922
IEEE’L‘KANSACTIONSON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995
The Performanceof Noncoherent OrthogonalIM-FSK
in the Presenceof Timing and FrequencyErrors
Sami Hinedi, Senior Member IEEE
Marvin Simon, Fellow IEEE
Dan Raphaeli, Student Member IEEE
ABSTRACT
The purpose of this paper is to evaluate this performance
of degradation
degradation, first by treating the two sources
separately, and thenby considering their simultaneous effect.In
particular, we shall present exact cxprcssionsfor thc symbol and
bit error probability performancesof noncoherent orthogonalM FSK conditioned on the presence of time and frequency errors.
These expressions involve integrals
of Marcum-Q functions and,
as such, their numerical evaluation is cumbcrsomc. Thus, for the
case of frequency error only, we present various upper o bounds
n
error probability performances that. because of their exponential
behavior, are simplerto evaluate. Numerical results are obtained
1.O INTRODUCTION
for cases ofpractical interest.
Before going into the details
of the analysis, we wish to point
NoncoherentorthogonalM-aryfrequency-shift-keying
out the existenceof several papers that
relate to the subjectat hand
(M-FSK) is a simple and robust formof digital communication [l-71. The paperthat, in principle, bears the closest resemblance
when the transmission channel is such that fast reliable carrier
is a papcr by Nakamoto,
to what wc arc trying to accomplish hcrc
recovery is difficult or impractical to achieve and
thc bandwidth Middlestead, and Wolfson [I]. Although the primary interest in
requirements are notoverlystringent.Moststudies
ofthis
[ 1] was frequency-hopped M-FSK, the authors
also attempted to
modulatioddemodulationtechniquefortheadditivewhite
usetheirresultstoobtainperformancedatafornoncoherent
Gaussiannoise(AWGN)channelhavefocusedontheerror
orthogonal M-FSK without frequency hopping (This
is discussed
probabilityperformancewhenthereceiver
is assumedto be in the section of their paper called Performance Without Nonperfectly time and frequency synchronized. Thatis, the receiver Coherent Combining). While the resultsin [ I ] are indeed correct
is assumed to have perfect knowledge of the instants of time at for predicting the performance of frequency-hoppcd M-FSK in
which the modulation can change state and also perfect knowl- the presence of time and frequency errors,
they are unfortunately
edge of the receivedcamer frequency. In practical systems, such incorrect for the unhopped case. It is important to understand
perfect knowledge is never available and thus the receiver must where the results in [l] fail to address the case of noncoherent
dcrive this information from the received signal imbedded in the
the
orthogonal M-FSK since isit indeed these issues that provide
AWGN. Since the estimatcs of the time epoch and received
basis for our papcr. We now explain thcsc diffcrcnccs in dctail.
carrier frequency derived atthe receiver are, in general, random
In 111 it is correctly recognized that in the presence of timing
variables (becauseof the presence of the AWGN), there
will exist and frequency errors two factors contribute to the perfomlance
an errorbetween these estimates and their true values. This lack degradation relativeto the perfectly synchronized case. First, the
of perfect time and frequency synchronization gives rise
to a signalcomponentofthecorrectcorrelator
is attenuatedand
degradation in errorprobabilityperformancerelative
to that second a portionof thc transmitted signal energy now appearsin
corresponding to the ideal case where perfect knowlcdgc
of time the outputs of each
of the M-1 incorrect correlators. This second
and frequency is assumed known.
source of degradation is referred to as the loss ofnrrhognnnlity. In
of
computingthedegradationfactorsforthesetwosources
degradation (see Eqs. (7) and (9) of [l]), the authors implicitly
Paper approved by 1. Korn, the Editor for Modulation of the IEEE Communications Society. Manuscnpt was received September 27. 1993. revised May 12,
assume that there exists a large difference between
the frequen1994. This aurk was performed at the Jet Propulsion Laboratory, Califomla
Space
Institute of Technology under a contract with the National Aeronautics and
cies of two successive transmissions, which in the frequencyAdministration.
hopped case corresponds to two successive hops and is thus
The authors are with the Jet Propulsion Laboratory, California Institute
of
Technoloev. Pasadena. CA 91 109
justified most of the time. In the caseof noncoherent orthogonal
Practical M-FSK systems experience a combination
of time and
frequency offsets (errors). This
paper assesses the deleterious effect
ofthese offsets, first individually
and then combined,on the average
hit error probability performance
ofthe system. Exact expressions
for these various error prohability performances are derived and
evaluatednumericallyforsystemparametersofinterest.
Also
presented are upper boundsonaverage
symbol error probability for
the case of frequency error alone which are
useful in assessing the
absolute and relative performanceof the system. Both continuous
and discontinuous phase M-FSK cases are consideredwhen timing
error is present, the latter being much less robust to this type of
offset.
HlNEDl
et
923
al . N O N C O H F . R E N T O R T H O G O N A I . M-FSK IN TTMING A N D FREQUENCY
M-FSK without frequency hopping, the frequency separation
between adjacent transmissions (M-FSK symbols in this case)
is
not necessarily large and thus the computation of the
two degradation factors analogous to(7) and (9) of [ 11 must take this fact
intoaccount.Furthermore,theissueofwhether
the phase is
continuous or discontinuous from symbolto symbol is critical in
evaluating these degradation factors whereas in the frequencyhopping case it is inconsequential because
of the large frequency
differencebetweenadjacenttransmissions(hops).Based
on
these observations alone, it is incorrect to compute the symbol
error probability for noncoherent orthogonal
M-FSK in the presenceoffrequencyandtimingerrorsusing(7)and(9)of[I]. Ifonly
frequency erroris prcscnt, i.c., thc timing error
is set equalto zero,
then the results in (7) and (9) of 111 can be uscd to correctly
compute symbol error probability.
sw, (11
A second incorrect approximationused in [l] is in computing
Figure I . M-ary Noncohercnl Rcccivcr for Equal Energy Signals
the bit error probability from the symbol error probability. The
authors assumethat thesignals remain orthogonal in the presence
use of a wcllof timing and frequency error which facilitates the
r(t)=1127;cos(2n(~+++Af)f+6')+n(t),O l f l T ( I )
known result [see [ 5 ] ,Eq. (5-54)] relating these two error probwhere P denotes the signal power
in Watts, Tdenotesthe symbol
abilities fororthogonalM-FSK.Unfortunately,however,this
4,
is
the
carrier
frequency in Hz,& = i/T is
time
in
seconds,
assumption is not valid and hence the bit crror probability results
the
transmitted
frequency
corresponding
to message mi,Afis the
found (see Figs. 1-5) in [ l ] for the special case of noncoherent
8
is
the
unknown canicrphase
error
in
the
carrier
frequency,
and
orthogonalM-FSK are incorrect.In fact, to properly evaluate the
to
be
uniformly
distributed.
Also,
n(t) denotesthe
assumed
bit errorprobability inthe presence of synchronization errors. one
No WIHz. AlAWGN
with
single-sided
power
spectral
density
must specify an appropriate mapping, for example,a Gray code
tcrnatively,
the
received
signal
can
be
interpreted
as a carrier at
of the symbols to bits. In the perfectly synchronized case, the bit
ji
shifted
by
the
appropriatc
signal
frcqucncy
6 + AL
frequency
error probability performance is completely independent of the
symbol-to-bit mapping since
all errors arcequally likely to occur. rather than 4.. The inphase integrator output, z,,~, matched to
The significance of these statements will become apparent latcr signal s , ( t ) = d%cos(2n(f, + f k ) f ) (corresponding to message
mk)bccomesl
on in the paper.
The organization of the paperis as follows. Section 2 exactly
treats the effect of frequency error alone (pcrfcct time synchronization is assumed) on orthogonal
M-FSK noncoherent detection.
= 2 P cos 2 n f,.
+ Af)t)cos(2n(f,. + f,)r)dr +
Section 3 exactly treats the effect of timing error alone (perfect
( (
frequency synchronization is assumed) on the same detection
scheme. Section 4 exactly trcats thc combined effect of timing where nc,Ais a zero-mean Gaussianrandom variable with variance
0
' = No / 2E,. Here, E,sg f T is the symbol energy in joules.
and frequency errors. Finally, Section 5 presents various upper
(2) yields
Simplifying
bounds on the performance in the presence of frequency error.
ld
+x
2.0 EFFECT OF FREQUENCY ERRORON ORTHOGON'AL
M-FSK NONCOHERENT
DETECTION
Consider the transmissmnof orthogonal M-FSK ovcr an
AWGN
a one-to-one correspondcncc
channel where the signal set has
,. ..,mM-,. The
with the sct ofM equiprobable messages rn,,rn,
is illusoptimum reccivcr (assuming perfect synchronization)
trated in Fig. I . Whcn the received frequency is not perfectly
known, the observed signal, assuming that message
mi was sent,
is given by
whcrcf;,, denotes the difference between the frequencies repreaenting messages m land mk, that is,
Similarly, the quadrature integrator output,z ? , ~is, given by
' Since we are dealing with noncoherent detection,wc can, in thc caseof perfect
timing synchronization. set f3= 0.
l E E E TRANSACTlONS ON COMMUNICATIONS. VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL
Y24
and signal orthogonalityis restored. Despite loss of orthogonal-
T
z,,~
r(t)d%sin(2n(fc
[cos(2n(Ae
+ f,)f)dt
+ Af)T)-I]
= E,
The envelope statistic
by
tributed
sin*(n(&
E,
-4
(5)
+ %%k
2n(A,k + Af)T
parameter with
1995
ity, the variables &,El,. . remain
thus
(and
independent since the Gaussian random variables resulting from
the noise integration are still independent as
the local signals
remain orthogonal.In this case, the probability
of correct symbol
detection, assuming that messagemi is transmitted, is givenby
will then be Rician dis-
=
rn=0,1, ..., M - I
k; given
+ Af)T)
=(Err
In
Q-function
terms
Marcum
of the
(6)
Normalizing z , . ~and z,,, by 1/ CJ =
the parameter p i
is then normalized by 2 / N& and since, as mentioncd above,
f; = z/Tfor orthogonal signals, then
[8] defined by
d m ,
1
2E, sin2(lr(i - k + p ) )
kk = -kk= ' 2 .(N:Es]
No
[ n ( i - k +p)p
[
(7)
sin2(x(i - k + p))
[+
detector
the
First,
that
note
matched
to
from signal attenuation equalto
-
k +P)p
the incoming
signal
suffers
Jr'
tgl( x m)dx,= I - Q
Hence, the conditional probability
of symbol error, assuming that
where p4 AjT denotes the frequency error normalized by the
fi(i.k)=
we have
(9)
message mi is transmitted, is given by
and the unconditional probabilityof symbol crror becomes
1 M-I
W)=--CP,(Elmi)
(16)
j d J
As previously mentioned, thc average bit error probability
cannot be obtained directly fromthe average symbol error prob(I0)
ability as is customary in perfectly synchronizedM-FSK systems,
the reason being that, for a given transmitted message, the symbol
which, as expected, reduces to
unity if p= 0. Simultaneously, loss
errors are not equally likely. To compute the average bit crror
of signal orthogonality occurs as
a result of signal spill-over into
of aparticular
probability we must first compute the probability
thc remaining M-1 detectors; hence, the nonzero means and the
(1 2),to
symbol error for agiven transmitted message. Analogous
to Rayleigh forp= 0) pdfs. Note that
resulting Rician (as opposed
the probability of choosing mkwhcn messagemi is transmitted is
for zero frequency error( p = 0), then
given by
M-FSK IK TIMING AND FKEQUkNCY
HINEDI er al.: NONCOHERENT
ORTHOGONAL
925
dump (I&D) circuits using its own estimate
of the symbol epoch
which is offsct fromthc truc epoch by At sec. This lack of time
synchronizationresultsinsignal attenuationinthedetectormatched
to the incoming frequency and moreover, loss of orthogonality
duetosignalspilloverintotheremainingdetectors.
In the
presence of timing error, the received signal canbe modeled as
If w(k,i) denotcs thc Hamming wcight
of the difference between
ml
the code words (bit mappings) assigned to messages (symbols)
and mk,that is, the number of bits in which the two differ,then the
average bit error probability is
where we have assumed that signals,( t ) is transmitted followed
by signal s , ( t ) and have allowed for the possibility of a carrier
phase discontinuity from symbol to symbol (so-called disconWe now discuss the mapping from which the set of Hamming tinuous phase M-FSK modulation). Since,fornoncoherent
detection, the absolute carrier phase is inconsequential, we can,
weights w ( k , i ) , k , i = O,l,. ..,M - I, k # i is computed.
It is clear that
if a symbol error occurs, it is more likely towithout
occur loss in generality, set 8, = 0 and e2 = 8 for i # J or 0, =
0
for
i = j . For so-calledcontinuous phase M-FSK (CPFSK), we
in an adjacent frequency than in any other. Thus, a Gray code
can,
in
addition, sct 8 = 0. Since the local epoch estimate isnot
mapping is appropriatetothistype
of modulation.Figure2
perfect,
the receiver I&Ds operate inthe interval (At,At + T ) to
Eb / No in dBwith
depicts the average bit error probability versus
obtain
at
the kth detector
p a s a parameter for binary,4-ary and 8-ary FSK and a conventional Gray code assignment.
M-I M-I
3.0 EFFECTOF TIMING ERROR
ON ORTHOGONAL
M-FSK
DETECTION
I
When the receivercamerfrequency
is precisely known
but the
symbol epoch is not, the receiver implements its integrate-and-
-,dB
No
(a) M
=
2
I
1
I
9
10
11
+ ~ ~ ' ' c o s ( 2 n (+
f ,f , ) r + Q)cos(2n(f;+ fk)t)dr
12
13
14
15
16
6
- , dB
NO
(b) M = 4
Figure 2. Bit Error Probability for M-FSK With Frequency Error
7
8
9
10
%,dB
No
(c) M = X
+ nc,k
11
12
13
IEEE TRANSACTIONS ON COMMUKICATIONS, VOL. 43, NO. 2/3/4. FEBRCARY/MARCH/APRIL 1995
926
f:(i,i,i)=
1, Vi
(24)
From (15), the conditional probabilityof symbol error, assuming
that message mi was scnt followed by message mi,is then given
by
can be expressed as
The unconditional (with respect
to the data) probability
of symbol
error is then
where
f; (i, j , k ) =
sin2(n(i-k)(1-d))
sin'(n(j-k)i)
z2(i- k)*
+
n*(j-k)Z
Aswas thecasefor frequencyerror in Section 2.0, the
presence of timing error produccsa lack of orthogonality which
reaults in the symbols error, not being equally likely. Hence to
compute the average bit error probability we must once again
compute the probability of a particulnr symbol error for a given
transmittedmcssagc.Analogous
to (17), theprobabilityof
choosing mk when message m iwas sent followed by message m.,
is given by
z*(i-k)(j-k)
+
sin'( n(i - j ) -
+) , i # k . j # k
n2(z- k ) ( j - k )
with d d A t i T denotingthetimingerrornormalized
symbol time. Some special cases of ( 2 3 ) are
f;(i,i,k)=
sin*(n(i-k))
1, i = k
z2(i-k)2 ={O, i # k
by the
(244
Finally,theaverage(overthedata)biterrorprobabilityis,
analogous to (1 8),
Here again the evaluationof (28) will depend on the mapping
of the symbols to bits. For a conventional Gray code mapping,
Eh i No in dB
Figure 3 depicts average bit error probability versus
for binary,4-ary and 8-ary FSK with i,as aparameter and the case
of continuous phascM-FSK. The numerical results in this figure
HINEDI
et ai..
927
NONCOHERENT ORTHOGONAL M-FSK IN TIMING AKD FREQUENCY
10-3
10.~
10-6
9
10
- , dB
NO
11
12
13
14
15
16
7
8
9
10
11
12
13
Eb
-,dB
N"
(a) M = 2
-
6
-,dB
NC
(c) M
(b) M = 4
=
8
Figure 3. Bit Error hobdbilily for Conlinuoub Phase M-FSK with Timing Error.
0.1
(28) over a uniform distribution for8. We observe that discontinuous phase M-FSK
is much more sensitive
to timing offsetthan
continuous phase M-FSKis. When the timing is perfect (A= 0),
the two performances are, of course, identical. This can be seen
by noting that (23) becomes independent of 8 when A = 0.
0.01
0 001
4.0 EFFECTOF TIMIKG AND FREQUENCY
ERRORS
ON
ORTHOGONAL M-FSK NONCOHERENT
DETECTION
-1u
10- 4
When both the incoming carrier frequency and symbol epoch
are unknown, then the received signal is still given by (1 9) but
with&, replacedby J;.+ Af . The inphase andquadrature outputs
now become
a"
10- 5
COS(~E(J;.
+~
f .6)r)cos(2~(&
+
+ fk)r)dr
(29)
10-6
+jT
dl+ I
10.~
9
10
11
12
13
14
15
Flgure 4. Blt Error Probability lor Discontinuous Phase M-FSK
with Timing Error; M = 4.
c o s [ 2 ~ ( & . + A f + ~ ) t + Q ) c o s ( 2+n&( J) f;) d r ] + r ~ ( . ~
and
z ! , ~= 2 P { / ~ c o s ( 2 z ( +Af+J;)r)sin(Zn(f,.
~
+f,)t)dt
(30)
+JT At+T c a s j ? i i ( i ; + A ~ + f ; ) ~ i R ) r i n ( 2 n ( l ; . + & ) f ) d r ~ + i l , , ,
arc obtainedby setting 8= 0 in (28). Digital computer simulations
were uscd to confirm somc of the cases,in particular, the results
Normalizing as before and following a similar procedure, we
corresponding toM=4in Figure 3b. Forpurposes ofcomparison,
obtain thc pdf of I$ given by
the corresponding results for the discontinuous phase case with
M = 4 are illustrated in Figure 4 and are obtained by avenging
ICEETKANSALTIONS Oh COMMUNICATIONS, VOL. 43, N O . 2/3/4, FEURUARY/MARCH/APRIL 1995
928
discontinuous phase case are illustrated in Figure
6 forM = 4. As
expected, discontinuous phaseM-FSK suffers a larger degradation than continuous phaseM-FSK.
5.0 BOUNDSON THE PERFORMANCE OF ORTHOGONAL M-FSK
DETECTION
IN THE PRESENCEOF FREQUENCY
ERROR
where
fi2
(i,j , k ) =
Because of the computational difficulty involved in evaluating expressions such as ( 1 5), it is advantageousto upper boundthe
average error probability performance. We shall consider only
the case of frequency errorby itself since itis the simplest of the
three cases treated in Sections2, 3, and 4.
Perhapsthe most well-known
upper boundon average symbol
error proabability performance is the union bound [9] which is
given by
sin*(z(i-k+p)(l-A))
sin2(z(j-k+p)A,)
2
x2(j-k+p)2
f
z2(i-k+p)
(
-
sin* x [ ( i - j ) - ( j - n + p ) ~ ] -Q- )
2
n*(i-k+p)(j-k+p)
sin’ z[(i- k + p ) l - ( j k + p ) ] -
-
+
+
(
-
-)e2
n*(i-kfp)(j-k+p)
i
sin2 x [ ( k - j - p ) - ( ; - i ) ~ ] - x*(i-k+p)(j-k+p)
‘I
(33)
@I
2
sin2 n(i - j ) - -
(
n2(i-k+p)(j-k+p)’
i#k,j#k
(32)
where Pji APr{choose mj[misent} is the so-called pairwise error
of
probability. For the case of orthogonal M-FSK in the presence
frequency error, using the approach taken in [9;Appendix 4B] it
is straightforward to show that
To emphasize a point made earlier concerning
the relation of our
results to those in [ 1 ] we observe that if (32) were evaluated for
1
= -2[ ~ - Q ( ~ , u ) + Q ( u , ~ ) ]
the case where i - j is large (corresponding to a large difference
in the frequency of adjacent transmissions), then the dominant where
term in this expression would be the first onc which depending
upon whether i = k (the correct correlator)or i f k (an incorrect
(35)
in notation, agree
correlator) would, making appropriate changes
with the analogous expressions in [l], namely, Eqs. (7) and (9). with $(i,i),
f;(i, j ) as in (9) anti (10). respectively. Such a
However, as previously pointed out,
the assumption of large i - j
bound is typically tight at high
S N R but loose at sufficiently low
is valid for the caseof frequency hoppingbut not here.
SNR.
If p = 0, then fin(i,j,k) reduces to j:(i,j,k) of (23) as
To simplify matters still further buttheatexpense of a loss in
cxpccted. Similarly, if A = 0, then fj,A(i, j , k ) reduces to $(i,k)
accuracy, one can
avoid thc computationof theMarcum Q-function
of (9). The probability of bit and symbol error arestill given by by Chernoff bounding the pairwise error probability leading to
(26)together with(25)and (28)together with (27),respectively, what is commonly called a union-Chemoff bound. Since from
with f : ( i , j , k ) replaced by &$(i,j,k) of (32). For a conventhe results of Section 2 , the pairwise error probabilityis given as
5 depictsaverage bit error
tional Graycodemapping,Fig.
Pji= Pr{gj > <.lmi},
then applying a Chernoff bound to
this
probability versus Eh / N , in dB for binary, 4-ary and 8-ary FSK probability we get
with p and l as parameters and the case of continuous phase
M-FSK. The numerical results in this figure are obtained by
setting 6’ = 0 in (32). Digital computer simulations were again
used to confirm someof the cases, in particular, those illustrated
in Figs. 5c and 5d. Note when
that the timing and frequency errors
which is expressed in terms of the product of the conditional
occur simultaneously, the losses are not additive. In particular,
characteristic functionsof the normalized envelopes correspondthe interaction of the two types of error results in a degradation
ing to the correct (ith) and incorrect correlator outputs. These
larger than the sumof the degradations dueto each error acting
to the Rician distribucharacteristic functions which correspond
alone. Forcomparison purposes,
the corresponding results forthe
tion of (8) are given by
uth)
929
HINEDI ei al.: NONCOHERENT ORTHOGONAL M-FSK IN TIMING AND FREQUENCY
10
11
12
13
=
2 and p
16
17
9
10
11
12
-
E.
K . dB
(a) M
15
14
-
13 15 14
16
No ' d B
~
(b)M=2andp=0.2
0. I
(~)M=4andp=0.1
10.~
10-4
\ u
~
10
11 13 12
14
15
16 6
7
8
9
10
I\
1
\
A.
11
=
0.05
12
13
6
7
8
-,dB
No
(d) M
=4
and p = 0.2
(e)M=Xandp=O.I
(t)M=Sandp=0.2
Figure 5. Bit Error Probabilily for Continuous Phase M-FSK With Both Frequency and Timing Errors
9
10
11
12
13
IEEE TRANSACTIONS OK COMMUNICATIONS, VOL. 43, N U . 2 / 3 / 4 , PEBRUARY/MARCH/APRIL
930
1995
Finally, substituting (37) into (36) gives
where UT havc Ict A. = 2iL. Evaluating (38) for each of the
M ( M - I ) Ci,s and substituting into (33) gives the desired unionChernoff bound2.
A number of years back, Omura [ 101 developed a Chernofftype bound on the error probability performance of M-ary communication systems which
in a sense is a compromise between the
union and theunion-Chernoffbound.Inourapplication,the
bound circumvents the needto compute the Marcum Q-function
(because of the use
of a Chemoffbound) but avoids the looseness
at low SNRs associated with the union bound. From an aesthetic
viewpoint, the result is obtained in a form that is similar to the
of noncoherent M-FSK with
exact error probability performance
no frequency crror which is exponential in behavior.
Since Omura's bound was never published but rather privately communicated to the authors, Appendix A presents the
derivation of the bound in its generalizcd form. Assuming that
signal n (rnessagern,J is transmitted, then the detectormatchedto
f, produces & with pdf as given by (8) with i = X- = n while the
remaining M - 1 detectors produce independent s with pdf as
given by (8) with i = n and k = i. As required by the results in
Appendix A, we need to evaluate the characteristic functionsof
the these two pdfs. These have been evaluated above in (37).
Using these results in (A.8) gives after some manipulations
10"
A
=
0.20:
2
lo-'$
10
\
12
11
E,
N,
13
\
14
15
dB
(b) M = 4 and 0 = 0.2
Figure 6. Bit Error Probability for Discontinuous Phase M-FSK
With Both Frequency and Timing Errors
HINEDI et a1 . NONCOHERENT ORTHOGONAL M-FSK IN TIMING AND FREQUENCY
931
Finally, the desired upper bound on average symbol error probability is given by
Figure 7 illustrates the upper bounds on C ( E ) as given by (33)
togetherwith (34), (33)together with (38) and (40)versus Eh / NO
in dB forM = 4 and various values
of normalized frequency error,
p, assuming minimum frequency spacing for orthogonality. It is
to be emphasized that the Chernoff and union-Chernoff bounds
are loose at highSNR and thus shouldnot be used to predict the
trueerrorprobabilityperformance.Rather,theirvalue
is for
making system comparisons and trade-offs.
The union bound,
however, is very tight at high SNR andbecan
used to predict the
absolute performance.
6.0 CONCLUSIONS
The error probability performance of noncoherentM-FSK is
quite sensitive to the presence of timing and frequency offsets
(errors) in the systcm. Fora given number of frequencies. M, and
fractionaloffset, the performance is muchmoresensitive
to
timing error than isit to frequency error.By studying these errors Figure 7. Bounds on the Symbol Error Probabillty of CFSK in the Presence of
Frequency Errur.
individually and thencombined, we arc to
able
note that the losses
due to these errors ,are not additive. In particular, the interaction
Simon, M. K., J . K. Omura, R. A. Schollz and €3.K . Levitt,
of the two types of error resultsin a degradation larger than the
Spread Spectrum Communications,Vol. 111, Computer Scisum of the degradations dueto each error acting alone. Furthermore, for the case of timing error (either or
alone
in combination
ence Press, Rockville, MD, 1984.
Lindsey, W. C. and M.K. Simon, TelecommunicationSyswith frequency error), the performance is much less robust for
temsEngineering,EnglewoodCliffs,NJ:Prentice-Hall,1973.
discontinuousphase M-FSK thanit is for continuousphase
M-FSK.
Fonseka, K . J. P. and N. Ekanayake, “Comparison of Three
Detection Techniques for M-ary CPFSK with Modulation
REFERENCES
IndexIIM,”GLOBECOM’84ConferenceRecord.pp.22.6.1-
22.6.6.
Nakamoto, F. S., R. W. Middlestead and C. R. Wolfson,
Dallel, Y.E. and S. Shamai, “An Upper Bound onthe Error
“Impact ofTime and Frcqucncy Errorson Processing SatelProbability of Quadratic Detection in Noisy Phase Chanlites with MFSK Modulation,” ICC ‘81 Conference Record,
nels,” IEEE Transactons on Communications, vol. 39,no.
June 14-18, 1981,
Denver,CO, pp. 37.3.1- 37.3.5.
1 1 . pp. 1635-1650,Novcmbcr 1991.
Chadwick, H., “TimeSynchronization in anMFSKReMarcum, J. E., “A Statistical Theory of Target Detection by
ceiver,” Pasadena, CA: Jet Propulsion Laboratory,SPS 37Pulse Radar,” Mathematical Appendix, Santa Monica, CA:
48, Vol. 111, 1967, pp. 252-64. Alsoprcscntcdatthe
RM-753,July I ,
RANDCorporation,TechnicalReport
Canadian Symposiumon Communications, Montreal, Que1948.
bec, Nov. 1968.
Proakis, J., Digital Communications.
2nd Ed., McGraw-Hill,
Wittke,P.andP.McLane,“Study
of theReception of
New York, 1989.
Frequency Dehopped M-ary FSK,” Research Report 83-1,
[IO]Omura, J. K.,private communication.
Queen’s University,Kingston, Ontario, March 1983.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL
932
Define
Appendix A
G e n e r a l i z e d M - a r y S y m b o l Error P r o b a b i l i t y
Bound
Consider an M-ary communication system whose
decisions are made based on a relation amongM out1 these
.
outputs be repreputs 5 0 , 5 1 , . ' ., 2 ~ ~ Lct
Then,
sented by independent random variables with pdfs as
follows :
-
zn
x,
fl(%)
i=O,l,...,n-l,
ni-l,...,M-l
(A.1)
f 0 ( x i ; [ i T L )for
and
T h a t is to say, forsomeparticular n, the random
variable x, has a fixed pdf where as the remaining
M - 1 r.v.'s xi,i # n,all have the identical form pdf Finally,
(perhaps different than thatfor x,) which, however,
depend on a parameterti, t h a t varies with bothi and
n. Assuming that signal sn(t) is transmitted, then a
correct decision is made at the receiver when x, > zi
for all a # n. Then, the conditional probability of a
correct decision is
Ps(C/ m,)
(-4.2)
= Prob{Correct decision/ m,}
= Probjz,
> x, for all i # n/ m,}
-L
rn
Prob {x. < a for all i
-
'r
=/m
-05
# n/ m,} fl(a)da
i1=0 i2=0
Prob{xi < a/ m,}fl(a)da
i l , i 2 ,...,
il<i2<
i=O,i#n
=Iy n
M-1
--03
[1- Prob{zi
2 a / m,}] f l ( c u ) d a
i=O,i#n
If Prob{z, 2 CY/mrL}is hard to evaluate, then use
the Chernoff bound
Prob{zi
-
2 a/ m,}
i*=o
j=1
...< i and
k
or using (A.4), simplifying and minimizing over the
Chernoff paxameter, we get
M-1
(A.3)
5 E{ex("'-a)/ m,}
-
e-XaE { e X X i
/ m,}
for m y X 2 0
L,,iZ,..+k+n
'If the two pdfs have identical form, then we shall ignore
t h e "0" and "1" subscripts on them and simply write
f(z)or
f ( z ,<), as appropriate. An cxample of where the two pdfs are,
in principle, different in form would correspond t o the case of
ideal (zero frequcncy error) noncoherent detection of A4-FSK
, in which case f1 (z) would be Rician and fo(z) wouldbe
Rayleigh. Also in this ideal situation, fo(z) would not dependent on a parameter ( which varics with the random variable
being characterized.
and
il<i2<,..<'h
Assumingequiprobablesignals,thentheaverage
probability of symbol error is given by
1995
HINEDI el al.: NONCOHERENT
ORTHOGONAL
M-FSK IN TIMING AND FREQUENCY
Note that if the parameter [in is independent of i,
].e., all x,,i # n, have identical pdfs fo(z),then
M-lM-l
M-1
'.'
i1=0
iz=o
ikZO
where
g&)
=
( M-1
)sa)
Srn
Thus, (A.9) together
with
ePX.fo(PPP
(A.ll)
(A.8) simplify t o
M-l
j=1
P,(E) 5
... and
,l,iz,...,lk#n
il<i2<
<il
(A.lO)
Sami Hinedi ("83)
wasborn in Aleppo,Syria, onMay 27,1963. He
received the B.S E.E., M.S E.E., and Ph.D. degrees from the Univenlty of
Southern California, Los Angeles, CA in 1983, 1984 and 1987, respectively.
Since June 1987, he has hem in the Communications Research Section of
theJetPropulsionLaboratory,Pasadena,CA,whereheiscurrentlythe
Group. His current
supervisor of theDigitalSignalProcessingResearch
in
interests include spread spectrum communications, parameter estimation
dynamic envmnments, modern rcceiver design and digital signal processing.
Hehas co-authored a book with M. K. Simon and W C. Lindsey entitled
Digital CommunicationsTechniques, Vo1.-I: Signal Design and Defectiun,
which will be published by Prentice-Hall.
Dr. Hinedi is a member of Eta Kappa Nu.
Daniel Raphaeli - biography unavailable
=
--m
k
J&o(4&3n)
933
mp
(-1)'++1(
k=l
Mi) g l ( - A k ) g t ( A )
(A.12)
Marvin K. Simon iscurrentlyaSeniorResearchEngineerattheJet
Propulsion
Laboratory,
Pasadena,
Californla
and
Lecturer
in
Communications attheCaliforniaInstitute
of Technology,Pasadena
California, Dr. Simon has worked extensively for the
last 25 years in the
for space,satellite, and
area of modulation,coding,andsynchronization
radiocommunications.Thefruitsofhisresearchhavebeensuccessfully
applied to thedesign of many of NASA'sdeepspaceandnear-earth
missions for which he holds R patents and over 15 NASA Tech Briefs. He is
a Fellow of the IEEE, Fellow of the IAE. and winner of a NASA Exceptional
Service Medalboth in recognitionof outstanding contributionsin analysis and
design of space communications systems. In addition, he is listed in Marquis
Who's Who in America.
He h a published over 110 papers on the above subjects and is co-author of
severaltextbooksincluding,
TelecommunicationSysremsEngineering
(Prentice-Hall, 1973 andreprinted by DoverPress, 1991), Phase-Locked
Loops andTheirApplication
(IEEEPress,
1978) SpreadSpectrum
Cummunicarions,Vuls. I, I,, and Ill (ComputerSciencePress, 1984). and
Trellis Coded Modulation with Applications (Macmillan, 1991). His work has
DeepSpaceTelecumrnunxatronSystems
alsoappearedinthetextbook
Ertgirteering (Plenum Press, 1984). He is the co-recipient of the 1986 Prize
Paper Award in Communicatlons for the IEEE Transactions on Vehicular
of books
Technology He is currentlycompletinganewtwo-volumeset
entltled Digital Communication Techniques.