naval postgraduate school moliterey, california ,•,, 0%l dtic · 1/ lcosati codels 18 subjct...
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AD-A260 976
NAVAL POSTGRADUATE SCHOOLMoliterey, California
DTIC,•,, 0%l ELECTEDfif,_. •W AR 8 1993 7
AN INTRODUCTION TO DOPPLER EFFECTAND FADING IN
MOBILE COMMUNICATIONS9
' y
Abdon B. do Paula
December 1992
Thesis Advisor: Tri T, tIlaSecond Reader: Ralph D. llippenstile
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AN INTRODUCTION TO DOPPLER EFFECT AND FADING IN MOBILE COMMUNICATION
"2 PERSINAL (,uTHCOR(S -De Paula, Abdon B,
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IIF,,D GRROUP SKRGO,POI Mobile communications with Doppler effect, fadingchannels
r', AiHTRA 'T 'onn-nip nn rvierre if necessAry and idenwify hy hlock rniirnhr)In this research we present an introductory analysis of a complex aspect of mobilecommunications: Doppler effect is evaluated in both Ricean and Rayleigh channels, Anollcoherent 2-FSK scheme Is selected to evaluate the behavior of the system under verystrong fading channel conditions. The analysis is conducted for the binary case due tothe possibility of developing closed form solutions, Therefore, the approach issimplified avoiding long lasting simulations that may obscure the concepts. Theprobability of bit error for the 2-FSK case can also be used as an initial bound for aM-FSK scheme. Diversity is evaluated as a means of combating fading and Dopplereffects. Error correcting codes, in the form of a convolutional codes, are also used
rnd applied to both effects,
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An Introduction to Doppler Effect and Fading in MobileCommunication
by
Abdon B. de PaulaLieutenant Commander, Engineer Corps, Brazilian Navy
B. S., Brarliiian Naval Ac-ademyB. S., Si5o Paulo University
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
December 1992
Author:Abdon B. de P ula
A 71 -rý _ULIApproved by: _ _ ___, I _-_____.
Tri T. IHa, Thasis Advisor
t~lph D. Ii ippenstiel, Second R-'ader
Michael A. Morgan, Clair i., Department of Electricaland Computer Engineering
11l
ABSTRACT
In this research we present an introdtici.ory analysis of a complex aspecL of
mobile communications: Doppler effect is ev..'aluated in both Ricean and Rayleigh
fading channels. A noncoherent 2-FSK scheme is selected to evaluate the behavior of
the system under very strong fading channel 3onditions. The analysis is conducted
for the binary case due to the possibility of d..cveloping closed form solutions. There-
forc, the approach is simplified avoiding loIny2; lasting simulations that may obscure
the concepts. The probability of bit error fc: the 2-FSK case can also be used as an
initial bound for a M-FSK scheme. Di-versi;.y is evaluated as a means of combating
fading and Doppler effects. Error corr,-tbg codes, in the form of convolutional
algorithms, are also used and applied to both effects.
Accestot, For
NTIS CRA&I
1.-) L JIcUliIdfMOLI'ctd 0J'ir:lifcc l~ot,
by
Distributiotn IAvailanltity Codes
ij AvdfI a1 11orcIJR,
TABLE OF CONTENTS
I. INTRODUCTION ............................. 1
II. DIVERSITY AND FADING CHANNELS ................ 7
III. THE PROBABILITY OF BIT ERROR CONDITIONED ON THEDOPPLER EFFECT IN A RICEAN FADING CHANNEL ....... 13
IV. CODED PERFORMANCE . ....................... 19
V. ANALYSIS OF RESULTS ........................ 21VI. CONCLUSION ............................... 27
APPENDIX A. EVALUATION OF THE INNER INTEGRAL
OF EQUATION 2.6 ...................... 29
APPENDIX B. BESSEL FUNCTION PROPERTIES ............... 31
Ai2PENDIX C. DEVELOPMENT OF PB EQUATION ........... 33
APPENDIX D. AVERAGE BIT ERROR PROBABILITY FOR A NON-COHERENT 2-FSK IN A RICEAN FADING CHANNEL . 35
APP.1.END"IX Uj ThCTiPT1 ER Er1, LECT COEF rICIEnT
T I.......... 139
APPENDIX F. FIGURES ............................ 41
REFERENCES .................................. 85
INITIAL DISTRIBUTION LIST ......................... 87
iv
LIST OF FIGURES
1. An L-fold diversity 2-FSK receiver ....... .................... 42
2. Probability of bit error conditioned on the Doppler effect coefficientin an AWGN channel as a function of the system diversity ....... .43
3. Probability of bit error conditioned on the Doppler effect coefficientin a Rayleigh fading channel (7" = 0) as a function of the systemdiversity .......... ................................... 44
4. Probability of bit error conditioned on the Doppler effect coefficient(fdT = 0) in a Rayleigh fading channel (r = 0) as a function of thesystem diversity ......... .............................. 45
5. Probability of bit error conditioned on the Doppler effect coefficient(fdT = 0.5) in a Rayleigh fading channel (r = 0) as a function of tiesystem diversity ......... .............................. 46
6. Probability of bit error conditioned on the Doppler effect coeflicient(fdT = 0) in a Riceain fading channel (r = 1) as a function of thesystem diversity ......... ............................. 47
7. Probability of bit error conditioned on the Doppler effect coefficient/~~ ~ -" I I-••"-
fJdT = 0.5) in a Ricean fading channel (r = 1) as a function of thesystem diversity ......... .............................. 48
8. Probability of bit error conditioned on the Doppler effect coefficient(fdT = 0) in a Ricean fading channel (r = 5) as a function of thesystem diversity .......... ............................. 49
9. Probability of bit error conditioned on the Doppler effect coefficient(fdT = 0.5) in a Ricean fading channel (r = 5) as a function of thesystem diversity ......... .............................. 50
10. Probability of bit error conditioned on the Doppler effect coefficient(fit = 0) in a Ricean fading channel (r 1 10) as a function of thesystem diversity ......... .............................. 51
11. Probability of bit error conditioned oi the Doppler effect coefficient(fdT = 0.5) in a Ricean fading channel (r = 10) as a function of thesystem diversity ......... .............................. 52
12. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 1) as a function of the systemdiversity. .......... .................................. 53
13. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 5) ab a function of the systemdiversity ......... .................................. 5,4
v
1l1. Probability of bit error conditioned on the Doppler effect cocilicientin a Ricean fading channel (r = 10) as a function of the systemdiversity .......... ................................... 55
15. Probability of bit error conditioned on the Doppler effect coefficientin an almost AWGN chiann.el (r = 10,000) and no Doppler effect fora rate 1/2, v = 2 convolutional code and system diversity L = 1. . .. 56
16. Probalbility of bit error conditione 1 on the Doppler effect coefficientin an almost AWGN channel (r = 10,000) and no Doppler effect fora rate 1/2, v = 2 convoiutional code and system diversity L = 2. . . . 57
17. Probability of bit error conditioned on the Doppler effect coefficientin a Rayleigh fading diannel (r = 0) and no Doppler effect for a rate1/2, z = 2 convolutional code and system diversity L = 2 ......... 58
18. Probability of bit error ,zonditioned on the Doppler effect coefficient ina Rayleigh, fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 2 convolutionalcode and system diversity L = 2 ....... ..................... ... 59
19. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading cbannel (r = 10) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 2 convolutionalcode and system diversity L = 2 ........ .................... 60
20. Probability of bit error conditioned on the Doppler effect coefficient inIa Rayleigh fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, iv = 2 conivoiutionalcode and system diversity L = 3 ....... ..................... 61
21. Probability of bit error conditioned on the Doppler effect coefficientin a R;.cean fading channel (r = 10) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 2 convoiutionalcode and system diversity L = ? ....... .................... 62
22. Probability of bit error conditioned on the Doppler effect coefficientin a Rayleigh fading channel (r = 0) and no Doppler effect for a rate2/3, iv == 2 convolutional code and system diversity L = 2 ....... .. 63
23. Probability of bit error conditioned on the Doppler effect coefficient ina Raylcigh fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v = 2 convolutionalcode and system diversity L = 2 ...... ..................... 64
24. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 10) and rmaximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, il = 2 convolutionalcode and system diversity L = 2 ....... ..................... 65
vi
I II
35. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v = 6 convolutionalcode and system diversity L = 2 ....... ..................... 76
36. Probability of bit error conditioned on the Dopplet effect coefficientin a Ricean fading channel•(r = 1) and maximujm Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, L = 6 convolutionalcode and system diversity L = 2 ....... ..................... 77
37. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 10) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v -- 6 convolutionalcode and system diversity L = 2 ....... ..................... 78
38. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect, i.e.,minima! Doppler effect coefficient for a rate 2/3, v = 6 convolutionalcode and system diversity L = 3 ........................... 79
39. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 1) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v = 6 convolutionalcode and system diversity L = 3 ....... ..................... 80
40. Probability of bit error conditioned on the Doppler effect cocficientin a Ricean fading channel (r = 10) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v = 6 convolutionalcode and system diversity L = 3 ....... ..................... 81
41. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v = 6 convolutionalcode and system diversity L = 4 ....... ..................... 82
42. Probability of bit error conditioned on the Doppler effect coefficientin a R.icean fading channel (r = 10) and 7 ax.m,un noppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, u = 6 convolutionalcode and system diversity L = 4 ....... ..................... 83
viii
25. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect. i.e.,minimal Doppler effect coefficient for a rate 2/3, v -z 2 convolutionalcode and system diversity L = 3 ...... ..................... 66
26. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 10) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 2/3, v, = 2 convolutionalcode and system diversity L = 3 ...... ..................... 67
27. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 2 ....... ..................... 68
28. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 1) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 2 ...... ..................... 69
"29. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 10) and maximum Doppler effect, i.e..minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 2 .......................... 70
30. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 3 ...... ..................... 71
31. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading channel (r = 1) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 3 ......................... 7
32. Probaoility of bit error conditioned on the Doppler effect coefficient.bk a . f•d, .. e channeltr = J - IAl-, ± L li .e . t : ie
minimal Doppler effect coefficient for a rate 1/2, v = U convolutionalcode and system diversity L = 3 .......................... 73
33. Probability of bit error conditioned on the Doppler effect coefficient ina Rayleigh fading channel (r = 0) and maximum Doppler effect, I.e.,minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 4 ......................... 74
34. Probability of bit error conditioned on the Doppler effect coefficientin a Ricean fading charnel (r = 10) and maximum Doppler effect, i.e.,minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 4 ......................... 75
vii
ACKNOWLEDGMENT
The author would like to express his appreciation to Professor Tri Ila at the
Naval Postgraduate School for the knowledge transmitted in various courses. Appre-
ciation is also due to Professor Ralph Hippenstiel for his comments on the present
work.
Gratitude is also due to LT Janet Stevens, his friend and partner in several
research projects, for reviewing the material and text corrections.
To Jim Allen for his editing and typesetting work.
To Bel, his lovely wife, for the continuous support and understanding with his
career.
ix
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"xi
I. INTRODUCTION
The term "mobile communication" is used to describe the radio communica-
tion link between two terminals of which one or both are in motion.
In urban areas, we observe the growth of cellular terrestrial mobile corn-
munication systems. Such systems are also employed where base stations can be
located relatively close to mobile users. However, in rural or remote areas, with low
population density, cellular systems are not economically feasible.
Satellite technology has reached the ability to bridge the gap of communica-
tions between mobile elements, complementing the existing cellular systems. Mobile
satellite systems are not restricted to land coverage. They include aeronautical and
maritime services as well.
Mobile land communication is greatly affected not only by the losses encoun-
tered in atmospheric propagation, but also by the general topography of uiw terrain.
The texture and roughness of the terrain tend to dissipate propagated energy, reduc-
ing the received signal strength at both mobile and fixed units. Shadowing caused
by trees and other natural or man made obstacles also affects the strength of tile
received signal.
In addition, there is multipath fading, which is caused by the reflecting of
various types of signal scatterers. The effects of multipath phenomena are more
significant in terrestrial communications than in air-to-base station or satellite-to-
earth station communications.
In this thesis, we assume that there are multiple propagation paths in the
model of the mobile link. A propagation delay and an attenuation factor are asso-
ciated with each path.
Tihe fading phenomenon is basically a result of time variations in the phase of
signals in each path. Sometimes, the components from tile paths add constructively
so that the received signal is large. Other times, they add destructively, resulting in
a very small or practically zero signal. Thus, `ading, the amplitude variation in the
received signal, is due to the time-variant multipath characteristic of the channel
[1].
"lo generalize the model for the practical cases of mobile communications, we
also consider the presence of a direct component of the signal. We define a direct-
to-diffuse ratio r = a 2!/2a2 usiirg the channel parameters. Physical interpretations
can be given to those parameters: a2 is associated with the strength of the direct
component and 29.2 is associated with the diffuse component.
Therefore, we consider ;t Ricean fading channel which is general enough to
allow us to solve the following types of problems:
a) Non-fading channel,
b) Rayleigh fading channel,
c) Ricean fading channel.
Another major concern in mobile communications is the Doppler effect. It is
well known that this effect occurs due to the relative speed between the elements
in the communication system. The effect of the Doppler is directly proportional to
the magnitude of the relative speed. Therefore, this effect can only be considered
significant when high speeds apply, i.e., in airp!anes.
The Doppler effect is modeled here as a contribution to the carrier frequency.
This contribution will be either positive or negative, according to the direction of
the relative movement between the comnunications elements.
2
Both fading and Doppler effects can impair the reception of the transmitted
signal. The error rate, as shown in [1], is only inversely proportional to the SNR in
a Rayleigh fading channel, in contrast with the exponential decrease in AWGN.
Diversity, as stated by Proakis [1], is an effective way of improving error rate
performance in fading channels. By supplyirg to the receiver s, reral replicas of the
same information bit over independently fading channels, the probability that the
signals will fade at the same time is considerably reduced.
We only use time diversity in the form of signal repetition or in an equivalent
form of sampling the same bit several times at a rate greater than thy bit rate.
To further improve the reception of a Doppler affected signal, we no longer use
matched filters, but use bandpass filters instead. We will verify that time diversity,
can be used to combat the fading characteristics of the channel and the Doppler
effect.
Finally, we consider the use of a convolutional code as a way of providing error
corrections. A convolutional code is generated by passing the information sequence
through a linear finite-state shift register. The input data is shifted into the shift
registers f bits at a time. As an output, we obtain n bits for each sat of f input bits.
Thus, the code rate is defined as R, = i/n.
Observing the tree that is generated by the convolutional encoder, we notice,
that the structure repeats itself after the 1 7 h stage, where v is the code constraint
length.
The distance properties and the crror rate performance of a convolutional
code can be obtained from its state diagram. The same diagram is used to compute
the code transfer function. From there, we obtain the minimum distance, d1 , arid
bounds for the probability of error rate.
3
Proakis has already shown that a convolutional code can be viewed as a type
of repetition scheme [1]. Thus, it is equivalent to time diversity. We will apply this
characteristic of the convolutional code over Doppler effect.
Due to the strong fading nature of the channels we want to analyze, we have
selected a non-col erent signaling scheme in which it is not required to estimate
the phase of the received signal. Nor will we be estimating the channel parame-
ters. Under these assumptions, our system can be simplified and has very robust
characteristics.
In our analysis, we consider a constant Doppler, i.e., we develop a bit error
probability for Riccan fading channels conditioned on the Doppler effect coefficiert.
In our approach, we obtain closed form solutions that may bring some insight and
avoid long lasting simulations.
The binary frequency shift keying (2-FSK) schemes proposed is well-suited
for further development of the initial base for future M-ary frequency shift keying
(M--FSK) base extension or to obtain M-FSK error performarn.ce bounds.
In Figure 1 we present the basic system on which we conduct our analysis.
It shows two energy detectors followed by a possible (non-coherent) average. The
two upper branches are responsible for the detection of the first signal of the binary
scheme and the two lower branches take care of the scn T.i.nl I our analysis
we will assume that the signal corresponding uo the frequency fo is sent. Therefore
the two upper branches will have the signal present and the two lower branches will
only have noise.
In Chapter II, we develop the probability of bit error for a square law detector,
2-FSK scheme, with diversity L, under Rayleigh and Ricean fading, and without
Doppler shifts. In Chapter III, the Doppler effect is added to the previous result in
the form of a Doppler effect coefficient.
4
N
In Chapter IV, we extend the already develojed conditional probability by
accounting for the convolutional error correcting code.
Finally, in Chapter V, we present verifications of the equations with some
numerical results, including some special cases, and compare them with results in
the literature. We evaluate the influence of the diversity size over the performance of
error rate. Also, we verify the performance of R, = 1/2 and R, = 2/3 convolutional
codes under the same conditions. In addition, we confirm the influence of the code
constraint length on the performance.
5
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6
II. DIVERSITY AND FADING CHANNELS
in this chapter we derive the error rate performance of binary FSK over a
frequency-nonselective, slowly fading Ricean channel. We assume that each diversity
signal fades independently. The signal can be considered as the sum of two com-
ponents, a nonfaded (direct) component and a Rayleigh-faded (diffuse) component,
hence the amplitude of the signal is a Ricean random variable.
We will analyze the bit error probability of the proposed system in Figure 1.
The Doppler effect will not be taken into consideration in this chapter but it will be
considered in Chapter 3.
For mathematical convenience we adopt the complex envelope notation for a
real signal. Thcrcforc, the surmmner outputs are (under signal fo present condit7 ion):
Yo IXk + nfk + jfklj , (2.1a)k=l
L
Y, = L Inlak + jnfkl2 , (2.1b)k=1
where Xk, assuming a Doppler shift of Wd, is given by
I
Xk = Ak k- Wdtdt (2.2),/2 I(k-1)T
and all the quadrature noise samples ncik, n,ik are independent and identically dis-
tributed (i.i.d.) zero-mean Gaussian random variables (RV) with variance No7'/2.
Each amplitude Ak of the signal is assumed to fade independently for each bit du-
ration T.
Since YK is the sum of squares of 2L i.i.d. Gaussian RVs with zero-mean and
variance aI = N0T/2, YK is chi-square distributed with the following probability
7
density function (pdf) [1]
1 L-I -l2~=(L y)!2Lo •L•Y• yi > 0 (2.3)
On the other hand, Yo is the sum of squares of L independert Gaussian RVs
with variance u 2 and mean RTe [Xk], and L independent Gaussian RVs with variance
kC but with mean Im [Xk]. Thus Y' is non-central chi-square distributed with the
following conditional pdf [1]
1-oyo\O/L))/2 v ( Y , Yo 0 0, (2.4)
J)'(oXl'OJ = -2c -V _I( / .ý 24
where IL-.(') is the (L - 1)th order modified Bessel function of the first kind and v
is the value assumed by the random variable V defined as
Lv E j• Xk12 (2.5)k=1
The conditional probability of error is given by
"Pb(V) = P,- I yoIVI = j 00 fy 1(yi/v)fy.(yojv)dyidyo . (2.6)
The inner integral can be evaluated as follows (see Appendix A)
fy 1 (yi)dyi -- °/12a' L I --- " (2.7)"- yu k=o t(9,72) I
Substituting (2.4) and (2.7) into (2.6) we obtain (see Appendix C)
S1 m (2.8)
where the random variable Z is defined as:
, - T2 L L
. E A =E A2 (2.9)k=lk k=1
where f# = TV/4a k.
8
Equation (2.8) has been derived in [2] for the case of fixed amplitude A.k =7 A.
For such cases equation (2.9) reduces to
Z LA2 -J(L L(A =L - (2.10)4k 422
0
where E/No is the signal energy-to-noise density ratio and E =z A2T/'2.
For the Ricean channel the signal amplitude Ik is Ricean-distributed with
fj a ' ,) - " -ka k -(a 2 + ,2k)/ 2 a 2 (a k -- 0fA a 2 / , a$2 1
aOk
where aC is the average power of the nonfaded (direct) component of the signal arid
2krU is the average power of the Rayleigh-faded (diffuse) component of the signal.
The total average received power in the interval (k - 1)7' < 1, • kT is
E (a+ 2ak) (2.12)2 2
and it is assumed to be constant for any integer k, I _< k S L.
Note that if k = 0, the channel is a, Rayleigh fading model and if 2o-- 0
there is no fading.
The random variable W47 defined below
Lw =ZEA] (2.1:3)k--
k=i
has a noncentral chi-square pdf given by [1]
I ( W (L- 1)/2 ;)22L (PI
fw(w/p) = - e-((+)/I) ) w > 0, (2.14)
whereL
(k(.15)k=1
9
Consequently, the pdf of the random variable Z = IW ihi (2.9) is
fz(z) =) 2P)/2"•'IL_, , z > 0 (2.16)
Taking the expected value of Pb(z) in (2.8) (see Appendix D) by using (2.16)
we obtain
~ e-(p/2c)[l-1/(l+•) 1 1 k k + L -1\' 3o'2Pb()= 2L (Il+ Or2)L r2,+L--1 1/oC72k (k=O m=0 +1
, e+ L - 1 1( + -1 (2.17)4-d Z + L - 1 2a 2(1 + /o ) i!
i=-0 k,
Let
2T +' 2r (2.18)
be the average signal energy with the assumption that ak and 2 A, are constant for
any 1 < k < L (hereafter we drop the 3 argument of Pb(f) to simplify the notation).
Also let the ratio r ot the direct, component power to the diffuse componient power
during a bit time be deined as follows:
2r = 2"(2.19)r24
Combining (2.10), (2.18), and (2.19) we obtain
P"k 2(r + 1)
k.Lu)
Substituting (2.15), (2.19), and (2.20) into (2.17) we get the average bit, error
probability (zero Doppler) as follows:
SC- eLrf[-(1+(E/No)/'2(r+I))j ' L-1- k /ko + L - 1
12(r+[1)J
( + L m+ -1 ( Lr 2 .2 1 )2(•r + 1) x i + L-1 )x ElNo .
1 + E/No 8=0 2(r+1)
10
As a check we observe that when L = 1 and r = oo (no fading condition)
equation (2.21) reduces to [1]
P 1 e-E/lN/2 (2,22)2
For a non-fading channel with diversity L, equation (2.21) can be reduced to
the next equation. Even though a similar equation is found in [2] we also developed
it in Appendix C. By replacing the value of Z given by (2.10) leads to:
C-(/2(r/o)L-1 1k
BN M
Pb = _l 2! E = k - 2- (2.23)
For a Rayleigh fading channel (r = 0) and assuming L = 1 we get the result
of [1]:1
Pb --- ,1;r (2.24)2 + •jv
For a Ricean fading channel and assuming L = 1 we obtain the result
r + 1_ (2.25)
2(r + 1) + E/No
Also for a Rayleigh fading channel and diversity L, equation (2.21) reduces
to the result obtained in [1, 4, 5]:
Pb ( )LL- Z ( k L -1 B FN 0 7
P=2-E/No k=O m= m+L -i) 2 + E/No) (2.26)
11
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12
III. THE PROBABILITY OF BIT ERROR
CONDITIONED ON THE DOPPLER EFFECTIN A RICEAN FADING CHANNEL
In a real mobile communication system, the signal may be distorted by
Doppler, which can seriously increase the bit error probability. In this chapter,
we investigate the system performance when Doppler is present.
The received signal affected by the Doppler in a fading channel with diversity,
at time k, can be represented as follows:
rk(t) -- Ak cos(27rft + 2lrfdt + 0,) , i = 0, 1 (3.1)
where fd is the Doppler frequency and 0, is the signal phase.
Equation (3.1) can be manipulated as follows:
rk(t) = Ak cos(27r[fi + fd]t + 0) , i =0,1. (3.2)
The orthogonality condition for 2-FSK (no Doppler shift) requires:
1Af = f, - f = -I (minimum frequency spacing) (3.3)
T
and1.
fA ± Jo = , k is a positive integer. (3.4)
In practice we may use an approximation to orthogonality condition by im-
posing that
f± + f»o > (3.5)
To study the Doppler effect, we assume a wideband 2-FSK, that is, Af >
1/7'. This ensures that the orthogonality condition is almost satisfied, and we can
analyze the performance as if we have an ideal orthogonal 2-FSK.
13
Returning now to the complex envelope notation, we see that equations (2.1)
and (2.2) remain the same with Xk given below:
Ak=ýk •kT
Xk=- i(k- )T ecldtdt (3.6)
Equation (2.5) will give us (see Appendix E):
Lv = ZIx~I
T2 sin(7rfdT) 2 L (3.)
2 (,(L rfdT) A , (3.7)
The definition of equation (2.9) remains the same but now the value of 3 is
given by:
-x ([ T (T4o2) sinc 2(fdt) , (3.8)
where sinc (x) = sin(7rx)
7rX
Thus equation (2.10) is transformed to:
Z = L ( ) sinc(fdT) . (3.9)
Equation (2.17) is still valid and if we use the value of Ocr' given below:
3c N00 sinc(fdT) . (3.10)2(r + 1)
14
We get the average bit error probability conditioned on a given I)oppler shift
fd:
1b ~-Lr[1-(1+(1/'No) sinc2(fdT)/2(r+i))-1] L- 1
2 [ + 77 sinc'(fdT) k=oL 2
2(r + 1)
k +L-1 1_, Lo 1+ 2(r
() sinc 2(fd T)\No)
& .+ L- I Lr( i + L - 1 (E3No) sinc(fT) f.11)
i =O + 2 (r + 1 ) .
Again we may now obtain from equation (3.11) the equation corresponding
to (2.23) with the Doppler shift accounted for:
t -(L/2)(P/No) SinC2 (ff) L-L1 k / (E + L (2 + )_- 1(fdT j
k=0r m + L - 1
(3.12)
For a ,aylcigh• fLadin cannel (_ u) and an L of i, we get the following
equation which corresponds to (2.24)
1
2 + (E/ATo) sinc2 (fdT) (3.13)
For a Ricean fading channel and an L of 1, we obtain the following equations
which corresponds to (2.25)
r + 1 e,-r(I-2(r+l)/(2(r+l)+(E/ND) sinc2 (fdT))) . ( 1
2(r + 1) + (-,/No) sinc2 (fdT)
15
For a Rayleigh fading channel and diversity L we get the following equation
which corresponds to (2.26)
{____L __ ___L-1 1 k k+L -] I (S/N0 ) sinc2 (fdt)Pb = (2 + (B/N0 ) sinc2(fdT) k- • ' + L- ( 2 + (//No) sinc2(fdT))
(3.15)
Again, as a fast check we observe that for L = 1, r = oo and ftT = 0 (no
Doppler effect) in equation (3.11), we obtain (2.22).
For a Rayleigh fading channel (r = 0), L = 1, and fdT = 0 we obtain (2.23).
For a Ricean fading channel, L = 1 and fdT = 0, we obtain (2.25).
Also from (3.11) we observe that the Doppler effect reduces the signal-to-
noise ratio (SNR) by a factor equal to sinc2 (fdT). SNR is defined to be /iNo.
For example, when the Doppler frequency fd is an integer multiple of 1/7', then
sinc2 ( fQT) = 0.
Therefore, when the Doppler frequency is large the receiver should sample
the signal faster than the bit time T to reduce the Doppler perturbatior. Such an
adaptive change in the sampling rate is equivalent to changing the diversity number
of the system. The receiver would modify the system resultant diversity.
We can also combine the diversity that we obtain by sending L replicas of
the signal with the diversity we obtain by sampling at a rate faster than the bit
rate, In other words, letting T' be the sampling interval (where T' = TIN and N
is a positive integer) and replacing the receiver of Figure 1 with a receiver whose
integration time (and sampling time) is T' and whose diversity is NL.
16
Letting E' be EIN, using equation (3.11) with L, B, and T replaced by NL,
E' and T', respectively, we get:
1 e-NLr[11-(-((,C/NNo) ainc2(fjT/N))/2(r+1))-1] NL-11
Ph 9NL E1+ (+- NL T ),Si[C k=O 2
+ No2(r- + 1)
S)
k+NL-1 1nm+ NL-1 1i+ 2(r+1)M 0E s i nc f d T )
~NNo N
S ._(( ) (3.16)sinc f
1+ , ,,/
2(r + 1)
again the same interpretation as before, we obtain the counterpart to
equation (3.12) for a non-fading channel with Doppler perturbation:
e-(L/2)(E/No) sinc2 (fdT/N) NL.-i 1 k NL-1 [ o sine2 f n
= 2 N L = i !
(3.17)
In that case we have:
Et 2E T .2 (dT\Z=NL ij sinc fd L 1 (3.18)
I(NNo) \No C JN . (.8
We can see in equation (3.18) the influence of increasing the sampling rate
(i.e., increasing N). Sampling at rate faster than the bit rate reduces the argument
of the sinc function. Therefore, the value of the sinc function is increased.
17
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18
IV. CODED PERFORMANCE
In this chapter the performance of the L-fold diversity receiver with error
correction code is investigated. Specifically we consider convoiutional codes with
Viterbi soft decision decoding.
For a rate f/n code the bit error probability Pb is upper bounded by the
following union bound [1, 6, 7]:
1 00Pb < - adPd (4.1)
where df is the fine distance of the code, ad is the information weight of a code path
of weight d, and Pd is the probability the all-zero path is eliminated by a path of
"weight d .. erging with it on th" " 0d o t""
We observe that Pd is exactly the probability that the sum of d samples of Y¾
is greater than the sum of d samples of Yo in Figure 1, that is, Pd is the probability
of error for noncoherent combining of d transmissions where each transrmssion has
L-fold diversity.
Thus the expansion for Pd is exactly the same as Pd in (3.11) with dL replacing
L, (C/n)EP/No replacing E/No and fT/n replacing T, that is
19
Pd ý dL r[-(±en(/o sic(taL 4 )2(+)
f ( in) ý-~(fdT)]2(r +1)
dL-1 1~~ k kdL-1 )N IxZEF 21: m+dL- )I. 2(r +1)
k=o [T= + If* E- sjnc 2 (• fdT)
r( m+L- dLr1Xti=\ + dL - 1 (~) Sin 2 ($ dT) z 42
LI± 2(r +1) j
20
V. ANALYSIS OF RESULTS
In this chapter, we analyze the performance (probability of bit error, Pb) for
the 2-FSK, square law detector system of Figure 1. We assume that the signal is
transmitted L times. We recall that by sampling faster than the bit rate we may
produce an equivalent diversity to sending the signal L times.
In Figure 2, the results for an additive white Gaussian noise (AWGN) channel
are presented. The curve for a 2-FSK signal, L = 1 and no Doppler perturbation
is shown as a reference. This plot provides us with a numerical check of the results
presevted in Proakis [1].
'Varying the diversity size has proven to be an efficient way to improve the
performance in the presence of Doppler. The curves we have obtained by setting
L = I to L = 10, show worse performance than the mentioned 2-FSK curve.
Sending the signal twice (i.e., L = 2) provides a Pb = 0.10 at an SNR of 10.
We can see an improvement when comparing this value to Pb = 0.50 which is the
bit error probability for an L of 1 (i.e., no diversity).
Note that a larger L may give poorer performance at, low SNR than a smaller
L would pcs,v'de. On the other hand, at high SNR, increasing L gives better perfor-
man,,:e Hence, there are crossovers among the various diversity rates in an AWGN
,dniei, as illustrated in Figure 2, for L = 3 and L = 4.
If. addUfion, we notice that the best relative improvement occurs when we
increase diversity from L = 1 to L = 2. The next best improvement is obtained by
increasir,,.g L f-om 2 to 3, representing a 2 dB gain at large SNR.
At large SNR, we can see that the improvement is less than 0.50 dB when
going fromn L = 4 to L = 10 in a non-fading channel.
21
1. Figure 3, the bit error probability for a Rayleigh fading channel (r = 0)
is prest.uted. Again we plot the 2-FSK, no Doppler perturbation curve, in order to
provide a numerical check with the results of [1].
Now, we can see that increasing diversity will result in even better perfor-
mance than the mentioned 2-FSK reference curve (L = 1 and no Doppler effect).
Again, the best improvement occurs when we change from L = 1 to L = 2.
The next best improvement is obtained by changing from L = 2 to L = 3 (i.e.,
aboat 4 dB at a large SNR).
Moving from L = 3 to L = 4 and from L = 4 to L = 10 gives us approximately
a gain of 2 dB at a large SNR.
In Figures 4 through 11, the probability of bit error as a function of diversity
size is presented. We show the results for two cases: zero Doppler and fdT = 0.5.
Also, wpresent four plots in cach figurc: - 3, 10, 30, 100.1
The same kind of behavior that we see in a Rayleigh fading channel [1] occurs
171 a icean fading channel, i.e., for low SNR there is a point of minimum Pb.
Besides, we notice that the Doppler perturbation reduces the value of L where
the minimum. appears. This is consistent with the fact that the Doppler coefficient
is a factor 'that reduces the SNR.
Thcrefore, increasing the diversity at low SNR may result in a reduction in
performance in a Rayleigh or Ricean fading channel.
By sampling the received signal at a rate faster than the bit rate we may
reduce the influence of the Doppler perturbation. We note that increasing the
diversity order results into an increased value of L where the minimum Pb is located.
'Those values are in linear units and allow easy comparison with [1, 2, 7, 9].
22
The behavior of the curves in a Ricean fading channel tends to resemble the
behavior of a non-fading channel as we increase the r value. This effect is presented
in Figures 12 through 15.
Figures 15 and 16 allow us a quick numerical verification of equation (4.2)with the results of [1] and [7]. WVe have used fdT = 0, which means the Doppler
coefficient (i.e., sinc part of equation (3.16)) equals to 1. By using r = 10,000 we
approach the AWGN channel. Therefore, the uncoded signal represents a 2-FSK in
an AWGN channel. Also, both figures present the comparison of the performance of
the system without convolutional code ('uncoded signal') and the rate 1/2, j = 2,
d= = 5 convolutional code. The systems have the same L. For the initial verification
we used L = 1, and no Doppler effect. Figure 16 provides a comparison for a
Rayleigh fading channel. By inspecting the values of Pb from Figures 15 and 16
"We notice thlat d UiVeiy d Ioe not improve performance in AWGN channel, thus
confirming Proakis' observations [1]. The results obtained for both the coded and
uncoded signal agree with the one presented by [7].
At large SNR we obtain a gain of about 2.5 dB over the uncoded signal in an
AWGN channel. 2 We recall Doppler has not been considered yet.
Thereafter, we consider the maximum reduction, due to Doppler (i.e., Doppler
coefficient equals zero). In our approach, we identify an effective and efficient way
of combating Doppler and fading. We compared sending L copies of the same
information bit (or sampling the received signal at a rate faster than the bit rate)
or the use of error correcting code (we also tried some combinations of diversity arid
coding).
We propose to use the information about the minimum 14 (see Figures 4-11)
to select candidate convolutional codes so as not to mae the systern diversity large.
2The gain is a bound due to equation (4.1).
23
The comparisons that follow are between a coded system and an uncoded
system. Of course, both systems have the same energy, SNR, and diversity L. We
want to verify how much improvement we may obtain by using a code over the
existent system that contains diversity only.
We evaluate two types of convolutional code: rate 1/2, v = 2, and rate 2/3,
-=2.
As we have seen before, the minimum values for Pb occur for diversity value
between 5 and 20. Choosing the rate 1/2, v = 2, convolutional code, we obtain
df = 5 [7]. We use L = 2. Therefore, observing equation (4.2) wve verify a reduction
to 1/4 of the original Doppler frequency in this case.
Figure 17 presents the results for zero Doppler. This resul, agrees with the
one presented in [7].
Figure 1i presents the details mentioned above in a Rayleigh fading chan-
nel (r = 0). At low SNR we see a possible loss. Since the code rate is 1/2 we
have expanded the system bandwidth two times over the bandwidth of the uncoded
system.
Figure 19 shows that for a Ricean fading channel (r = 10) the bound on the
gain is very much reduced in comparison with the one obtained in the Rayleigh case.
There is still a chance of loss at row SNR. Figures 19 and 21 show that an increase
in diversity size improves the performance of the uncoded system more than the one
of the coded system in a Ricean fading channel with r = 10 (which is getting close
to a practical AWGN channel). For Rayleigh fading, the improvement is about the
same as for Ricean fading.
Figure 22 provides a numerical check with the literature [1, 7] for the rate
2/3, v =: 2, convolutional code. The same type of performance improvement using
diversity is also observed in this code (Figures 23 to 26).
24
We analyze two other convolutional codes that are more suitable for practical
use (due to the fact that they have larger value for df): rate 1/2, v = 6 with d- 10
and rate 2/3, v = 6 with df = 6.
We consider a Rayleigh fading (r = 0) and Ricean fading channels for some
values of L. The results are presented in Figures 27 to 42.
The gains we obtain here are much greater than the ones obtained before.
We point out that such selected codes have greater free distance than the previous
ones.
The coding gains are greater in a Rayleigh fading channel than in a Riceari
fading channel, confirming what we have seen in the other codes. P6 can be reduced
by increasing the system diversity L and the direct-to-diffuse ratio r, but the relative
gain is reduced as r increases, i.e., as the channel tends to an AWGN channel.
The rate 1/2 codes presented here perform better than the rate 2/3 codes,
although the gain per bandwidth expansion ratio in the rate 2/3 code is greater
than the rate 1/2 code.
Finally, we notice that there are crossovers among the various uncoded and
coded schemes. This is illustrated in Figures 15 through 41.
25
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26
VI. CONCLUSION
A closed form expression for the probability of bit error was independently
derived for 2-FSK, square law detector, conditioned on Doppler in a Ricean fading
channel. We have shown how to obtain the Rayleigh fading channel and AWGN
cases (with Doppler) from that expression. The equations agree with the ones
presented in the literature for the zero Doppler cases.
The equation can also be used in a Jensen's inequality [9] to obtain a lower
bound if the Doppler frequency is assumed to be a random variable. In this situation,
it may also be used to estimate Pb by some statistical simulation method.
The probability density function that was developed can be used to get an
expression for a P6 in a M-FSK scheme. We can directly apply the equation for Pb
for the binary case in a bound for the M-ary case [1].
In addition. we have demonstrated the use of convolutional codes to address
both fading and Doppler effects. We have verified that a bandwidth expansion, due
to coding rate, is the price that we pay for obtaining the SNR gains.
Finally, we suggest the use of non-linear schemes, similar to Trellis Coded
Modulation, to keep the SNR gAin and provide a savings in terms of bandwidth
expansion.
In addition, it seems a natural extension of the present research is to study
the application of a dual-k convolutional code [1, 7] coupled with a noncoherent
detection M-FSK scheme.
27
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28
APPENDIX AEVALUATION OF THE INNER INTEGRAL OF
EQUATION 2.6
Evaluation of the inner integral of Equation (2.6):
00 ________ L-1 -y1/eJ fy1 (y,)dy, = J -!2Lo2Ly- C dyl
Yi (L - ' l)2~ L-1 I2 2
( L) L/2 - / y, (2 2 - (/2)') 4- (A.(1)'
Yo 20-2(L - 1)! C )di L (L- 1)! 20-.2)
Changing variables in (A.1):
Y-2-° (A.2)
yoXo = 2o.2
1 (dy 1 AA)dx = .-- dy, = i-- 2A.
S.. .. L-I I- I,- A.\ "1 o
2a2o
--Jx,- (L - 1)! e-d -•1 Tk•__° ! :oL '
li - -(-~ (A.5)-1 kOk!kOk k=O k!
Substituting (A.3) in (A.5):(Yo)!
= y ,o,2, 2 k (A.6)fIoo k=O
29
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30
APPENDIX B
BESSEL FUNCTION PROPERTIES
00 xM+L'/ 2 gf.JL (2IIV'x) dx (B. )
Jo
-m!ove- 2/Y 1~(~ [m + v > -1] (B. 2)
whiert-~
L~(b) = E )f(-] 1*VI (B.3)
],(X) =j-'J,(j~x) (B. )
().5
J~, (23V,-x) = J, (j2a,4xI) = -.1-h, (2cex/-) (13.6)
Substituting (B.6) in (B.1)
Jo Xmnt/ 2 e--YW 1-, (2av/) dx = j n+P/2 c 'Cv I" (2a,/x) dx
Ijcxm+v/2e-rxv(X o' dx - " rnjaeO 2/>ytnvl1L' ( -,2) (13.7)
Titus:
Lx 7nv2+ "e-J- (2- v dx -~'"%y--L -a ( 2i) (B3.8)
31
wherem (n \1- 2 V
n0O
Substituting (B.9) in (13.8)
x (~/ -YI,2a\/T) dx = 7n!avc"I/my n T+ v
32
APPENDIX C
DEVELOPMENT OF Pb EQUATION
Substituting Equation (A.4) and Equation (A.6) of Appendix A in Equation (2.6)
of Chapter 2:
'Pb = j0 e- Yo/20- 1 (~)k- { 0~ () (L2/ a22 dL
(C.1)
Letyo 1
x = o- dx -•a-dyo.2u2 2o 2
,pb _x e-k X
k=Qoc-• k! (L-2,)/2
Cz. L-1= 1 x((L-1)/2+k)-2vll 2 dx. (C.2)Z(1,71)/2 E_ k' Jo L1 2z)d
Using E•uation.. (A.l0) of Append X A, let
v L-1,
n k
y= 2,
33
e -z L-1"e z(L_) 2 Z {k!z(L-1)/2ez/22-k-L
k (k+ L- 1 7 i , Z "
X ()o kE- -I)")" (C.3)
_____ L-1Ik k+L-1\(z/2)rn(C4P -2 L Ek 2- E k (C.4)
-z/ 1- 1 k+ 1N(/Tb m+L-1 m2 (C.5)
Equations (C.4) and (C.5) agree with the results presented in [2] and [7].
34
APPENDIX D
AVERAGE BIT ERROR PROBABILITY FOR ANON-COHERENT 2-FSK IN A RICEAN FADING
CHANNEL
Pb = JP(z,(,)fz(z)dz = 'Pb(z,L)fz(z)dz (D.1)
.- • )- 2 L k + L -1 m 2) (L-)/2
J(~ zm(L)/2( +1)/2Je4P/-3o 'L1('h)d
k L-1 ]z&( L-I \-
Pb0 2Yr = + 0- 2 n(2a,)1pp(-)2'
"- Io L~ 1 e T . 1 J "<t"nn \(tP ) '-, " x F(z) (D.2)
2 k=U f0 2 -a ~ L m! 2#~,LU tjp
F(z) j" i zm4-I(L-1)/2e-(l/2+l/2B')zIL- dz (P3
2-• d z D
/
Using Equation (B.8) of Appendix B and making the following changc of
variables:
= L-1
i 1"+ -- - I- . 222•o•
2L k=0 k n= k 25
V157
we get:
F((z) = !L[V(] L-Pe-,f/o•[i+I :• 1 1 L , /. 0-2J 122-[1 + 0,• ](DA.1)
Further applying (B.10) of Appendix B to (D.4):
1 1 -m-Lm / \ 1)1V i
(z) = m"( 2 / 3 2)L-1 2ec/2,&'+\+ 1 .=o ,• k i • 1+rk] <(D.5)
Substituting (D.5) in (D.2):
Pb( 2_)_1 ( 1k+k-1 e-/'12m,• C)+Pb~f2k ?n m+ -1 (2fa)(0lp)L_,/22, .- •
k=O 0 =O )
pm+v G(D.6)
(2pO))L Y0 kL Jm + L-1 2 0-
TL m+L--i P( p
r = n-i Ji !±3c]
n ko ..- + L - 1
2mi=0 ' 2O
- k + O-1 lO2],,,m '-' Lk-- ~ +--P2'[+l~''- a L I [I + rj- L( k
2L 72~kO m -- L2O(ý-)
k )( k
Se + L - 3 I pX,= -1 -i ,!. (2 .2[1 + 00-2'T
36
iL-i +
-1 )-- (-(+-L- 1)! = (iE ( + L -) -±L-i
(D.8)
Substituting (D.8) in (D.7):
P•f)=2L (+ •)L k= / - ,, =o + L + 1 01+,2 •
2u'[1 + po.2
x [1++ L-1 k (k (D.7
i=O
37
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38
APPENDIX E
DOPPLER EFFECT COEFFICIENT
Doppler effect coefficient:'Ak kT jItd
-Xk (- fk-1)T elwdtdt (E.)
L 12 L 2 kT e~dd 2
V =E Xk =E A (E.2)k=1 k=l k-1)T
kT Cwk 2 - e Jw(k-1)T
(klTCjdWtdt =H(-j) (for Wd 0 )Jk-1)T Wd
jwd(kl)T[e - 1] (E.3)
1kT 2 T 2 ejweT -1
J(k-1)T Wd dI waT
cos wdTJsin wdT - 1 2 7 )2 (cos wdT - 1) + jsin wdT 2
I WdT I IWd T
12COS 2WdT + sin 2WdT± 1 - 2cos WdT T 72 2 - 2cos WdT
(wdT) 2 I 2IT(wdT) 2
2si.2 --5- 2 sin 2 W Tsin 2 ( s, n (,,----)=2T2 •'wd• f =4T 2 -- ( ) =2-. { T(22-T}2)
(Wd T) 2 (WdT) 2 (WdT) 2 (WdT)
(E-4)
'Note that here Wd = 21rfd.
39
When Wd = 0:rkT 2
)k-)T eodt = T2 (E.5)
Taking the limit of (E.4) when Wd -* 0:
lim T 2 siI T 2 (E.6)
Thus the expression (E.4) is valid also for Wd 0.
Substituting (E.4) in (E.2): T~~ ~ 22i dV 2 (f) (E.7)
2 k=1
40
APPENDIX F
FIGURES
41
____---________________ I--_________ ________-____
(1) U)
o •o oo0 .0)
-44
o
C~i C' C~J C
C)
Figure 1: An L-fold diversity 2-FSK receiver.
42
00
* ~ 00
• . -.• - -r-
II- ' o +.÷..•.•• " r-
,- NJ•°°-
+•+
,I)
SI I I l I I I . - •;
qd
Figure 2: Probability of bit error conditioned on the Doppler effectcoefficient in an AWGN channel as a function of the system diversity.
43
4,
+
-fn
.+ II
+ -/ •I -
"4- , 1)
e'.//
Figur 3:. Poaityf ii error codtoe nteDplrefc
/"
. . - -VI'
coefficient in a Rayleigh fading channel (r == 0) as a function of the systemdiversity,
44
TF T-r--T - TT-,T-- 1 TF•1 I ITf-l -F-!TT TrT1I I -!TTI I I T 11 1 1 I .Ti I I I I
li I/ II
' I0 I.
CDz',i
(
"Fop e
.. ... .... . ...... . I-.
• ..//0
coefficient (fdT =0) in a Rayleigh fading channel (r = 0) as a function ofthe system diversity.
45
iiiI 1 m I r 1 ll1l I 1 urTn I I mmII m•- rr- rl'--T} ri-] ---- - •:
I II II IlC/) (J1
r°A
42-
0 C
4d
Figure 5: Probability of bit error conditioned on the Doppler effectcofCefficientth yte(fdTdie iy= 0.5) in a Rayleigh fading channel (r = 0) as a function
of te sytem iverity
460
J_
I I I I I I nII I I- II- I I- Ii a l I I lI • , II I l l I I I I I , , • t
..-I
II II II I
z z z
C) ] C=) CD~
II-
- -j
r . -I
qd
Figure 6: Probability of bit error conditioned on the Doppler effectcoefficient (fdT = 0) in a Ricean fading channel (r = 1) as a function ofthe system diversity.
47
I l l t l I I it T Tl l l T1-t M T T I I I m i n l l t i i " t i i t i t I I I C )-
Q CDII II .II • Ii -- "
z z
-.1
. -o
L L L L/ I K I A11., I I III I I - o.i 1.JiLJAJo
qd
Figure 7: Probability of bit error conditioned on the Doppler effectcoefficient (fdT = 0.5) in a Ricean fading channel (r = 1) as a function ofthe system diversity.
48
TpTT-n,- ~-i~fTFFnFT IIIIII I rrrvI7T7--F--F ITTý
'-
II II IIv• Ifii f-
z z z'4
"C 0
VI-
qd
Figure 8: Probability of bit error conditioned on thle Doppler effectcoefficient (fdT = 0) in a Ricean fading channel (r = 5) as a function ofthe system diversity.
49
.jJ
H IM I 1 1111111 I I .. . .. .I '.l I-1 111I 1 ll I I l Tl T-" .
* **4II... ..... ..
II II II .I
Z Z r4
qd
Figure 9: Probability of bit error conditioned on the Doppler effectcoefficient (fdT = 0.5) in a Ricean fading channel (r =5) as a function ofthe system diversity.
50
l ll I I 1111111 I I lll n7t I I II#I11 I 11 1 II I I I I il - I
CD C.. .. D
L .. .. . "
Fiue1: Poaityobierrcndtioe onteDplrefc
,, 0i
III
qd
Figure 10: Probability of bit error conditioned on the Doppler effectcoefficient (fdT 0 ) in a Ricean fading channel (r = 10) as a function of
the system diversity.
51
"fI-MlfTF7ffl1TFFTT71UTj-yTm--IT-FT-r-T'
lIE-I-
M ee
II II II I
z z
th ytm divrsty
.5 2. ,.- * . ... . .
-. ,, &. ..
0 0 0
'-4 .4 ,- '-4 ~ -4
qdi
Figue 1: Pobablit ofbit rro coditinedon he Dpplr efeccoeficint (dT 0.) ina Rcea fadng hanel ( 10 asa fuctin othe sstemdivesity
i 52
V /+
+- N
II + •
S- +-4.
+ ,/
.1.
/ /t' b /
+ / , 7
.- "z
qd0
Figre 2: robbilty f bt erorconitine -nteDplrefc
coemcient in aRicean fading hannel (r --- 1) safnto ftesse
III
/fn
-kn
54+
~//
1--
+ 0
,./.7
- i .7C,
/ ..
/
/"
/ ,
,J • " II .
/ / .
* /"
//,
/zJ,.
/•
qd
Figure 13: Probability of bit error conditioned on the Doppler effectcoefficient in a Ricean fading channel (r =-5) as a function of the systemdiversity.
54
++
+
4.
.--
¢N
tn~II 0I- -J z
. / 7
7/.
I" "" 0
coffcin in al Ria fain chne"r:;1)a fnto ftesse
7.-
o o. 0 0 ..
qd
Figure 14: Probability of bit error conditioned on the Doppler effectcoefficient in a Ricean fading channel (r 10) as a function of the system
diversity.
55
IIIII I '
00
-I
00
00
//
1-4 11Sd :IIj )! -I~lq q j
i i a s oDop
, -4 -4 1 I-I I -I I -I
qd :.1o13 P J!• .I/[[!Jqeqo~d
Figure 15: Probability of bit error conditioned on the Doppler effectcoefficient in an almost AWGN channel (r = 10,000) and no Doppler
effect for a rate 1/2, v = 2 convolutional code and system diversity L = 1.
56
I- - c'I
00
CCA
C11 C.D
oo
CC-
0-0
U >
C)D
C:) C) CDC
II- I
*0 II
II II u '' '0
//
r-V
• .L _ IIIJI I __I I _
C-I r~ '- 4 . -~ ' ~
qd :1o1j3 1!S jo Afl!iqeqolj
Figure 16: Probability of bit error conditioned on the Doppler effectcoefficient in an almost AWGN channel (r = 10,000) and no Dopplereffect for a rate 1/2, v = 2 convolutional code and system diversity L = 2.
_57
Ir1 i " I I
C/)
'-4-
II
Vx
II1 u v•II
II II H
qd~~~~~ :.jj I!I-,Mi~uqj
8 58
0 ,()'/ ,/
/7•"
0 -
/..7
i'"7
/J/
- I l [ I I
'- ""4 '-- "- "" ' r"i -" '-4 l
qd :Jo•Ja 1]I JO £1lqtq~
F~igure 17: Probability of bit error conditioned onl the Doppler effectcoe~Jfrient in a IRayleigh fading channel (r -0) and no Doppler effect fora rate 1/2, i = 2 convohitional code and system diversity IL = 2.
Q to
77
0a
z .7
8 /
q-/
F 1
.7..,
//
coefficient in a Rayleigh fading channel (r = 0) and maximum Dopplereffect, i.e., minimal Doppler effect coefficient for a rate 1/2, v =- 2 convo-lutional code and system diversity L ==2.
59
"-In
Czl
U,
4)
0
¢..)
- -F
1" -41.-
600
//U--
SIII I . _i . . . <
,-; £, e, , "+' "'a r- o. --
qd :J2iJq 1!U: jo ,{r~lqqlqo4d
Figure 19: Probability of bit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r =10) and maximum Doppler effect,i.e., minimal Doppler effect coefficient for a rate 1/2, v = 2 convolutionalcode and syrstem diversity L -- 2.
60
1II I I 1II"Ic
C'4-
CO
,
ii
CD I- C_-7
--1
-U U
.q.,d ,-, /ýjoK~~qq~
C
Figure 20: Probability of bit, error conditioned on the Doppler effectcoefficient in a Ra~yleigh fading channel (r =- 0) and maximumn Dopplereffect, i.e., minimal Doppler effect coefficient for a rate 1/2., v -= 2 convo-lutional code and system diversity 1,-= 3.
61-SIx
,, I
~-~~----f------T -------TI I I
II
C:))
--
0
J. I I lI
qd :loldl~ j!q jo 4j!l!q:qo~jd
Figure 21, Probability of bit error conditioned on the Doppler effect co-efficient in a Ricea•n fading channel (r -= 10) and maximum Doppler effect,i.e., minimal Doppler effect coefficient for a rate 1/2, v = 2 convolutionalcode and system diversity L = I.
62
IIIII I I [Tc'
CISl
/
~//
a-)
S /
/
0S//
8///
//
/
0 ' 0)
qd :ioi04 jIg jo ,(j!j!cuqOJd
Figure 22: Probability of bit error conditioned on the Doppler effectcoefficient in a Rayleigh fading channel (r = 0) and no Doppler effect fora rate 2/3, v = 2 convolutional code and system diversity L = 2.
63
//
CNI
C1
CC
U.U
//
.• L
//
//
/, II/
LIj1 1 I L I I I . .
qId :IwiJE lf!-l JO A1
I I I I I I - v'
,- 1 I•I
CA
Um(3, j
di[
o -"
UU
C',ý
d)
o.-
qd :ioj, 1!g jo ,1!1!qegojd
Figure 24: Probability of bit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r = 10) and maximum Doppler effect,i.e., minimal Doppler effect coefficient for a rate 2/3, v = 2 convolutionalcode and system diversity L = 2.
65
L 1 II I
/ 0
C)
/7
"Z"X7
Czz/ IJI
/--
//
7' C4
//
//
, IIlI I I I I I -
qd :dioij )!q jo A)!I!quqoJd
Figure 25: Probability of bit error conditioned on the Doppler effectcoefficient in a Rayleigh fading channel (r = 0) and maximum Dopplereffect, i.e., minimal Doppler effect coefficient for a rate 2/3, v = 2 convo-lutional code and zystem diversity L = 3.
66
I I 1 I I I I I - •'
Cii
ICI
CD b
Fu 2o oIn te
z
effiiguen in: aPronadingt chane (ierr condi)ioned maximum Doppler effecto
i.e., minimal Doppler effect coefficient for a rate 2/3, v = 2 convolutional
code and system diversity L = 3.
67t
I I I I I I I I
C;"
r I
L) I 4 , -
I I I I I II 0 2
P %0 i0
4)4
0
I ~C>
qj :iol•Jg i!• jo A[
I I IT T I I I '
II c
-Pý
Uzo r•
c3 m
II I
r7.
• 1 •J II 1 I Iv
qd :ionIl I!U jo Aqq I I
Figure 28: Probability of bit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r = 1) and maximum Doppler effect,i.e., minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity Li = 2.
69
Cn3
SIiI I I I I I •
II
o M
''0
z
4'))
(U
kr)
CD)
qd :101J3 II!E jo k!l!quqold
Figure 29: Probability of bit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r -- 10) and maximum Doppler effect,i.e., minimal Doppler effect coefficient for a rate 1/2, v = 6 convolutionalcode and system diversity L = 2.
70
I!II 1 I 1 1 I "
II
II "U '-4
o Ii
Qj~CD
qd :1on-l I!•I jo Xl!l!quqcold
Figure 30: Probabil'ity of bit error conditioned on the Doppler effectcoefficient in a Riyleigh fading channel (r = 0) and maximum Dopplereffect, i.e., minimal Doppler effect coeffcient for a rate 1/2, v, = 6 conlvo- •l-ational code and systemn diversity L == 3. •
71~
-~ LIPn
I Ln~
Km
90I
]r-i
l 1ll:- C 1 ; 4 C.)
qj "Jo ij i )!q jo 4j!l!qeqo~d
Figu:,e 31: Probability of bit error conditioned on the Doppler effect co-O-Ticient in a Ricean fading channel (r = 1) and maximum Doppler effect,i.e., minimal Doppler effect coerficient for a rate 1/2, v = 6 convolutionalcod,, ,ad system diversity L = 3.
72
Cfo
III
0 II 11
;>
, T3
(i ,--t1 I-IT
0_
C---
oC
qd :•jL• 12 J)i ,o Xl!quqo~d
Figure 32:: Probability of" bit etror -ýanditioned on the Doppler effect co-effcient in. a Ricean fading channel (r = 10) and maximum Doppler effect,i.e., minimal Doppler effect coefflicient for a rate 1/2, v := 6 canvolutionalcode and system diversity L, = 3.
73
cAJ I
IfI
if"
00UU
0*
~1 V
-4 ._ -4
I¢'
I I J_ 1 . W IIL :
qd :,joJ l!ig ii o ApplqprqoiJ
Figure 33: Probability of bit error conditioned on the Doppler effectcoefficient in a Rayleigh fading channel (r = 0) and maximum Dopplereffect, i.e., minimal Doppler effect coefficient for a rate 1/2, v = 6 convo-lutional code and system diversity L = 4.
74
7U
C)
'-'0
OF C
III
-1 1ý
tlir
qd :jojJa ")!S jo A)
I ~ ~ ~ I i i L I ] I I •
CU%
SIP.
r -4 - 4r
oo(J
"6j-
CU-
II I I I 1 I :
qd:oiJ~ l JO• . Io !!q~qoid
Figure 35: Probability of bit error conditioned on the Doppler effectcoefficient in a Rayleigh fading channel (r 0 ) and maximum Dopplereffect, i.e., minimal Doppler effect coefficient for a rate 2/3, u --6 convo-lutional code and system diversity L = 2.
76 •
pp
1ý Cq
1-41
Fgr 36 rbblt fbterrcniioe on th Dope ef .,cco
l II II I.. I
efficient in a Ricean fading channel (r = 1) and maximum Doppler effect,i.e., minimalI Doppler effect coefficient for a rate 2/3, v =- 6 convolutionalcode and system diversity L = 2.
77 I
I FI I T 1 F "I I I "u
o II U
0 IIo >
OC
Figure 37: Proba.bilit~y of bit erro•r conditioned on the Doppler effect: co--efficient in a Ricean fading channel (r = 10) and maximumn Doppler effect,i.e., mrinima}. Doppler effect coefficient for a rate 2/3, v 6 convolutionalcode and system diversity L 2.
78
"• ,--1 • q I (:I
j....l"i
01 -4 -1-
Figure 38: Probability of bit error conditioned on the Doppler effect •.coefficient in a Rayleigh fading channel (7, = 0) and maximum Doppler -eff-ect: i~e., minimal Doppler effect coefficien.t for a rate 2/3, w• 7= 6 convo-lutior-a] code and system diversity L = 3.
719
11II I I I I ' -
CII)
II,
SU ii- , z
VV
Cnil
CI)
_____
qd:ljwjg I!U jo A4!!qeqo~j
Figure 39: Probability of bit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r = 1) and maximum Doppler effect,
i.e., minimal Doppler effect coefficient for a rate 2/3, v = 6 convolutionalcode and system diversity L = 3.
80
SI I I I I I I
I~i
SII
4 kr)
81~
'a'
oo
U
I I I I I I -- IA .
qd:joi i' J!to i(1il~qLqo.1d
Figure 40: Probability of Dit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r =- 10) and maximum Doppler effect,i.e., nmrinimal Doppler effect coefficient for a rate 2/3, i.' 6 convolutionalcode and system diversity L = 3.
81
•CD
I II l II II
°n
020
"OI
C z
0.))o •
qd :joJll 1!g JO Aj!l!quqoid
Figure 41: Probability of bit error conditioned on the Doppler effect
coefficient in a Rayleigh fading channel (r = 0) and maximum Dopplereffect, i.e., minimal Doppler effect coefficient for a rate 2/3, v, = 6 convo-
lutional code and system diversity L = 4.
82
4.1~
rr
00
-III
q-d :J011 IIIj X~~nq,
oJ II If
Fige 4: P
e(
ien=cd
JI --. I I I I
Figure 42: Probability of bit error conditioned on the Doppler effect co-efficient in a Ricean fading channel (r =10) and maximum Doppler effect,i.e., minimal Doppler effect coefficient for a rate 2/3, v 6 convolutionalcode and system diversity L 4.
83
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84
REFERENCES
1. J. G. Proakis, Digital Communications, McGraw-Hill Book Company, NewYork, NY, 1989.
2. J. I. Marcum, "Statistical Theory of Target Detection by Pulsed Radar," IH:EETrans. Inf. Theory, Vol. IT-6, pp. 56-267, April 1960.
3. G. L. Turin, "Communication Through Noisy Random-Multipath Channels,"IRE Convention Record, Part 4, pp. 159-166, 1956.
4. G. L. Turin, "On Optimal Diversity Reception," IRE Trans. on Com1nunica-tions Systems, Vol. CS-10, pp. 22-31, March 1962.
5. J. N. Pierce, "Theoretical Diversity Improvement in Frequency Shift Keying,"Proc. IRE, Vol. 64, pp. 903-910, May, 1958.
6. A. J. Viterbi and J. K. Omura, Principles of Digital Comn-nunication andCoding, McGraw-Hill Book Company, New York, NY, 1979.
7. G. C. Clark, Jr. and J. R. Cain, Error-Correction Coding for Digital Commu-nications, Plenum Press, New York, NY, 1981.
8. S. M. Ross, Stochastic Processes, Wiley, New York, NY, 1983.
9. W. C. Lindsey, "Error Probabilities for Riceart Fading Multichannel Receptionof Binary and M-ary Signal," IEEE Trans. Inf. Thcory, Vol. IT-10, pp. 339--350, October 1969.
85
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