
0( ) =
£ £ÏÌÓ
,
, otherwise(5)
For Manchester encoded data, p t( ) is a unit square-wavepulse of duration Tb , i.e,
p tt T
T t Tb
b b2
1 0 2
1 2( ) =
£ £- £ £
ÏÌÓ
, /
, /(6)
Thus, defining the received observables
r l r t p t kT tdt
r t tdt l
r t tdt r t tdt l
k l b ckT
k T
ckT
k T
ckT
k T
ck T
k T
b
b
b
b
b
b
b
b
( ) = ( ) -( )
=( ) =
( ) - ( ) =
Ï
ÌÔ
ÓÔ
+( )
+( )
+( )
+( )
+( )
Ú
Ú
Ú Ú
D cos
cos ;
cos cos ;/
/
w
w
w w
1
1
1 2
1 2
1
1
2
(7)
then a classification choice between the two pulses shapesbased on the LLF would be to choose Manchester if
ln cosh ln cosh2 2
12 2
200
1
00
1P
Nr
P
Nrk
k
K
kk
Kb b
( )ÊËÁ
ˆ¯
< ( )ÊËÁ
ˆ¯=
-
=
-
  (8)
Otherwise, choose NRZ.
REDUCED COMPLEXITY DATA FORMAT BPSKCLASSIFIERS
To simplify the form of the classification rule in (8), wereplace the ln cosh ◊( ) function by its small and largeargument approximations. In particular,
ln cosh
/ ;
ln ;x
x x
x x@
-ÏÌÓ
2 2
2
small
large(9)
Thus, for low SNR, (8) simplifies to
r t tdt
r t tdt r t tdt
ckT
k T
k
K
ckT
k T
ck T
k T
k
K
b
bb
b
b
b
bb
( )ÊË
ˆ¯
< ( ) - ( )ÊË
ˆ¯
+( )
=
-
+( )
+( )
+( )
=
-
ÚÂ
Ú ÚÂ
cos
cos cos/
/
w
w w
1 2
0
1
1 2
1 2
1 2
0
1(10)
or, equivalently
r t tdt r dckT
k T
ck T
k T
k
K
b
b
b
bb
( ) ( ) <+( )
+( )
+( )
=
-
Ú ÚÂ cos cos/
/w t w t t
1 2
1 2
1
0
1
0 (11)
For high SNR, (8) reduces to
r t tdt r t tdt
r t tdt r t tdt
ckT
k T
ck T
k T
k
K
ckT
k T
ck T
k T
k
K
b
b
b
bb
b
b
b
bb
( ) + ( )
< ( ) - ( )
+( )
+( )
+( )
=
-
+( )
+( )
+( )
=
-
Ú ÚÂ
Ú ÚÂ
cos cos
cos cos
/
/
/
/
w w
w w
1 2
1 2
1
0
1
1 2
1 2
1
0
1(12)
Note that while the optimum classifier of (8) requiresknowledge of SNR, the reduced-complexity classifiers of(10) and (12) do not. Figure 1 is a block diagram of theimplementation of the low and high SNR classifiersdefined by (11) and (12).
PROBABILITY OF MISCLASSIFICATION FORCOHERENT BPSK
To illustrate the behavior of the misclassificationprobability, PM , with signal-to-noise ratio (SNR), weconsider the low SNR case and evaluate first theprobability of the event in (11) given that the transmitteddata sequence was in fact NRZ encoded. In particular, werecognize that given a particular data sequence of Kb bits,
X r t tdt Y r tdck ckT
k T
ck ck T
k T
b
b
b
b= ( ) = ( )+( )
+( )
+( )Ú Úcos , cos ;
/
/w t w t
1 2
1 2
1
k Kb= -0 1 1, ,..., are mutually independent and identically
distributed (i.i.d.) Gaussian random variables (RVs).Thus, the LLF
D r t tdt r d
X Y
ckT
k T
ck T
k T
k
K
ck ckk
K
b
b
b
bb
b
= ( ) ( )
=
+( )
+( )
+( )
=
-
=
-
Ú ÚÂ
Â
cos cos/
/w t w t t
1 2
1 2
1
0
1
0
1(13)
is a special case of a quadratic form of real Gaussian RVsand the probability of the event in (11), namely, Pr D <{ }0can be evaluated in closed form by applying the results in[1, Appendix B] and the additional simplification of thesein [2, Appendix 9A]. To see this connection, we define thecomplex Gaussian RVs X X jX Y Y jYk ck c k k ck c k= + = ++ +, ,,1 1.
Then, the complex quadratic form X Y X Yk k k k* *+ . is equal
to 2 1 1X Y X Yck ck c k c k+( )+ +, , . Arbitrarily assuming Kb is
even, then we can rewrite D of (13) as
3
D X Y X Yk k k kk
Kb
= +( )=
-
Â12 0
2 1* *
/
(14)
Comparing (14) with [1, Eq. (B.1)] we see that the formeris a special case of the latter corresponding toA B C= = =0 1 2, / . Specifically, making use of the firstand second moments of Xk and Yk given by
X Y a ja P T
E X X N T
E Y Y N T
E X X Y Y
k k k k b
xx k k b
yy ck ck b
xy ck ck ck ck
= = +( )= -{ } =
= -{ } =
= -( ) -( ){ } =
+1
2
0
2
0
8
12
8
12
8
12
0
/
/
/
*
m
m
m
(15)
then from [2, Eq. (9A.15)]
PK
K kQ a b Q b aM K
b
bk
K
k kb
b
112
12
1
211
2
( ) = +-
-ÊËÁ
ˆ¯
( ) - ( )[ ]-=
 /, ,
/
(16)
where Q a bk ,( ) is the kth-order Marcum Q-function and
av v
bv v
=-( ) =
+( )x x x x1 2 1 2
2 2, (17)
with
vN T
X Y KPT N
X Y KPT
xx yy b
ck yy ck xxk
K
bb
ck ckk
K
bb
b
b
= =
= +( ) =
= =
=
-
=
-
Â
Â
1 8
12 64
8
0
1
2 2
0
2 1 30
20
2 1 2
m m
x m m
x
/
/
(18)
Substituting (18) into (17) gives
a b K PT N K E Nb b b b= = ( ) = ( )0 0 0, / / (19)
However,
Q bb b
n
Q b
k
n
n
k
k
02
2
0 1
2 2
0
1
, exp/
!
,
( ) = -ÊËÁ
ˆ¯
( )
( ) ==
-
 (20)
Thus, using (19) and (20) in (16) gives the desired result
PK
K k
K E
N
K E N
n
M K
b
bk
K
b b b b
n
n
k
b
b
112
12
1
2
2
21
11
2
0
0
0
1
( ) = +-
-ÊËÁ
ˆ¯
¥ -ÊËÁ
ˆ¯
( ) -È
ÎÍÍ
˘
˚˙˙
-=
=
-
Â
Â
/
exp/
!
/
(21)
Noting thatK
K kb
bk
KK
b
b-
-ÊËÁ
ˆ¯
==
-Â1
22
1
22
/
/
(22)
then (21) further simplifies to
PK
K kK E
N
K E N
nM K
b
bk
Kb b b b
n
n
k
b
b
11
2
1
2 2
21
1
2
0
0
0
1
( ) =-
-ÊËÁ
ˆ¯
-ÊËÁ
ˆ¯
( )-
= =
-
 Â/exp
/
!
/
(23)
To compute the probability of choosing NRZwhen in fact Manchester is the true encoding, we need toevaluate Pr PrD D≥{ } = - <{ }0 1 0 when instead of (15)we have
X a ja P T
Y a ja P T
k k k b
k k k b
= +( )= - +( )
+
+
1
1
8
8
/
/(24)
Since the impact of the negative mean for Yk in (24) is toreverse the sign of x2 in (18), then we immediatelyconclude that for this case the values of a and b in (19)merely switch roles, i.e.,
a K E N bb b= ( ) =/ ,0 0 (25)
Substituting these values in (16) now gives
PK
K k
K E
N
K E N
n
M K
b
bk
K
b b b b
n
n
k
b
b
2 112
12
1
2
12
2
11
2
0
0
0
1
( ) = - +-
-ÊËÁ
ˆ¯
ÏÌÓ
¥ - -ÊËÁ
ˆ¯
( )È
ÎÍÍ
˘
˚˙˙
¸˝ÔÔ
-=
=
-
Â
Â
/
exp/
!
/
(26)
which again simplifies to
PK
K kK E
N
K E N
nM K
b
bk
Kb b b b
n
n
k
b
b
21
2
1
2 2
21
1
2
0
0
0
1
( ) =-
-ÊËÁ
ˆ¯
-ÊËÁ
ˆ¯
( )-
= =
-
 Â/exp
/
!
/
(27)Since (23) and (27) are identical, the average probability ofmismatch, PM is then either of the two results.
Illustrated in Fig. 2 are numerical results for themisclassification probability obtained by computersimulation for the optimum and reduced-complexity dataformat classifiers as given by (8), (11) and (12). Alsoillustrated are the numerical results obtained from theclosed-form analytical solution given in (23) for the lowSNR reduced-complexity scheme. As can be seen, theagreement between theoretical and simulated results isexact. Furthermore, the difference in performancebetween the optimum and reduced-complexity classifiersis quite small over a large range of SNRs.
MAXIMUM-LIKELIHOOD NONCOHERENTCLASSIFIER OF DATA FORMAT FOR BPSK
Here we assume that the carrier has a random phase, q ,that is unknown and uniformly distributed. Thus, thereceived signal of (1) is now modeled as
r t P a p t nT t n tn bn
c( ) = -( )ÊËÁ
ˆ¯
+( ) + ( )=-•
•
Â2 cos w q (28)
and the corresponding CLF becomes
p r t a p t
CP
Na r t p t kT t dt
n
k b ckT
k T
k
K
b
bb
( ) { } ( )( )= ( ) -( ) +( )Ê
ËÁˆ¯
+( )
=
-
Ú’
, ,
exp cos
q
w q2 2
0
1
0
1(29)
4
At this point we have the option of first averaging over therandom carrier phase and then the data or vice versa.Considering the first option, we start by rewriting (29) as
p r t a p t
CP
Na r a r
a r
a r
n
k ckk
K
k skk
K
k skk
K
k ckk
K
b b
b
b
( ) { } ( )( )
=ÊËÁ
ˆ¯
+ÊËÁ
ˆ¯
+( )Ê
ËÁÁ
ˆ
¯˜
=
=
-
=
-
- =
-
=
-
 Â
Â
Â
, ,
exp cos ;
tan
q
q h
h
2 2
0 0
1 2
0
1 2
1 0
1
0
1
(30)
Averaging over the carrier phase results in (ignoringconstants)
p r t a p t IP
Na r a rn k ck
k
K
k skk
Kb b
( ) { } ( )( ) = ÊËÁ
ˆ¯
+ ÊËÁ
ˆ¯
Ê
ËÁÁ
ˆ
¯˜
=
-
=
-
 Â, 0
0 0
1 2
0
1 22 2
(31)
where I0 ◊( ) is the zero order modified Bessel function ofthe first kind. Unfortunately, the average over the datasequence cannot be obtained in closed form. Hence, theclassification algorithm can only be stated as follows:Given that NRZ was transmitted, choose the Manchesterformat if
E IP
Na r a r
E IP
Na r a r
k ckk
K
k skk
K
k ckk
K
k skk
K
b b
b b
a
a
00 0
1 2
0
1 2
00 0
1 2
0
2 21 1
2 22 2
( )ÊËÁ
ˆ¯
+ ( )ÊËÁ
ˆ¯
Ê
ËÁÁ
ˆ
¯˜
ÏÌÔ
ÓÔ
¸˝Ô
Ô
< ( )ÊËÁ
ˆ¯
+ ( )
=
-
=
-
=
-
=
 Â
Â--
ÂÊËÁ
ˆ¯
Ê
ËÁÁ
ˆ
¯˜
ÏÌÔ
ÓÔ
¸˝Ô
Ô
1 2
(32)
where Ea◊{ } denotes expectation over the data sequence
a = ( )-a a aKb0 1 1, ,..., . Otherwise, choose NRZ.
Consider now the second option where we firstaverage over the data sequence. Then,p r t p t
C EP
Na r t p t kT t dt
CP
Nr t p t kT t dt
ak b ckT
k T
k
K
b ckT
k T
k b
bb
b
b
( ) ( )( )
= ( ) -( ) +( )ÊËÁ
ˆ¯
ÏÌÓ
¸˝˛
= ( ) -( ) +( )
+( )
=
-
+( )
Ú’
Ú
,
exp cos
exp ln cosh cos
q
w q
w q
2 2
2 2
0
1
0
1
0
1ÊÊËÁ
ˆ¯
È
ÎÍ
˘
˚˙
=
-
Âk
Kb
0
1
(33)Thus, a classification between NRZ and Manchesterencoding would be based on a comparison of
EP
Nr t p t kT t dtb ckT
k T
k
K
b
bb
qw qexp ln cosh cos
2 2
01
1
0
1
( ) -( ) +( )ÊËÁ
ˆ¯
È
ÎÍ
˘
˚˙
ÏÌÔ
ÓÔ
¸˝ÔÔ
+( )
=
-
ÚÂwith
EP
Nr t p t kT t dtb ckT
k T
k
K
b
bb
qw qexp ln cosh cos
2 2
02
1
0
1
( ) -( ) +( )ÊËÁ
ˆ¯
È
ÎÍ
˘
˚˙
ÏÌÔ
ÓÔ
¸˝ÔÔ
+( )
=
-
ÚÂTo simplify matters, before averaging over the carrierphase, one must employ the approximations to thenonlinearities given in (9). In particular, for low SNR wehavep r t p t
EP
Nr t p t kT t dt
P
Nr r
I
b ckT
k T
k
K
ck skk
K
b
bb
b
( ) ( )( )
= ( ) -( ) +( )ÊËÁ
ˆ¯
È
ÎÍÍ
˘
˚˙˙
ÏÌÔ
ÓÔ
¸˝Ô
Ô
= +( )È
ÎÍ
˘
˚˙
¥
+( )
=
-
=
-
ÚÂ
Â
qw qexp cos
exp
12
2 2
2
0
12
0
1
02
2 2
0
1
0002
2 2
0
1 2
2 2
0
1 22
2 2P
Nr r r rck sk k
k
K
ck sk kk
Kb b
+( )ÊËÁ
ˆ¯
+ +( )ÊËÁ
ˆ¯
Ê
ËÁÁ
ˆ
¯˜
=
-
=
-
 Âcos sinh h
(34)where
hksk
ck
r
r= -tan 1 (35)
Thus, since
cos , sin2 222 2
2 2 2 2h hk
ck sk
ck sk
kck sk
ck sk
r r
r r
r r
r r= -
+=
+(36)
we finally have
p r t p tP
Nr r I
P
Nrck sk
k
K
kk
Kb b
( ) ( )( ) = +( )È
ÎÍ
˘
˚˙
ÊËÁ
ˆ¯=
-
=
-
 Âexp ˜2 2
02
2 2
0
1
002
2
0
1
(37)
where
r r jr r t p t kT e dtk ck sk bj t
kT
k Tc
b
b
= + = ( ) -( )+( )ÚD w
1
(38)
Finally then, the classification decision rule analogous to(32) is: Given that NRZ data was transmitted, decide onManchester coding if
exp ˜ ˜
exp ˜ ˜
21
21
22
22
02
2
0
1
002
2
0
1
02
2
0
1
002
2
0
1
P
Nr I
P
Nr
P
Nr I
P
Nr
kk
K
kk
K
kk
K
kk
K
b b
b b
( )È
ÎÍ
˘
˚˙ ( )
ÊËÁ
ˆ¯
< ( )È
ÎÍ
˘
˚˙ ( )
ÊËÁ
ˆ¯
=
-
=
-
=
-
=
-
 Â
 Â(39)
Equivalently, normalizing the observables to
˜¢=( )
-( )+( )Úr
T
r t
Pp t kT e dtk
bb
j t
kT
k Tc
b
bD 1
2
1w (40)
then (39) becomes
exp ˜ ˜
exp ˜
21
21
22
2
0
22
0
1
00
2
2
0
1
0
22
0
1
0
E
Nr I
E
Nr
E
Nr I
E
N
bk
k
Kb
kk
K
bk
k
Kb
b b
b
ÊËÁ
ˆ¯
¢( )È
ÎÍÍ
˘
˚˙˙
ÊËÁ
ˆ¯
¢ ( )Ê
ËÁ
ˆ
¯˜
<ÊËÁ
ˆ¯
¢( )È
ÎÍÍ
˘
˚˙˙
=
-
=
-
=
-
 Â
Â00
2
2
0
1
2ÊËÁ
ˆ¯
¢ ( )Ê
ËÁ
ˆ
¯˜
=
-
 rkk
Kb
(41)
Since we have already assumed low SNR inarriving at (41), we can further approximate the
5
nonlinearities in that equation by their values for smallarguments. Retaining only linear terms, we arrive at thesimplification
˜ ˜¢( ) < ¢( )=
-
=
-
 Âr rkk
K
kk
Kb b
1 22
0
12
0
1
(42)
or, equivalently
˜ ˜r rkk
K
kk
Kb b
1 22
0
12
0
1
( ) < ( )=
-
=
-
  (43)
which again does not require knowledge of SNR. On theother hand, if we retain second-order terms, then (41)simplifies to
˜ ˜ ˜
˜ ˜
¢( ) +ÊËÁ
ˆ¯
¢( )ÊËÁ
ˆ¯
+ ¢ ( )È
ÎÍÍ
˘
˚˙˙
< ¢( ) +ÊËÁ
ˆ¯
¢
=
-
=
-
=
-
=
-
  Â
Â
rE
Nr r
rE
Nr
kk
Kb
kk
K
kk
K
kk
Kb
b b b
b
114
22 1 1
214
22
2
0
1
0
2
2
0
1 2
2
0
1 2
2
0
1
0
2
kkk
K
kk
Kb b
r2 22
0
1 2
2
0
1 2
( )ÊËÁ
ˆ¯
+ ¢ ( )È
ÎÍÍ
˘
˚˙˙=
-
=
-
  ˜
(44)
which is SNR-dependent.Expanding (43) in the form of (10), we obtain
r t tdt r t tdt
r t tdt r t tdt
ckT
k T
ckT
k T
k
K
ckT
k T
ck T
k T
k
K
b
b
b
bb
b
b
b
bb
( )ÊË
ˆ¯ + ( )Ê
ˈ¯
< ( ) - ( )ÊË
ˆ¯
+
+( ) +( )
=
-
+( )
+( )
+( )
=
-
Ú ÚÂ
Ú ÚÂ
cos sin
cos cos/
/
w w
w w
1 2 1 2
0
1
1 2
1 2
1 2
0
1
rr t tdt r t tdtckT
k T
ck T
k T
k
K
b
b
b
bb
( ) - ( )ÊË
ˆ¯
+( )
+( )
+( )
=
-
Ú ÚÂ sin sin/
/w w
1 2
1 2
1 2
0
1
(45)
or
Re/
/r t e dt r e dj t
kT
k T j t
k T
k T
k
Kc
b
bc
b
bb
( ) ( )ÏÌÓ
¸˝˛
<+( ) -
+( )
+( )
=
-
Ú ÚÂ w wt t1 2
1 2
1
0
1
0 (46)
which is the analogous result to (11) for the coherent case.For high SNR, even after applying the
approximations to the nonlinearities given in (9), it is stilldifficult to average over the random carrier phase.Instead, we take note of the resemblance between (46) and(11) for the low SNR case and propose an ad hoc complexequivalent to (12) for the noncoherent high SNR case,namely,
r t e dt r t e dt
r t e dt r t e dt
j t
kT
k T j t
k T
k T
k
K
j t
kT
k T j t
k T
k T
k
K
c
b
bc
b
bb
c
b
bc
b
bb
( ) + ( )
< ( ) - ( )
+( )
+( )
+( )
=
-
+( )
+( )
+( )
=
-
Ú ÚÂ
Ú ÚÂ
w w
w w
1 2
1 2
1
0
1
1 2
1 2
1
0
1
/
/
/
/
(47)
Figure 3 is a block diagram of the implementation of thelow and high SNR classifiers defined by (46) and (47).
PROBABILITY OF MISCLASSIFICATION FORNONCOHERENT BPSK
To compute the probability of misclassification, we notethat (46) is still made up of a sum of products of mutuallyindependent real Gaussian RVs and thus can still bewritten in the form of (14) with twice as many terms, i.e.,
D X Y X Yk k k kk
Kb
= +( )=
-
Â12 0
1* * (48)
where now the complex Gaussian RVs are defined asX X jX Y Y jYk ck sk k ck sk= + = +, . The means of the terms aregiven by
X Y a j P Tk k k b= = -( )cos sin /q q 8 (49)whereas the variances and crosscorrelations are the sameas in (15). Thus, since the magnitude of the means in (49)is reduced by a factor of 2 relative to that of the meansin (15), we conclude that the probability ofmisclassification is obtained from (23) by replacingE Nb / 0 with E Nb / 2 0 and Kb with 2Kb resulting in
PK
K kK E
N
K E N
nM K
b
bk
Kb b b b
n
n
k
b
b
=-
-ÊËÁ
ˆ¯
-ÊËÁ
ˆ¯
( )-
= =
-
 Â12
2 1
2
22 1
1 0
0
0
1
exp/
!
(50)Fig. 4 illustrates numerical results for the
misclassification probability obtained by computersimulation for the low SNR and high SNR reduced-complexity data format classifiers as specified by (46) and(47), respectively, as well as the optimum classifierdescribed by the comparison below Eq. (33). Alsoillustrated are the numerical results obtained from theclosed-form analytical solution given in (50) for the lowSNR reduced-complexity scheme (which are in exactagreement with the simulation results). As in the coherentcase, the difference in performance between the low andhigh SNR reduced-complexity classifiers is again quitesmall over a large range of SNRs. Furthermore, we seehere again that the performances of the approximate butsimpler classification algorithms are in close proximity tothat of the optimum one. Finally, comparison between thecorresponding coherent and noncoherent classifiers isillustrated in Fig. 5 and reveals a penalty of approximately1 dB or less depending on the SNR.
ACKNOWLEDGEMENT
This research was carried out at the Jet PropulsionLaboratory, California Institute of Technology, undercontract with the National Aeronautics and SpaceAdministration.
REFERENCES
1. J. Proakis, Digital Communications, New York:McGraw-Hill, 4th ed., 2001.2. M. K. Simon and M.-S. Alouini, Digital Communicationover Fading Channels: A Unified Approach toPerformance Analysis, 2nd ed., New York: John Wiley &Sons., 2005.
6
r t( )
cos wct
◊( )k+1/2( )Tb
Ú dtk+1( )Tb
Delay
Tb / 2
< 0
Manchester
NRZ> 0
DataFormat
Decision
(a) Low SNR
(b) High SNR
◊( )k+1/2( )Tb
Ú dtk+1( )Tb
Delay
Tb / 2
r t( )
cos wct
◊( )k+1/2( )Tb
Ú dtk+1( )Tb
Delay
Tb / 2
+
-
+
+
+
◊
◊
Fig. 1. Reduced Complexity Coherent Data Format Classifiers for BPSK Modulation
-
◊( )k= 0
Kb -1
Â
< 0
Manchester
NRZ> 0
DataFormat
Decision◊( )
k= 0
Kb -1
Â
Pro
b. o
f M
iscl
assi
fica
tio
n
Low SNR ApproximationSimulations and analysis
High SNR APPROXIMATIONSimulations
OPTIMUM ML CLASSIFIERSimulations
Eb/N0, dB6543210-1-2
10-6
10-5
10-4
10-3
10-2
10-1
100
Kb=8
Kb=16
Kb=32
Coherent Classifier
Fig. 2. A Comparison of the Performance of Coherent Data Format Classifiers for BPSK Modulation.
7
r t( )◊( )
k+1/2( )TbÚ dtk+1( )Tb
Delay
Tb / 2
< 0
Manchester
NRZ> 0
DataFormat
Decision
(a) Low SNR
(b) High SNR
◊( )k+1/2( )Tb
Ú dtk+1( )Tb
Delay
Tb / 2
r t( )
◊( )k+1/2( )Tb
Ú dtk+1( )Tb
Delay
Tb / 2
+
-
+
+
+
◊
◊
Fig. 3. Reduced Complexity Noncoherent Data Format Classifiers for BPSK Modulation
-
◊( )k= 0
Kb -1
Â
< 0
Manchester
NRZ> 0
DataFormat
Decision◊( )
k= 0
Kb -1
Â
ejwc t
Re ◊{ }
ejwc t
*
Fig. 4. A Comparison of the Performance of Noncoherent Data Format Classifiers for BPSK Modulation.
Pro
b. o
f M
iscl
assi
fica
tio
n
Kb=8
Kb=16
Kb=32
Noncoherent classifier
Eb/N0, dB
Sim Low SNR
Analysis Low SNR
Sim optimum
109876543210-1-2-310
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Sim High SNR
Pro
b. o
f M
iscl
assi
fica
tio
n
Kb=16
Analysis Low SNRcoherent
Sim High SNRcoherent
Sim Low SNRcoherent
Sim optimum coherent
Sim Low SNRnoncoherent
Sim High SNRnoncoherent
Analysis Low SNR
noncoherent
876543210-1-210
-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Sim optimumnoncoherent
Fig. 5. A Comparison of the Performance of Coherent and Noncoherent Data Format Classifiers for BPSK Modulation.
Eb/N0, dB