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!Internal Report

# 3036/2006

Statistical Characterization of the Sum of

Squared Complex Gaussian

Random Variables

Goncalo N. Tavares and Luis M. Tavares

March, 2006

Goncalo N. Tavares is with the Department of Electrical and Computer Engineering, Instituto Superior

Tecnico (IST) and with Instituto de Engenharia de Sistemas e Computadores – Investigacao e Desen-

volvimento (INESC-ID), Lisbon, Portugal (email: goncalo.tavares@inesc-id.pt). Luis M. Tavares is

with the Department of Engineering, Escola Superior de Tecnologia e Gestao (ESTIG), Beja, Portugal

and with INESC-ID, Lisbon, Portugal (email: miguel.tavares@estig.ipbeja.pt).

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Important note

To the best of the authors knowledge, the results in this report are correct and

accurate. However and due to the its preliminary status, some errors may still

subsist. Permission to use the results in this report is granted provided the

results are duly acknowledged and referred.

Abstract

In this report we derive new results for the statistics of the random variable z ,∑N

n=1 x2n =

zI + jzQ = rejφ where the {xn} are a set of mutually independent complex-valued Gaussian

random variables with either zero or non-zero means and equal variance. Each random variable

xn is assumed to have independent real and imaginary components with equal variance for all n.

In the zero mean case, expressions are derived for the joint probability density function (p.d.f.)

and cumulative distribution function (c.d.f.) of (zI , zQ) and for the marginal p.d.f. and c.d.f.

of zI , zQ and r. In the non-zero mean case, the joint p.d.f. of (zI , zQ) and of (r, φ) and the

marginal p.d.f. of zI , zQ and r are presented. An useful Fourier series expansion for the p.d.f.

of the phase φ is also derived. As a practical application of the non-zero mean results, a theo-

retical performance analysis of the well-known non-data-aided (NDA) Viterbi & Viterbi (V&V)

feedforward carrier phase estimator operating with BPSK signals is presented. In particular,

an expression for the exact p.d.f. of the carrier phase estimates is derived.

Keywords: Quadratic forms, complex Gaussian random variables, carrier phase estimation,

equivocation, cycle-slipping.

i

ii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Problem Statement and Derivation Methodology . . . . . . . . . . . . . . . . . . 2

3 Results for the zero mean case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 Joint probability density function fZI ,ZQ(zI , zQ) . . . . . . . . . . . . . . 4

3.2 Marginal probability density functions fZI(zI) and fZQ

(zQ) . . . . . . . . 4

3.3 Marginal cumulative distribution functions FZI(zI) and FZQ

(zQ) . . . . . 5

3.4 Joint cumulative distribution function FZI ,ZQ(zI , zQ) . . . . . . . . . . . . 5

3.5 Probability density function of the modulus fR(r) . . . . . . . . . . . . . 6

3.6 Cumulative distribution function of the modulus FR(r) . . . . . . . . . . 6

3.7 Moments of the modulus Er[rα] . . . . . . . . . . . . . . . . . . . . . . . 6

4 Results for the non-zero mean case . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1 Joint probability density function fZI ,ZQ(zI , zQ) . . . . . . . . . . . . . . 7

4.2 Marginal probability density functions fZI(zI) and fZQ

(zQ) . . . . . . . . 7

4.3 Marginal cumulative distribution functions FZI(zI) and FZQ

(zQ) . . . . . 7

4.4 Joint probability density function fR,Φ(r, φ) . . . . . . . . . . . . . . . . 8

4.5 Probability density function of the modulus fR(r) . . . . . . . . . . . . . 9

4.6 Cumulative distribution function of the modulus FR(r) . . . . . . . . . . 9

4.7 Probability density function of the phase fΦ(φ) . . . . . . . . . . . . . . 9

5 Analysis of the NDA V&V estimator with BPSK signals . . . . . . . . . . . . . 12

5.1 Exact estimate statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Exact equivocation probability . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3 Approximate cycle-slip probability . . . . . . . . . . . . . . . . . . . . . 20

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

iii

iv

List of Figures

1 The coefficients cn(γ) of the Fourier series expansion for the p.d.f. of the phase

φ: (a) N = 4, (b) N = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Viterbi & Viterbi carrier phase synchronizer: (a) NDA carrier phase estimator,

(b) phase-unwrapping post-processing. . . . . . . . . . . . . . . . . . . . . . . . 13

3 The analytical and simulated p.d.f. of BPSK carrier phase estimates using the

V&V feedforward estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Variance of V&V carrier phase estimates from BPSK signals as a function of the

operating SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Equivocation probability conditioned on the value of the true carrier phase offset

φT : (a) Es/N0 = −5 dB and (b) Es/N0 = 0 dB. . . . . . . . . . . . . . . . . . . 18

6 Total (unconditional) equivocation probability as a function of the operating SNR. 19

7 Cycle-slip probability of the V&V carrier phase estimator for BPSK as a function

of the operating SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

v

vi

1 Introduction

The statistical characterization of quadratic forms involving complex-valued, possibly jointly

distributed Gaussian random variables is a challenging problem which has been the sub-

ject of significant research. The main reason for this interest is the fact that this type of

sum arises naturally in many diverse scientific and engineering contexts. For example, in

the context of determining the probability of error for digital signaling over slowly Rayleigh

and Rice fading channels using N diversity paths, Proakis statistically characterizes the sum

z ,∑N

n=1 xny∗n = zI + jzQ = rejφ [9]. The {xn, yn} are assumed to be correlated zero-mean

complex-valued Gaussian random variables, statistically independent but identically distributed

with any other {xl, yl}, l 6= n. When the {xn, yn} have zero mean (Rayleigh fading channel) [9]

reports expressions for the characteristic function (c.f.) of z, for the joint probability density

function (p.d.f.) of (zI , zQ) and (r, φ) and also for the marginal p.d.f. of φ. In the context of eval-

uating the performance of multichannel reception with differentially coherent and noncoherent

detection, Proakis derives the c.f. of the quadratic D ,∑N

n=1 [A|xn|2 + B|yn|2 + 2<{Cxny∗n}]

and evaluates the probability of error Pe = Pr {D < 0} in closed form [10]. More recently,

Simon & Alouini determine the c.f. of the quadratic D = A∑N1

n=1 |xn|2 + B∑N2

n=1 |yn|2 with

possibly N1 6= N2 where {xn} and {yn} are mutually independent complex-valued Gaussian

random variables and evaluate the outage probability Pout = Pr{D < 0} of multichannel cel-

lular systems subject to independent, identically distributed (i.i.d.) interfering signals using

maximal-ratio combining [13]. In the context of radar target detection and Doppler shift es-

timation, Lank et al. study the sum zk ,∑N

n=k+1 xnx∗n−k, k = 1, . . . , N − 1 where the {xn}

are independent complex-valued Gaussian random variables [5]. In particular, the c.f. of zk is

derived (when the {xn} have either zero or non-zero mean). When the {xn} have zero-mean

the p.d.f. of zk is also reported. The statistical characterization of a number of other useful

quadratic forms in Gaussian random variables may be found in [8], [12] and [6].

The sum z =∑N

n=1 xMn arises naturally in the context of non-data-aided (NDA) synchro-

nization parameter estimation in digital communications e.g., the nonlinear estimation of the

carrier phase offset [14] and the nonlinear least squares estimation of the carrier frequency

offset [15]. In an attempt to remove the random modulation from data symbols belonging

to a 2π/M rotationally-symmetric constellation (which perturbs the estimation process), the

received matched filter output samples {xn} are raised to the Mth power. In this work we

study this sum with M = 2, assuming the {xn} are independent complex-valued Gaussian

random variables, each having independent real and imaginary components. To the best of

1

the authors knowledge, and despite the significant amount of available results concerning the

statistical characterization of quadratic forms involving complex-valued Gaussian random vari-

ables, the study of this sum has not been previously reported in the open literature. While in

the particular context of synchronization parameter estimation only the case when the {xn}have non-zero mean is of interest, we also study the sum z when the {xn} have zero mean.

As an application example, the results for the non-zero mean case are used to theoretically

analyse the performance and characteristics of the well-known NDA Viterbi & Viterbi (V&V)

carrier phase estimator [14] operating from BPSK signals received over an AWGN channel. In

particular, the new results allow the determination of the exact probability density function

of the carrier phase estimates (and therefore of moments of any order, including the variance)

and of the equivocation probability of the feedforward synchronizer (based on this estimator),

as given in [1]. In addition, the cycle-slip probability of this synchronizer is also theoretically

assessed using an approximate formula proposed in [3].

The organization of this report is as follows. In Section 2 the problem is formally stated

and the strategy used in the statistical characterization is outlined. The results for the zero

mean and non-zero mean cases are presented in Section 3 and Section 4 respectively. The

results in Section 4 are then used to statistically characterize the V&V feedforward carrier

phase estimator. This characterization is presented in Section 5. Finally, in Section 6 some

concluding remarks are presented.

2 Problem Statement and Derivation Methodology

Let {xn}Nn=1 be a set of mutually i.i.d. complex Gaussian random variables, each with complex

mean µn and variance 2σ2 (for all n) and with independent real and imaginary components xI(n)

and xQ(n) respectively, each with variance σ2n = σ2 for all n. Define the set of complex-valued

random variables

yn = yI(n) + jyQ(n) , x2n = x2

I(n) − x2Q(n) + j2xI(n)xQ(n) n = 1, . . . , N. (1)

The aim of the work reported in this document is to statistically characterize the complex-valued

random variable (r.v.)

z = zI + jzQ ,

N∑

n=1

yn =N∑

n=1

x2n (2)

2

when the {xn} have either zero or non-zero mean. In the sequel we will need the mean and the

variance of z which are easily computed as

E[z] =N∑

n=1

E[x2n] =

N∑

n=1

µ2n , M = |M |ejφ = MI + jMQ (3a)

and

var{z} = 8Nσ4 + 8NPσ2 (3b)

where P , 1N

∑Nn=1 |µn|2. Since each yn , x2

n is the result of a memoryless nonlinear function

of the i.i.d. random variables {xn}, the {yn} are also i.i.d.. However, yI(n) and yQ(n) are

non-Gaussian, correlated random variables. The joint characteristic function of the random

variables yI(n) and yQ(n) is found to be

ΦyI (n),yQ(n)(tI , tQ) =

∫ ∞

−∞

∫ ∞

−∞

ej(x2I(n)−x2

Q(n))tI+j2xI(n)xQ(n)tQfXI(n),XQ(n)(xI(n), xQ(n))dxI(n)dxQ(n)(4a)

=1

[1 + 4σ4(t2I + t2Q)

] 12

exp

{

−2σ2|µn|2(t2I + t2Q) + j<{µ2n}tI + j={µ2

n}tQ1 + 4σ4(t2I + t2Q)

}

(4b)

where in (4a)

fXI(n),XQ(n)(xI(n), xQ(n)) =1

2πσ2e−

(xI (n)−<{µn})2+(xQ(n)−={µn})2

2σ2 (5)

is the joint probability density function of xI(n) and xQ(n). Since the yn are independent, if

follows that the joint c.f. of zI and zQ is given by

ΦzI ,zQ(tI , tQ) =

N∏

n=1

ΦyI(n),yQ(n)(tI , tQ)

=1

[1 + 4σ4(t2I + t2Q)

]N2

exp

{

−2σ2NP (t2I + t2Q) + jMItI + jMQtQ

1 + 4σ4(t2I + t2Q)

}

. (6)

The joint probability density function of zI and zQ may now be obtained by Fourier transforming

the c.f. in (6) i.e.,

fZI ,ZQ(zI , zQ) =

1

(2π)2

∫ ∞

−∞

∫ ∞

−∞

ΦzI ,zQ(tI , tQ)e−j(zI tI+zQtQ)dtIdtQ. (7)

3 Results for the zero mean case

In this section we address the case when all of the {xn} have zero mean i.e., {µn = 0 + j0}Nn=1.

This implies P = 0 and M = 0 + j0 and the c.f. in (6) reduces to

ΦzI ,zQ(tI , tQ) =

1[1 + 4σ4(t2I + t2Q)

]N2

. (8)

3

Because E[x2I(n)] = E[x2

Q(n)] = σ2 and E[xI(n)xQ(n)] = 0 we conclude from (1) and (2) that

the random variables {yn} and z also have zero mean. Interesting, the c.f. in (8) can be obtained

from [10, eq. (9)], which is the c.f. of the r.v. u ,∑L

n=1 xny∗n where {xn}L

n=1 and {yn}Ln=1

are complex zero-mean correlated Gaussian r.v.’s with variance mxx and myy respectively and

correlation mxy = E[xny∗n] equal for all n. For this it should be considered that yn = x∗

n for

all n = 1, 2, . . . , L so we should set mxx = myy = 4σ2, mxy = 0 and L = N/2 (because by

considering yn = x∗n the number of different r.v.’s is L/2; also, to keep the noise power the same

in both analysis, twice the variance must be considered).

3.1 Joint probability density function fZI ,ZQ(zI , zQ)

Inserting (8) in (7), making the change of variables tI = u cos θ and tQ = u sin θ (a transfor-

mation with Jacobian u) and then using [2, eq. (6.565-4)] one finds the joint p.d.f. for (zI , zQ)

as

fZI ,ZQ(zI , zQ) =

1

(2π)2

∫ ∞

0

∫ π

−π

u

(1 + 4σ4u2)N2

e−ju(zI cos θ+zQ sin θ)dθdu

=1

2π

∫ ∞

0

u

(1 + 4σ4u2)N2

J0

(

u√

z2I + z2

Q

)

du

=1

2π

(z2

I + z2Q

)N−24

(2σ2)N+2

2 2N−2

2 Γ(

N2

)KN2−1

(1

2σ2

√

z2I + z2

Q

)

, −∞ < zI , zQ < ∞ (9)

where Jν(x) is the Bessel function of the first kind and order ν, Γ(x) is the gamma function and

Kν(x) is the modified Bessel function of the second kind and order ν (Macdonald function).

When ν is of the form ν = n ± 12

with n integer, the function Kn± 12(x) may be expressed as a

finite sum of elementary functions. This is the case in (9) when N is odd and is also the case

with other results reported in this Section.

3.2 Marginal probability density functions fZI(zI) and fZQ

(zQ)

The marginal probability density functions for zI and zQ are easily determined by suitable

integration of (9) over the unwanted random variable i.e., fZI(zI) =

∫∞

−∞fZI ,ZQ

(zI , zQ)dzQ, and

fZQ(zQ) =

∫∞

−∞fZI ,ZQ

(zI , zQ)dzI. Using [11, eq. (2.16.3-8)] it may be shown that these densities

are given by

fZ(u) =1√2π

1

2N2−1Γ

(N2

)2σ2

( |u|2σ2

)N−12

KN−12

( |u|2σ2

)

, −∞ < u < ∞ (10)

with u = zI or u = zQ.

4

3.3 Marginal cumulative distribution functions FZI(zI) and FZQ

(zQ)

The marginal cumulative distribution function (c.d.f.) for zI and zQ may be determined by

integration of (10) as FZI(zI) , Pr{ZI ≤ zI} =

∫ zI

−∞fU(u)du and FZQ

(zQ) , Pr{ZQ ≤ zQ} =∫ zQ

−∞fU(u)du. Using [11, eq. (1.12.1-3)] it can be shown that

FU(u) =1

2+ sgn(u)G1(|u|), −∞ < u < ∞ (11)

with u = zI or u = zQ and with

G1(x) =1

√π2

N−32

( x

2σ2

)N+12

[

1

Γ(

N2

)KN−12

( x

2σ2

)

1F2

(

1;3

2,N

2;1

4

( x

2σ2

)2)

+1

2

( x

2σ2

)2 1

Γ(

N+22

)KN−32

( x

2σ2

)

1F2

(

1;3

2,N + 2

2;1

4

( x

2σ2

)2)]

, x ≥ 0 (12)

where 1F2(α; β1, β2; x) is the generalized hypergeometric function. When N is even, we may

use [11, eq. (1.12.1-2)] to find the following simpler representation

G1(x) =Γ(

N−12

)

2√

πΓ(

N2

)

( x

2σ2

)

1F2

(1

2;3 − N

2,3

2;1

4

( x

2σ2

)2)

+Γ(

1−N2

)

√π N2NΓ

(N2

)

( x

2σ2

)N

1F2

(N

2;N + 1

2,N + 2

2;1

4

( x

2σ2

)2)

, x ≥ 0, N even.(13)

Using [2, eq. (8.467)] we can find an even simpler representation for G1(x) when N is even

G1(x) =1

2− e−

x

2σ2

N2−1∑

k=0

2−(N2

+k)(

N2− 1 + k

k

) N2−1−k∑

m=0

1

m!

( x

2σ2

)m

, x ≥ 0, N even. (14)

3.4 Joint cumulative distribution function FZI ,ZQ(zI , zQ)

The joint c.d.f. for zI and zQ may be determined by suitable integration of (9) as

FZI ,ZQ(zI , zQ) , Pr{ZI ≤ zI , ZQ ≤ zQ} =

∫ zI

−∞

∫ zQ

−∞fZI ,ZQ

(u, v)dudv. Using [11, eq. (2.16.3-

8)] one finds

FZI ,ZQ(zI , zQ) =

FZI(zI) + FZQ

(zQ) − 1 + G2(|z|), zI ≥ 0, zQ ≥ 0

FZQ(zQ) − G2(|z|), zI ≥ 0, zQ < 0

FZI(zI) − G2(|z|), zI < 0, zQ ≥ 0

G2(|z|), zI < 0, zQ < 0

(15)

where |z| =√

z2I + z2

Q and

G2(x) =1

(2σ2)N2 2

N+22 Γ

(N2

)

( x

2σ2

)N2

KN2

( x

2σ2

)

. (16)

5

3.5 Probability density function of the modulus fR(r)

Defining r , |z| and φ , arg{z} we may write the joint p.d.f. of zI and zQ in (9) in polar form

as

fR,Φ(r, φ) =1

2π︸︷︷︸

fΦ(φ)

1

2N−2

2 2σ2Γ(

N2

)

( r

2σ2

)N2

KN2−1

( r

2σ2

)

. (17)

which, as expected [recall that z has zero mean and that xI(n) and xQ(n) are uncorrelated for

all n], reveals that φ is uniformly distributed in [−π, π]. The marginal p.d.f. of r is thus

fR(r) =1

2N−2

2 2σ2Γ(

N2

)

( r

2σ2

)N2

KN2−1

( r

2σ2

)

. (18)

3.6 Cumulative distribution function of the modulus FR(r)

Using [11, eq. (2.16.3-8)] we find that the cumulative distribution function FR(r) , Pr{R ≤r} =

∫ r

0fR(u)du with fR(u) given by (18) may be expressed as

FR(r) = 1 − 1

2N−2

2 Γ(

N2

)

( r

2σ2

)N2

KN2

( r

2σ2

)

. (19)

3.7 Moments of the modulus Er[rα]

Using the p.d.f. (18) and [2, eq. (6.561-16)] one can show that the moments of r , |z| of order

α are given by

Er[rα] ,

∫ ∞

0

rαfR(r)dr =2α(2σ2)α

Γ(

N2

) Γ

(N + α

2

)

Γ(α

2+ 1)

(20)

which is valid for any real α > −min{2, N}.

4 Results for the non-zero mean case

In this section we address the case when the {xn} have arbitrary, possibly non-zero means.

While the results in this Section are also valid for zero-mean {xn}, this case has already been

addressed in the previous Section. Therefore we consider that at least one of the xn has non-zero

mean1. Under this assumption it is possible to define an average signal-to-noise-ratio (SNR)

1It is worth noting that if at least two of the {xn} have non-zero means, the complex r.v. z may have zero

mean, M , E[z] = 0 + j0.

6

just before the square operation as

γ =

∑Nn=1 E[|xn|2]

∑Nn=1 var{xn}

=

∑Nn=1 |µn|22σ2N

=P

2σ2. (21)

For convenience, the results presented in this Section are expressed in terms of the normalized

SNR γ , γ/P = 1/(2σ2).

4.1 Joint probability density function fZI ,ZQ(zI , zQ)

Inserting (6) in (7) and making the change of variables tI = u2σ2 cos θ and tQ = u

2σ2 sin θ one

finds the joint probability density function for (zI , zQ) as

fZI ,ZQ(zI , zQ) =

γ2

(2π)2

∫ ∞

0

∫ π

−π

f(u) e−juγ

h“

zI−MI

1+u2

”

cos θ+“

zQ−MQ

1+u2

”

sin θi

dθdu (22a)

=γ2

2π

∫ ∞

0

f(u)J0

uγ

√(

zI −MI

1 + u2

)2

+

(

zQ − MQ

1 + u2

)2

du (22b)

−∞ < zI , zQ < ∞

where we have defined

f(u) ,u

(1 + u2)N2

exp

(

−γNPu2

1 + u2

)

. (23)

4.2 Marginal probability density functions fZI(zI) and fZQ

(zQ)

The marginal probability density functions for zI and zQ are determined by suitable integration

of (22b) over the unwanted random variable i.e., fZI(zI) =

∫∞

−∞fZI ,ZQ

(zI , zQ)dzQ, and fZQ(zQ) =

∫∞

−∞fZI ,ZQ

(zI , zQ)dzI . Using [11, eq. (2.12.4-17)] it may be shown that these densities are given

by

fZI(zI) =

γ

π

∫ ∞

0

f(u)

ucos

[

uγ

(

zI −MI

1 + u2

)]

du, −∞ < zI < ∞ (24)

and

fZQ(zQ) =

γ

π

∫ ∞

0

f(u)

ucos

[

uγ

(

zQ − MQ

1 + u2

)]

du, −∞ < zQ < ∞. (25)

4.3 Marginal cumulative distribution functions FZI(zI) and FZQ

(zQ)

The marginal cumulative distribution functions for zI and zQ may be determined by integration

of (24) and (25) respectively as FZI(zI) , Pr{ZI ≤ zI} =

∫ zI

−∞fZ(z)dz and FZQ

(zQ) , Pr{ZQ ≤zQ} =

∫ zQ

−∞fZ(z)dz. It can be shown that

FZI(zI) =

1

2+

1

π

∫ ∞

0

f(u)

u2sin

[

uγ

(

zI −MI

1 + u2

)]

du, −∞ < zI < ∞ (26)

7

and

FZQ(zQ) =

1

2+

1

π

∫ ∞

0

f(u)

u2sin

[

uγ

(

zQ − MQ

1 + u2

)]

du, −∞ < zQ < ∞. (27)

4.4 Joint probability density function fR,Φ(r, φ)

With z = rejφ the joint p.d.f. of r and φ may be written as

fR,Φ(r, φ) =γ2r

2π

∫ ∞

0

f(u)J0

uγ

√(

r cos φ − MI

1 + u2

)2

+

(

r sin φ − MQ

1 + u2

)2

du,

=γ2r

2π

∫ ∞

0

f(u)J0

uγ

√(

r cos(φ − φ) − |M |1 + u2

)2

+ r2 sin2(φ − φ)

du,

0 ≤ r < ∞, |φ − φ| ≤ π.(28)

The joint p.d.f. in (28) is an integral representation which may be computationally costly to

evaluate. It is therefore useful to find some alternative representation, particularly to allow

efficient computation of the marginal densities of r and φ. To pursue this goal we derive a

Fourier series representation for this joint density which will prove to be very useful. We start

by expressing the joint p.d.f. (22a) with z in polar form

fR,Φ(r, φ) =γ2r

(2π)2

∫ ∞

0

∫ π

−π

f(u) e−juγ

h“

r cos φ−MI

1+u2

”

cos θ+“

r sin φ−MQ

1+u2

”

sin θi

dθdu

=γ2r

(2π)2

∫ ∞

0

∫ π

−π

f(u) ej uγ

1+u2 (MI cos θ+MQ sin θ)e−jruγ cos(φ−θ)︸ ︷︷ ︸

,g(φ)

dθdu. (29)

The function g(φ) is periodic with period 2π and may therefore be expressed as a Fourier series

g(φ) =∑∞

n=−∞ an(ruγ)ejnφ with coefficients

an(ruγ) ,1

2π

∫ π

−π

ejruγ cos(φ−θ)e−jnφdφ = e−jnθ(−j)nJn (ruγ) . (30)

Making this substitution in (29) and recalling from (3a) that MI , <{M} = |M | cos φ and

MQ , ={M} = |M | sin φ, yields

fR,Φ(r, φ) =γ2r

(2π)2

∫ ∞

0

∫ π

−π

f(u) ej uγ

1+u2 |M | cos(θ−φ)∞∑

n=−∞

e−jnθ(−j)nJn (ruγ) ejnφ dθdu

=∞∑

n=−∞

ejn(φ−φ) γ2r

2π

∫ ∞

0

f(u)(−j)nJn(ruγ)1

2π

∫ π

−π

ej u

1+4σ4u2 |M |γ cos(θ−φ)−jn(θ−φ)dθ

︸ ︷︷ ︸

jnJn

“

|M |γ u

1+4σ4u2

”

du

=∞∑

n=−∞

bn(r, γ)ejn(φ−φ) (31)

8

where bn(r, γ) are the Fourier coefficients given by

bn(r, γ) ,γ2r

2π

∫ ∞

0

f(u)Jn

(

|M |γ u

1 + u2

)

Jn(ruγ) du. (32)

Finally, noting that b−n(r, γ) = bn(r, γ) we may write

fR,Φ(r, φ) = b0(r, γ) + 2∞∑

n=1

bn(r, γ) cos[n(φ − φ)],0 ≤ r < ∞|φ − φ| ≤ π

. (33)

4.5 Probability density function of the modulus fR(r)

The marginal p.d.f. fR(r) may be obtained by integrating (28) over the support of φ. However,

it is simpler to use (22a) with z expressed in polar form and integrate first over φ

fR(r) =γ2r

(2π)2

∫ ∞

0

∫ π

−π

∫ π

−π

f(u) e−juγ

h“

r cos φ−MI

1+u2

”

cos θ+“

r sinφ−MQ

1+u2

”

sin θi

dφdθdu

=γ2r

2π

∫ ∞

0

∫ π

−π

f(u) e−j uγ

1+u2 (MI cos θ+MQ sin θ)J0(ruγ) dθdu

= γ2r

∫ ∞

0

f(u)J0

(

|M |γ u

1 + u2

)

J0(ruγ) du. (34)

Alternatively, one may integrate the Fourier series representation for fR,Φ(r, φ) in (33) over

φ ∈ [−π, π] and obtain fR(r) = 2πb0(r, γ) which equals (34), as it should.

4.6 Cumulative distribution function of the modulus FR(r)

Using [11, eq. (1.8.1-21)] we find that the cumulative distribution function FR(r) , Pr{R ≤r} =

∫ r

0fR(u)du with fR(u) given by (34) is

FR(r) = γr

∫ ∞

0

f(u)

uJ0

(

|M |γ u

1 + u2

)

J1(ruγ) du. (35)

4.7 Probability density function of the phase fΦ(φ)

The marginal p.d.f. fΦ(φ) may be obtained by integration of the joint p.d.f. in (28) over r ≥ 0.

This leads to a double integral representation which is not very useful in practice. Instead, we

use the series representation in (33) to obtain

fΦ(φ) =1

2π

[

1 + 2∞∑

n=1

cn(γ) cos[n(φ − φ)]

]

, |φ − φ| ≤ π (36)

9

where the Fourier coefficients are (n ≥ 1)

cn(γ) , γ2

∫ ∞

0

∫ ∞

0

rf(u)Jn

(

|M |γ u

1 + u2

)

Jn(ruγ) dudr (37a)

= n

∫ ∞

0

f(u)

u2Jn

(

|M |γ u

1 + u2

)

du. (37b)

The representation in (37b) is proved in the Appendix and is important because it allows the

coefficients to be computed by means of a single improper integral [instead of the double one in

(37a)]. In the Appendix it is also shown that the coefficients cn(γ) have the alternative series

representation

cn(γ) =n

2

∞∑

k=0

(−1)k

k!(n + k)!

Γ(

n2

+ k)Γ(

N+n2

+ k)

Γ(

N2

+ n + 2k)

( |M |γ2

)n+2k

1F1

(n

2+ k,

N

2+ n + 2k;−γNP

)

(38)

where 1F1(a, b; z) is the confluent hypergeometric function, which is readily available in many

currently available mathematical software packages, thus avoiding the custom numerical inte-

gration in (37b). From (36) we conclude that fΦ(φ) is a symmetric function around the mean φ

i.e., fΦ(φ−φ) = fΦ(φ−φ). This is a consequence of the uncorrelateness (independence) assump-

tion for the real an imaginary components of the {xn}. It can be shown that this symmetry

property implies that the cn(γ) are strictly positive.

The coefficients cn(γ) (expressed in dB) are plotted in Figure 1(a) for N = 4 and in Fig-

ure 1(b) for N = 16, as a function of the order n, for several values of the SNR. When γ is low,

the coefficient decay is very fast with increasing order, meaning that only a small number of

coefficients needs to be considered in the series representation (36) for the phase p.d.f. fΦ(φ).

This is true even for N = 16 [see Figure 1(b)] provided γ is small (γ ≤ 0 dB) and is justified

because in these circumstances, the phase p.d.f. is a smooth, low-peaked function of φ and

thus few coefficients are required for accurate Fourier series representation. However, as the

product γ × N increases, the p.d.f. of the phase φ becomes increasingly sharp and peaked

around its mean φ and accordingly, accurate representation of fΦ(φ) by the series (36) requires

many coefficients. For example, with γ = 10 dB the ratio c1/c21 exceeds 90 dB for N = 4, but

is only about 25 dB for N = 16.

10

100−

80−

60−

2 0−

4 0−

0

()

Coefficient

(dB)

nc

γ

order, n

1 5 13 19 2 1171511973

10 dB

5 dB

0 dB

5 dB

10 dB

γ

γ

γ

γ

γ

= −

= −

=

=

=

4N =

(a)

100−

80−

6 0−

2 0−

40−

0

()

Coefficient

(dB)

nc

γ

order, n

16N =

1 5 13 19 2 1171511973

10 dB

5 dB

0 dB

5 dB

10 dB

γ

γ

γ

γ

γ

= −

= −

=

=

=

(b)

Figure 1: The coefficients cn(γ) of the Fourier series expansion for the p.d.f. of the phase φ:

(a) N = 4, (b) N = 16.

11

5 Analysis of the NDA V&V estimator with BPSK sig-

nals

As an application example, we will use the results reported in Section 4, pertaining to

the non-zero mean case, to analyse a particular implementation of the well-known NDA

Viterbi & Viterbi (V&V) feedforward carrier phase estimator, proposed in [14] for operation

with M -ary phase-shift-keying (M -PSK) linearly modulated signals over AWGN channels. This

estimator is typically used to estimate the carrier phase in TDMA burst-mode transmission sys-

tems. We consider the transmission of a M -PSK signal and the following received signal model:

firstly, any carrier frequency offset is accurately estimated and removed from the received noisy

signal, leaving a frequency-corrected signal that remains affected by an unknown carrier phase

offset. This signal is then passed through a matched filter and sampled at the symbol rate 1/T

with perfect symbol timing. The carrier phase offset is denoted φT and is assumed to remain

constant throughout the length of the observation interval which, in burst-mode systems, usu-

ally coincides with the frame duration. However, if operation is with long frames it is possible

that φT changes throughout the data-burst. In this case we consider that the observation vector

x = {xn} is segmented into consecutive non-overlapping blocks xm , {xn +mN}Nn=1, each with

N samples (NT seconds) long, such that φT remains approximately constant over each block.

The complex envelope of the matched filter output may thus be written

xn = anejφT + wn, n = 1, 2, . . . , N (39)

where an is a random symbol from the M -PSK alphabet {Ak = ej 2πM

k}M−1k=0 and wn is a sample of

a complex-valued white Gaussian zero-mean noise process with independent real and imaginary

parts, each with variance σ2 = N0/(2Es) where N0 is the power spectral density of the noise and

Es is the average symbol energy. The symbol signal-to-noise ratio is SNR = E[|an|2]/(2σ2) =

1/(2σ2) = Es/N0 which coincides with the previous definition of γ and also with γ = γP in (21)

because P = 1N

∑Nn=1 |anejφT |2 = 1. For each block xm, a feedforward carrier phase estimate is

computed as [14]

φm =1

Marg

{N∑

n=1

xMn+mN

}

. (40)

The feedforward nonlinear V&V carrier phase estimator for BPSK signals (M = 2) is repre-

sented in Figure 2(a).

12

( )x tn

x

1sf

T=

BWAm

z

2( )i1a r g { }

2i

1

N

n

n

y

=

∑

ny ˆ

mφ

: 1N

(a)

2π−

Delay

mφ�ˆ

mφ

NT

mε

π

mod mε π

mε

2π

0

2π−

1mφ

−

�

(b)

Figure 2: Viterbi & Viterbi carrier phase synchronizer: (a) NDA carrier phase estimator, (b)

phase-unwrapping post-processing.

The matched filter output samples are squared (to remove the BPSK modulation due to

the random symbols an = ±1) and filtered by a block window accumulator (BWA) averaging

filter which simply computes the (scaled) arithmetic mean of the set {yn+mN = x2n+mN}N

n=1

and outputs samples {zm =∑N

n=1 yn+mN} every NT seconds [3]. Because successive blocks are

non-overlapping, the samples {zm} are statistically independent. Further, since each zm is the

sum of a set of non-zero mean complex-valued Gaussian random variables, the results presented

in Section 4 may be applied to statistically characterize the carrier phase estimates {φm}. This

is done in the next subsections where we compute the exact p.d.f. and variance of φm as well

as other important statistical measures which completely characterize the performance of the

carrier phase estimator/synchronizer.

5.1 Exact estimate statistics

Probability density function

For BPSK, the carrier phase estimates are obtained as

φm =1

2arg

{N∑

n=1

x2n+mN

}

. (41)

Comparing (41) with a generalized version of (2) i.e., zm ,∑N

n=1 x2n+mN = rmejφm, we conclude

that φm = 12arg{zm} = 1

2φm. The p.d.f. of the carrier phase estimate φm is thus fΦm

(φm) =

13

2fΦ(φ)|φ=2φm. Using the series representation (36) for fΦ(φ) one finds the exact p.d.f. [Note

that φ = arg{E[z]} = 2φT , see (3a)]

fΦm(φm) =

1

π

[

1 + 2∞∑

n=1

cn(γ) cos[2n(φm − φT )]

]

, |φm − φT | ≤π

2. (42)

The analytical p.d.f. of the carrier phase estimates {φm} is represented in Figure 3 for several

values of the SNR together with a set of simulated points obtained by histogram computation

using 107 trials per simulated SNR value. It is seen that the simulated points match closely

the theoretical values. As expected, the p.d.f. is symmetrical around E[φm] = φT (= 0) and

becomes sharply peaked as Es/N0 increases.

()

ˆˆ

Probability den

sity function,

mm

fφ

Φ

2

0

1

9 0− 9 00 6 030−

ˆE s t i m a t e ( d e g r e e )m

φ

60− 30

BPSK

1 6

0 T

N

φ

=

=

3

4

0

0

0

0

Simulation points

E x ac t v alue s

1 0 d B

5 d B

0 d B

5 d B

s

s

s

s

E N

E N

E N

E N

→

→

= −

= −

=

=

Figure 3: The analytical and simulated p.d.f. of BPSK carrier phase estimates using the V&V

feedforward estimator.

Variance

Using the p.d.f. (42) the analytical value of the estimate variance is found as

var{φm} , E[(φm − φT )2] =

∫ π/2+φT

−π/2+φT

(φm − φT )2fΦm(φm) dφ

=

∫ π/2

−π/2

φ2mfΦm

(φm + φT ) dφm

14

=1

π

∫ π/2

−π/2

φ2m

[

1 + 2∞∑

n=1

cn(γ) cos(2nφm)

]

dφm

=π2

12+

∞∑

n=1

(−1)n

n2cn(γ). (43)

{}

2ˆ

Estim

ate variance, var

(rad)

mφ

110−

210−

310−

15− 10− 5− 0 5 10

0 (dB)sE N

1

V & V e s t i m a t o r

B P S K

1 6N =

Simulation

C R B ( )

E x ac t v ar ianc eTφ

Figure 4: Variance of V&V carrier phase estimates from BPSK signals as a function of the

operating SNR.

Plots of the exact and simulated variance are shown in Figure 4, as a function of the

operating SNR. Also plotted is the Cramer-Rao lower bound (CRLB), which is a funda-

mental lower bound on the variance of any unbiased carrier phase estimate and is given by

CRLB(φT ) = (2NEs/N0)−1. This bound is obtained under the assumption that data modula-

tion is absent from the received signal and thus is strictly valid for data-aided (DA) estimation

or for estimation from an unmodulated carrier. Nevertheless, it may be used to bound the

variance of NDA estimates (as is the case under consideration) when the SNR is high. For

the simulated points, feedforward carrier phase estimates were obtained using N = 16 samples

15

(107 trials per simulated SNR point). As can be seen, the agreement between the analytical

(exact) and simulated variance is very good. Also worth noting, as Es/N0 → ∞ the variance

approaches the CRLB and the estimator is thus efficient.

5.2 Exact equivocation probability

The nonlinear carrier phase estimator in (40) suffers from an anomaly known as equivocation [1].

If the phases {φm} , {arg{zm}} of the BWA output samples are supported in [−π, π], the carrier

phase estimates {φm} will be restricted to the interval [− πM

, πM

]. When the true carrier phase

φT is near ± πM

,±3πM

,±5πM

, . . . then, due to the Mth power operation, {φm} will be close to

the ±π boundary and successive phase samples φm will most likely exhibit positive or negative

jumps of ≈ 2π. Therefore, estimates will also exhibit jumps of ≈ 2πM

. This anomaly is due to the

noise or to a residual frequency offset in the received signal samples {xn} and will occur even

when the SNR is high, causing an unacceptable symbol error rate degradation [1, 3]. Even in

the absence of noise and frequency offset, equivocation will preclude the possibility of estimates

to follow the dynamics of the true carrier phase. Equivocation occurs whenever the phase of

{φm} crosses the ±π boundary or equivalently when {zm, zm+1} represent a crossing from the

second to the third or from the third to the second quadrant [1]. From the previous discussion

we may conclude that the probability of equivocation depends heavily on the true carrier phase

φT and should thus be defined as a conditional probability on this parameter. It will be denoted

P (EQ|φT ) and following [1, eq. (12)], is given by

P (EQ|φT ) = Pr{[ π

2≤ φm ≤ π

∣∣∣φT

]

∩[

−π ≤ φm+1 ≤ −π

2

∣∣∣φT

]}

+ Pr{[

−π ≤ φm ≤ −π

2

∣∣∣φT

]

∩[ π

2≤ φm+1 ≤ π

∣∣∣φT

]}

. (44)

Because samples {zm, zm+i} are statistically independent for all i 6= 0, (44) becomes

P (EQ|φT ) = 2 Pr{ π

2≤ φ ≤ π

∣∣∣φT

}

· Pr{

−π ≤ φ ≤ −π

2

∣∣∣φT

}

(45)

where φ denotes any of the φm. Note that (44) and (45) are valid for M -PSK. For BPSK, using

(36) and recalling that φ = 2φT it is easy to show that

P (EQ|φT ) = 2

[

1

4+

1

π

∞∑

n=1

1

ncn(γ)

(

− sin[n(2φT − π)] + sin[

n(

2φT − π

2

)])]

·[

1

4+

1

π

∞∑

n=1

1

ncn(γ)

(

sin[n(2φT + π)] − sin[

n(

2φT +π

2

)])]

. (46)

16

The unconditional (total) probability of equivocation is given by [1, eq. (13)]

P (EQ) =

∫ π

−π

P (EQ|φT )fΦT(φT )dφT . (47)

Assuming φT is uniformly distributed in [−π, π], the total equivocation probability in (47)

becomes after some algebra

P (EQ) =1

8− 4

π2

∞∑

n=1

1

(4n − 2)2c24n−2(γ). (48)

We will compare the exact result in (46) with the Gaussian approximation given by [1,

eq. (14)] with correlation coefficient2 ρ = 0

P (EQ|φT ) ≈ 1

4

[

1 − 1

2erfc

(

MQ√

2σ2z

)]

erfc

(

MQ√

2σ2z

)

erfc2

(

MI√

2σ2z

)

. (49)

Here, MI + jMQ = E[zm] and 2σ2z = var{zm} represent the mean and the variance of the BWA

output samples. From (3a) and (3b) these moments are given by

E[zm] = MI + jMQ = Nej2φT (50a)

and

var{zm} = 2σ2z = 8Nσ4 + 8Nσ2 = 2N

(

1 + 2Es

N0

)/(Es

N0

)2

. (50b)

The results for P (EQ|φT ) with N = 16 are presented in Figures 5(a) and 5(b) for Es/N0 =

−5 dB and Es/N0 = 0 dB respectively. The simulated result agrees very well with the exact

result for the conditional equivocation probability in (46). It is seen that the Fitz approximation

(49) is more accurate when the SNR is lower. This may have been anticipated because this

approximation is based on the central limit theorem which is more accurate when the component

random variables exhibit smoother p.d.f.’s, as is the case for lower SNR. Also as expected,

P (EQ|φT ) is maximum for φT = ±π2

because for this particular value of φT the phases {φm}of the BWA output samples will be near the boundary ±π and estimates will equivocate much

often than for any other value of φT .

The exact total equivocation probability (48) is plotted in Figure 6 together with the ap-

proximation in (47) which has been computed using the Fitz approximation in (49) for the

conditional equivocation probability P (EQ|φT ) .

2The parameter ρ represents the correlation coefficient between successive BWA output samples, which is

zero under our assumptions.

17

Conditional equivocation probab

ility,

(EQ

|)

TP

φ

1

210−

110−

180− 1800 9090−

True carrier phase, (degree)Tφ

Simulation

G aus s ian ap p r ox .

E x ac t p r ob ab ility

0

BPSK

1 6

5 d B s

N

E N

=

= −

(a)

1

510−

310−

180− 1809090−

0

BPSK

1 6

0 d B s

N

E N

=

=

Simulation

G aus s ian ap p r ox .

E x ac t p r ob ab ility

0

410−

210−

110−

True carrier phase, (degree)Tφ

Conditional equivocation probab

ility,

(EQ

|)

TP

φ

(b)

Figure 5: Equivocation probability conditioned on the value of the true carrier phase offset φT :

(a) Es/N0 = −5 dB and (b) Es/N0 = 0 dB.

18

0.1

0.0110− 5− 0 5 10 15

0 (dB)sE N

0.2

V&V estimator

B P S K

1 6N =

Total equivocation probability,

(EQ)

P

Simulation

G aus s ian ap p r ox .

E x ac t p r ob ab ility

Figure 6: Total (unconditional) equivocation probability as a function of the operating SNR.

As can be seen, the simulated results match closely the theoretical values. Also, we conclude

that the Fitz approximation is indeed very good. It is worth noting the slow decay of P (EQ)

with the SNR: for a 20 dB increase in Es/N0, the total equivocation probability decreases less

than one order of magnitude (from 0.121 for Es/N0 = −5 dB to 0.014 for Es/N0 = 15dB).

Therefore, one may expect equivocation to occur even when operation is with very high Es/N0.

To eliminate equivocation a number of different post-processing techniques have been proposed

in the literature [1,3,4], [7, sec. 6.4.4]. We consider a simple post-processing structure in which

phase estimates {φm} are unwrapped to produce the final carrier phase estimates {φm} accord-

ing to

φm = φm−1 +(

φm − φm−1

)

mod 2πM

. (51)

Note that |φm − φm−1| =∣∣∣(φm − φm−1)mod 2π

M

∣∣∣ < π

Mi.e., successive final carrier phase estimates

always differ by less than πM

(in absolute value). In this way equivocation is eliminated and the

final estimates φm are able to correctly follow the dynamics of the true carrier phase φT . The

phase unwrapping signal processing in (51) is represented in Figure 2(b) for BPSK.

19

5.3 Approximate cycle-slip probability

The post-processing required to eliminate the equivocation phenomenon gives rise to the possi-

bility of cycle-slips to occur. Due to the 2πM

phase ambiguity of carrier phase estimates, there are

infinitely many stable points of operation {φT ± 2πM

k} with k any integer. Whenever estimates

leave the vicinity of the current stable point and fluctuate around another adjacent stable point

a cycle-slip occurs [7]. This may happen due to the noise, residual frequency offset or the actual

dynamics of the true carrier phase φT . It can be shown that a cycle-slip of 2πM

occurs whenever

the trajectory of the BWA output samples {zm = |zm|ejφm} encircles the origin of the complex

plane [3]. Exact analysis of the cycle-slip phenomenon is very difficult. However, an approxi-

mation for the cycle-slip probability P (SLIP) of the feedforward V&V synchronizer in Figure 2

has been derived by De Jonghe and Moeneclaey [3,4]. Assuming φT = 0, this approximation is

given by

P (SLIP) ≈ 4

∫ π

π/2

P (SLIP|φm) fΦ(φm) dφm (52)

where

P (SLIP|φm) ≈(∫ φm−π

−π/2

fΦ(φm−1) dφm−1

)

·(∫ π/2

φm−π

fΦ(φm+1) dφm+1

)

(53)

and will be referred to as J&M approximation. The probability (52) is approximate because

it only accounts for cycle-slips involving three consecutive samples φm−1, φm, φm+1. However,

when Es/N0 increases these will be the predominant cycle-slips and therefore the approximation

should improve in this circumstance [4]. Using (36) the approximate conditional probability in

(53) may be evaluated for BPSK. The result is

P (SLIP|φm) ≈[

3

4− φm

2π+

1

π

∞∑

n=1

1

ncn(γ)

[

sin(nπ

2

)

− (−1)n sin (nφm)]]

·[

−1

4+

φm

2π+

1

π

∞∑

n=1

1

ncn(γ)

[

sin(nπ

2

)

+ (−1)n sin (nφm)]]

. (54)

The approximate BPSK cycle-slip probability may now be evaluated inserting (36) and (54)

in (52) and performing the integration3 over φm ∈ [π/2, π]. The result is plotted in Figure 7

together with a set of simulation results (108 trials per simulated point). When Es/N0 is low the

J&M approximation underestimates the actual cycle-slip probability (simulated points). This

is expected because under this operation condition, many slips involve more than three BWA

samples and these are not accounted for by (52). However, as Es/N0 increases the approximation

becomes very tight because most cycle-slips involve only three successive filter output samples,

as considered in the approximation.

3This integral may be evaluated in closed form but the result is too long to report here.

20

Cycle-slip probab

ility,

(SLIP

)P

310−

510−

710−

10− 8− 4− 0 2 4

0 (dB)sE N

6− 2−

110−

210−

410−

610−

V&V estimator

B P S K

1 6

0T

N

φ

=

=

Simulation

J & M ap p r ox imation

Figure 7: Cycle-slip probability of the V&V carrier phase estimator for BPSK as a function of

the operating SNR.

6 Conclusions

In this report we have derived the exact statistics of the sum z ,∑N

n=1 x2n = zI + jzQ = rejφ

where the {xn} are i.i.d. complex-valued Gaussian random variables having either zero or non-

zero means. In the zero mean case, expressions were derived for the joint p.d.f. and c.d.f. of

(zI , zQ), for the marginal p.d.f. and c.d.f. of zI , zQ and r and also for the moments E[rα].

In the non-zero mean case, the joint p.d.f. of (zI , zQ) and of (r, φ) and the marginal p.d.f.

of zI , zQ and r were derived. An useful Fourier series expansion for the p.d.f. of φ was also

obtained. The non-zero mean results were applied to theoretically characterize the performance

of the well-known V&V carrier phase feedforward estimator operating from BPSK signals. In

particular, exact expressions were obtained for the p.d.f. and variance of estimates. Using the

exact p.d.f. of the carrier phase estimates and available results from [1] and [3], expressions for

the exact equivocation probability and approximate cycle-slip probability of the V&V carrier

synchronizer were presented.

21

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23