922 IEEE’L‘KANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995

The Performance of Noncoherent Orthogonal IM-FSK in the Presence of Timing and Frequency Errors

Sami Hinedi, Senior Member IEEE Marvin Simon, Fellow IEEE

Dan Raphaeli, Student Member IEEE

ABSTRACT

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 evaluated numerically for system parameters of interest. Also presented are upper boundsonaverage symbol error probability for the case of frequency error alone which are useful in assessing the absolute and relative performance of the system. Both continuous and discontinuous phase M-FSK cases are considered when timing error is present, the latter being much less robust to this type of offset.

1 .O INTRODUCTION

Noncoherent orthogonal M-ary frequency-shift-keying (M-FSK) is a simple and robust form of digital communication when the transmission channel is such that fast reliable carrier recovery is difficult or impractical to achieve and thc bandwidth requirements are not overly stringent. Most studies of this modulatioddemodulation technique for the additive white Gaussian noise (AWGN) channel have focused on the error probability performance when the receiver is assumed to be perfectly time and frequency synchronized. That is, the receiver is assumed to have perfect knowledge of the instants of time at which the modulation can change state and also perfect knowl- edge of the received camer frequency. In practical systems, such perfect knowledge is never available and thus the receiver must dcrive this information from the received signal imbedded in the AWGN. Since the estimatcs of the time epoch and received carrier frequency derived at the receiver are, in general, random variables (because of the presence of the AWGN), there will exist an error between these estimates and their true values. This lack of perfect time and frequency synchronization gives rise to a degradation in error probability performance relative to that corresponding to the ideal case where perfect knowlcdgc of time and frequency is assumed known.

cations Society. Manuscnpt was received September 27. 1993. revised May 12, Paper approved by 1. Korn, the Editor for Modulation of the IEEE Communi-

1994. This aurk was performed at the Jet Propulsion Laboratory, Califomla

Administration. Institute of Technology under a contract with the National Aeronautics and Space

The authors are with the Jet Propulsion Laboratory, California Institute of Technoloev. Pasadena. CA 91 109

The purpose of this paper is to evaluate this performance degradation, first by treating the two sources of degradation separately, and then by considering their simultaneous effect. In particular, we shall present exact cxprcssions for thc symbol and bit error probability performances of noncoherent orthogonal M - 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 bounds on error probability performances that. because of their exponential behavior, are simpler to evaluate. Numerical results are obtained for cases of practical interest.

Before going into the details of the analysis, we wish to point out the existence of several papers that relate to the subject at hand [l-71. The paper that, in principle, bears the closest resemblance to what wc arc trying to accomplish hcrc is a papcr by Nakamoto, Middlestead, and Wolfson [ I ] . Although the primary interest in [ 1 ] was frequency-hopped M-FSK, the authors also attempted to use their results to obtain performance data for noncoherent orthogonal M-FSK without frequency hopping (This is discussed in the section of their paper called Performance Without Non- Coherent Combining). While the results in [ I ] are indeed correct for predicting the performance of frequency-hoppcd M-FSK in the presence of time and frequency errors, they are unfortunately incorrect for the unhopped case. It is important to understand where the results in [l] fail to address the case of noncoherent orthogonal M-FSK since it is indeed these issues that provide the basis for our papcr. We now explain thcsc diffcrcnccs in dctail.

In 11 1 i t is correctly recognized that in the presence of timing and frequency errors two factors contribute to the perfomlance degradation relative to the perfectly synchronized case. First, the signal component of the correct correlator is attenuated and second a portion of thc transmitted signal energy now appears in the outputs of each of the M-1 incorrect correlators. This second source of degradation is referred to as the loss ofnrrhognnnlity. In computing the degradation factors for these two sources of degradation (see Eqs. (7) and (9) of [l]), the authors implicitly assume that there exists a large difference between the frequen- cies of two successive transmissions, which in the frequency- hopped case corresponds to two successive hops and is thus justified most of the time. In the case of noncoherent orthogonal

HlNEDl et al . NONCOHF.RENT ORTHOGONAI . M-FSK IN TTMING AND 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 degra- dation factors analogous to (7) and (9) of [ 11 must take this fact into account. Furthermore, the issue of whether the phase is continuous or discontinuous from symbol to symbol is critical in evaluating these degradation factors whereas in the frequency- hopping case it is inconsequential because of the large frequency difference between adjacent transmissions (hops). Based on these observations alone, it is incorrect to compute the symbol error probability for noncoherent orthogonal M-FSK in the pres- enceoffrequencyandtimingerrorsusing(7)and(9)of[I]. Ifonly frequency error is prcscnt, i.c., thc timing error is set equal to zero, then the results in (7) and (9) of 111 can be uscd to correctly compute symbol error probability.

A second incorrect approximation used in [l] is in computing the bit error probability from the symbol error probability. The authors assume that the signals remain orthogonal in the presence of timing and frequency error which facilitates the use of a wcll- known result [see [5] , Eq. (5-54)] relating these two error prob- abilities for orthogonal M-FSK. Unfortunately, however, this assumption is not valid and hence the bit crror probability results found (see Figs. 1-5) in [ l ] for the special case of noncoherent orthogonalM-FSK are incorrect. In fact, to properly evaluate the bit errorprobability in the presence of synchronization errors. one must specify an appropriate mapping, for example, a Gray code of the symbols to bits. In the perfectly synchronized case, the bit error probability performance is completely independent of the symbol-to-bit mapping since all errors arc equally likely to occur. The significance of these statements will become apparent latcr on in the paper.

The organization of the paper is as follows. Section 2 exactly treats the effect of frequency error alone (pcrfcct time synchroni- zation is assumed) on orthogonal M-FSK noncoherent detection. 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 and frequency errors. Finally, Section 5 presents various upper bounds on the performance in the presence of frequency error.

2.0 EFFECT OF FREQUENCY ERROR ON ORTHOGON'AL M-FSK NONCOHERENT DETECTION

Consider the transmissmn of orthogonal M-FSK ovcr an AWGN channel where the signal set has a one-to-one correspondcncc with the sct of M equiprobable messages rn,,rn, ,. . ., mM-, . The optimum reccivcr (assuming perfect synchronization) is illus- trated in Fig. I . Whcn the received frequency is not perfectly known, the observed signal, assuming that message mi was sent, is given by

923

sw, (11

Figure I . M-ary Noncohercnl Rcccivcr for Equal Energy Signals

r( t )=1127;cos(2n(~+++Af)f+6')+n(t) , O l f l T ( I )

where P denotes the signal power in Watts, Tdenotes the symbol time in seconds, 4, is the carrier frequency in Hz,& = i/T is the transmitted frequency corresponding to message mi, Afis the error in the carrier frequency, and 8 is the unknown canicr phase assumed to be uniformly distributed. Also, n(t) denotes the AWGN with single-sided power spectral density No WIHz. Al- tcrnatively, the received signal can be interpreted as a carrier at frequency j i shifted by the appropriatc signal frcqucncy 6 + AL rather than 4.. The inphase integrator output, z,,~, matched to signal s , ( t ) = d%cos(2n(f, + f k ) f ) (corresponding to message mk) bccomesl

= 2 P cos 2 n f,. + x + Af)t)cos(2n(f,. + f,)r)dr + ld ( (

where nc,A is a zero-mean Gaussianrandom variable with variance 0' = No / 2E,. Here, E,s g f T is the symbol energy in joules. Simplifying (2) yields

whcrcf;,, denotes the difference between the frequencies repre- aenting messages ml and mk, that is,

Similarly, the quadrature integrator output, z ? , ~ , is given by

' Since we are dealing with noncoherent detection, wc can, in thc case of perfect timing synchronization. set f3 = 0.

Y 24 l E E E TRANSACTlONS ON COMMUNICATIONS. VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995

T and signal orthogonality is restored. Despite loss of orthogonal-

z , , ~ r(t)d%sin(2n(fc + f,)f)dt ity, the variables &,El,. . (and thus remain ( 5 ) 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 message mi is transmitted, is given by

[cos(2n(Ae + Af)T)-I] = E,

2n(A,k + Af)T + %%k

The envelope statistic E, = 4- will then be Rician dis- tributed with parameter k; given by rn=0,1, ..., M - I

sin*(n(& + Af)T) = ( E r r In terms of the Marcum Q-function [8] defined by

(6) Normalizing z , . ~ and z,,, by 1 / CJ = d m , the parameter p i is then normalized by 2 / N& and since, as mentioncd above, we have f; = z/Tfor orthogonal signals, then

2E, sin2(lr(i - k + p ) ) Jr' tgl (xm )dx, = I - Q kk = - kk= - (7)

' 2 .(N:Es] [ No 1 [ n ( i - k +p)p Hence, the conditional probability of symbol error, assuming that where p4 AjT denotes the frequency error normalized by the message mi is transmitted, is given by

f i ( i . k ) = sin2(x(i - k + p))

(9) and the unconditional probability of symbol crror becomes

[ + - k + P ) p 1 M - I

W)=--CP,(Elmi) (16) First, note that the detector matched to the incoming signal suffers j d J

from signal attenuation equal to As previously mentioned, thc average bit error probability

which, as expected, reduces to unity if p= 0. Simultaneously, loss of signal orthogonality occurs as a result of signal spill-over into thc remaining M-1 detectors; hence, the nonzero means and the resulting Rician (as opposed to Rayleigh forp= 0) pdfs. Note that for zero frequency error (p = 0), then

cannot be obtained directly from the 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 errors are not equally likely. To compute the average bit crror probability we must first compute the probability of aparticular symbol error for agiven transmitted message. Analogous to (1 2), the probability of choosing mk whcn message mi is transmitted is given by

HINEDI er al.: NONCOHERENT ORTHOGONAL M-FSK IK TIMING AND FKEQUkNCY 925

If w(k, i ) denotcs thc Hamming wcight of the difference between the code words (bit mappings) assigned to messages (symbols) ml and mk, that is, the number of bits in which the two differ, then the average bit error probability is

M-I M-I

We now discuss the mapping from which the set of Hamming 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 to occur in an adjacent frequency than in any other. Thus, a Gray code mapping is appropriate to this type of modulation. Figure 2 depicts the average bit error probability versus Eb / No in dB with p a s a parameter for binary, 4-ary and 8-ary FSK and a conven- tional Gray code assignment.

3.0 EFFECT OF TIMING ERROR ON ORTHOGONAL M-FSK DETECTION

When the receivercamerfrequency is precisely known but the symbol epoch is not, the receiver implements its integrate-and-

-,dB No

(a) M = 2

dump (I&D) circuits using its own estimate of the symbol epoch which is offsct from thc truc epoch by At sec. This lack of time synchronizationresultsinsignal attenuationinthedetectormatched to the incoming frequency and moreover, loss of orthogonality due to signal spillover into the remaining detectors. In the presence of timing error, the received signal can be modeled as

where we have assumed that signal s, ( 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 discon- tinuous phase M-FSK modulation). Since, for noncoherent detection, the absolute carrier phase is inconsequential, we can, without loss in generality, set 8, = 0 and e2 = 8 for i # J or 0, =

0 for i = j . For so-called continuous phase M-FSK (CPFSK), we can, in addition, sct 8 = 0. Since the local epoch estimate is not perfect, the receiver I&Ds operate in the interval (At ,At + T ) to obtain at the kth detector

+ ~ ~ ' ' c o s ( 2 n ( f , + f , ) r + Q)cos(2n(f; + fk)t)dr + nc,k I

I 1 I 9 10 11 12 13 14 15 16

- , dB NO

(b) M = 4

6 7 8 9 10 11 12 13

%,dB No

(c ) M = X Figure 2. Bit Error Probability for M-FSK With Frequency Error

926 IEEE TRANSACTIONS ON COMMUKICATIONS, VOL. 43, NO. 2/3/4. FEBRCARY/MARCH/APRIL 1995

can be expressed as

where

f; (i, j , k ) = sin2(n(i-k)(1-d)) sin'(n(j-k)i)

z2(i - k)* n * ( j - k ) Z +

z*(i-k)( j -k)

sin'( n(i - j ) - +) n2(z - k)( j - k )

+ , i # k . j # k

with d d A t i T denoting the timing error normalized by the symbol time. Some special cases of (23) are

f;(i,i,k)= sin*(n(i-k)) 1, i = k

z2( i -k)2 = { O , i # k (244

f:(i,i,i)= 1, Vi ( 2 4 )

From (15), the conditional probability of symbol error, assuming that message mi was scnt followed by message mi, is then given

by

The unconditional (with respect to the data) probability of symbol error is then

As was the case for frequency error in Section 2.0, the presence of timing error produccs a 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 transmitted mcssagc. Analogous to (17), the probability of choosing mk when message mi was sent followed by message m., is given by

Finally, the average (over the data) bit error probability is, analogous to (1 8),

Here again the evaluation of (28) will depend on the mapping of the symbols to bits. For a conventional Gray code mapping, Figure 3 depicts average bit error probability versus Eh i No in dB for binary, 4-ary and 8-ary FSK with i, as aparameter and the case of continuous phasc M-FSK. The numerical results in this figure

HINEDI et ai.. NONCOHERENT ORTHOGONAL M-FSK IN TIMING AKD FREQUENCY 927

10-3

10.~

10-6

- 1u 10- a"

10-

- , dB NO

(a) M = 2

9 10 11 12 13 14 15 16

- ,dB N"

(b) M = 4

- 6 7 8 9 10 11 12 13

Figure 3. Bit Error hobdbilily for Conlinuoub Phase M-FSK with Timing Error.

0.1

0.01

0 001

4

5

10-6

10.~ 9 10 11 12 13 14 15

Flgure 4. Blt Error Probability lor Discontinuous Phase M-FSK with Timing Error; M = 4.

arc obtained by setting 8= 0 in (28). Digital computer simulations were uscd to confirm somc of the cases, in particular, the results corresponding toM=4in Figure 3b. Forpurposes ofcomparison, the corresponding results for the discontinuous phase case with M = 4 are illustrated in Figure 4 and are obtained by avenging

- , d B E b

NC

(c) M = 8

(28) over a uniform distribution for 8. We observe that discon- tinuous phase M-FSK is much more sensitive to timing offset than continuous phase M-FSK is. 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.

4.0 EFFECT OF TIMIKG AND FREQUENCY ERRORS ON ORTHOGONAL M-FSK NONCOHERENT DETECTION

When both the incoming carrier frequency and symbol epoch are unknown, then the received signal is still given by (1 9) but with&, replaced by J;. + Af . The inphase and quadrature outputs now become

COS(~E(J;. + ~ f + .6)r)cos(2~(& + fk)r)dr (29)

dl+ I +jT 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 )d t (30)

+JT At+T cas j? i i ( i ;+A~+f ; )~ iR) r in (2n( l ; .+&) f )dr~+i l , , ,

Normalizing as before and following a similar procedure, we obtain thc pdf of I$ given by

928 ICEETKANSALTIONS Oh COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEURUARY/MARCH/APRIL 1995

discontinuous phase case are illustrated in Figure 6 forM = 4. As expected, discontinuous phase M-FSK suffers a larger degrada- tion than continuous phase M-FSK.

5.0 BOUNDS ON THE PERFORMANCE OF ORTHOGONAL M-FSK DETECTION IN THE PRESENCE OF FREQUENCY ERROR

where

f i 2 (i, j , k ) = sin*(z(i-k+p)(l-A)) sin2(z(j-k+p)A,)

z 2 ( i - k + p ) 2 f x 2 ( j - k + p ) 2

sin* ( x [ ( i - j ) - ( j - n + p ) ~ ] - - ) Q

n * ( i - k + p ) ( j - k + p ) - 2

sin’ z[(i - k + p ) l - ( j - k + p) ] - -) n * ( i - k f p ) ( j - k + p )

( e - 2

sin2 x [ ( k - j - p ) - ( ; - i ) ~ ] - -

x * ( i - k + p ) ( j - k + p ) + i @I 2

+ ( ‘I i # k , j # k sin2 n(i - j ) - -

n 2 ( i - k + p ) ( j - k + p ) ’ (32)

To emphasize a point made earlier concerning the relation of our results to those in [ 1 ] we observe that if (32) were evaluated for the case where i - j is large (corresponding to a large difference in the frequency of adjacent transmissions), then the dominant term in this expression would be the first onc which depending upon whether i = k (the correct correlator) or i f k (an incorrect correlator) would, making appropriate changes in notation, agree with the analogous expressions in [l], namely, Eqs. (7) and (9). However, as previously pointed out, the assumption of large i - j is valid for the case of frequency hopping but not here.

If p = 0, then fin(i,j,k) reduces to j:(i,j ,k) of (23) as cxpccted. Similarly, if A = 0, then fj,A(i, j , k ) reduces to $(i,k) of (9). The probability of bit and symbol error are still given by (26) together with (25) and (28) together with (27), respectively, with f : ( i , j , k ) replaced by &$( i , j , k ) of (32). For a conven- tional Gray code mapping, Fig. 5 depicts average bit error probability versus Eh / N , in dB for binary, 4-ary and 8-ary FSK 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 some of the cases, in particular, those illustrated in Figs. 5c and 5d. Note that when the timing and frequency errors occur simultaneously, the losses are not additive. In particular, the interaction of the two types of error results in a degradation larger than the sum of the degradations due to each error acting alone. Forcomparison purposes, the corresponding results forthe

Because of the computational difficulty involved in evaluat- ing expressions such as ( 1 5) , it is advantageous to upper bound the average error probability performance. We shall consider only the case of frequency error by itself since it is the simplest of the three cases treated in Sections 2, 3, and 4.

Perhapsthe most well-known upper bound on average symbol error proabability performance is the union bound [9] which is given by

(33)

where Pji APr{choose mj[mi sent} is the so-called pairwise error probability. For the case of orthogonal M-FSK in the presence of

frequency error, using the approach taken in [9;Appendix 4B] it is straightforward to show that

1 = - [ ~ - Q ( ~ , u ) + Q ( u , ~ ) ] 2

where

(35)

with $(i,i), f;(i, j ) as in (9) anti (10). respectively. Such a bound is typically tight at high S N R but loose at sufficiently low SNR.

To simplify matters still further but at the expense of a loss in accuracy, one can avoid thc computation of the Marcum Q-function by Chernoff bounding the pairwise error probability leading to what is commonly called a union-Chemoff bound. Since from the results of Section 2 , the pairwise error probability is given as Pji = Pr{gj > <.lmi}, then applying a Chernoff bound to this probability we get

which is expressed in terms of the product of the conditional characteristic functions of the normalized envelopes correspond- ing to the correct (ith) and incorrect uth) correlator outputs. These characteristic functions which correspond to the Rician distribu- tion of (8) are given by

HINEDI ei al.: NONCOHERENT ORTHOGONAL M-FSK IN TIMING AND FREQUENCY 929

K. dB

10 11 12 13 14 15 16 17

No ' d B E.

9 10 1 1 12 13 14 15 16 -

- 14 15 16

(a) M = 2 and p ~ 0. I ( b )M=2andp=0 .2 ( ~ ) M = 4 a n d p = 0 . 1

1 0 . ~

- 10-4

~ \ u 1 I \

\ A. = 0.05

10 11 12 13 14 15 16 6 7 8 9 10 1 1 12 13 6 7 8 9 10 1 1 12 13

- , d B 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

930 IEEE TRANSACTIONS OK COMMUNICATIONS, VOL. 43, NU. 2/3/4, PEBRUARY/MARCH/APRIL 1995

10"

A = 0.20:

lo-'$ 10 11 12 13 14 15 \ \

E, - N, dB

(b) M = 4 and 0 = 0.2

Figure 6. Bit Error Probability for Discontinuous Phase M-FSK With Both Frequency and Timing Errors

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 union- Chernoff bound2.

A number of years back, Omura [ 101 developed a Chernoff- type bound on the error probability performance of M-ary com- munication systems which in a sense is a compromise between the union and the union-Chernoff bound. In our application, the bound circumvents the need to compute the Marcum Q-function (because of the use of a Chemoff bound) 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 exact error probability performance of noncoherent M-FSK with no frequency crror which is exponential in behavior.

Since Omura's bound was never published but rather pri- vately 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 detectormatched to f, produces & with pdf as given by (8) with i = X- = n while the remaining M - 1 detectors produce independent 2 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 functions of the these two pdfs. These have been evaluated above in (37). Using these results in (A.8) gives after some manipulations

HINEDI et a1 . NONCOHERENT ORTHOGONAL M-FSK IN TIMING AND FREQUENCY 931

Finally, the desired upper bound on average symbol error prob- ability 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 for M = 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 high SNR and thus should not be used to predict the true error probability performance. Rather, their value is for making system comparisons and trade-offs. The union bound, however, is very tight at high SNR and can be used to predict the absolute performance.

6.0 CONCLUSIONS

The error probability performance of noncoherent M-FSK is quite sensitive to the presence of timing and frequency offsets (errors) in the systcm. For a given number of frequencies. M, and fractional offset, the performance is much more sensitive to timing error than it is to frequency error. By studying these errors individually and thencombined, we arc able to note that the losses due to these errors ,are not additive. In particular, the interaction of the two types of error results in a degradation larger than the sum of the degradations due to each error acting alone. Further- more, for the case of timing error (either alone or in combination with frequency error), the performance is much less robust for discontinuous phase M-FSK than it is for continuous phase M-FSK.

REFERENCES

Nakamoto, F. S., R. W. Middlestead and C. R. Wolfson, “Impact of Time and Frcqucncy Errors on Processing Satel- lites with MFSK Modulation,” ICC ‘81 Conference Record, June 14-18, 1981, Denver,CO, pp. 37.3.1- 37.3.5. Chadwick, H., “Time Synchronization in an MFSK Re- ceiver,” Pasadena, CA: Jet Propulsion Laboratory, SPS 37- 48, Vol. 111, 1967, pp. 252-64. Also prcscntcd at the Canadian Symposium on Communications, Montreal, Que- bec, Nov. 1968. Wittke, P. and P. McLane, “Study of the Reception of

Frequency Dehopped M-ary FSK,” Research Report 83- 1, Queen’s University, Kingston, Ontario, March 1983.

Figure 7. Bounds on the Symbol Error Probabillty of CFSK in the Presence of Frequency Errur.

Simon, M. K., J . K. Omura, R. A. Schollz and €3. K . Levitt, Spread Spectrum Communications, Vol. 111, Computer Sci- ence Press, Rockville, MD, 1984. Lindsey, W. C. and M. K. Simon, Telecommunication Sys- temsEngineering,EnglewoodCliffs,NJ:Prentice-Hall, 1973. Fonseka, K . J. P. and N. Ekanayake, “Comparison of Three Detection Techniques for M-ary CPFSK with Modulation IndexIIM,”GLOBECOM’84ConferenceRecord.pp.22.6.1- 22.6.6. Dallel, Y.E. and S. Shamai, “An Upper Bound on the Error Probability of Quadratic Detection in Noisy Phase Chan- nels,” IEEE Transactons on Communications, vol. 39, no. 1 1 . pp. 1635-1650, Novcmbcr 1991.

Marcum, J. E., “A Statistical Theory of Target Detection by Pulse Radar,” Mathematical Appendix, Santa Monica, CA: RAND Corporation, Technical Report RM-753, July I , 1948. Proakis, J., Digital Communications. 2nd Ed., McGraw-Hill, New York, 1989.

[IO] Omura, J. K., private communication.

932 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995

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 among M out- puts 5 0 , 5 1 , . ' ., 2 ~ ~ 1 . Lct these outputs be repre- sented by independent random variables with pdfs as follows :

zn - fl(%) x, f 0 ( x i ; [ i T L ) for i=O, l , . . . , n - l ,

n i - l , . . . , M - l (A.1)

Tha t is to say, for some particular 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 (perhaps different than that for x,) which, however, depend on a parameter ti, tha t varies with both i 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,}

- Prob {x. < a for all i # n/ m,} fl(a)da

=/m 'r Prob{xi < a/ m,}fl(a)da

=Iy n [1 - Prob{zi 2 a / m,}] f l (cu)da

rn - L -05 i=O,i#n

M - 1

--03 i=O,i#n

If Prob{z, 2 CY/ mrL} is hard to evaluate, then use the Chernoff bound

Prob{zi 2 a/ m,} (A.3) 5 E{ex("'-a)/ m,} - - e-XaE { e X X i / m,} for m y X 2 0

'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 to the case of ideal (zero frequcncy error) noncoherent detection of A4-FSK , in which case f1 (z) would be Rician and fo(z) would be Rayleigh. Also in this ideal situation, fo(z) would not depen- dent on a parameter ( which varics with the random variable being characterized.

Define

Then,

and

Finally,

i1=0 i2=0 i*=o j = 1 - i l < i 2 < ... < i k il,i2 ,..., and

or using (A.4), simplifying and minimizing over the Chernoff paxameter, we get

M-1

L , , i Z , . . + k + n and i l < i 2 < , . . < ' h

Assuming equiprobable signals, then the average probability of symbol error is given by

HINEDI el al . : NONCOHERENT ORTHOGONAL M-FSK IN TIMING AND FREQUENCY 933

Note that if the parameter [in is independent of i, where ].e., all x,, i # n, have identical pdfs fo(z), then g&) = Srn ePX.fo(PPP (A. l l ) M - l M - l M-1 k

--m

' . ' J&o(4&3n) = ( ) s a ) M - 1 Thus, (A.9) together with (A.8) simplify t o - i1=0 iz=o i k Z O j=1 M-l

,l,iz,...,lk#n i l < i 2 < ... < i l and P,(E) 5 m p (-1)'++1( Mi ) gl ( -Ak)gt (A)

(A.lO) (A.12) k = l

Sami Hinedi ("83) was born in Aleppo, Syria, on May 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 the Jet Propulsion Laboratory, Pasadena, CA, where he is currently the supervisor of the Digital Signal Processing Research Group. His current interests include spread spectrum communications, parameter estimation in dynamic envmnments, modern rcceiver design and digital signal processing. He has co-authored a book with M. K. Simon and W C. Lindsey entitled Digital Communications Techniques, 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

Marvin K. Simon is currently a Senior Research Engineer at the Jet

Communications at the California Institute of Technology, Pasadena Propulsion Laboratory, Pasadena, Californla and Lecturer in

California, Dr. Simon has worked extensively for the last 25 years in the area of modulation, coding, and synchronization for space, satellite, and radio communications. The fruits of his research have been successfully applied to the design of many of NASA's deep space and near-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

design of space communications systems. In addition, he is listed in Marquis Service Medal both in recognition of outstanding contributions in analysis and

Who's Who in America.

several textbooks including, Telecommunication Sysrems Engineering He h a published over 110 papers on the above subjects and is co-author of

(Prentice-Hall, 1973 and reprinted by Dover Press, 1991), Phase-Locked Loops and Their Application (IEEE Press, 1978) Spread Spectrum Cummunicarions, Vuls. I, I,, and Ill (Computer Science Press, 1984). and Trellis Coded Modulation with Applications (Macmillan, 1991). His work has also appeared in the textbook Deep Space Telecumrnunxatron Systems Ertgirteering (Plenum Press, 1984). He is the co-recipient of the 1986 Prize Paper Award in Communicatlons for the IEEE Transactions on Vehicular Technology He is currently completing a new two-volume set of books entltled Digital Communication Techniques.