Error probabilities of FFH/BFSK with noise normalization and soft decision Viterbi decoding in a fading channel with partial-band jamming

Calhoun: The NPS Institutional Archive

Theses and Dissertations

Thesis Collection

1994-03

Error probabilities of FFH/BFSK with noise

normalization and soft decision Viterbi decoding in a

fading channel with partial-band jamming

Chua, Anthony Y. P.

Monterey, California. Naval Postgraduate School

NAVAL POSIGRADUA1E SCHOOL

Monterey, California

THESIS

ERROR PROBABILITIES OF FFH/BFSK WiTH NOISE NORMALIZATION AND SOFT DEOSION

V]TERBI DECODING IN A FADING CHANNEL WITH PARTIAL-BAND JAMMING

Thesis Advi,or: Thesis Co-Advisor: by Anthony Y. P _ Chua March \994 Tri T. Ha R.ClarkRoberlSOn Approved for public release; distribution is unlimited.

DUDlEY KNOX LIBRARY NAVAL POSTCRADUATE SCHOO! MONTER6Y CA 9394;),5101

WITH NOISE NORMALIZATION AND SOFT DECISION VITERBI DECODING IN A FADING CHANNEL \VlTH PARTIAL-BAND JAMMING

6 AUTIIOR(S) CHUA, ANTHONY, y, I'

10 SPONSORING':MONITORrr-c AGEr--cy REPORT , __ UMBER I: SUPI'LI;'\H'NTARY'IUTES Tht vitw> txprtssed in this thesis are tho'ie of the author and do not reflect

the official policy or po>ilion of tlie Department of Defense or the US. Government 12a. DISTRIIlU'I'IOr-.;'AYAILABILlTY STATE\lENT 112~'ADISTRIBUTION CODE

Apr:roved for public release; d~.::ib:::"':::ion",i,-," u:::c''''irn::::''=,d'---_ _ _ ---'----'''---_ _ __ _

-jl

17. S[CURITY CLASSIFI CATION OF REf'ORT Unclassified

I~, SECURITY CLASSIFICA TION 01' ABSTRACT Unclassified

Author

Approved for public reJease; distribution is unlimited Error Probabilities of FFH/BFSK with Noise Normalimtion

and Soft Decision Viterbi Decoding in a Fading Channel with Partial-Band Jamming

by Anthon)' Y.P. }thua Ministry of Defense, Singapore B.S.E.E., NationaJ University of Singapore, 1986

Submitted ill partial fulfillmenl of the requirements for the degree of MASTER OF SCIENCE IN ELECTRlCAL ENGINEERING

from the NAVAL POSTGRADUATE SCHOOL

March 1994

-Anthony Y. P. Chua Approved by

R Clark RoOcrtson, Thesis Co-Ad vi SOl

Michael A. Morgan, Chaifman Department of Electrical and Computer Engineering

ABSTRACT

An error probability analysis of a communications link employing convolutional coding ""'ith soft decision Viterbi decoding implemented on a fast fre.quency-hopped. bin~ry frequency-shift keying (FFH/BFSK) spread spedrum system is petfonnect. The signal is transmitted through a frequency non-selective. slowly fading channel with partial-hand jamming, );"oise n0I111alization combining is employed at the receiver to alleviate the effects of panial-band jamming, The rl~ceivcd signal ampli.tude of each hop is llIodekd as a Rician process, and each hop is assumed to fade independently.

It is found that ""ith the implementation of soft decision Viterui decoding that tile pett"Ol1nani.T of the receiver is improved dramatically when the coded bit energy to paJ1iat-band noise power spcctrdl density rdtio (E,,/N,) is greater tiJan lOdE, At higher R,.iN:, the asymptotic error improves dramatically and varies fronl W·j to 10.1• depending on the constraint length (1'). number of hops/bit (1..), and the strength of the direct signat (al/2al ), In addition, nearty worst case jamming oceUT<; when the jammer uses a fuJI band jamming "tmtegy, even when L-J and there is a very strong direct signal (f>."/2r? = 100), Due to non-coherent cOl11hining losses, when thc hop per hit rat.io i, increased, there is ,ome degradation at moderate F,,/N,. FurthemlOre, when a stronger code is used (i.e" the constraint length longer), performance improves, ~specially for high SiNI

where the asymptotic: error is rOOuced. Finally, soft decision decoding improves perfonnance over hard decision decoding from 4 to 8dB at moderate r1>/);": depending on the code rate and with a much lower asymptotic error at high E,/X,

TABLE OF CO~TE~TS

L INTRODUCTION

n

ANALYSIS

A. PROBABILITY OF BIT ERROR, Pb

PROBABIUTY DENSITY FUNCTION OF THE DECISION VARIABLE Z,

J. Probability Density Function of

Z,.

2. Probability Den,ity Function of Z, J 5 3. Probability Density Function of Z, with Rayleigh Fading 23 C. PROBABILITY DENSITY FUNCTION OF THE DECISION

VARIABLE Z, 28

ill NUMERICAL Rf":.SULTS . 32

A N1.!t.-1ERICAL PROCEDURE 32

B. PERFORMANCE WITH RAYLEIGH FADING

:

n

C PERFORMANCE WITH RICIAN FADING 35

NAVAl P()$TGRADl);.1·i: SCHOOl

MONTEREY CA 93&43-5101

UST OF REFERENCES 39

LIST OF SYMBOLS

0 ' Average Power of the Non-faded (direct) componenl of the 'iignal

20-' Average Power of the Rayleigh-fadeD (diffuse) component of signal

u. Average Power of the Noise at hop k

Fraction of hops jammed

(202

+ O"~?)

I (20",' cr)

v{(2zl') I u).}

{3, I 1(2(1

+ ':,)

Wi Number of paths for event of weight j A," n(n-l)(n-2) ... (n-i+1)

a, RMS amplilude of signal

Baud""idlh of the signal

dl,,, Hamming free distance

E., Uncoded bit energy E, Cooed bit energy

N] Power Spectral Density of the Jamming Noise

No Power Spectral Density of the Thermal Noise

NT Power Spectral Density of the Total Noise

PI Probability of an error event of weight j

PJQ) Conditional probability of an error event given that i hops are jammed P1 Total Noise Power

Rt Infonnalion Bit RaTe

R"

Hop RaTe

X,~,X,. Receive(! Signal at Branch I and 2 Z .. ,Z,~ Nonnalized received signal at Branch I and 2

L INTROUUCTlON

An important ch~racteristic of milit~ry communications is the requirement to be

robust against hostile interft::rt::nce or other unimentional interference. Fasl frequency -hopping spread spectnJIll systems are widely used in miJitary communications to mitigate the effects of hostile intert'erence. Fast frequency-hopping implies that each bit is transmitted multiple limes at differell1 carrier frequencies detennined by pseudorandom sequence known only by the transmitter and intended receiver AS a result, the hostile interference musl he spread over a much wider bandwidth which results

in a lower intt::rferenee noise power spectral density

Past frequency-hopped syslems are susceptible 10 partial-band interfert::ncL: when

the demodulator assigns equal weight 10 each of the mullipie tmnsmissions, or hops. As a result, various methods of signal normalization to overcome the advt::T>c effects of

partial-band jamming have been considered Previous works have studied the perfonnance of receivers utilizing noise n0n11alization [Ref I), ratio-statistic Tlormalil~ation [Ref. 21, and self-nonnalization [Ref. 3J

For receivers utilizing noise nonnaiilA1tion, convolutional coding with hard decision decoding [Ref. IJ has been studied, It was found that with hard decision decoding th~t perfonnanee is improved significantly as compared to the uncodt::d system. The

impli:Olentiition of soft decision decoding is e:\pectoo to fun her illlprove perfonnance [Ref.

41.

The analysis and actual implementation of a receiver utilizing soft decision

decoding requires much more computing power than a receiver with either no coding or one with hard decision decoding since the decision on the coded transmissions arc not made until they are in the Viterbi decoder. With improved computing capability, soft decision convolutional coding is now practical and will be implemented in the new generation of frtxluency-hopping rJdios. The investigation of the pctformance of systems using convolutional coding and soft. decision Viterbi decoding is an important problem as such systems will be installed in satellite and other military systems.

An error probability analysis of a communications link employing convolutional coding with soft decision VitertJi dlX:oding of a fast frequency-hopped. binary freq uency-shift keying (FFH/BFSK) spread spectmm system with non-coherent, noise-normalized detection and a channel with Rician fading and partial-band jamming is pcrfonned in this thesis This is a continuation of the research conducted in [Ref. I] with the implementation of convolutional coding with soft decision Viterbi decoding.

The data are first enwded by convolutional coding. After that, the transmitter transmits L hops for each encoded bit. The channel is modeled as a Rician fading channel. Both partial- band intelference and thermal noise are also assumed to affect the channel The partial-band interference is assumed to be unaffected by the fading channel

TIle FPHIBFSK receiver with noise nomlalization and soft decision Viterbi decoding is shown in Fig. I. At the receiver, perfect dehopping is assumed and the Signals are demodulated by a pair of quadrature receivers. Before going to the soft -decision Vilerbi docoder, the signals are passed through a noise n0n11alization combiner.

AT the noise normalization combiner, the noise power, estimated using a noise-only channel estimator, is inverted and used to multiply each hop of an encoded bit

F~1Ch hop is assumed to fade independently, and the channel for each hop is

modeled as a frequency non "selecTive, slol'.,'ly fading Rician process. Given these

assumptions. Ihe implication is that the channel·s cohen:nce bandwidth is smaller than the smallest frcquency hop spacing but larger than the dehopped signal bandwidth, and the channel coherence time is larger than the hop duration. The amplitude of the dehopped signal is modeled as a Rician random variable and consists of a non-faded (direct) component and a Rayleigh-faded (diffuse) component [Ref.

41-There are two types of intelference considered in Ihis study. The first is partiai

-band noise jamming which can be caused either by a partial-band jammer or some unintended narrowband interference. If the fraction of the spread bandwidth jammed is -y, then the panial band noise jamming, modeled as additive Gaussian noise and assumed to be present in both branches of the BFSK demodulator when present. affects each bit with probability -yo If the average power spe-ctral densily of the interference over the entire spread bandwidth is N/2, then the power spectral density of the partial·band

interference is N/(2-y) when it is present. The second type of interference considered

is wideband interference and is assumed to be caused by themla! noise or other wideband interferences. The power spectral density of this widehand noise is defined as No/l. Of course, when, = 1, the partial-band interference is aho essenrially wideband in nature from the perspective of the receiver

when jamming is pro;:scnt

(1) when jamming is absent

If the noise normalized BFSK demodulator has a noise equivalent bandwidth of B

Hz. thell the rr-.ccivcd noise power for a given hop k is

(2)

,

_1[""

.

No] B Uk - Y NoB with probability y witb probability (l-y)

The bit rate is assumed to be Rho With L hops/bit, the hop rate

R"

= L~. Therefore, the noise equivalent bandwidth, B, must satisfy B ;:;:

Ro.

For this analysis.

H is assumed to be (',qual to

Ro.

However, the overall spread bandwidth is assumed to

be very largo;: as compared to the hop rate. As the bit energy is assumed to be constant,

the coded bit energy, E .. is related to the uncoded bit energy,

E"

,

and the code rale, r,

by E. = r F"

It is well known that in order to maintain the same data bit rate for a coded system

as for the uncoded system that the transmission bandwidth must be increased by llr. For

conventional system this may be a disarlvi\nt.lge. bllt ror sprei\d spectrum systems where

n.

A,\'ALYSIS

In this thesis, an analysis of the bit error probability for (he receiver illustrated in Fig I is ptTfonned. This analysis requires the statistics of thL: sampkd outputs, Xc>.,

i=l,l of the quadrature receiver for each branch and the noise power, for cach hop k of a bil

A. PROBABIJ,ITY OF BIT ERROR, P"

It is well known that for ~fl-dlX:isi()1I Vilcrbi dccouing that the probability of bit eHor is upper bounded by [Ref. 4-61

(3)

where k is the numher of input bits to the encooeJ

"'J

is the loral infonnatioll ",,'eight of a path of weight

i

[Ref 5J

I' is the probabi.!ity that [he an-zero path is climinat.ccl by a path of \veight j,Jnd

d"" is the fro::: distance of the code:

This is the standard union ooundllsecl to evaluate the peli"ormance ot coded systems. The lightness of the bound can only be e~lablistJed by cornparisoll witll a simulation of the system under consideration since an analytic solution doe$ nOI exist. The bound is

reasonably tight for a conventional MFSK system [Ref. 5]. Consequently, it is expected thaI The bOlJnd will also be tight for rhe noise normalized receiver, especially when the number of hops per bit is small

Since all the hops are independent and identically distributed, P, is derived from

(4)

where Pr(ip) is probability of error given t.hat ~ of L hops in pLil coded transmission are jammed. This can be simplified to

,

"

°L

;,,0 ;;

L

_1,0

L

i,..{)

(~)

(~)

I;;,

,

(5)

JL-L" y'" (I y)

,

.

,

P/i)

where i = [i, i.1 ~]. and P/D is the conditional probability that the all-zero path is

eliminated by a path of weigbtj given (hat

i

hops are jammed and equals (pl(i,) .... pl~l

Finally, it is possible 10 simplify (5) to a simple hinomial expression of the rom]

From lRef. 6 pp 34], the conditional probability PJj} can be calculated for binary

orthogonal signaling and noncohcrent detection and is equal to the probability that the

sum of j samples from the signal-absent quadratic detector is gn:ater than the SUIIJ of j

sampks from lht': signal present detector, Hence,

P

,W

~

1

-

]

f,,

(

',IU

[

i

f~(,,)

d,

,1

<u,

" 0

(7)

where ZJ and Z, arc the random variable of the noise lloffilalized outputs of the branches I and 2 rcspcctiv::ly and is sUlllmcd over j codr-d transmissions. It is assumed, withoul loss of gcm:rality, lhat (he signal is present in hranch I.

From (7), it is observed that in order to detennine the bound on bit error

probabilities given by (3) for a noise uonnaiizcd receiver and a Rician fading channel,

we need to firs! detennine the conditional probability density function of ZI and the

probability density function of

z.."

the noise nonnalized decision stati,(ics \>"hich arc

fonm:d by summing the noise nonnalized, sampled outputs for each hop over j coded

transmissions [Ref. 4,6]. This summation of the j coded transmissions is made inside

tho: Vit..:rbi decoder. It should be noted, with reference to Fig. 1, that no (kcision is

8. PRORARILITY DENSITY nH\TTION OF THE DECISIOK VARIABLE Z,

1. I>robabilit}· Densit)' .Fllndion of Z!';

To detennine the prohahility density function of the noise nomlalized, sampled outputs for each hop, we leI

0-/ re

present the noise power in a given hop k of a bit. For this thcsis, we assume that

0"/

is detemlined from the noise-only channel

estimator and is known exactly. Thi, results in an ideal performance analysis. Funhermore, without loss of generality, the signal is assumed to he in hmnch I of the

BFSK demodulator To detennim: the prohability density fUllction of the noise normalized random variable Z", we stan with the conditional probability density timction of the random variable X" at the output of the quadratic detector. This output is equivalent to the envelope squared of a sine wave plus a narrowband process_ The probability density function of the random variable used to model this ,ignal is [Ref. 7

. pp 112, eq. 4-69J

(8)

for q > 0 where A = mlS signal amplitude,

cr

= variance of the narrowband process, and

laCe) = modificd Bessel function of the first !;ind and order zero

If

tT,'

is the average noise power in a given hop k and the signa! amplitudl:

(9)

where lice) is lhe unit step function.

Next, since the signal amplitude for each hop is modeled as a Rician rJndom

variable, then [Ref. 8 : pp. 592, eq. 7·S3]

(10)

where the tOial average signal power is 3.l , which is a summation of 0/ {avcmge power of the nOll-faded (direct) component} and 2t? {average power of the Rayleigh faded

(diffuse) component}. The tOlal average signal power is assumed to remain constant

from hop to hop. It should be noted that when

ci

= 0, the channel is a Rayleigh fading

model, and if 2r? = 0, there is no fading

FurthemlOre, the noise nomlalved received signal is related to the sampled

outputs by

(11)

In order 10 determine the conditional probability density function of Ihe noise nonnalized

(12)

Substituting (9) into (12), we get the conditional probability density function of Zl.

(13)

The uncondit.ional probahility density function of Z" is now found from

(14)

Substituti.ng (10) and (13) into (14), we get

(15)

which can be evaluated to yield

(16)

Here, we make the substitutions

=

}i?=!~

(17)

",

to get (18) From [Ref. 9 : pp 23:<:, eq 4.436J, 11

(19)

If we let p = 0 and since 10(.lt) = Jo(ix) , (19) becomes

(20)

If we let";' = ia, ~. = ib, A = p2, and at = x, the integrJ.1 in (18) becomes

(21)

o - ' -_2),

e)

1

~""l

_4),

If

_,

""'I

_2)'

·

since Jo(-x) '" lo(x)

(22)

Now subsliluting in (17), we get

(23)

Next, we substitule Ihe signal-to-noise ratio of the diffuse component, h = 201/0/, to get

(25)

which can be simplified to

(26)

Finally, substituting (he signal-to-noisc ratio of the direct component, p. = Ct.' I

0",

'

, we

ge<

(27)

Therefore, the probability density function of Z" has been detemlined.

2. Pmoaoility Density Function of ZI

Lei Z"r1l1 and ZI"ill denote the random variable Z!,cp when hop is jammed or not jammed. respectively _ For convolutional coding with soft decision Viterhi dewding.

we need to determine the random variable for thc sum of j coded transmissions [Ref. 9

-10], Assuming that for a coded tnlnsmission p of the j transmissions, ~ hops arc

Jammed, we obtain the resultant decision random variable Z,

(28)

where it can be seen that ~ is not necessarily the same for each of the j coded

transmissIOns

Since ZI':,(O', n = I ,2 are independent and identically distrihuted random variahles.

'" 1,2 ₍₂₉₎

we can simplify (2R) to

to,

-

.

~

Z;;l

I

E

,{~+,

ti

l'

jL-I>,

=

r

~ zi~)

t

~

Z;;)

(30)

In order to detennine the prohahilit.y density function of Z" we need to find the L<tplace lransfonu

or

lht: probability density function of Z,,'''' rhe probability density function is obtained frOlu (27) by ft'piacing p, and ~. with p,'n) and respectively, to gel

(31)

The Laplace transfoml is found from

(32)

This is simplified 10

If we substitute

then

Fz::J(s) " ~~n) exr{_2p~n)~~n)) [c;q:{-Zi~\P~D)+S)) lo(2p~n)j2p~~)z{;))

dz:

:)

Next, we let z,,{n) = u2 in (35) which implies

z;~)

"

0

z;~)

-

00

Making these substitutions in (35), we get = U" 0 Fz,~(s) = 2p~n) exr{-2pr)p~~):

[

cxr{_u2(p~n)+S):1

l

o

(2p~n)u/2-;;-~

)

1)

du

17 (33) (34) (35) (36) (37)

From [Ref. \0 : pp 486, eq 11.4.291,

[ e-·'" 1,,1 J.(bt) dl =

(2a~

:'']

e-b'14.' (38)

If we let p = 0, and since 10 = JoCix), (38) becomes

(39)

In addition, we let

a 2 " j}~n) +s

b

;2jj}

~)M

(40)

Substituting (39) and (40) into the integral portion of (37), we gel

(41)

PUlling (41) into (37). we get

F ()

p

k

"

)

{

2 (n)p(n)

2(P

"

k

'

)

' "']

Pk

z~:) s ; p~o) ~s e}( ~ P. k ~ p~n) ..-s

(42)

Since all the hops are indcpendcnt, we obtain the conditional probability density function forthe decision variable Z, given thaI il,i2, .•... ij hops of the j bilS have interference as

(43)

where a®b implies the convolution of a and b, and represents a c-fold convolution of a Equation (43) simplifies to

(44)

In order to deten11ine the convolution, we will multiply in the s domain. That is, from (42) v.,'e get the Laplace transfonn of the conditional probability density fUllction of Zl

giVen that i" i,. ., ~ hops have interference

(45)

The above Laplace transfonn cannot he inverted analytically except when either all hops are jammed or no hops are jammed In these two caSeS. the inverse Laplace transform of (45) is

From [Ref. J I : pp 59, eq 81], we gel Le. k " 2jL(pio)2 p~n) ~ ; jL (47) (48)

Substituting this into the inverse Laplace lrnnsfoml on the right hand side of (46). we gel

Therefore, substituting (49) into (46), we get the probability density fUllction of Z, for either no jamming or fulJ hand jamming as

(5())

(51)

fhis can he simplified to

(52)

l'herefore, the probability density function of Zi has been deriveD

3. Probability ~Ilsity Function of Z, with Rayleigh Fading

For the spe(:ial case of Rayleigh fading, p~ --> 0, and Ihe Laplace transfoml

of the conditional probability density function of Z, given ii, '" i, given by (46)

reduces to

(53)

Depending on the values of i", the powers of the bracketed tenllS can theoretically vary up to infinity. Fuoheonore, we kIlOw that is a constant for each hop and cacti bit.

Therefore. we let

and (53) simplifies to

a " pk(l) b " p);(2)

F, (,:0

•

[-,-)m

[

--,,-)"

, s~a s~b m

b"

(

I

I

)

" a (s+a)m (s+b)"

By using partial fractions for the above c:qualion, we get

23

(54)

F7/:;lil

-amb"[~

(s+a)

Kz,

~2

(sib) + (s+b)"

(56)

LTsing IIeavj,ide's Expansion Theorem Cor parlial fraclions [Ref. [2 pp 174, eq 7-831,

we get

(57)

where j = 1,2

r - highc~t POWL:f of the; denominators (i,(; , if j -I, r-Ill; and ifj =2, I'=n) x = the tcnn of the: expansion

T1Wf(:fOI"c. l'urLilerrnon:,

Rl(~)

= ' -(s-a)'" ~,,(s) = R,.(S): •• b 24 I (b -a)~ 1 (a_b)m (58) (59)

K1(m I) =

(m

-

-

-

~'-d-)1

~Rl

(s

t

-

-

(":rt

.

(60) (b-a)',·1 (61) K'(m_;)= .11 d'R1(S)1 1. ds' •...•

=

,!

[(-I)' n(n+I)(nT21 ... (nti-I))

i1 (b-at" (62) K I

I

f

(

1)t:1-1 ll(n+I)(1l+2) ... (Il+nl 2)) ll= (ro-I)I \ (b-arrn-1 (63) Similarly (64) 25

(65) K1(ni) =

-J-

~RlS)l

It dsi ._h = ~ [(_I); m(m+l)(m+2) .. (llloi-l)) i! (a_b)m.: (66) 0_ 1_. [<_l)n .. l m(m+l)(m+2) ...

(n+m-2»)

~

I

(n-l)1 (a-b)""" 1 (67)

Using the following Laplace tmnsforrn pairs.

f(t) =tn ....

f(t) = e-" - pes) = ~

(68)

we get,

(69)

fZ,(ZIIi.) = [Kll .Klll l" ..

~

K

ln

,

(~l_

l

)!

l

exp( -I\~) ll) U(Zj) +

Using III = I and n=2 in (70), we get

Kn =

(;j~b)

K2l = -

(a

~

b)

2

Next. using III =2 and n=2 in (70). we get

Kll '"

(b~a?

K21 =

(a~bP

Kzl

= -

(3~W

It can be secn from the above that thc results are as expected in [Ref. I : eq 15].

27

(70)

(71)

C. PROBABILITY DEKSITY F1J~CTION OF THE DECISION VARJABLE Zl In order to get the probability density function of

z"

,

we let the direct (Pl.) and

diffuse (~l.) components be zero and (l,I"~ = '/2 in (42). This result is then raised to the (jL)"' power \0 yield

[ 1 ]" = 2(s +

·

i)

(73)

which is inverted easily to obtain

(74)

The dramatic simplification in obtaini.ng the probability density function for Z2 occurs due to the normalization which makes the output of the branch thaI does not contain the signal independent of whether the hop is jammed or not. Next, we need to find

(75)

If we let x = z,l2, this gives

ill = dz2 2

Z2=O - x=o

Slibstit.uting the ahove into (75), we ohr.ain the simpliflrd iIltrgral

o

'I'12

xjl.-I _{e ' 2 dx.} o 2(jL-1)!

l;rom [Ref. 9 pp e;q 2.321.2], Vie ge;t

(76)

(77)

rx~e""d.x

= e; ....

l~

It

n(n l) ...

(n-k+l)x~-kl

(7R)

• it k_l al-I

If wr: Ie! n = (jL-l) and a co--1 in (78) and u,e lhis rr:sult in (77), wr: get

',,12

dz~

=

(iL~I)1

[ XjL-1 e->-dx =

lGL~)!

(_XjL-1

.. i:E

(_l)~GL-l) ... GL-k) k 1 (-l)~ -

j~{jL-t)

If we substitute the limits of the integral, we get

G~-~'~)!

I

-

j~

UL-ll {jL-k)

This (an be simpliuc,d to obtain

-~

(jL-l)1

(79)

(81)

Therefore, rhe inregral of rhe probabili1y densi1y func1ion for

0.

is

(82)

m. :VmvlERICAL RESULTS

A. NU\1ERlCAL PROCEDLRE

The comJlur.ation of the bit error probability requires numerical ccmpulation of (7)

with the probability density function of Z: being computed by fwding the inverse Laplace tranSfUJlll of (45) numerically cxn:pt for thc two special case~ of eilher alJ hops jalllmed or no hops jammed. The remainder of the integrand of (7) is given by (fl2l. The results of thi, numerical computation arc then substitutNI into (5)

This result for PJ is then substituted into (3) where the wcights (?'),), the coding free distance (dr,.J, and the number of input bits (k) are used ro detennine the probability of bit error.

System pcrfonnance is evaluatcd for various value~ of jamming fraction )'. diversity, fading conditions, and values of the bit energy-to-jamming noise density ratio (E,,/:'\:l For the Rician case, the ratio of direct-to-diffuse· signal energy ((>.li2al ) is assumed to be the same for each hop k of the bit

In thi~ ~tudy, E,/N u= 20dB is used when (l,"/2,i'~ I Othenvise, E"iN,,= 11.:15dB is lIsed. It is expected that when E,/N" i~ nxlucw, pcrfoJll1ance will improve at lowel values of E,)N, since E,,'/Nc-' is more dominant. However, the improvement is inconsequential since for low E,,lNj system perfol1nance with convolutional coding is too

p<Xlr to be llseful. It seems reasonable to assume that the trends obtaiJled with these values of E,IN" will not he significanrly affect('",d as E,)No increases.

B. PERFORMANCE WITH RAYLEIGH FADL"IG

Figures 2 to JO shows tht: perfonnancc of the noise nonnalized rcceiver with

Ray](;igh fading. It is observed [rom these figures that coding improves the performance

when E,JN, is greater (han lOdE

Figures 2 and 3 shows the perfonnance of the reco;:ivcr when 'Y = ].0 (full band

jamming), 0.25, 0.1 and D.OI and the number of hops per bit L = I with constraint

length 1/ = 2 and 4, respectively. The figures show (har the asymptotic probability of

bit errors vary from about J06 (v = 2) to about 10-" (v = 4), These asymptotic

probability of bit errors arc much better than that obtained for ullcoded performance

(10.2). Comparing Figs. 2 and 3 with Figs. 4 and 5, 1;','hcrc L is increased to 2,

perfonnance is seen to have improved with the asymptotic hit error probability being 10-9

(v = 2) and 10-'1 (v ~ 4) as compared to 10-'; and 10-8 when L = 1. It is also seen that

for Rayleigh fading the worst ease jamming stratcgy to be used is full hand jamming

By increasing ~, performam::e is further improvt:d. This improvement is much

lllore dramatic at high ~!Nl as can he scen in Fig. 6. Comparing Fig. 6 with Fig. 7,

it is observed that the asymptotic bit error probability is improved eithcr by increasing J' or L. The asymptotic bit error probabilities for various valul;s of v and L are given

in Table I

TARLE t

Constraint Length, P Asymptotic Probability of Hi: Error

L ~ 1 L ~ 2

- ₂ 10-" 10"

- 4 IU ~ 10"-'

"

~ ₆ 10.:1 < 10:<

It can also be seen in Fig. 8 that when F~)N, is small

«

15dB) and L is increased thar the non-coherenr combining losses arc dominant and degrade the perfonnance However, for Iligller E"iN" the divenity gain i, dominant and improves the performance of the rcn.:ivcr.

In Figs. 9 and 10, soft-decision Viterbi decoding is compared to harddecisioll Viteri)i decoding (for p = 6). It is observed that soli decision decoding improvts pcrfon1lance over hard decision decoding by about 7 to lOdE when E"iN~ is alDund 10

to 20ctB (rig. 9). The asymptotic bit error probability for rate = 1/2 is seen to be 10-1 '

a<; compared to 10---. Purthennore, for rate = 1l3, the asymptotic bit error probability is 10-'4 as compared to 10' As expceted, there is [urlher improvement when L is incrtased (Fig. 10).

C. PERFORl\1A.t~CE WITH RTCIA.."\l FADIN(i

Figures II to \7, Figs. 18 to 24, and Figs. 25 to 26 show the performance of the receiver for a strong direct signal (n1/2r?-= 10), a weak direct signal (0:2/20" = I), and a very strong dir<"-ct signal ((il/2ti = 100), respectively. It is observed Iha\ coding

improves performance when E"iN) is greater than about lOdE.

Figures II and 12 show the perfonnance of the receiver when"f = 1.0 (full hand jamming). 0.25, 0. I and 0.01 and p - 2 for L = 1 and 2. respectively. The asymptotic

bit error prohahility is about 1O-~ as compared to 10-2 for the ullcoded case. The

comparahle situation, hut wilh a weak direct signal component (Q';l2rl = I), is shown in Figs. 18 all(] 19. In this case, perfOn1lallCe is slightly worse as compared \0 lhe case

with 0.1.120) = 10. It is ohserved from all the above cases that lhe worst case jamming

strategy is quite close to full band jamming, evcn when L "'" 1, except when o/l2tJ = 100 (Figs. 25 and 26). Even then, partial-band jamming does not significantly degrade perfonnance Therdore, it is observed that convolutional coding improves the pcrfonnance of frequency hopping systems by forcing the jammer to spread its resources

over a much wider band as compared to the non-coded system. This conclusion remains

valid even when L= I. It should also be not('-<:i that with a very strong direct signal, the effect of the jammer is much less significant than when the direct signal is weaker for

the same~/Nl

By increasing ~ (Figs. 13, 14.20, and 21), it is observed that the perfomlance is

improved by about IdE as ~ is increased from 2 to 4 and from 4 to 6. This improvement

reduces the asymptotic bit error probability for strong direct signal from ahout IO-j _to

about 10-1 and for the weak direct signal from about 10-4 to 10-7 when ~ = G.

Figure 15 shows the pcrfomlance of the receiver when there is an increase in

diversity. It is observed that for the strong direct signal (cil2tl = 10) that the non·

coherent combining loss is greater than Ihe diversity gain since peli"Ol1llanCe is degradoo

with an increas(~ in diversity. For the weak direct .signal case (cil2tl = I) (Fig. 22),

diversity gain is greater than the non-coherent combining losses when E,,/NJ is greater

than 15dB

With ~ = 6 (Figs 16,17,23 and 24), when the ]'"dle is reduced from 11210 1/3,

it is observed that there is minimal increase in pcrtol1llance when there is a strong direct

signal. When there is a weak direct signal, the improvement in perfol1llance is about 0.5

to I.OdE Comparing these to hard decision decoding, there is an improvement of about 5dB

From the above discussions, it is concluded that with the implementation of

convolutional coding with soft decision Viterbi decoding that pcrtol1llance is generally improved when EjN] is higher than about lOdE. Another conclusion is that

convolutional coding and soft decision Viterbi decoding render fast frequency-hopping

with L> I largely unnecessary.

IV. CONCLUSION

When convolutional coding with soft decision Virerbi decoding is employed in conjunction with FFHiBFSK utilizing noise l1onmJization combining, system perfonmmce is superior to the peri"ormance of the equivalent uncoded system when

E"iN,

is greater than lOdE. Below IOdB the uncoded system peri'onllarlce is too poor for coding to be effective. Thi.s improvement in perfonnance is particularly true for Rayleigh fading channels. In addition. at high

E"r:'-l"I,

the asymplOtic prohahility of bil error improves dramatically as compared to the uncoded system with the probability of bit error val)'ing from 10-6(0 10-" depending on P, L, and (0:'/2,;') as compared to about IO-J with no coding. The jamming strategy required to cause worst case receiver performance is full band jamming, even when L= I, e:-::cept for very strong direct signal (c/.l2rr = 1(0). Even thcn. partial-band jamming does not significantly degrade peri"onnance. This is a signit1cant depar1ure from what is ohservcd for the uncoded sysTeTll. This mean, that the enemy jammer must spread its jamming power over a much wider bandwidth. It abo means that fasl frequency-hopping willi L> I is largely unnecessary when convolutional coding and soft decision Vilerhi decoding are used.

Due 10 non-coherent combining losses, there is some degradarion in pcrfomlance

when th~ hop pa bit ratio is illcreased and the received signal is at moderate E"iN,. l! is found, as expectf'Al, that when a stronger code (higher constraint length) is used

that perfomlance is improved, especially for high F..JN1. Finally, soft decision decoding

improves the pcrfonnance over the hard decision decoding by about 4 to 8 dB (depending on the code rate) at moderate ~INl and has a much lower asymptotic error for high

LIST OF REFERENCES

[IJ Robertson. R. C., and Ha, T. T., "Error Probabilities of Fast Frequency-Hopped

MFSK with Noise·Nonllalization ComhiniJlg in a Fading Channel with Partial-Band

Interference," IEEE Transac/iOiIS on Communiclllions, v. 40, no. 2, pp. 404·412,

Febntary 1992

[2J Robertson, R. C., Riley, 1. R .. and Ha. T. T .. "Error Probabilities of Fast

Frequency-Hopped FSK with Ratio-statistie Combini,ng in a Fading Channel with

Partial-band Interferenee," Proc. IEEE Miliral)' Commllnicatiofl.l COl!!, v. J, pp.

865-S69. 1992.

[J] Robertson, R.

c.,

and Lee, K. Y., "Perfonnanee of Fast Frequency-Hopped

MFSK Receivers with Linear and Self-Nonnalization Combining in a Rician

Fading Channel with Partial-Band iEEE jOlmwl on Seleeted Areas

in Commllnicatiom, v. 10, no. 4, pr. May 1992.

[4] Proakis.1. G., Digital Communicatiofl.l, 2nd ed., McGraw Hill. [989

[51 Olden walder. J. P., Optimal

Univ. California, Los Angeles, CA,

[6] Clark. G.

c.

,

Jr.. and Cain, J 8., Error Correction Coding for Digital

Communications, Plenum Press, 1981

[71 Whalen, A. D., Defeaion of Signals in NoiJe, Academic Press, 1971

[8] Couch, L. \V , Digital atld Analog COn/nmnications S;'!ifCmS, 4th ctl., Macmillan.

1993

[9] Ryshik. I. M., and Gradstein.l. S.,

Deutscher Verlag Der Wissenschaften,

Products alld Imegra/s, Veb

[101 Abramowitz, M., and Stcgun, I. A., Handbook of Mathematical Functions, Dover,

1972

[II] Beyer. W. H., Handb()ok ojMaritematica! Punetiow, 6th ed., CRC Press, 1987

[12] Van VaJkenbmg, ,\-1. E., Network Analysis, 2nd eel., Prentice Hall. 1955.

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Fig. 4 ; Performance of :'Il"uisc-Normalil.ation Receiver with Suft-Decision Viterbi Decuding fur y == 1.0, U.25, U.I and 0.01 with Rayleigh fading, and E/No = 20JldH, L = 2 and v = 2.

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Fig. 5: Perfurmancf' uf Noise·Nurmalizatiun Receiver with Suft-Decision VilertJi Decoding for Y= 1.11, U.25, 0.1 and U.O! wilh Rayleigh fading, and E/No = ZO.OdB, L = 2 and v = 4.

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Fig. ~: Performance of Nuise-Normalizatiun Receiwr willi Soft·Decision Vilcrbi Decuding

rur nllmUl'f of hops/bit of L::" 1,2,3 and 4 with Rayleigh fading, full band jamming, Eb/N"

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Fig.9: Performance of Nuise-Normalillltioll Rctciv{:r wilh Soft-Decision Viterhi Decoding and H:Hd-Dccision Viterhi Decoding for code rate, r = LIz and 1/3 with Rayleigh fading, foil band jamming, £I/N" = 20.0dll, L = I, and v = 6.

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Fig. 10: Perfurmance of Noise-Normalization Receiver with Soft-DecisiOIl Viterbi Decoding and Hard-Decision Vitcrbi Decoding for (;00(' rate, r = lf2 and 1(3 wilh Raylt'igh fading, full band jamming, 1\IN" = 20.1MB, L::; 2, and v = 6.

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Fig. II ; Performance of l\oise-NurmalizaliOlI Receiver with Soft-Derision Vitcrbi Decoding

fur Y= 1.0,0.25,0.1 and (WI wilh Rician fading, a strong direct signal (a2/2o-1 = HI), E/No

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Fig. 16 ; Performance of Noise-Normalizatiun Receiver with Soft-Decision Vilcrbi Decoding and Hard-DC<'isiun Vi!crbi Decoding for corle rate, r = 112 anil 1/3 wilh Rician fading, a strong direct signal (u"/2crl = 10), full band jamming, Eo/No = 13.35dB, L = I, allli v = 6.

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Fi/.!o 17 : Performance

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'"

]

1.0E-4

~

.g

_1.010-5

0:

1.0E-6

Fig. 23 ; Perfurmance of ~oisc-I\urm:llization Rccci\'cr with Soft-Decision Viterbi [lecoding lind Hard-Decision Viterbi Decoding for code rate, r = 1/2 and 1/3 with Rician fading, II wcali: direct siJ:,:nat (a,'"/Zol = I), full band jamming, Ebil\() = ZO.OdB, L = 1. and v = 6.

l.OEO

l.OE-l

2

l.OE-2

iii

iii

l.OE-3

'C

_1.0E-4

.~

~

1.0E-5

JO 0

c:

1.0E-6

-o-Soft Dec, Rate = 1/2

-+-Soft Dec, Rate = 1/3

-i:I Hard Dec, Rate = 1/2

l.OE

-

7

EQard Dec, Rate = 1/3

~ Uncodcd

1.0E-8

~~~~~~~~~~~~~---l

5

15

Fig. 24 : Performance uf l\uise-l\urmaliz:Jliull Receiver with Sul'l-Decision Viterbi Decoding and Hard-Decision VilcrlJi Dl'Coding fur cude rate, r :: Lll and LI~ with niciall fading, a w{'ak direct signal (al/2cr" = I), full band jamming, E_/No

=

20.0dll, L :: 2, and v = 6.

1.0EO

01= 100 -+-1 = 0.2'i

1.0£-1

0'(:= 0.10 -6-",/ = 0.01 -... Uncoded, "'I '" 100

2

1.0£-2

Uncolled,l = 0.01

Pi

iii

1.0£-3

'0

_1.0£-4

:E

:0

_1.0E-5

-§

.t

1.0£-6

1.0£

-

7

1.0£-8 - - -

-

-

- -

-

-

-

-o

ill

~ ~

E/NI

(dB)

Fig. 2:' ; Perfurmance of Nuisl'-Nurmalil.aliun Receiver wilh Suft-Decision Vitcrbi Decoding

fur 'Y = 1.0,0.25, (1.1 and 11.111 wilh Rici:m fading, a very sl.rong direcl. signal (01/201 := 100),

E,/No = B.35dH, L := I, and \' = 2.

1.0£0

1.0£-1

e

1.0£-2

ob

Cl

1.0£-3

a

1.0£-4

G'

'"

~

1.0E-5

.c

2

1.0E-6

"-1.0E-7

Fig. 26; Performance of Noise-I\'ormalization Recei.'er with Suft-Decisiun Vitcrbi Decoding

fur Y= LO, 0.25, 0.1 and 0.01 with Rician fading, a very strong direct signal (al/lcJI = IOU),

E"iN" 1.I.35dll, L = 2, and v = 2.

L"IITIAL DISTRIBCTlON LIST

Defense Technical Infonnatlon Center Cameron Station Alexandria, VA 22304-6145 2. Dudley Knox Naval Postgraduate Monterey, CA 93943 Chainnan, Code DC

Electrical and Computer Engineering Department Naval Postgrauuate School

Momerey. CA 93943 4. Tri T. lIa, Code EC/Ha

Electrical and Computer Engineering Department

Naval Postgraduate School

Monterey, CA 93943 R. Clark Rolx'~rtson, Code DC/Rc Electrical and Computer Engineering Department Naval Postgraduate School

Monterey, CA 93943 6. Director, DMO

Level I, LEO Building, Paya Lebar Airport. Singapore 1953 Republic of Singapore

Anthony Y.P. Chua

Level I, LEO Builuing, Paya Lebar Airport, Singapore 1953. Republic of Singapore.

---~~---~

DUDLEY KNOX LIBRARY

NA

ljA.

r'i.Ol.i:'~·~ ~CHOOI