Performance Analysis of M-PSK SignalConstellations in Riemannian Varieties

Rodrigo Gusmao Cavalcante and Reginaldo Palazzo Jr.

Departamento de Telematica, FEEC-UNICAMP, Campinas, Brazil{rgc,palazzo}@dt.fee.unicamp.br

Abstract. In this paper we consider the performance analysis of a dig-ital communication system under the hypothesis that the signal con-stellations are on Riemannian manifolds. As a consequence of this newapproach, it was necessary to extend the concepts related to signal con-stellations, symbol error probability and average energy of the signalconstellation to the theory of differentiable manifolds. The importantresult coming out of this formalism is that the sectional curvature ofthe variety is a relevant parameter in the design and in the performanceanalysis of signal constellations.

1 Introduction

The main objectives to achieve when designing a new communication system arethat the new system be less complex and have a better performance under theerror probability criterion. Equivalently, for a fixed error rate, the signal-to-noiseratio be less than that of the known systems.

In this direction, we consider each one of the block diagrams, Fig. 1, as a set ofpoints Ei together with a metric di. This allows an interpretation of each one ofthe blocks as a metric space (Ei, di). As an example, the source encoder consistsof a set of codewords E1 with an associated distance d1, where this distancemay be the chi-squared distance. The channel encoder has associated a set ofcodewords E2, in general, with the Hamming distance d2. The channel consistsof a set of random points with the Euclidean distance (when the additive whitegaussian noise is taken into consideration). In general, each one of the decodersmake use of the same metric as the corresponding encoders with the purpose ofmatching the corresponding metric spaces.

Therefore, the objective of the metric space point of view is to determinethe geometric and algebraic characteristics associated to each one of the set ofpoints Ei as well as the properties and conditions that must be satisfied by thetransformations that will connect the distinct metric spaces such that a betterperformance be achieved by the new communication system.

In general, the metric space (E3, d3), consisting of the blocks modulator,channel, and demodulator, is associated to the Euclidean space with its usualmetric. This is the reason the Euclidean geometry is used in the design of thesignal constellations in the modulator in order to minimize the noise action whichin general is the gaussian noise.

M. Fossorier, T. Hoeholdt, and A. Poli (Eds.): AAECC 2003, LNCS 2643, pp. 191–203, 2003.c© Springer-Verlag Berlin Heidelberg 2003

192 R. Gusmao Cavalcante and R. Palazzo Jr.

Decoder Demodulator

ModulatorEncoder

Demodulator

NoiseChannel

ModulatorChannel

ChannelSource

SourceSource

Receptor

(E1, d1) (E3, d3)

(E3, d3)(E2, d2)(E1, d1)

(E2, d2)

Fig. 1. Communication system model

On the other hand, the dashed block associated with (E3, d3) may be charac-terized as a discrete memoryless channel. Under this characterization, the mainmotivation of this paper comes from the results shown in [6], where it is iden-tified and analyzed a substantial number of discrete memoryless channels andtheir embedding on compact surfaces with and without borders. Still in [6] it isconsidered some channels, for instance the C8[3] channel with 8 input, 8 outputand 3 transitions per input, having the property that its embedding occurs in aminimal surface (catenoid). To the best of our knowledge, minimal surfaces werenot considered previously in the literature. Since there are real channels meetingthe previous condition, this motivates us to design signal constellations and torealize the performance analysis of the corresponding communication system.

Based on these facts, naturally comes the following question: What is theperformance of a communication system when the geometric structure associ-ated with a signal constellation is not the Euclidean one? A partial answer to thisquestion can be found in [1] and [2], for the particular case of signal constellationsin hyperbolic spaces. In [8] it was considered the case when the signal constel-lations are Riemannian varieties, however focusing on signal constellations onminimal surfaces.

In this paper, we focus on the analysis of the metric space consisting ofthe three blocks, namely: the modulator, channel and demodulator as shown inFig. 1, aiming at the construction of signal constellations in Riemannian varietiesand the corresponding performance analysis. In order to achieve this goal, wehave to extend the concepts of error probability, average energy of the signalconstellation and of the noise to the context of Riemannian varieties.

2 Review of Riemannian Geometry

To the best of our knowledge, the problem of designing signal constellations inRiemannian variety has not been considered previously in the literature. Thus,in order to do that we review some important concepts needed to what follows.

Performance Analysis of M-PSK Signal Constellations 193

Definition 1. A differentiable variety of dimension n consists of a set M anda family of bijective mappings xα : Uα ⊂ R

n → M of open sets Uα of Rn in M

such that

1. ∪αxα(Uα) = M .2. For every pair α and β, with xα(Uα) ∩ xβ(Uβ) = W �= φ, the sets x−1

α (W )and x−1

β (W ) are open sets in Rn and the mappings x−1

β ◦xα are differentiable.3. The family {(Uα,xα)} is maximal with respect to (1) e (2).

In order to deal with certain quantities such as length, area, angles, and soon, with respect to differentiable varieties we have to establish the concept of aRiemannian metric.

Definition 2. A Riemannian metric in a differentiable variety M is a corre-spondence associating to each point p of M a dot product 〈 , 〉p (that is, apositive definite, symmetric, bilinear form) in the tangent space TpM , varyingdifferentiably in the following sense: If x : U ⊂ R

n → M is a system of localcoordinates at p, with x(x1, x2, . . . ,xn) = q ∈ x(U), then 〈 ∂

∂xi(q), ∂

∂xj(q)〉q =

gij(x1, . . . ,xn) is a differentiable function in U .

A differentiable variety with a given Riemannian metric is called a Rieman-nian variety. We may use the Riemannian metric to determine the length of acurve c : I ⊂ R → M constrained to the closed interval [a, b] ⊂ I as follows

lba(c(t)) =∫ b

a

⟨dc

dt,dc

dt

⟩1/2

dt , (1)

where dc(t)dt denotes a vector field originating from c(t), and called the tangent

field of c(t).At this point it is worthwhile introducing a class of curves in M , called

geodesics, having the property of minimizing the length of a segment joiningtwo closest points in M .

A parameterized curve γ : I → M is a geodesic γ(t) = (x1(t), . . . ,xn(t)) in asystem of coordinates (U,x) if and only if it satisfies the following second orderdifferential equations

d2xk

dt2+∑i,j

Γ kij

dxi

dt

dxj

dt= 0, k = 1, . . . , n , (2)

where Γmij are the Christoffel symbols of a Riemannian connection M given by

Γmij =

12

∑k

{∂

∂xigjk +

∂

∂xjgki − ∂

∂xkgij

}gkm ,

where gkm is an element of the matrix Gkm, whose inverse is Gkm.Next, we present the definition of sectional curvature, a parameter which has

been shown to be relevant in the design of signal constellations in Riemannianvarieties.

194 R. Gusmao Cavalcante and R. Palazzo Jr.

Definition 3. [4] Given a point p ∈ M and a bi-dimensional subspace Σ ⊂ TpMthe real number K(x, y) = K(Σ), where {x, y} is any basis for Σ, is calledsectional curvature of Σ in M , defined by

K(x, y) =〈R(x, y)x, y〉

|x ∧ y|2 ,

where R(x, y) denotes the curvature of a Riemannian variety M , which intu-itively measures how close is the Riemannian variety to the Euclidean one, and|x∧y| =

√|x|2 + |y|2 − 〈x, y〉2 denotes the area of a bi-dimensional parallelogramdetermined by the pair of vectors x, y.

3 Signal Constellations in Riemannian Varieties

In designing signal constellations in a Riemannian variety we have to consider aRiemannian metric in each one of the blocks, namely: modulator, channel anddemodulator, as shown in Fig. 1, such that the combination of them leads to ametric space (E3, d3). These metrics do not need to be the same, however for thecommunication system to achieve a better performance it is necessary to satisfythe metric matching condition.

Definition 4. A signal constellation X = {x1, . . . , xm} in an n-dimensionalRiemannian variety M with a coordinate system (U,x) is a set of n-dimensionalpoints.

{x1 = (x11, . . . ,xn1), . . . , xm = (x1m, . . . ,xnm)} ⊂ U.

Since the signal constellations are defined in a Riemannian variety M , thenwe have to use the Riemannian metric Gij , in order to construct and to realizethe performance analysis of such signal constellations. Therefore, the distance tobe used between any two given points in M will be the least geodesic distancebetween the points in M .

We may think of the noise action in the channel as a transformation thattakes the transmitted signal xm ∈ M to the received signal y ∈ M . We considerthat this transformation is given by

y = expxm(v) , v ∈ TxmM , (3)

where TxmM is the tangent plane of M at xm, and the mapping expxm

:TxmM → M is called an exponential mapping. Geometrically, expxm

(v) de-notes a point in M when it moves a distance equal to |v| from xm on a geodesicpassing by xm with velocity v

|v| .Notice that the way we have defined the noise in (3) requires that the ex-

ponential mapping be defined for every v ∈ V ⊂ TxmM , where V is an open

set of TxmM . Therefore, if we assume that V is the sample space of the random

vector v = (v1, . . . , vn), then the noise is characterized by the probability density

Performance Analysis of M-PSK Signal Constellations 195

function of v. Hence, the probability density function of y, given that xm wastransmitted, is given by

pY (y/xm) = pV (v = (exp−11 (y), . . . , exp−1

n (y))|J | , (4)

where |J | denotes the absolute value of the Jacobian of J , given by

J =

dexp−11 (y)

dy1

dexp−11 (y)

dy2. . .

dexp−11 (y)

dyn

......

. . ....

dexp−1n (y)

dy1

dexp−1n (y)

dy2. . .

dexp−1n (y)

dyn

.

Let Rm be the decision region of xm, that is, the set representing all thepoints which are decided as xm. Thus, if xm is the transmitted signal, then theprobability that the decision will be in favor of a signal which is not in Rm, isgiven by

Pe,m = 1 −∫

Rm

pY (y/xm) dx1 . . . dxn . (5)

It is important to realize that the error probability associated to xm, (5),does not depend on a particular coordinate system (U,x) and that it is invariantby isometry. The average symbol error probability, Pe, of a signal constellationin a Riemannian variety may be written as

Pe =∑X

P (xm)Pe,m , (6)

where P (xm) denotes the a priori probability of the signal xm.The average energy of a signal constellation X is given by

Et =∑X

P (xm)d2(xm, x) , (7)

where d2(xm, x) is the squared geodesic distance between the signal xm and thecenter of mass of the signal constellation x. It is known that the center of massof a signal constellation is the one that minimizes the average energy. Therefore,to determine x we have to take the derivative of the average energy with respectto x, and assume that it is zero at x = x. Hence, x is the unique solution to

∂Et

∂xj

∣∣∣∣∣x=x

=∑X

p(xm)d(xm, x)∂d(xm, x)

∂xj

∣∣∣∣∣x=x

= 0 , j = 1, . . . , n . (8)

In order to have an explicit equation for the signal-to-noise ratio, we haveto determine the noise power in a Riemannian variety M given that xm wastransmitted. This is accomplished by defining the noise power as

σ2 =∫

M

d2(y, xm)pY (y/xm) dx1 . . . dxn . (9)

196 R. Gusmao Cavalcante and R. Palazzo Jr.

As previously mentioned, the noise probability density function in a Rieman-nian variety M is associated to the probability density function of the randomvector v ∈ TxmM . For instance, if v = (v1, . . . , vn) is a random vector such thatits components vj , j = 1, . . . , n, are gaussian random variables, then v is alsogaussian, since TxmM besides being an n-dimensional vector space the transfor-mation is linear. In order for the probability density function of y be well definedwe have to assume that the Riemannian variety M is complete, that is, for everyxm ∈ M the exponential mapping expxm

(v), is defined for every v ∈ TxmM .In the particular case when the random variables vj , j = 1, . . . , n, are gaus-

sian with zero mean and equal variances, we define the probability density func-tion of y given that xm was transmitted, as

p(y/xm) = k1e−k2d2(y,xm)

√det(Gij) , (10)

where d2(y, xm) is the squared geodesic distance between the received signal yand the transmitted signal xm,

√det(Gij) is the volume element of the variety

M , and k1, k2 are constants satisfying the condition∫

M

k1e−k2d2(y,xm)

√det(Gij) dx1 . . . dxn = 1 . (11)

As in the case of finding the noise power given by (9), the constants k1 andk2 given by (11) depend on the transmitted signal xm. However, if there existsan isometry taking one signal to the other, then σ2, k1 and k2 assume the samevalues independently of the signal in consideration. This fact was used in [8]in the performance analysis of two 4-PSK signal constellations on the Enneperminimal surface.

Among the Riemannian varieties, the ones with constant sectional curvatureare the simplest to deal with when considering the construction and the per-formance analysis of the signal constellations. This is a consequence of the factthat these spaces are homogeneous, that is, every point is on an equal footingfrom every other point. As a consequence, the noise always acts in the sameway independent of the point in consideration. Hence, σ2, k1 and k2 assumethe same values for every point in M . Since the spaces with constant curvaturehave a great number of local isometries, it is clear that we may naturally havegeometrically uniform signal constellations in such spaces.

On the other hand, determining all the geometrically uniform signal con-stellations in the space of constant curvature implies that we have to find allthe subgroups that act transitively in these spaces. However, this is not an easyproblem to solve.

4 Establishing the PDF in Spaces with ConstantCurvature K

In this section, we consider an example of a signal constellation of the type M-PSK in a bi-dimensional Riemannian variety. Due to the geometric structure

Performance Analysis of M-PSK Signal Constellations 197

of this type of signal constellation, we employ the geodesic polar coordinatesystem, (ρ, θ), [3]. In this system, the coefficients g11(ρ, θ), g21(ρ, θ) = g12(ρ, θ)and g22(ρ, θ) of the Riemannian metric must satisfy the following conditions

g11 = 1, g12 = 0, limρ→0

g22 = 0, limρ→0

(√

g22)ρ = 1 . (12)

According to (2), a geodesic γ : I ⊂ R → M in polar coordinates, γ(t) =(ρ(t), θ(t)) must satisfy the following system of second order partial differentialequations {

ρ′′ − 12 (g22)ρ(θ′)2 = 0 ,

θ′′ + (g22)ρ

g22ρ′θ′ + 1

2(g22)θ

g22(θ′)2 = 0 .

From (1) the arc length of the geodesic γ(t), from t1 to t2, is given by

lt2t1 (γ(t)) =∫ t2

t1

√(ρ′)2 + g22(θ′)2 dt .

If the coordinate system is the polar one, then g22 may be found as thesolution to the following second order differential equation, [3],

(g22)ρρ + Kg22 = 0 , (13)

where K denotes the sectional curvature of the variety M . In the particular casewhen M is bi-dimensional, K is known as the gaussian curvature. For simplicity,we assume that K is constant. Next we consider the solutions to (13) for K = 0,K > 0 and K < 0. Since g22 must also satisfy (12), we finally have

g22(ρ, θ) =

ρ2 , if K = 0 ,

1K sin2 (

√Kρ) , if K > 0 ,

1−K sinh2 (

√−Kρ) , if K < 0 .

We associate to the Euclidean space E2 the curvature K = 0, to the sphericalspace S2 the curvature K > 0 and constant, and to the hyperbolic plane H2 thecurvature K < 0 and constant.

When considering the Euclidean space E2, the geodesic distance between anytwo given points z1, z2 ∈ E2 is given by

dE = |z1 − z2| , (14)

where | | denotes the absolute value of zi = riejθi , i = 1, 2.

When considering the spherical space S2, we can show that the geodesicdistance between any two given points z1, z2 ∈ S2 is given by

dS =2πl√

K± 1

j√

Klog

|1 + z1z2| + j|z1 − z2||1 + z1z2| − j|z1 − z2| , (15)

198 R. Gusmao Cavalcante and R. Palazzo Jr.

where l is the number of times that a geodesic passes by the point z1 or itsantipodal, until arriving at z2, zi = rie

jθi and ri = −j(ejρi

√K −1)/(ejρi

√K +1),

i = 1, 2.When considering the hyperbolic space H2, we can show that the geodesic

distance between any two given points z1, z2 ∈ H2 is given by

dH =1√−K

log|1 − z1z2| + |z1 − z2||1 − z1z2| − |z1 − z2| , (16)

where zi = riejθi and ri = (eρi

√−K − 1)/(eρi

√−K + 1), i = 1, 2.By use of (10), (11) and (9) we can find the probability density function,

pY (y/xm), as well as the noise average energy, σ2, for each one of the spacesE2, S2 and H2. As previously mentioned, these spaces are homogeneous and,therefore, the values of k1, k2 and σ2 are the same independent of the trans-mitted signal xm. Thus, in order to simplify the calculations, we assume thatthe transmitted signal is xm = (0, θ). Hence, for y = (ρ, θ), we end up with thefollowing cases:

1. For K = 0, the probability density function is given by

pE(ρ, θ) = k1e−k2ρ2

ρ ,

where k1 = k2/π and σ2 = 1/k2. Since the coordinate system is polar, theresulting probability density function pE(ρ, θ) is Rayleigh.

2. For K < 0, the probability density function is given by

pH(ρ, θ) =k1√−K

e−k2ρ2sinh (

√−Kρ) ,

with

k1 =π−3/2eK/4k2

√−Kk2

erf(√−K/2

√k2)

,

where erf(x) denotes the error function defined for every w ∈ C as

erf(w) =2√π

∫ w

0e−z2

dz .

The noise average energy is given by

σ2 =2√−Kk2e

K/4k2 +√

πerf(√−K/4k2)(2k2 − K)

4k22√

πerf(√−K/4k2)

.

3. For K > 0, the probability density function is given by

pS(ρ, θ) =k1√K

e−k2ρ2 | sin (√

Kρ)| ,

where

k1 =√

K

2π

( ∞∑i=0

∫ (i+1)π

iπ

(−1)ie−k2ρ2sin (

√Kρ) dρ

)−1

.

Performance Analysis of M-PSK Signal Constellations 199

For k2 � K, then k1 may be approximated by

k1 ≈ iπ−3/2eK/4k2√

Kk2

erf(i√

K/2√

k2).

Consequently, the noise average energy is given by

σ2 ≈ 2i√

Kk2eK/4k2 +

√πerf(i

√K/4k2)(2k2 − K)

4k22√

πerf(i√

K/4k2).

In order to show that (10) is well defined, we consider the probability densityfunction for K = 1, K = 0, K = −1, and the noise average energy equal to 1.Substituting these values into the previous equations, we end up with

pS(ρ, θ) = 0.31264e−0.8013ρ2 | sin (ρ)| ,

pE(ρ, θ) = e−ρ2ρ/π , (17)

pH(ρ, θ) = 0.31572e−1.1492ρ2sinh (ρ) .

As expected, we notice from (17) that the random variable θ is uniformlydistributed in the three spaces, that is pS(θ) = pE(θ) = pH(θ) = 1/2π. Althoughthe analytical expressions in (17) may seem different, graphically, Fig. 2(a), theyseem to be the same up to the fourth decimal digit, as can be seen in Fig. 2(b).However, the most important fact can be observed in Figs. 2(c) and 2(d), wherethe cumulative distribution functions PS(ρ), PE(ρ) and PH(ρ) are practicallythe same although there is a difference from the fourth decimal digit on. Webelieve that this difference is related to the approximations to the error functioncalculations. Due to the consistence of the results obtained, we conclude that(10) is well defined.

5 Performance Analysis of M-PSK in Spaces with K �= 0

Figure 3 shows the average error probability versus the signal-to-noise ratio forthe 4-PSK signal constellation in four different spaces with constant sectionalcurvature equal to 1, 0, -1, and -2. As may be seen, the average error probabilitydiminishes when the sectional curvature get smaller and negative. Figure 4 showsthe average error probability versus the signal-to-noise ratio for the 8-PSK and16-PSK signal constellations in spaces whose sectional curvature equals 0 and -1.As noticed for the 4-PSK, the average error probability diminishes for decreasingvalues of K. This fact shows that the curvature K is an important parameter tobe considered in the design of signal constellations.

As mentioned previously, the average error probability of an M-PSK sig-nal constellation in spaces with constant negative sectional curvature is upper-bounded by the average error probability of the M-PSK signal constellation inthe Euclidean space. This may be explained by the fact that the Pe decreases

200 R. Gusmao Cavalcante and R. Palazzo Jr.

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ρ

pS(ρ)

pE(ρ)

pH

(ρ)

(a) PDF

0 0.5 1 1.5 2 2.5 3 3.5−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

ρ

pE(ρ)−p

S(ρ)

pE(ρ)−p

H(ρ)

(b) Difference

0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ρ

PS(ρ)

PE(ρ)

PH

(ρ)

(c) CDF

0 0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2

3

4

5x 10

−3

ρ

PE(ρ)−P

S(ρ)

PE(ρ)−P

H(ρ)

(d) Error

Fig. 2. PDF and CDF in spaces with curvatures 1, 0, and -1

−6 −4 −2 0 2 4 6 8 10 1210

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Pe

S/N (dB)

K=1K=0K=−1K=−2

Fig. 3. Pe × S/N for 4-PSK in spaceswith curvatures 1, 0, -1 and -2

−6 −4 −2 0 2 4 6 8 10 1210

−5

10−4

10−3

10−2

10−1

100

Pe

S/N (dB)

8−PSK K=016−PSK K=08−PSK K=−116−PSK K=−1

Fig. 4. Pe×S/N for 8-PSK and 16-PSKin spaces with curvatures 0 and -1

when the minimum distance of the constellation, dmin, increases. On the otherhand, the minimum distance increases when the sectional curvature decreases,for a fixed average energy of the signal constellation.

Performance Analysis of M-PSK Signal Constellations 201

2 4 6 8 10 12 14 160.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

M−PSK

d K/d

0

K=1K=0K=−1K=−3

Fig. 5. dK/d0 × M for Et = (π/2)2

0

K1

K2K3

x2x1 . . . X

K

xM

Fig. 6. Sectional curvature versus signalconstellation X

For instance, in Fig. 5 we notice that for an M-PSK constellation withEt = (π/2)2, the ratio between the minimum distance in a space with cur-vature K, dK , and the minimum distance in the Euclidean space, K = 0, d0, isgreater than one for K < 0 and less than one for K > 0. Next, we compare theperformances achieved by the same signal constellation in different Riemannianvarieties. For that, we use Rauch’s Theorem, [4], which intuitively states that ifthe gaussian curvature increases, the length decreases. Furthermore, if K1 < K2,we may conclude that the performance of a communication system with a signalconstellation in a Riemannian variety with sectional curvature K1 is better thanthe performance of a communication system with the same signal constellationin another Riemannian variety with sectional curvature K2. Equivalently,

K1 < K2 ⇒ Pe(K1) < Pe(K2) . (18)

From(18), we take into consideration the performance analysis of a communi-cation system when it is subjected to the same signal constellation X in threedifferent Riemannian varieties Mi, with corresponding sectional curvatures Ki,for i = 1, 2, 3. Figure 6 depicts the sectional curvature, K, versus the signalconstellation, X . From it, we may draw the following considerations:

– The performance of X in M1 is worse than the performance of the samesignal constellation in M2 and M3;

– We can not say anything based on (18) about the performance of X in M2being worse or better than the performance of the same signal constellationin M3;

– In order to compare Pe(K2) and Pe(K3) we have to calculate the values ofPe,m(K2) and Pe,m(K3), for each signal xm and we have to consider thepossible values of P (xm).

When the variety does not have constant sectional curvature the determinationof the symbol error probability, Pe,m, is quite difficult. However, if the varietyM has non-positive sectional curvature for all of its points, then from Hadamard

202 R. Gusmao Cavalcante and R. Palazzo Jr.

0 0.5 1 1.5 2 2.5 3

−3

−2

−1

0

1

2

3

Geodesics

r1=(ρ,π/4)

r2=(ρ,−π/4)

R1

x1

(a) Hyperbolic space, H2

−1.5 −1 −0.5 0 0.5 1 1.5

−3

−2

−1

0

1

2

3

x

y

Equivalent to r

1

Equivalent to r

2

R1eq

(b) Tangent space, Tx1H2

Fig. 7. Decision region of the signal x1 = (π/2, 0) in a 4-PSK signal constellation

−5 −4 −3 −2 −1 0 110

−3

10−2

10−1

100

Pe

4−PSK8−PSK

Curvature (K)

Fig. 8. Error probability versus sectional curvature

Theorem, [4], we have that the exponential mapping is a global diffeomorphismof TpM over M . Therefore, for this particular case, a way of obtaining Pe,m is tointegrate the probability density function of the random vector v ∈ TxmM (asso-ciated to the noise in the variety) over an equivalent decision region Req

m in TxmM

obtained from the decision region Rm in M . Therefore, Reqm = exp−1

xm(Rm).

As an example, we consider the decision region of the signal x1 = (π/2, 0)belonging to the 4-PSK signal constellation in the hyperbolic plane H2, Figs. 7(a)and 7(b). Observe that Req

1 contains the region delimited by the dashed line,representing the decision region of the same signal x1, however in E2. Thisis an alternative way of justifying the better performance achieved by the 4-PSK signal constellation in H2 than that in E2. Figure 8 shows the averageerror probability versus the sectional curvature of the 4-PSK and 8-PSK signal

Performance Analysis of M-PSK Signal Constellations 203

constellations for a signal-to-noise ratio equal to 4dB. As it was expected, Pe

diminishes with decreasing values of K.From the previous arguments and facts, the goal of this paper has been

achieved, that is, we aimed at showing the importance of the sectional curvaturein the design and performance analysis of signal constellations in Riemannianvariety. Hence, the designer of a communication system has to take into consid-eration the sectional curvature, or equivalently, the genus of the surface wherethe signal constellation is on. The starting of the whole process of finding thegenus or the sectional curvature comes from the embedding of the discrete mem-oryless channel on compact surfaces with or without borders. Therefore, surfaceswith genus g ≥ 2 or K < 0 are strong candidates for designing communicationsystems with better performance.

6 Conclusions

In this paper we have shown a dependence of the average error probability of asignal constellation in a Riemannian variety with its sectional curvature. Thisrelationship establishes that for sectional curvatures K1 and K2 such that K1 <K2, then the performance a communication system with a signal constellationin a Riemannian variety M1 with constant sectional curvature K1 is better thanthe performance of a communication system with a signal constellation in aRiemannian variety M2 with constant sectional curvature K2.

Acknowledgement. The authors would like to thank the financial supportfrom the Brazilian Agencies FAPESP, CNPq and CAPES.

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