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Internal Report
# 3036/2006
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Statistical Characterization of the Sum of
Squared Complex Gaussian
Random Variables
Gonçalo N. Tavares and Luis M. Tavares
March, 2006
Gonçalo N. Tavares is with the Department of Electrical and Computer Engineering, Instituto Superior
Técnico (IST) and with Instituto de Engenharia de Sistemas e Computadores – Investigação e Desenvolvimento (INESC-ID), Lisbon, Portugal (email: [email protected]). Luis M. Tavares is
with the Department of Engineering, Escola Superior de Tecnologia e Gestão (ESTIG), Beja, Portugal
and with INESC-ID, Lisbon, Portugal (email: [email protected]).
'
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 ,
PN
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 theoretical 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 [r α ] . . . . . . . . . . . . . . . . . . . . . . .
6
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
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
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4
5
6
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Viterbi & Viterbi carrier phase synchronizer: (a) NDA carrier phase estimator,
(b) phase-unwrapping post-processing. . . . . . . . . . . . . . . . . . . . . . . .
3
14
Variance of V&V carrier phase estimates from BPSK signals as a function of the
operating SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
13
The analytical and simulated p.d.f. of BPSK carrier phase estimates using the
V&V feedforward estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
11
15
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
21
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 subject 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
P
∗
jφ
z , N
[9]. The {xn , yn } are assumed to be correlated zero-mean
n=1 xn yn = zI + jzQ = re
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 evaluating the performance of multichannel reception with differentially coherent and noncoherent
P
2
2
∗
detection, Proakis derives the c.f. of the quadratic D , N
n=1 [A|xn | + B|yn | + 2< {Cxn yn }]
and evaluates the probability of error Pe = Pr {D < 0} in closed form [10]. More recently,
P 1
P N2
2
2
Simon & Alouini determine the c.f. of the quadratic D = A N
n=1 |xn | + B
n=1 |yn | 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 esP
∗
timation, Lank et al. study the sum zk , N
n=k+1 xn xn−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].
P
M
The sum z = N
n=1 xn 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 M th 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 variables, 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 {x n }
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 }N
n=1 be a set of mutually i.i.d. complex Gaussian random variables, each with complex
mean µn and variance 2σ 2 (for all n) and with independent real and imaginary components xI (n)
and xQ (n) respectively, each with variance σn2 = σ 2 for all n. Define the set of complex-valued
random variables
yn = yI (n) + jyQ (n) , x2n = x2I (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
X
n=1
2
yn =
N
X
n=1
x2n
(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
X
E[x2n ]
=
n=1
n=1
and
N
X
µ2n , M = |M |ej φ̄ = MI + jMQ
(3a)
var{z} = 8N σ 4 + 8N P σ 2
where P ,
1
N
PN
n=1
(3b)
|µn |2 . Since each yn , x2n 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
Z ∞Z ∞
2
2
ΦyI (n),yQ (n) (tI , tQ ) =
ej(xI (n)−xQ (n))tI +j2xI (n)xQ (n)tQ fXI (n),XQ (n) (xI (n), xQ (n))dxI (n)dxQ (n)
(4a)
−∞ −∞
(
)
−2σ 2 |µn |2 (t2I + t2Q ) + j<{µ2n }tI + j={µ2n }tQ
1
(4b)
= 12 exp
4 (t2 + t2 )
1
+
4σ
2
2
4
I
Q
1 + 4σ (t + t )
I
Q
where in (4a)
fXI (n),XQ (n) (xI (n), xQ (n)) =
(x (n)−<{µn })2 +(xQ (n)−={µn })2
1
− I
2σ 2
e
2πσ 2
(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
Y
ΦyI (n),yQ (n) (tI , tQ )
n=1
1
= N exp
1 + 4σ 4 (t2I + t2Q ) 2
(
−2σ 2 N P (t2I + t2Q ) + jMI tI + jMQ tQ
1 + 4σ 4 (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.,
1
fZI ,ZQ (zI , zQ ) =
(2π)2
3
Z
∞
−∞
Z
∞
ΦzI ,zQ (tI , tQ )e−j(zI tI +zQ tQ ) dtI dtQ .
(7)
−∞
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}N
n=1 .
This implies P = 0 and M = 0 + j0 and the c.f. in (6) reduces to
ΦzI ,zQ (tI , tQ ) = 1
1 + 4σ 4 (t2I + t2Q )
3
N2 .
(8)
Because E[x2I (n)] = E[x2Q (n)] = σ 2 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
PL
∗
L
L
from [10, eq. (9)], which is the c.f. of the r.v. u ,
n=1 xn yn where {xn }n=1 and {yn }n=1
are complex zero-mean correlated Gaussian r.v.’s with variance mxx and myy respectively and
correlation mxy = E[xn yn∗ ] 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 = 4σ 2 , 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 transformation with Jacobian u) and then using [2, eq. (6.565-4)] one finds the joint p.d.f. for (zI , zQ )
as
Z ∞Z π
u
1
−ju(zI cos θ+zQ sin θ)
dθdu
fZI ,ZQ (zI , zQ ) =
N e
2
4
2
(2π) 0
−π (1 + 4σ u ) 2
Z ∞
q
1
u
2
2
J0 u zI + zQ du
=
2π 0 (1 + 4σ 4 u2 ) N2
N 4−2
2
zI2 + zQ
1
1 q 2
2
=
zI + zQ , −∞ < zI , zQ < ∞ (9)
K N2 −1
2π (2σ 2 ) N2+2 2 N2−2 Γ N
2σ 2
2
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 ±
1
2
with n integer, the function Kn± 1 (x) may be expressed as a
2
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
R∞
integration of (9) over the unwanted random variable i.e., fZI (zI ) = −∞ fZI ,ZQ (zI , zQ )dzQ , and
R∞
fZQ (zQ ) = −∞ fZI ,ZQ (zI , zQ )dzI . Using [11, eq. (2.16.3-8)] it may be shown that these densities
are given by
1
1
fZ (u) = √
N
−1
2π 2 2 Γ N2 2σ 2
|u|
2σ 2
N 2−1
with u = zI or u = zQ .
4
K N −1
2
|u|
2σ 2
,
−∞ < u < ∞
(10)
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
R zI
integration of (10) as FZI (zI ) , Pr{ZI ≤ zI } = −∞
fU (u)du and FZQ (zQ ) , Pr{ZQ ≤ zQ } =
R zQ
f (u)du. Using [11, eq. (1.12.1-3)] it can be shown that
−∞ U
FU (u) =
1
+ sgn(u)G1 (|u|),
2
−∞ < u < ∞
(11)
with u = zI or u = zQ and with
"
x N2+1
x 1
1
3 N 1 x 2
K N −1
G1 (x) = √ N −3
;
1 F2 1; ,
2
2σ 2
2σ 2
2 2 4 2σ 2
Γ N2
π2 2
#
x 1
1 x 2
3 N + 2 1 x 2
K N −3
;
+
,
1 F2 1; ,
2
2 2σ 2 Γ N 2+2
2σ 2
2
2
4 2σ 2
x ≥ 0 (12)
where 1 F2 (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
Γ N2−1 x 1 3 − N 3 1 x 2
;
, ;
G1 (x) = √
1 F2
2σ 2
2
2
2 4 2σ 2
2 πΓ N2
x N
Γ 1−N
N N + 1 N + 2 1 x 2
2
;
,
;
+√
,
1 F2
2σ 2
2
2
2
4 2σ 2
π N 2N Γ N2
x ≥ 0, N even.
(13)
Using [2, eq. (8.467)] we can find an even simpler representation for G1 (x) when N is even
N
N
−1
N
−1−k
2
X
N
− 1 + k 2 X 1 x m
1
− x2
−
+k
(
)
2
2 2
,
G1 (x) = − e 2σ
2
2
m!
2σ
k
m=0
k=0
3.4
x ≥ 0, N even.
(14)
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
R zI R zQ
FZI ,ZQ (zI , zQ ) , Pr{ZI ≤ zI , ZQ ≤ zQ } = −∞
f
(u, v)dudv. Using [11, eq. (2.16.3−∞ ZI ,ZQ
8)] one finds



FZI (zI ) + FZQ (zQ ) − 1 + G2 (|z|),





FZ (zQ ) − G2 (|z|),
Q
FZI ,ZQ (zI , zQ ) =
F (z ) − G (|z|),


ZI I
2




G (|z|),
2
q
2
where |z| = zI2 + zQ
and
zI ≥ 0, zQ ≥ 0
zI ≥ 0, zQ < 0
zI < 0, zQ < 0
x N2
x .
K
N
G2 (x) =
N +2
N
2
2σ 2
2σ 2
(2σ 2 ) 2 2 2 Γ N2
1
5
(15)
zI < 0, zQ ≥ 0
(16)
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
r r N2
K N −1
.
N
2
2σ 2
2σ 2
1
1
fR,Φ (r, φ) =
N −2
2π 2 2 2σ 2 Γ
|{z}
(17)
2
fΦ (φ)
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
r r N2
K N −1
fR (r) = N −2
.
2
2σ 2
2σ 2
2 2 2σ 2 Γ N2
1
3.6
(18)
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 ≤
Rr
r} = 0 fR (u)du with fR (u) given by (18) may be expressed as
FR (r) = 1 −
3.7
r N2
r .
K
N
N −2
2
2σ 2
2σ 2
2 2 Γ N2
1
(19)
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
α
Er [r ] ,
Z
∞
N +α
α
2α (2σ 2 )α
Γ
+1
Γ
r fR (r)dr =
2
2
Γ N2
α
0
(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)
1
It 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
PN
PN
2
P
|µn |2
n=1 E[|xn | ]
γ̄ = PN
= n=12
= 2.
2σ N
2σ
n=1 var{xn }
(21)
For convenience, the results presented in this Section are expressed in terms of the normalized
SNR γ , γ̄/P = 1/(2σ 2 ).
4.1
Joint probability density function fZI ,ZQ (zI , zQ )
Inserting (6) in (7) and making the change of variables tI =
u
2σ 2
cos θ and tQ =
u
2σ 2
sin θ one
finds the joint probability density function for (zI , zQ ) as
Z ∞Z π
h“
”
“
”
i
M
M
γ2
−juγ zI − I2 cos θ+ zQ − Q2 sin θ
1+u
1+u
dθdu
f (u) e
fZI ,ZQ (zI , zQ ) =
(2π)2 0
−π
 s

2 2
2 Z ∞
γ
MQ
MI
 du
=
f (u)J0 uγ
+ zQ −
zI −
2
2π 0
1+u
1 + u2
(22a)
(22b)
−∞ < zI , zQ < ∞
where we have defined
f (u) ,
4.2
u
N
(1 + u2 ) 2
u2
exp −γN P
1 + u2
.
(23)
Marginal probability density functions fZI (zI ) and fZQ (zQ )
The marginal probability density functions for zI and zQ are determined by suitable integration
R∞
of (22b) over the unwanted random variable i.e., fZI (zI ) = −∞ fZI ,ZQ (zI , zQ )dzQ , and fZQ (zQ ) =
R∞
f
(z , z )dzI . Using [11, eq. (2.12.4-17)] it may be shown that these densities are given
−∞ ZI ,ZQ I Q
by
and
4.3
γ
fZI (zI ) =
π
Z
γ
fZQ (zQ ) =
π
Z
∞
0
∞
0
f (u)
MI
cos uγ zI −
du,
u
1 + u2
−∞ < zI < ∞
(24)
f (u)
MQ
cos uγ zQ −
du,
u
1 + u2
−∞ < zQ < ∞.
(25)
Marginal cumulative distribution functions FZI (zI ) and FZQ (zQ)
The marginal cumulative distribution functions for zI and zQ may be determined by integration
R zI
fZ (z)dz and FZQ (zQ ) , Pr{ZQ ≤
of (24) and (25) respectively as FZI (zI ) , Pr{ZI ≤ zI } = −∞
R zQ
zQ } = −∞ fZ (z)dz. It can be shown that
Z
MI
1 1 ∞ f (u)
sin uγ zI −
du, −∞ < zI < ∞
(26)
FZI (zI ) = +
2 π 0
u2
1 + u2
7
and
4.4
1 1
FZQ (zQ ) = +
2 π
Z
∞
0
f (u)
MQ
sin uγ zQ −
du,
u2
1 + u2
−∞ < zQ < ∞.
(27)
Joint probability density function fR,Φ (r, φ)
With z = rejφ the joint p.d.f. of r and φ may be written as
 s

2 2
2 Z ∞
MI
γ r
MQ
 du,
f (u)J0 uγ
r cos φ −
fR,Φ (r, φ) =
+ r sin φ −
2
2π 0
1+u
1 + u2
 s

2
2 Z ∞
γ r
|M |
=
r cos(φ − φ̄) −
f (u)J0 uγ
+ r 2 sin2 (φ − φ̄) du,
2π 0
1 + u2
0 ≤ r < ∞, |φ − φ̄| ≤ 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
Z ∞Z π
h“
”
“
”
i
M
M
γ2r
−juγ r cos φ− I2 cos θ+ r sin φ− Q2 sin θ
1+u
1+u
f (u) e
dθdu
fR,Φ (r, φ) =
(2π)2 0
−π
Z ∞Z π
γ2r
j uγ 2 (MI cos θ+MQ sin θ) −jruγ cos(φ−θ)
1+u
f
(u)
e
=
{z
} dθdu.
|e
(2π)2 0
−π
(29)
,g(φ)
The function g(φ) is periodic with period 2π and may therefore be expressed as a Fourier series
P
jnφ
g(φ) = ∞
with coefficients
n=−∞ an (ruγ)e
Z π
1
ejruγ cos(φ−θ) e−jnφ dφ = e−jnθ (−j)n Jn (ruγ) .
(30)
an (ruγ) ,
2π −π
Making this substitution in (29) and recalling from (3a) that MI , <{M } = |M | cos φ̄ and
MQ , ={M } = |M | sin φ̄, yields
Z ∞Z π
∞
X
γ2r
j uγ 2 |M | cos(θ−φ̄)
1+u
e−jnθ (−j)n Jn (ruγ) ejnφ dθdu
fR,Φ (r, φ) =
f
(u)
e
(2π)2 0
−π
n=−∞
Z π
∞
2 Z ∞
X
u
1
j
|M |γ cos(θ−φ̄)−jn(θ−φ̄)
jn(φ−φ̄) γ r
n
=
e
f (u)(−j) Jn (ruγ)
e 1+4σ4 u2
dθ du
2π
2π
0
−π
n=−∞
{z
}
|
“
”
j n Jn |M |γ
=
∞
X
bn (r, γ)ejn(φ−φ̄)
u
1+4σ 4 u2
(31)
n=−∞
8
where bn (r, γ) are the Fourier coefficients given by
γ2r
bn (r, γ) ,
2π
Z
∞
f (u)Jn
0
u
|M |γ
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
∞
X
n=1
4.5
bn (r, γ) cos[n(φ − φ̄)],
0≤r<∞
|φ − φ̄| ≤ π
.
(33)
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
Z ∞Z π Z π
”
“
”
i
h“
M
M
γ2r
−juγ r cos φ− I 2 cos θ+ r sin φ− Q2 sin θ
1+u
1+u
dφdθdu
fR (r) =
f (u) e
(2π)2 0
−π −π
Z Z
γ2r ∞ π
−j uγ (M cos θ+MQ sin θ)
=
f (u) e 1+u2 I
J0 (ruγ) dθdu
2π 0
−π
Z ∞
u
2
f (u)J0 |M |γ
= γ r
J0 (ruγ) du.
1 + u2
0
(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 ≤
Rr
r} = 0 fR (u)du with fR (u) given by (34) is
FR (r) = γr
4.7
Z
∞
0
u
f (u)
J0 |M |γ
J1 (ruγ) du.
u
1 + u2
(35)
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
"
#
∞
X
1
1+2
cn (γ) cos[n(φ − φ̄)] ,
fΦ (φ) =
2π
n=1
9
|φ − φ̄| ≤ π
(36)
where the Fourier coefficients are (n ≥ 1)
cn (γ) , γ
2
= n
Z
Z
0
∞
0
∞
Z
∞
u
rf (u)Jn |M |γ
1 + u2
0
u
f (u)
Jn |M |γ
du.
u2
1 + u2
Jn (ruγ) dudr
(37a)
(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
n+2k
∞
+k
n
N
|M |γ
n X (−1)k Γ n2 + k Γ N +n
2
+ k, + n + 2k; −γN P
cn (γ) =
1 F1
N
2
k!(n + k)!
2
2
2
Γ
+
n
+
2k
2
k=0
(38)
where 1 F1 (a, b; z) is the confluent hypergeometric function, which is readily available in many
currently available mathematical software packages, thus avoiding the custom numerical integration in (37b). From (36) we conclude that fΦ (φ) is a symmetric function around the mean φ̄
i.e., fΦ (φ− φ̄) = fΦ (φ̄−φ). This is a consequence of the uncorrelateness (independence) assumption 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 Figure 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 φ̄ 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
0
N =4
γ = −10 dB
γ = −5 dB
−2 0
Coefficient cn ( γ ) (dB)
γ = 0 dB
γ = 5 dB
γ = 10 dB
−4 0
−60
−80
−100
1
3
5
7
9
11
order, n
13
15
17
19
2 1
(a)
0
N = 16
Coefficient cn ( γ ) (dB)
−2 0
−40
−6 0
γ = −10 dB
γ = −5 dB
−80
γ = 0 dB
γ = 5 dB
γ = 10 dB
−100
1
3
5
7
9
11
order, n
13
15
17
19
2 1
(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 signals
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 systems. 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, usually 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 }N
n=1 , each with
N samples (N T seconds) long, such that φT remains approximately constant over each block.
The complex envelope of the matched filter output may thus be written
xn = an ejφT + wn ,
n = 1, 2, . . . , N
(39)
2π
−1
where an is a random symbol from the M -PSK alphabet {Ak = ej M k }M
k=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 σ 2 = 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 ]/(2σ 2 ) =
1/(2σ 2 ) = Es /N0 which coincides with the previous definition of γ and also with γ̄ = γP in (21)
P
jφT 2
because P = N1 N
| = 1. For each block xm , a feedforward carrier phase estimate is
n=1 |an e
computed as [14]
( N
)
X
1
M
φ̂m =
arg
xn+mN .
M
n=1
(40)
The feedforward nonlinear V&V carrier phase estimator for BPSK signals (M = 2) is represented in Figure 2(a).
12
x (t )
BWA
yn
xn
N :1
N
2
(i)
∑ yn
zm
1
a r g { i}
2
φˆm
n =1
fs =
1
T
(a)
εm mod π
π2
φˆm
φm
εm
0
π
εm
−−ππ 22
φm −1
Delay
NT
(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
PN
and outputs samples {zm = n=1 yn+mN } every N T 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 { φ̂m }. This
is done in the next subsections where we compute the exact p.d.f. and variance of φ̂m 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
( N
)
X
1
x2n+mN .
(41)
φ̂m = arg
2
n=1
P
jφm
2
, we conclude
Comparing (41) with a generalized version of (2) i.e., zm , N
n=1 xn+mN = rm e
that φ̂m =
1
2
arg{zm } = 21 φm . The p.d.f. of the carrier phase estimate φ̂m is thus fΦ̂m (φ̂m ) =
13
2fΦ (φ)|φ=2φ̂m . Using the series representation (36) for fΦ (φ) one finds the exact p.d.f. [Note
that φ̄ = arg{E[z]} = 2φT , see (3a)]
#
"
∞
X
1
cn (γ) cos[2n(φ̂m − φT )] ,
1+2
fΦ̂m (φ̂m ) =
π
n=1
|φ̂m − φT | ≤
π
.
2
(42)
The analytical p.d.f. of the carrier phase estimates {φ̂m } 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[φ̂m ] = φT (= 0) and
becomes sharply peaked as Es /N0 increases.
4
→ Simulation points
→ E x ac t v alue s
BPSK
P r o b a b i l i t y d e n s i t y f u n c t i o n , fΦˆ m ( φˆm )
N =1 6
Es N 0 = − 1 0 d B
φT = 0
Es N 0 = − 5 d B
3
Es N 0 = 0 d B
Es N 0 = 5 d B
2
1
0
−9 0
−60
0
30
−30
ˆ
E s t i m a t e φm ( d e g r e e )
6 0
9 0
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
Z π/2+φT
2
var{φ̂m } , E[(φ̂m − φT ) ] =
(φ̂m − φT )2 fΦ̂m (φ̂m ) dφ̂
−π/2+φT
=
Z
π/2
−π/2
φ̂2m fΦ̂m (φ̂m + φT ) dφ̂m
14
1
=
π
Z
2
=
π/2
−π/2
"
φ̂2m 1 + 2
∞
X
(−1)n
π
+
12 n=1
n2
∞
X
#
cn (γ) cos(2nφ̂m ) dφ̂m
n=1
cn (γ).
(43)
1
10−1
E s t im a t e v a r ia n c e , v a r
{ φˆm }
( r a d 2)
Simulation
C R B ( φT )
E x ac t v ar ianc e
10−2
V & V
e s t im a t o r
B P S K
N =1 6
10−3
−15
−10
−5
0
Es N 0 (dB)
5
10
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 Cramér-Rao lower bound (CRLB), which is a fundamental lower bound on the variance of any unbiased carrier phase estimate and is given by
CRLB(φT ) = (2N Es /N0 )−1 . This bound is obtained under the assumption that data modulation 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
π π
, M ]. When the true carrier phase
phase estimates {φ̂m } will be restricted to the interval [− M
π
φT is near ± M
, ± 3π
, ± 5π
, . . . then, due to the M th power operation, {φm } will be close to
M
M
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 ≈
2π
.
M
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
i h
π io
≤ φm ≤ π φT ∩ −π ≤ φm+1 ≤ − φT
P (EQ|φT ) = Pr
2
2 io
nh
π i h π
≤ φm+1 ≤ π φT .
+ Pr −π ≤ φm ≤ − φT ∩
2
2
nh π
(44)
Because samples {zm , zm+i } are statistically independent for all i 6= 0, (44) becomes
P (EQ|φT ) = 2 Pr
o
n
π o
≤ φ ≤ π φT · Pr −π ≤ φ ≤ − φT
2
2
nπ
(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 φ̄ = 2φT it is easy to show that
"
#
∞
h 1 1X1
π i
P (EQ|φT ) = 2
+
cn (γ) − sin[n(2φT − π)] + sin n 2φT −
4 π n=1 n
2
"
#
∞
h 1 1X1
π i
+
cn (γ) sin[n(2φT + π)] − sin n 2φT +
·
.
4 π n=1 n
2
16
(46)
The unconditional (total) probability of equivocation is given by [1, eq. (13)]
Z π
P (EQ|φT )fΦT (φT )dφT .
P (EQ) =
(47)
−π
Assuming φT is uniformly distributed in [−π, π], the total equivocation probability in (47)
becomes after some algebra
∞
1
4 X
1
P (EQ) = − 2
c24n−2 (γ).
2
8 π n=1 (4n − 2)
(48)
We will compare the exact result in (46) with the Gaussian approximation given by [1,
eq. (14)] with correlation coefficient2 ρ = 0
!#
"
1
1
MQ
P (EQ|φT ) ≈
erfc
1 − erfc p
4
2
2σz2
MQ
p
2σz2
!
erfc2
MI
p
2σz2
!
.
(49)
Here, MI + jMQ = E[zm ] and 2σz2 = 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 = N ej2φT
(50a)
and
var{zm } =
2σz2
4
2
= 8N σ + 8N σ = 2N
Es
1+2
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 approximation in (47) which has been computed using the Fitz approximation in (49) for the
conditional equivocation probability P (EQ|φT ) .
2
The parameter ρ represents the correlation coefficient between successive BWA output samples, which is
zero under our assumptions.
17
1
Conditional equivocation probability, P(EQ
| φT )
Simulation
G aus s ian ap p r ox .
E x ac t p r ob ab ility
BPSK
N =1 6
E
s
N0 = − 5 d B
10−1
10−2
−180
−90
0
90
True carrier phase, φT (degree)
180
(a)
1
Conditional equivocation probability, P(EQ
| φT )
Simulation
G aus s ian ap p r ox .
E x ac t p r ob ab ility
10−1
10−2
10−3
BPSK
10−4
N =1 6
E
10−5
−180
s
N0 = 0 d B
0
−90
90
180
True carrier phase, φT (degree)
(b)
Figure 5: Equivocation probability conditioned on the value of the true carrier phase offset φ T :
(a) Es /N0 = −5 dB and (b) Es /N0 = 0 dB.
18
0.2
Total equivocation probability, P (EQ)
Simulation
G aus s ian ap p r ox .
E x ac t p r ob ab ility
0.1
V&V estimator
B P S K
N =1 6
0.01
−10
−5
0
5
Es N 0 (dB)
10
15
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 = 15 dB).
Therefore, one may expect equivocation to occur even when operation is with very high E s /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 {φ̂m } are unwrapped to produce the final carrier phase estimates {φ̃m } according to
φ̃m = φ̃m−1 + φ̂m − φ̃m−1
.
(51)
mod 2π
M
π
i.e., successive final carrier phase estimates
Note that |φ̃m − φ̃m−1 | = (φ̂m − φ̃m−1 )mod 2π < M
M
always differ by less than
π
M
(in absolute value). In this way equivocation is eliminated and the
final estimates φ̃m 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 possibility of cycle-slips to occur. Due to the
2π
M
phase ambiguity of carrier phase estimates, there are
infinitely many stable points of operation {φT ±
2π
k}
M
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
the trajectory of the BWA output samples {zm = |zm |e
jφm
2π
M
occurs whenever
} 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
where
P (SLIP|φm ) ≈
Z
Z
π
P (SLIP|φm ) fΦ (φm ) dφm
φm −π
−π/2
(52)
π/2
fΦ (φm−1 ) dφm−1 ·
Z
π/2
fΦ (φm+1 ) dφm+1
φm −π
!
(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
"
#
∞
i
h nπ 3 φm 1 X 1
− (−1)n sin (nφm )
−
+
cn (γ) sin
P (SLIP|φm ) ≈
4
2π
π n=1 n
2
#
"
∞
h nπ i
1X1
1 φm
+
cn (γ) sin
+ (−1)n sin (nφm ) .
· − +
4 2π
π n=1 n
2
(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.
3
This integral may be evaluated in closed form but the result is too long to report here.
20
10−1
Simulation
J & M ap p r ox imation
Cycle-slip probability, P(SLIP)
10−2
10−3
10−4
10−5
V&V estimator
−6
10
B P S K
N =1 6
φT = 0
10−7
−10
−8
−6
−4
−2
Es N 0 (dB)
0
4
2
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 ,
PN
n=1
x2n = zI + jzQ = rejφ
where the {xn } are i.i.d. complex-valued Gaussian random variables having either zero or nonzero 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[r α ].
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