On the Selection of a Two-Dimensional Signal Constellation in the Presence of Phase Jitter and Gaussian Noise

Copyright O 1973 Ameriosn Telephone and Telegraph Company THE BELL SYBTEU TECHNICAL JOURNAL

Vol. 52. No. , July-Augut, 1973 PrinUd in U.S.A.

On the Selection of a Two-Dimensional Signal Constellation in the Presence of Phase Jitter and Gaussian Noise

By G. J . F O S C H I N I . R. D. G I T L I N . and S. B . W E I N S T E I N

(Manuscript received February 1, 1973)

A long-standing communications problem is the efficient coding of a block of binary data into a pair of in-phase and quadrature components. This modulation technique may be regarded as the placing of a discrete number of signal points in two dimensions. Quadrature amplitude modulation (QAM) and combined amplitude and phase maduMion (AM-PM) are two familiar examples of this signaling format. Subject to a peak or average power constraint, the selection of the signal coordinates is done so as to minimize the probability of error. In the design of high-speed data communication systems this problem becomes one of great practical significance since the dense packing of signal points reduces the margin against Gaussian noise. Phase jitter, which tends to perturb the angidar location of the transmitted signal point, further degrades the error rate. Previous investigations have considered the signal evaluation and design problem in the presence of Gaussian noise alone and within the framework of a particular structure, such as conventional amplitude and phase modulation. We present techniques to evaluate and optimize the choice of a signal constellation in the presence of both Gaussian noise and carrier phase jitter. The performance of a number of currently used or proposed signal constellations are compared.

The evaluation and the optimization are based upon a perturbation analysis of the probability density of the received signal given the transmitted signal. Laplace's asymptotic formula is used for the evaluation. Discretizing the signal space reduces the optimal signal design problem under a peak power constraint to a tractable mathematical programming problem.

Our results indicate that in Gaussian noise alone an improvement in signal-to-noise ratio of as much as 2 dB may be realized by using quadrature amplitude modulation instead of conventional amplitude and phase

927

928 B E L L S Y S T E M T E C H N I C A L J O U B N A L , J U L Y - A U G U S T 1973

modulaiion. New modulation formats are proposed which perform very well in Gaussian noise and addionay are quite insensitive to moderate amounts of phase jitter.

I. I N T R O D U C T I O N

A very a t t rac t ive modulat ion forma t for coheren t high-speed d a t a transmission is the family of suppressed-carrier , two-dimensional signal constellations of which quadra tu re ampli tude modulat ion (QAM ) and combined ampli tude and phase modulat ion (AM-PM ) are two examples . In this pape r wc will consider using this more general signal format , which is equivalen t to t he a rb i t rary placemen t of a discrete numbe r of signal points in the plane , subjec t only to a peak or average power constraint . T h e object will be to mit igate t he major statistical transmission impairments encountered on t he voice-grade telephone channel , such as carrier frequency offset, carrier phase j i t ter , and addi t ive noise. Ou r a t ten t ion is focused on constellations of 16 points , since this seems to be t he larges t constellation which is practica l for t h e typica l telephone channel . However , t he techniques we develop are applicable to constellations of a rbi t rary size.

The placing of signal points in the plane is a long-standing problem t ha t has received considerable a t tent ion in t he past . Previous investig a t i o n s ' ' ' have considered t h e signal evaluation and design problem in the presence of Gaussian noise alone and within the framework of a part icular s t ruc ture such as combined ampli tude and phase modulation . When Gaussian noise is the only transmission impairment , it is well known t h a t a t high signal-to-noise ratios ( > 2 5 dB ) the signal points should be placed as far apa r t from each o ther as possible ( the circle-packing problem) . In the application to high-speed digital communication systems , t h e TVO-dimensional signal design problem becomes one of great practical significance because the dense packing of a large number of signal points markedly reduces the margin against random noise and phase j i t ter . The signal design problem in the presence of both phase j i t te r and Gaussian noise has not been solved before and is the subject of our discussion.

Coherent receiver s tructures have recently been proposed which employ an adapt ive equalizer^ to compensate for a n y linear distortion, low-frequency phase j i t ter, or small amounts of frequency offset introduced by the channel. A phase-locked loop* may be used to suppress high-frequency phase j i t te r and frequency offset. Of course, t he ou tpu t of a phase-locked loop will still deviate somewhat from the op t imum demodulat ing phase angle. One purpose of this s tudy is to assess the

TWO-DIMENSIONAL SIGNAL CONSTELLATION 929

effect of such phase errors on the system error ra te and to indicate how this knowledge can be incorporated into the system design.

By system design we have in mind the selection of both a part icular two-dimensional signaling format and the decision device placed a t the demodulator output . In order to pursue these objectives, we discuss:

() the relative immuni ty of various signal constellations as a function of the degree of noncoherency (i.e., t he size of the phase error) ;

(ii) an efficient i terat ive procedure for determining op t imum signal formats unde r a peak power constraint ;

(Hi) system performance for the following hierarchy of decision de vices: easily implementable , op t imum in Gaussian noise, and a maximum-likelihood detector which uses the statistics of t he phase error ;

(it;) the accuracy required in any phase-locked loop to a t ta in a satisfactory error r a te ; and

(v) the resulting error ra te when no a t t e m p t is made to t rack the jitter.*

Our approach is to assume t h a t intersymbo l interference has been effectively eliminated by the equalizer while the phase-locked loop, if there is one, has only partially removed t he phase j i t ter . Thus t he equalizer ou tpu t will be the sum of the partially coherent^ t ransmi t ted signal and addit ive Gaussian noise. We adop t a phenomenological mode l which assumes t h a t the (slowly varying) phase error has a Tikhonov density. The Tikhonov densi ty is associated with a conventiona l first-order phase-locked loop whose inpu t is the sum of a sinusoid (whose phase is being t racked by t he loop) and Gaussian noise. Under our assumed operating conditions of high signal-to-noise rat io , this density will closely approximate the actua l phase density . When no a t t e m p t a t t racking is made , ' the j i t te r is modeled as being uniformly distr ibuted in a reasonable peak-to-peak range . For each of these j i t te r densities, the probability density of the demodulator output , conditioned on the t ransmit ted symbol , is used to es t imate the error ra te . This est imate is of the minimum distance type , where the "d is tance ," which reflects the presence of phase j i t ter , is measured in

* In the sequel, we will use the term jitter as a catch-all when referring to phase jitter and/or frequency offset.

^ A partially coherent signal is one whose carrier phase is jittered by a random component which is not uniformly distributed in the range (, ) .

' Since the passband equalizer* will determine the optimum static demodulating phase, the absence of a phase-locked loop does not imply the use of an arbitrary demodulating phase.

9 3 0 T H E B E L L S Y S T E M T E C H N I C A L J O U R N A L , J U L Y - A U G U S T 1973

a non-Euclidean manner . The error ra te is given for various signal constellations and detector s tructures imder peak and average power constraints. B y discretizing t h e received signal space, a n i terat ive procedure is developed to determine (locally) op t imum signal formats under a peak power constraint. This technique, which assumes a maximum-likelihood detector, makes use of an efficient search procedure developed by Kernighan and Lin.*

T h e system model and the problem formulation are presented in Section I I . An asymptot ic est imate and an upper bound on the error ra te are developed in Section I I I . A comparison of the relative immuni ty of some popular signal constellations and detectors to phase j i t ter is described in Section IV. Section V discusses a technique to obtain locally opt imum signal s tructures under a peak power constraint. *

I I . S Y S T E M M O D E L A N D P R O B L E M F O B M U L A T I O N

2.1 Preliminaries

We consider the two-dimensional synchronous d a t a communicat ion system shown in Fig. 1. Binary d a t a are first grouped into blocks of bi ts , and each block of bits is then mapped into one of 2*^ two-tuples (a, b). The sequences {*} and {bk] ampli tude modulate , respectively, an in-phase and quadra tu re carrier to generate the t ransmi t ted signal

m(t) = akp(t - kT) cos + >*(< - kT) sin wt, (1) k k

where 1/T is t he symbo l rate,^ p ( ) represents t he t ransmit te r pulse shaping , and Wc is the carrier frequency. I t will be assumed t h a t t h e two-tuples are equiprobable . The received signal a t the ou tpu t of t he bandpass filter is given by

r ( 0 = ( akx{t -kT)- bky{t - kT)) cos ( ( . + ) -|- ()) k k

- ( auy{t -kT) + bk(t - kT)) k k

X s m ((o). + )t + 9(t)) + n(t), ( 2 ) where x(t) and y{t) are the system (baseband) in-phase and quadra tu re impulse responses,^ is the carrier frequency offset, e{t) is t he random

' T h e present authors have recently treated the two-dimensional signal design problem under an average power constraint.^

* Note that the data rate is M/T bits/second. ^ These pulses represent the cascade of the transmitter shaping filter, the channel,

and the receiving filter.


BINARY DATA

sin

TRANSMITTER

\Dlt) R(t \Dlt) CHANNEL

R(t CHANNEL

1 1

r i 1

1 1 1

TRACKING LOOP

1 1 I i

R(t)^ BPF

r(t) O-^O-^ ADAPTIVE

EQUALIZER D E

MODULATOR DETECTOR

I

11 RECEIVER

Fig. 1An in-phaae and quadrature data transmiseion system.

phase j i t ter , and n(t) is addit ive Gaussian noise. The received signal, which is sampled a t the symbo l ra te , is adaptively equalized and then coherently demodulated mth the aid of a phase-locked loop. ' The demodulated ou tput , denoted by the sequence of two-tuples zt = Ki t , ik)}, is t hen processed by t h e (simple) detector t o give t h e ou tpu t sequence |(d*, 5*)}. The system error ra te is jus t t he probabil-i ty t h a t (dk, bk) differs from the t ransmit ted two-tuple (a*, 6*).

2.2 Basic Model

Fo r the purposes of this s tudy , it will be convenien t to assume t h a t t h e equalizer has completely eliminated t h e intersymbo l interference presen t in x(t) and y{t), bu t t h a t the phase-locked loop has only par-tially compensated for the carrier phase j i t ter . The in-phase and quadra tu re demodula to r ou tpu t s , a t t h e fcth sampling ins tant , a re then given by

z(kT) = a* cos k - bk sin * -|- nc(kT), i{kT) = ak sin -I- 6* cos k + n.{kT), ^

932 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1973

Fig. 2^Effect of GauBsian noise and phase jitter on transznitted symbol.

where * is the (slowly varying) phase error in the tracking loop, and n,{kT) and n,{kT) are, respectively, the in-phase and quadra ture Gaussian noise components. * Dropping t h e t ime index, eq. (3) can be rewrit ten to give the basic model

= As + n, (4)

where the vectors are given by*

^-(0 - ( : ) and the matr ix R is the rotat ional (by an angle ) t ransformation

^ ^ / c o s * - s i n ^ y (5) Vsm coa /

' Recall that n.(,kT) and n.(kT) are independent Gaussian random variables with equal variance, No. It should be noted that 2No is the noise power contained in the bandwidth of the received signal.

* We denote the values that a and b can assume by ^) and bO, respectively, and the values of the transmitted symbols by = ( a^&U) ) .

TWO-DIMENSIONAL SIGNAL CONSTELLATION 9 3 3

- 3 -1

-1

- 3

1 3

AVERAGE P O W E R - 3 . 0 0

PEAK P O W E R - 4 . 2 4

Fig. 3Quadrature amplitude modulation.

As we show in Fig. 2 , t he effect of the phase j i t te r is t o ro ta te the t ransmi t ted symbol, s, by an angle ; thus the demodulator output , z, is dispersed in an angular manner due to phase j i t ter and in a circularly symmetric way due to the Gaussian noise. The receiver will make an error when these perturbat ions move the demodulator ou tpu t across the decision boundary associated with the t ransmit ted symbol. For a particular t ransmit ted d a t a sequence, the demodulated sequence will be scattered abou t the t ransmit ted points in a manner which reflects the combined effects of phase j i t te r and Gaussian noise. For the t ransmit ted signal constellation of Fig. 3 , which is known as QAM, a typical scattered demodulated sequence is shown in Fig. 4 . As one might expect, for those signal points further away from the origin, the angular displacement becomes more apparent . An est imate of this effect is given by the mean-square error between the t ransmit ted and demodiated symbols . Fo r small values of j i t ter , this error is obtained from ( 4 ) by noting t h a t

cos 1 sin

sin cos

averaging the norm-squared of both sides gives*

E\\z - S^T = No + 5||8">||., (6)

'The notation ||z|| denotes the Euclidean norm of z; additionally the notation (z, ) will be used to denote the inner product of and .

934 THE BELL SYSTEM TECHNICAL JOURNAL , JULY-AUGUST 1973

0 - i l i i i . t t t

t : '

=!

t : .

111..

I i : , m . 4 f Fig. 4Received signal points in a QAM system.

where is the vajiance of and denotes the statistical average. Thus , because of phase j i t ter , signal points located further from the origin are subjected to a larger meannsquare error.

2.3 Probabily Density of the Demodtdator Ovlput I n order to evaluate the system error rate , t he probabili ty density

function (pdf) of the phase error mus t be specified. T h e pdf of t he phase error in a phase-locked loop t h a t is t racking t he angle of t he two -dimensional d a t a signal given b y (2) is no t ye t known , b u t as explained below i t can be approximated by the following (Tikhonov) density*:

I


2No 1

(10a)

[ | | z - s O ) | | + 2 (z , sO ' )

- 2(z, i e s f ' ) ) ] ! - (10b)

Subst i tu t ing the Tikhonov den.sity into (9) gives

^ '^ '^ = 2 ^ ' ' ^ { " Wo " ^ " ^ " + ^'"^^

(/o(e))-> /_ '^ exp i - [ ( (z , 8 ' ) -I- aNo) cos

+ {, s, is given by

p(z|s


j :

O - 3 0

- 0

1

Fig. 5Tikhonov phase jitter density () = exp ( cos )/2-/.().

t h a t for large values of a rgument /o(x) /1 Vi|, and assummg a is sufficiently large and Nt is sufficiently small, we get

: exp ( 2 + , s> + ) +


10 1-10

Fig. 6Constant distance contours: values of for which

2,Nm(j) = e x p { - ( , 8 / ) } = 1 0 - , 10-, 10-. P given by eq. (14).

are able to give a very suggestive geometric interpretat ion of the j i t te r phenomenon as well as accurate estimates of the error ra te . In Fig . 6 we show some contours of constan t "d is tance" (i.e., points which are equiprobable) abou t a given point . Note the angular orientation and similarity of these " b a n a n a " shaped contours to those obtained ex-perimentally (Fig. 4) .

Our procedure will be to use (13) and (14) to es t imate t he error ra te for various constellations and detector s t ructures ; however , before we do this , we first wish to discuss some properties of the function d(z, s) which will be useful in est imating t he system performance , and then to consider the conditional density p>(z) in the absence of a phase-locked loop.

2.4 Some Properties of the Jitter Distance d(z, s) (i) A requisite property t ha t d(z, s) should possess is t h a t for vanish-

ingly small j i t ter the distance between points becomes Euclidean . This is easily verified by expanding (14) in terms of 1/a and observing t h a t

lim d(z, s) = ||z - s||. I / a - O

(15)

SNR-25dB. AND RMS PHASE JITTER- DEGREES


(z, 8) + |||||||8|| + 2(, a) + (aNo)'\ (19)

since squaring gives

(z, 8 ) ' + 2


convex metric in any circle about the origin for any value of 2No. The question remains open as to whether or not d(z, s) admits an accurate convex metric approximation in a neighborhood of the origin. Appendix reports the results of an investigation of this question.

By considering the above properties, i t is apparen t t h a t a signal constellation will be relatively immune to small amounts of phase j i t ter either if the error ra te (or minimum distance) is determined by a point a t the origin and any other signal point, or if signal points on the same circle are widely separated (the more circles the greater the Gaussian noise penal ty) .

2.5 Probability Density in the Absence of a Phase-Locked Loop In a subsequent section we will compare the performance of various

signal constellations, and the following question natural ly arises: Can we, by judicious signal design, eliminate the need for a phase-locked loop?* Prel iminary to answering this question we mus t obtain the densi ty of the demodulated signal in the absence of a phase-locked loop. For simplicity we model the j i t te r as arising from the single tone

(i) = A cos (, - ), (20)

where 2A is t he peak-to-peak j i t ter , , is the j i t ter frequency, and is a uniformly distr ibuted random phase. For this model, t he j i t ter densi ty is given by

1 \\ A

O

(21)

\\ > determine the density of the demodulated samples we use (9) and (10) to write

Pj(Z) 1

2tJV( exp 1 _ i _ Q 2 _ 8 0 ) | | . + 2(z ,S s i n ] | which for small ( < 12 degrees) peak-to-peak j i t ter becomes

PZ) 1

2, exp 2No llz - sO'll .(>),

p()d

(22)

* The equahzer* is capable of determining the optimum demodulating static phase, so that the demodulation is noncoherent only to the degree that the untracked jitter degrades the error rate.

940 THE BELL SYSTEM TECHNICAL JOUBNAL, JULY-AUGUST 1973

where

^ )(,8


< -DECISION REGION

BOUNDARY

MIN. \ DISTANCE \ _ (LARGE PHASE JlTTERl \

\

/ /

/

/

Fig. 7^Typical decision region about signal point s., with minimum distances to boundary shown for Gaussian noise alone (no phase jitter) and for large jitter (angular displacements highly likely).

This quan t i ty may be writ ten , using t he conditional densi ty pj(z), as

Pei = f ^ Pz)dz. (29 )

Le t * denote a point of global minimum for the fimction d ' or D* on the decision region boundary and let M, be the number of times this minimum is achieved on the boundary. Let (u, v) denote an orthogonal coordinate system erected a t z* with the positive u axis pointed along the boundary line in a clockwise direction and pointed outside >. Then , as is shown in Appendix A, for a high signal-to-noise rat io {No > 0) with aNt = k, the conditiona l error r a te is given by

Mj I No d^d^

J 2 T du*

exp 2Ni (30a)

Fo r the case of Gaussian noise alone (no phase j i t ter) d ' becomes t he ordinary Euclidean distance and the partais in (30a) are easily evaluated . In this case, with s" ' the 8ignal(s) closest to s''', we have

s) -I- s


dv

d'd'

' - ' 5 = ||s> - s^'ll

= 2. au'l

Hence , for the Gaussian case.

M y /Fo - Is"' - 8^11

a useful formula in i ts own r ight . In t e rms of t h e distance functions d() and 0 ( ) . t he asymptot ic

error ra te is determined by t he poin t on t he decision boundary "closest" to the t ransmi t ted signal. Of course, in the presence of phase j i t ter , this poin t will generally differ from t he closest poin t according to a Euclidean measurement . In Fig . 7, t he indicated poin t on t he vertica l boundary segmen t is t he closest poin t to Sy in t he EucUdean sense. As phase j i t te r increases, those points with a radia l coordinate nearly equa l to t h a t of 8y becomes closer to Sy. So for large phase j i t te r , t he poin t indicated on t he boundary segmen t above 8y is the "closest" point . Thu s t he exponential decay in error r a te is qui te similar to t h e a8}anptotic Gaussian result since it is of the form

exp -dLn{j)/2Nol (31)

where dmin(i) is t he minimum distance (measured in a non-Euclidean manner ) to t he jth decision boundary . The minimum distance can sometimes be obtained analytically , b u t mos t often m u s t be obtained by a compute r search of the boundary .

The min imum distance to the jth decision boundary is part icularly easy to determine if t he function d( , ) is a metric which possesses t he midpoin t property , i.e., for each distinc t pai r of points 8


d(s , s('>) (32b)

where z* is the poin t on the boundary closest to s

944 THE BELL SYSTEM TECHNICAL JOUBNAL , JULY-AUGUST 1973

If we le t CTi denote the first contour which touches t he boundary , then i t is clear t h a t

P.i = P r [z $ i|s


Our comparisons will be made by varjdng the following quant i t i es :

(t) signal constellations () signal-to-noise ratios

(in) rms j i t ter (iv) decision boundaries (v) phase-error density.

Clearly the pie may be sUced several ways, so let us first say a few words abou t each of the above variables.

(t) Signal constellations: For the purposes of signal evaluation we will consider the existing 16-point constellations QAM, 8-8, and (4,90) shown in Figs. 9a through 9c. T h e circular constellation (4, 90) has signal points equally spaced (i.e., 90 degrees apar t ) on four circles. This large angular spacing of points on the same circle suggests t h a t this constellation will be insensitive to small amounts of phase j i t ter . The rat io of outer radius to inner radius (r j / r i ) for the 8-8 constellation is 1.59, found by Lucky ' to minimize the error ra te in Gaussian noise. 8-8 is an optimized form of A M - P M in which signal points on the outer circle do not lie on the same radial lines as those on the inner circle. I t offers an order-of-magnitude improvement (over A M - P M ) in error r a t e in the presence of Gaussian noise. We will also consider the new circular modulat ion formats 1-5-10, 1-6-9, 5 -11 , shown in Fig. 10. The opt imum ratio ( r j / r i ) is very close to 2 for these constellations as we have determined by equating the three smallest nearest-neighbor distances.

(ii) We consider peak and average S N R ' s , Ppk and , respectively, chosen so t h a t P, (no j i t ter) is < 10"*. These quanti t ies are

T 1

- T -1

- - t 1 1

t 1 1 1 1

f 1

1

- f - ' -

1

1

1 1

i 1 1

1-3 f - -

i 3|

4 1 1

1

1 1

- 4

(a)

(a) quadrature amplitude modulation ^ , - , ; . . . 1-8; (b) modified AM-PM (8-8), pp*/p.r = 1.43; (c) circular

constellation, (4, 90) pp.a/p.T = 1.85.

Fig. 9^Existing signal constellations: (QAM),Pp^/i

946 THE BELL SYSTEM TECHNICAL JOUBNAL , JULY-AUQUST 1973

(b)

Fig. 10New signal consteUatons: (a) 1-5-10, Vv^/V = 1-42; (b) 1-6-9, PPWP .T = 1.525; (c) 5-11, ,/. = 1.31.

defined by Ppk = max ||s) The Tikhonov density will be taken as representat ive of those systems which use a t racking loop, while the peak-to-peak densi ty will be used to model t h e raw (untracked) phase j i t ter .

T h e error r a t e curves a re grouped as follows: Figure 11 shows t h e error ra te vs rms residual phase j i t te r (Tikhonov density) under average and peak power constraints for the six constellations described above. For each constraint , a Gaussian noise power (and thus an S N R ) is assumed which places the curves in a useful operat ing range, and the receiver is presumed to use t h e Gaussian op t imum decision region boundaries. These boundaries are piecewise-linear contours constructed from segments of perpendicular bisectors of lines joining signal points as shown in Fig. 12a for the 1-5-10 constellation. This is equivalent to deciding in favor of the signal point closest in Euclidean distance to the demodulated point. Figure 12b shows a more "prac t ica l" set of decision boundaries for the 1-5-10 constellation. Figure 11 indicates the immuni ty of the (4, 90) constellation to small amoun t s of phase j i t t e r a t t he expense of an error ra te more t han an order of magni tude greater


10-

10-

s u. o

i:

- '

(a)

/

(4.90) 9 ^ 8-8 9 ^ 6 - 1 1-6-1( 6 -9

SI J R - 2 2 dB

0.6 1 1.6 2.6 RMS JITTER IN DEGREES

(b)

(4,90)

Q A M ^

f -

8-8

1-6-9

1-5-1C 6 6-11

s i (R-23 dB

Fig. 11Error rate va jitter with a phase-locked loop: (a) average power constraint; (b) peak power constraint.

t han those of the 1-5-10, 1-6-9, and QAM constellations. I n practical operation wi th a t racking loop, any of these al ternat ive constellations will almost always outperform the (4, 90) constellation.

Figure 13 shows the error rates vs S N R , again under average and peak power constraints, in the presence of Gaussian noise alone (no phase j i t ter) and with 1.5 degrees rms residual phase j i t ter in addit ion

(a) (b)

Fig. 12^Decision region boundaries for part of 1-6-10 constellation: (a) Gaussian-optimum decision region boundaries; (b) practical decision region b o u n d a r i e s .


20 20.5 21 21.5 22 22.5 23 21.5 22 22.5 23 23.5 24 24.5 SNR IN DECIBELS

Fig. 13Error rate vs SNR for channels with and without phase jitter; Gaussian-optimum receiver is assumed to have a phase-locked loop: (a) no jitter, average power constraint; (b) no jitter, peak power constraint; (c) rms jitter = 1.5 degrees, average power constraint; (d) rms jitter = 1.5 degrees, peak power constraint.


10-

10-

lE

10- Li.

o

I I 10-

s 8E

10-

10 -

(a)

/ (4,90)

1-5-1

*aAM

-

SI R-22 IB

0.S 1.5 2.5 3 0

(b)

(4,9 1)

OA / 1- i-10

s H-23i

RMS JITTER IN DEGREES

Fig. 14Probability of error for receiver with "practical" decision region boundaries for the 1-6-10 constellation and with phase-locked loop: (a) average power constraint; (b) peak power constraint.

to the Gaussian noise. Although some of t h e curves shift their relative positions (a t least under the average power constraint) when phase j i t ter is added, the good performances of 1-5-10 and 1-6-9 are maintained. QAM performs respectably and the (4, 90) constellation comes in last.

Figure 14 is similar to a reduced Fig. 11 except t h a t the "prac t ica l" set of decision region boundaries is presumed for the 1-5-10 constellation. As can easily be seen, the Gaussian opt imum boundaries for QAM are also practical boundaries, and the j i t ter - immune (4, 90) constellation is shown to best advantage by presuming Gaussian opt imum boundaries. Under the average power constraint, QAM is superior t o 1-5-10 below about 1.5 degrees rms j i t ter . Under the peak power cons t ra int , 1-5-10 is uniformly superior to QAM. The (4, 90) constellation does not show an advantage until the rms j i t ter reaches 2.5 to 3 degrees.

Figure 15 is a set of error ra te vs S N R plots with practical decision region boundaries for 1-5-10. Curves are shown for Gaussian noise alone (no j i t ter) and 1.5 degrees rms j i t ter . Only the average power


22.5 SNR IN DECIBELS

\ (b) \

90)

1 -5-10

QAM^

Fig. 15^Probability of error vs SNR for Gauaeian-optimum receiver with "practical" decision region boundaries for the 1-5-10 constellation and with phaso-locked loop: (a) no jitter, average power constraint; (b) rms jitter 1.5 degrees, average power constraint.

constraint is presumed. As before, Q A M shows an advantage in the absence of phase j i t ter and still does well in the presence of moderately severe residual phase j i t ter .

Figure 16 presents some da ta for receivers which do not use t racking loops. In this case the peak-to-peak density of eq. (22) describes the raw j i t ter . Curves are plotted vs peak j i t ter imder average and peak power constraints. Some interesting features are the resistance of 1-5-10 up to a threshold of abou t 8 degrees peak-to-peak j i t t e r and the rapid deterioration of the performance of QAM.

Figure 17 shows t h e performance vs S N R for t h e receivers wi thout t racking loops when the peak-to-peak j i t ter is 12 degrees. This is the only instance for which the (4, 90) consteUation looks relatively good, b u t here, too, the 1-6-9 constellation performs slightly bet ter . Q A M , of course, does ra ther poorly.

T h e advan tage of using a phase-locked t racking loop with QAM and 1-5-10 can be seen from the above da t a and a simple calculation. If t he raw phase j i t te r is modeled by

Q(0 = A cos [,< -I-


(b)

(4.1 01 4 OA

= ^ 1

1-6-9

1-5-10

11

s MR-23 dB

8 10 12 PEAK-TO-PEAK JITTER IN DEGREES

Fig. 16Probability of error va. peak-to-peak jitter for Gaussian-optimum receiver without a phase-locked loop: (a) average power constraint; (b) peak power constraint.

where ^ is uniformly distributed from to ir and 2A is t he peak-to-peak ji t ter, then a rule of t h u m b ' suggests t h a t t he residual rms j i t ter out of a t racking loop is of the order of 0.1 X 2A. For i4 = 6 degrees, this rms value is 1.2 degrees. A comparison of the curves of Figs. 11 and 16 shows t h a t substantial ly lower error rates are achieved by the receiver with a t racking loop. The performance of constellation (4, 90) is relatively unaffected by the introduction of a t racking loop.

A further conclusion t h a t can be drawn from the numerical da t a is t h a t t he QAM constellation, which is simple to generate and to demodulate , performs quite well in Gaussian noise alone or (with the aid of t racking loop) Gaussian noise plus phase j i t ter . More circular constellations, such as 1-5-10 and 1-6-9, appear to offer a moderate furthe r advan tage a t t h e expense of greater complexity.

V. OPTIMUM SIGNAL CONSTELLATIONS UNDER A PEAK POWER CONSTRAINT

In this section we discretize the received signal space to obtain a t ractable optimization problem. T h e discretizing is such t h a t t he signal points are selected from a circle containing L points while the received points lie in a circle oi (N > L) points (no te : < L). We make the following two comments concerning this approach to solving


20^ 20.5 21 21.5 22 22.5 23 21.5 22 22.5 23 23.5 24 24.6 SNR IN DECIBELS

Fig. 17Probability of error vs SNR for Gaussian-optimum receiver without a phase-locked loop: (a) peak-to-peak jitter = 12 degrees, average power constraint; (b) peak-to-peak jitter = 12 degrees, peak power constraint.

t he problem:

(i) the level of discretization mus t be fine enough to provide a good approximation to the continuous problem, and

(n) the outer radius mus t be chosen so t h a t for all practical purposes the probabiUty t h a t a received point lies outeide the outer circle is negligible. T h e peak power constraint s imply means t h a t no signal points can be selected outside the L circle.

5.1 Discrete Maximum-Likelihood Formidation If we denote the received point, z, by "i" and the t ransmit ted signal,

s


P r [receive " i " ] (40)

Subst i tu t ing (40) in (39) and recalling t h a t t he t ransmit ted signals are equiprobable gives

P r [ co r rec t ] = P r [receive "z" | s end " A " ] t'_i = p ( | A ) , (41)

and t he op t imum constellation is the signals (or columns) t h a t maximize

( | ) . (42) -1

Note t ha t since p{i\ti) is the maximum entry in the i t h row of the transi t ion matr ix, the problem is one of selecting out of L columns such t h a t t he sum of the row maxima is maximized. T h e error ra te may be determined from (41).

S.2 Optimum Constellations A heuristic program for solving the combinational optimization

problem posed by (42) has been developed by Kemighan and Lin . ' Their process is based upon i terat ive improvement of either known initial constellations or random initial s ta r t s . For each s tar t , a "locally

tional densities p,{z) over an appropriate region. I t is convenient to work with the maximum-likelihood receiver (i.e., t he opt imum decision boundaries are used) which receives "i" and declares t h a t " " was t ransmit ted, where

p ( i | ) > P ( t | j), j 9^ j , ti = \ , 2 , - M . (38)

Note t h a t we have (temporarily) fixed the points in the signal constellation. I t is easy to see t h a t t he probabili ty of being correct is given by

P r [cor rec t ] = P r [correct | receive " i " ] P r [receive " t " ] , (39)

bu t by Bayes rule

P r [correct I receive " t " ]

= P r [send receive " t " ]

_ P r [receive " t " [send " A " ] P r [send " t " ]


Rg. 1 B. W.

itimum signal constellation: Peak SNR = 27 dB, no jitter (courtesy of and S. Lin).

o p t i m u m " solution is found in the sense t h a t no change of position of a single signal can improve the criterion. The heuristic process is very fast and, for the resolution we require, 20 random s t a r t s can be pursued to completion in 25 seconds. Fo r part icular values of rms phase j i t t e r and noise power we find, among the best of 20 local opt ima, reaffirmation of known solutions and in some instances new competit ive constellations. For example, for a peak signal energy of S N R = 22 d B , Figs. 18, 19, and 20 give the best among the 20 local opt ima for an rms of 0, 1.5, and 3 degrees, respectively. As expected, the 0-degree solution has a 5-11 character and the 1.5-degree solution has a 1-5-10 character. On t h e other hand, t h e 3-degree solution is somewhat of a surprise; i t has a 1-6-9 character. The (4, 90) consteUation, which is


Fig. 19Optimum signal constellation: Peak SNR = 27 dB, 1.5 degrees rms jitter (courtesy of B. W. Kemighan and S. Lin).

best a t 3 degrees among heretofore considered designs, has an error ra te only a few percent worse t han t h a t of the 1-6-9 constellation.

A byproduct of the development of the above procedure is the demonstrat ion of the fact t ha t numerical quadra tu re routines offer a competit ive al ternat ive to asymptot ic techniques and bounding methods for the estimation of system error rates.

VI. CONCLUSIONS

Comparisons have been made of several well-known two-dimensional signal formats in the presence of Gaussian noise and phase j i t ter , a t high signal-to-noise ratios and imder bo th peak and average power constraints. I t has been demonstra ted tha t , under an average

956 THE BELL SYSTEM TECHNICAL JOUBNAL, JULY-AUOUST 1973

Fig. 20Optimum signal constellation: Peak SNR = 27 dB, 3 degrees rms jitter (courtesy of B. W. Kemighan and S. Lin).

power constraint for systems which have a high-quaUty phase-locked loop (rms residual j i t te r < 1 degree), QAM had the lowest error r a t e of all candidate constellations. If t he residual j i t ter is < 1 . 5 degrees rms, t he 1-5-10 constellation becomes extremely a t t rac t ive since it is immune to phase j i t ter in this range and provides the same asymptot ic (no-jitter) error ra te as Q A M . For small amounts of j i t ter , 1-5-10 and QAM have a 2-dB S N R advantage over the (4,90) constellation which is " i m m u n e " to phase j i t ter . Both these constellations offer a 0.5-to-l-d B advantage in S N R over conventional A M / P M signaling techniques. Thus , under an average power constraint, bo th QAM and 1-5-10 meri t consideration.

Fo r a peak power constraint , in addit ion to making comparisons similar to the above, we have been able to a t t a in the op t imum signal


20

dz [ z = (u, o ) ]

constellations for various levels of j i t ter . Our comparisons indicate t ha t , for j i t te r < 1 degree, QAM suffers a 1.5-dB S N R penal ty with respect to the 5-11 constellation, while 1-5-10 suffers a 0.1-dB penal ty. At 1.5 degrees rms j i t ter , 1-5-10 again is superior to both QAM and 5-11 (by 4 and 1 d B respectively).

Based upon the peak and average power constraints, the new modulation format 1-5-10 appears to make extremely efficient use of available signal power and for a slight increase in modula t ion/demodulation complexity offers considerable immuni ty to moderate ( < 1 . 5 degrees rms) residual phase j i t ter . QAM systems which employ high-quali ty phase-locked loops will also be operating very efficiently, provided t h a t the residual phase error is < 0 . 8 degree rms.

VII. ACKNOWLEDGMENTS

We would like to thank J. Salz and J. E . Mazo for valuable discussions concerning this investigation.

APPENDIX A

Method of Laplace Let g(x) and h(x) be continuous real fimctions on [a , 6 ] where A(a;)

is also twice continuously differentiable. Then, if h(x) a t ta ins a single maxima a t c (a < c < 6), we have t h a t

j ' g(x)enikmodj. ^ p(c)e

958 THE BELL SYSTEM TECHNICAL JOUBNAL, JULY-AUQUST 1973

Rewriting / as

l i m / / e x p { , U ( z ) } / / ^ L

Gr

e(tu 0

/ ^ e(i/*)*(0)y* yjexp . 1 i:A(z) - ( 0 ) ] | We shall proceed to integrate with the exponent in the integrand replaced by its local representation

g(i/t)(o) J' j'^ exp IJ [A,(0)t -f- A(0 )uV2 + A.(0)wi/] dvdu.

The absence of A(0 ) and A , , ( 0 ) follow directly from (i). In tegra t ing (dv) we get

/ ^ e 0,


k*'* A . ( O ) V A . .(0)\(0)

Since for t small enough A,(0) [A2,(0)/ABU(0)]' < O, we conclude

A,(O) VA(O )

In error ra te computat ions one is often integrat ing over t he exterior of a convex polygon . In t he body of this paper we encounter the case where h{z) has a finite numbe r of global maxima on the boundary , a t mos t one on each side, and none a t the vertices . Le t zH be t he n t h local maximum . M a p t he exterior half-space containing z into the upper-half plane via a rotat ion , composed with a t ranslat ion taking z * 0. Then the method of the las t paragraph can be applied . T h e process is repeated for each maxima and the results sum to the required a symp totic es t imate of the exterior integral . T h e fact t h a t the exterior half-spaces containing distinc t points of maxima may overlap is of no consequence . Fur thermore , in our applications , A(z) is symmetr ic with respec t to each zi and the process need only be completed once and t he answer multiplied by t he multiplicity of t he maxima .

APPENDIX

The Nature of (, y) . 1 Introduction

Le t V be a vector space endowed with a scalar produc t (x, y) and a norm derived therefrom . I t is no t known whether dy(x, y):VX V * \R\ defined by

dy{x, y) = (||x|| + ||y||' + 2- ' - 2|||||%||-|-2- + 7 - ) " is a metric for any values of (0 < < >). No r is it known whether


Notice dy(x, y) > 0 unless = y.lf dy(x, y) is a metric on V X F for a part icular value of y then i t is a metric for all (0 < y < ) ; this is easily obtained by emplojng the mapping | Vrlz. B.2 One-Dimensional Case

In the special case F = we can show di{x, y) is a metric . In one dimension we have ( the " 1 " subscrip t will now be suppressed)

Mr V) ^ xy + i^o ^ '^' UV(i + j/)' + 2 ' | xy + 1 ^ 0 '

T o show t he tr iangle inequality , first notice tha t , if three points a, b, and c are on the same side of zero, the distances are all EucUdean . So for t he remaining cases to be investigated we assume one poin t ha s a different sign t han the o ther two . Notice dix, y) = d(x, y) so we lose no generaUty by assuming c ^ 6 ^ 0 ^ a. Two subcases remain : (i) only d(c,a) is non-Euclidean ; (ii) d(c, a) and d{b, a) are non -Euclidean . Fo r (i) the distances involved are (c 6), (b a), and | [ ( o -F c) -t- 2 ] " | .Now | [ ( o -I- c ) - | - 2 ' ] " | g c - since, by squaring, this is equivalent to 1 + ac 0. To show

(c - 6) (6 - a) -I- I [ (a -|- c) H- 2 ] ' / ' | and

(6 - o) (c - 6) + I [ (o -I- c) -I- 2]"| ,

i t is enough to show

[(c -I- a) - 25] g C(a -|- c ) ' + 2 ' ] since squaring both sides when t he right-hand side is positive can only weaken t he inequaUty . The last inequality can be simplified to b* be ^ 1 + ab for which t he left-hand side is negative and the right-hand side is not . On t he other hand , (ii) is immediate since c b, V(o -I- c) -I- 4, and V(6 + a)* + A can be identified as sides of a t r iangle with apex (a, 2) and base points c and b. Notice d does no t have the midpoin t property since d(10, 10) = 2, ye t t he only points y sa t i s fy ing d (10 , y) = I are 9 and 11 , b u t bo th d (9 , - 1 0 ) d ( l l , - 1 0 ) exceed 2 .

This last observation shows t h a t the open spheres in this metric space are no t all connected . In two dimensions , t he boundaries of certa in open spheres are disconnected ; specifically, it can be shown t h a t for certain values of g > 0

{y\d(x,7)=q] is a disconnected set if ||y||'7 > 1.


\^\\xnyr + 2y-^{x,j) + y - ' \ . - ( i + .


. V a + b c

and the infimum is over all triangles for which the bracketed expression is positive. This geometric extermal problem is disposed of by subst i tu t ing a triangle of the form depicted in Fig. 2 1 . I t follows easily lim, ^ 0 = 0 . Thus = 0 and the only d(x, y) metric is t he obvious one. A straightforward subst i tut ion of the three vectors depicted above into t h e "tr iangle inequal i ty" for (, y) yields again t h a t for no fixed 7 > 0 can the triangle inequaUty hold for this family of triangles.

Strikingly, for the class of triplets in Fig. 2 1 , it is easily demonstrated, via subst i tut ion, t h a t t h e triangle inequali ty for dy(x, y) holds for an open interval including 7 = 0 !

B.4 Metric on the Boundary of the Disc We end on a positive no te : (, y) is a metric on the boundary of the

uni t disc for a nontrivial interval [ 0 , ] . * I t is enough to show inf

* We anticipate applications in phase-modulation systems.

For (, y ) , the midpoint proper ty and all requirements for a metric hold except for the triangle inequality. The triangle inequali ty question is easier for us to investigate for S{x,j) t h a n for (, y) as it develops into a geometrical extremal problem which we can handle. In solving the extremal problem, a class of vector tr iplets in the uni t disc emerge for which no value of > 0 exists such t h a t the triangle inequality holds uniformly for the class. Although these tr iplets arise in the analysis of (X, y) , they serve jus t as ruinously for (, y) .

In order to discuss this geometric extremal problem, we require some notat ion. Suppose we have a triangle (nontrivialpositive area) in the uni t disc with side lengths a, b, and c and opposing vertices A, B, and C, respectively. Let ha, ht, and A . denote the distance from t h e origin to t he line containing the side of length a, b, and c, respectively.

Wri te the "tr iangle inequal i ty" expression for 5(x, y ) :

U-7\\(l-hl,)k ||x - z|| ( l - I A I . )

+ l | z - y | |

Changing to the notat ion jus t introduced , we have directly t h a t the triangle inequality holds for 0 ^ ^ where = inf (, b, c).


Fig. 21Infimizing family of vectors.

(, b,c) > 0 where the infimum is over those triangles circumscribed by t h e uni t circle for which is positive. Now A? = 1 - oV4 and similarly for hi and so

Dividing gives

(a, 6, c) = ( l - ^

So > 0 if and only if

o ' + 6 + c - o6 + ac + >c -

abe sup . < 00 . ^ a + b c

3abc a + b c

A t th is point , we mus t digress and recall some plane geometry from Ref. 12. Fi rs t abc/i is t he area of the tr iangle and 2" ' (a + 6 + c) is called a semiperimeter . Given any t r iangle ABC, extend t h e two lines emanat ing from t he apex A. Construc t bisectors to the two exterior angles complementary to the angles and C respectively. The bisectors meet in a point equidistant from the three lines containing a, b, and c. The circle tangent to these three lines is called an excircle. See Fig. 22. T h e center of the excircle (where the bisectors meet) is called the excenter and of course the radius is called the exradius. Each t r i angle has th ree excircles. Finally from Ref. 12 we need:

9 6 4 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUQUST 1973

C

Fig. 22Exdrde tangent to c.

10,11

UPPER BOUND ON EX RAD 11

I

Fie. 23Upper exradius bound (or exdrdes tangent to c. Since 0 ^ c ^ 2, the exami are uniformly bounded.


Theorem: An exradius of a triangle is equal to the ratio of the area to the difference between the semiperimeter and the side to which the excircle is tangent internally.

So if the set of exradii of triangles circumscribed by the unit circle is uniformly bounded away from infinity, then > 0 and f 7 | o < T < T metrics .

To complete t he proof, consider any triangle with vertices on t h e uni t disc boundary . With an isometric t ransformation of the circle in to itself, we can s i tuate c horizontally with apex C above c in the left half-plane. T h e more obtuse the exterior angles a t A and B, the larger the excircle t angen t to c. So take a horizontal line o' through and a line b' going through A and (0, 1) and replace a and b with a' and b'. Clearly, bisectors constructed using a' and b' will intersect a t a point further away from c than the excenter of any excircle tangent to c. By similar triangles, the primed bisectors intersect a t a distance 2(1 dr Vl cV4) from c where the sign is plus if c lies in the lower half-plane and negative otherwise (Fig. 23).

REFERENCES

1. Lucky, R. W., and Hancock, J. C , "On the Optimum Performance of N-&Ty Systems Having Two Degrees of Freedom," Trans. Conunun. Syst. (March 1962).

2. Salz, J., Sheehan, J. R., and Paris, D. J., "Dato Transmission by Combined AM and PM," B.S.T.J., SO, No. 7 (September 1971), pp. 2399-2419.

3. Thomas, C. M., "Amplitude Phase-Keying with iV-ary Alphabets: A Technique for Bandwidth Reduction," Int. Telemetering Conf. Proc, VHI (October 1972).

4. GitUn, R. D., Ho, E. Y., and Mazo, J. E., "Passband Equalization of Differentially Phase-Modulated Dato Signals," B.S.T.J., St, No. 2 (February 1973), pp. 129-238.

5. Viterbi, A. J., Principles of Coherent Communication, New York: McGraw-Hill, 1966, pp. 86-96 and 118-119.

6. Kernighan, . W., and Lin, S., "Heuristic Solution of a Signal Design Optimization Problem," Proc. Seventh Annual Princeton Conf. on Information Sciences and Systems, March 1973.

7. Foschini, G. J., Gitlin, R. D., and Weinstein, S. B., "Optimization of Two-Dimensional Signal Constellations in the Presence of Gaussian Noise," to be published in IEEE Transactions on Communications.

8. Mazo, J. E., "A Note on Metrics and Metric Convexity," unpublished work. 9. Falconer, D. D., unpublished work.

10. Papoulis, ., The Fourier Integral and lU Application, New York: McGraw-Hill, 1962.

11. Jones, D. S. "Asymptotic Behavior of Integrals," SIAM Review, I4, No. 2 (April 1972) pp. 286-317.

12. Davis, D. R., Modem College Geometry, Reading, Mass.: Addison-Weeley, 1957.

Documents

On the Selection of a Two-Dimensional Signal Constellation in the Presence of Phase Jitter and Gaussian Noise