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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998
The Modified Cramér–Rao Bound
in Vector Parameter Estimation
Fulvio Gini, Member, IEEE, Ruggero Reggiannini, and Umberto Mengali, Fellow, IEEE
Abstract— In this paper we extend the scalar modified
Cramér–Rao bound (MCRB) to the estimation of a vector of
nonrandom parameters in the presence of nuisance parameters.
The resulting bound is denoted with the acronym MCRVB,
where “V” stands for “vector.” As with the scalar bound, the
MCRVB is generally looser than the conventional CRVB, but
the two bounds are shown to coincide in some situations of
practical interest. The MCRVB is applied to the joint estimation
of carrier frequency, phase, and symbol epoch of a linearly
modulated waveform corrupted by correlated impulsive noise
(encompassing white Gaussian noise as a particular case),
wherein data symbols and noise power are regarded as nuisance
parameters. In this situation, calculation of the conventional
CRVB is infeasible, while application of the MCRVB leads to
simple useful expressions with moderate analytical effort. When
specialized to the case of white Gaussian noise, the MCRVB
yields results already available in the literature in fragmentary
form and simplified contexts.
Index Terms— Cramér–Rao bound, non-Gaussian noise, parameter estimation.
I. INTRODUCTION
A
SCALAR lower bound to the error variance of unbiased
parameter estimators has been proposed in [1], with the
name modified Cramér–Rao bound (MCRB). This bound has
two major features: 1) unlike the conventional Cramér–Rao
bound (CRB), it can be easily calculated in the presence
of unwanted (or nuisance) parameters and 2) it is generally
looser than the CRB, but in some cases of practical interest
it approaches the CRB as computed for known nuisance
parameters.
In many situations, however, the need arises for jointly
estimating two or more parameters in the presence of nuisance
terms [2, p. 333], [3, p. 329], [4]. A typical example is
the simultaneous estimation of carrier frequency, phase, and
symbol epoch of a modulated waveform, wherein the unknown
data symbols represent nuisance terms. In this situation, a
vector bound (as opposed to a scalar bound) is required to
assess the joint estimation performance.
In the next section, we extend the MCRB to the joint estimation of a vector of nonrandom parameters in the presence
of random nuisance parameters. The resulting bound will be
denoted with the acronym MCRVB, where “V” stands for
Paper approved by E. Eleftheriou, the Editor for Equalization and Coding
of the IEEE Communications Society. Manuscript received August 26, 1996;
revised May 8, 1997.
The authors are with the Department of Information Engineering, University
of Pisa, I-56126 Pisa, Italy (e-mail: gini@iet.unipi.it; reggiann@iet.unipi.it;
mengali@iet.unipi.it).
Publisher Item Identifier S 0090-6778(98)01063-0.
“vector.” Like the conventional CRVB [2, p. 79], [3, Ch. 3],
the derivation of the MCRVB relies on the definition of a
properly modified Fisher information matrix (MFIM), which is
the expectation (with respect to the nuisance parameters) of the
conventional FIM as computed for fixed nuisance parameters.
The diagonal elements of the inverse of the MFIM represent
lower bounds to the error variance in the estimation of the corresponding parameters. Although it is generally looser than the
CRVB, the MCRVB can be calculated with moderate effort.
Conversely, in most practical situations the CRVB cannot be
derived in closed form, especially in the presence of data modulation [1], [5] and in the presence of non-Gaussian noise [6].
In Section III the MCRVB is applied to the problem of joint
estimation of frequency, phase, and symbol epoch of a linearly
modulated signal corrupted by correlated impulsive noise,
where the latter is modeled as a spherically invariant random
process (SIRP), i.e., a complex-valued process whose statistics
are invariant to phase rotations [7]–[9]. The above model
has been proposed to account for the effect of atmospheric
noise [9], [10] and encompasses stationary white Gaussian
noise as a particular case. The tails in the probability density
function (pdf) of impulsive noise are typically higher than
those exhibited by a Gaussian pdf, thus accounting for the
quality of the noise being “impulsive.” Other authors have
dealt with symbol detection in the presence of non-Gaussian
noise, one instance being in [11], where the first-order pdf of
the assumed noise model belongs to the SIRP class. Closedform calculation of the conventional CRVB for the proposed
estimation problem turns out to be impossible. Conversely,
evaluation of the MCRVB can be readily managed and leads
to closed-form expressions that are useful when dealing with
synchronization issues for channels affected by impulsive
noise. To the authors’ knowledge, no specific bounds for the
above noise model have been proposed so far in the literature.
When specialized to the important case of white Gaussian
noise (Section IV), the MCRVB yields results similar to
those available in the literature in the simplified context of
a purely sinusoidal signal model [12], [13]. Similarities are
also found with the results in [1], which deals with the
separate estimation of one parameter at a time, the others being
regarded as nuisance terms with a priori noninformative pdf’s
(i.e., uniformly distributed random variables). The approach
followed here, however, is more general and provides insight
into the interactions between estimation errors. In particular,
we discuss the relation between the choice of time origin
in the observation interval and the mutual coupling between
estimation errors. Further, in Section IV we show that in
0090–6778/98$10.00  1998 IEEE
GINI et al.: MODIFIED CRAMÉR–RAO BOUND
53
certain situations the MCRVB approaches asymptotically (as
the observation length grows large) the CRVB calculated for
known nuisance parameters.
It is worth noting that the MFIM satisfies Fisher’s five
properties and thus qualifies as an information quantity [14, p.
60]. We also observe that a necessary and sufficient condition
for the equality in (2) to hold is
II. THE MODIFIED CRAMÉR–RAO VECTOR BOUND
Denote
a noise-corrupted waveform observed in the
interval
where
is a vector
parameter to be estimated in the presence of the random
nuisance vector
Assume further that
the following “regularity conditions” hold [14].
does not depend on .
1) The domain
2) The partial derivatives
exist
and satisfy the relations
(1)
for all .
3) The partial derivative of
with respect to
exists and has a finite second-order moment. Also, the
second partial derivative exists and has finite first-order
moment.
4) The pdf of the nuisance parameters
does not
depend on .
denote an unbiased estimator
Then, letting
of , the MCRVB to the error covariance matrix
of can
be formulated as
(2)
where
means that the matrix
is positive
semidefinite and the matrix
is the MFIM
(3)
Note that the derivatives are computed at the true value of
and the expectation is taken with respect to
To prove (2) it is sufficient to show that the following
inequality holds:
(4)
where
denotes the FIM [14, p. 60]
(9)
From (4) it turns out that, as with its scalar version, the
MCRVB is looser than the CRVB. This fact might raise some
concern about the tightness of the MCRVB. However, the
MCRVB can often be derived in closed form while the CRVB
can not. Moreover, as discussed in Section IV, the MCRVB
coincides with the true CRVB, provided that the latter is
calculated under the assumption of known nuisance parameters
and a sufficiently long observation interval is considered.
III. JOINT ESTIMATION OF FREQUENCY,
PHASE, AND SYMBOL EPOCH
A. Signal and Noise Models
Using complex-envelope notation, the observed waveform
is modeled as the sum of signal
plus noise
(10)
The former is modeled as
(11)
is a known amplitude, denotes the offset of the
where
carrier frequency from its nominal value,
is the carrier
phase at the time origin,
represents the symbol epoch,
is the symbol spacing, the
’s are complex-valued
independent identically distributed (i.i.d.) zero-mean random
data, and, finally,
is the signaling pulse. For simplicity we
regard
as approximately time-limited so that the signal
component (11) may be written as
(12)
where
and
integer part of .”
The noise component is modeled as
means “the
(13)
(5)
and
(6)
In fact, assuming (4) is true, (2) follows immediately as
(7)
where we used the CRVB inequality
(8)
The proof of (4) is given in Appendix A.
is a positive random variable representing the
where
local noise power, while
denotes a complex-valued
Gaussian process with unit-power zero-mean i.i.d. real and
imaginary components of arbitrary spectral shape. Depending
on the statistics of
, the above model encompasses several
important distributions, such as the contaminated normal, the
Middleton class A, the generalized Laplace, the generalized
Gaussian, the generalized Cauchy, the K-law, the Weibull, etc.
[8]–[10]. As the statistical properties of
are not affected
by a fixed arbitrary phase rotation, it follows that
is a
SIRP [8]. Equation (13) is viewed as a realistic model for
some communications channels affected by additive impulsive
noise [9]–[11].
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998
B. Discussion
C. Calculation of the MCRVB
In the following we concentrate on the joint estimation of
the synchronization parameters
in the presence
of the nuisance terms
where
is the
vector of transmitted symbols. Application of the conventional
is hindered by the presence
CRVB to the estimation of
of the nuisance vector . As is seen from (5), calculation
of the CRVB involves knowledge of the function
,
which in principle could be obtained by averaging out from
. As will be shortly demonstrated, however, such
an approach is analytically intractable. An alternative route
consists of seeking the CRVB for all the parameters and
, and then restricting our attention to . Once again, this
attempt proves vain because the FIM cannot be formulated in
the presence of discrete parameters, as are the data symbols
involved in (11).
The FIM for has been calculated in [12] and [13] for the
simple case of a pure sinewave corrupted by Gaussian noise. In
[15], where a data modulated carrier is assumed, the presence
of the symbols is overlooked by considering a reduced-size
FIM and so returning to a sinewave. Simultaneous estimation
of phase and frequency of a sinewave corrupted by multiplicative and additive noises is dealt with in [6]. The paper shows
that the CRVB can be calculated in closed form only under
the assumption of Gaussian noises.
As will be soon shown, calculation of the MCRVB is
feasible if equally spaced samples of
are taken as the
elements of the observed vector . To this purpose, we assume
that both signal and noise have one-sided bandwidth
and
that sample spacing is
. As is known, such
conditions ensure an exact equivalence between discrete-time
and continuous-time signal models. Using the subscript to
denote a sample taken at the instant
, the th
element of has the form
Conversely, the MCRVB can be computed through a
straightforward procedure which is detailed in Appendix B and
briefly outlined in the following. Start from the conditional
pdf
, which is easily seen to be Gaussian,
(14)
is the number of samples in
where
the observation interval. Letting
and
and assuming independence of signal
and noise, the pdf of can be written as
(17)
where the superscript denotes transpose conjugate,
is
is the cothe vector representation of the signal (12), and
variance matrix of the noise sequence
defined as
(18)
and
being the real and imaginary part of , respec, we find that its
tively. Recalling the assumptions on
where
is such that
covariance matrix is
.
Substituting (17) into (3) allows us to compute the MFIM as
are first written as
follows. The elements of the MFIM
expectations with respect to of the corresponding elements
of the FIM for the estimation of when is perfectly known,
i.e.,
(19)
where
(20)
Second, recalling the definition of FIM and noting that the
vector is Gaussian for a fixed , results in [3, Appendix 3C]
(15)
Observe that
is indispensable in the calculation of
the conventional CRVB for the estimation of , since it is
involved in the expectations in (5). On the other hand, the
derivation of
implies separate knowledge of
and
. The former has to be derived from
, taking
the relationship between and into account, while the latter
can (in principle) be found from
(16)
is a multivariate Gaussian pdf (see [8] and
where
Appendix B). Clearly, the above procedure is prohibitively
complex and this explains why the true CRVB can hardly be
computed.
(21)
and, eventually,
Finally, inserting (21) into (19) yields
the MCRVB .
Equations (19)–(21) are fully developed in Appendix B for
is a first-order autoregressive
the important case in which
process (i.e., the noise is exponentially correlated), whereby
becomes
being the
the matrix
one-lag correlation coefficient
. Under
these conditions the desired MFIM is found to be
where
and
is the MFIM computed under the assumption of Gaussian noise
.
with power
GINI et al.: MODIFIED CRAMÉR–RAO BOUND
55
Collecting the above results produces
(29)
where
bottom of the page.
(22)
is the mean-square value of the symbols,
where
denotes the Fourier transform of the signaling pulse
,
is the convolution
, and
is
the mean-square bandwidth of
and (30)–(32), shown at the
IV. JOINT ESTIMATION WITH A WHITE GAUSSIAN CHANNEL
The MCRVB for a white Gaussian channel is obtained from
(22), letting
and, correspondingly,
, i.e.,
(23)
Also,
and
(33)
are parameters defined as
(24)
(25)
being the one-lag correlation coefficients of
, respectively, i.e.,
and
where
, and
spectral density of the noise.
Bearing in mind that
is the (one-sided) power
(34)
(26)
,
denoting the Fourier
where
.
transform of
has root-raised-cosine shape with
For example, if
, and
are expressed
rolloff factor , the parameters ,
by
(27)
(28)
is the received energy per symbol, from (33) the following
bounds to the variance of the estimates , , and are found
(35)
(36)
(37)
is the observation length.
where
It is worth noting that bound (35) coincides with the CRB
calculated in [12] for a pure (nonmodulated) sinewave under
the assumption that the phase is unknown and uniformly
distributed in
. Furthermore, both the CRB and (35) do
not depend on the time origin. Conversely, the scalar MCRB
on derived in [1] under the assumption that the time epoch
is uniformly distributed in
, as well as the CRB for
(30)
(31)
(32)
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998
a pure sinewave of known phase [2, p. 280], [12] do depend
on the time origin, since they vary as the inverse of
.
In particular, they achieve a maximum, given by (35), only
when the observation window is centered on the time origin
. This result suggests that frequency
estimation may be aided by knowledge of the signal phase at
an instant different from the midpoint
of the
observation interval, with an estimation accuracy improving
with the distance
. This is intuitively explained
considering the plot of the total unwrapped phase (i.e., not
constrained modulo
) of a noisy sinusoid as a function
of time. The observed phase samples will appear to be spread
about a straight line (representing the true phase), and a
sensible estimate of the sinewave frequency is the slope of
the straight line that best fits the samples [16]. As is easily
recognized, the uncertainty in slope determination decreases
if the straight line is constrained to pass through a given
point (known phase) at an instant
. In the limit, as
, the slope estimate becomes infinitely accurate.
From (36), we note that the bound for phase estimation does
depend on the time origin. This behavior has been already
pointed out in [12] and [13] in discussing the pure sinewave
case. For instance, letting
or
we find
Bound (37) is identical to the scalar MCRB derived in [1]
for random frequency, phase, and data symbols. This result
can be explained noting that the entries in the last column and
last row of the MFIM (see Appendix B) are all zero except
for element
. From (33) we observe that the bound
on is decoupled from those of and , so it is not surprising
that (37) coincides with the MCRB derived in [1] under the
assumption that and are random nuisance parameters.
Some situations are mentioned in [1] in which the scalar
MCRB calculated under the assumption of random nuisance
parameters coincides with the CRB evaluated for known
nuisance parameters. Under the same assumptions, it is easily
inferred that the above property also holds for the case of
joint parameter estimation. Specifically, the assumptions used
in [1] are: 1) the modulation is M-level phase—shift keying
(actually this assumption is not strictly necessary but it was
made to simplify the proof); 2) the data sequence is ergodic;
3) the ratio
is large; and 4) the observation length
grows to infinity. To ease comparison with the results in [1],
we now use the continuous-time signal and noise models (10),
(11), and (13) where
is white Gaussian noise of one-sided
power spectral density
. If
is known, the FIM for the
parameters
is easily calculated as
(38)
while the choice
bound
leads to the smallest
(39)
The limit (39) is identical to the scalar CRB for the estimation
of the phase of a sinewave of known frequency [3, p. 33]. Such
a bound does not depend on the time origin, however. This
result does not come unexpected since, choosing
and
makes the off-diagonal elements in the MFIM
vanish (see Appendix B). Thus, the bounds in the estimation
of and are decoupled: any a priori knowledge on does
not affect the accuracy in the estimation of , and vice versa.
In [1], it is shown that the scalar bounds for and are,
respectively,
(42)
and, for simplicity, we have
where
assumed the origin as the center of the observation interval.
the main
In [1, Appendix B] it is shown that as
diagonal elements
,
, and
approach the following
expressions: P
(43)
(44)
(40)
(45)
Letting
and
denote the bounds for
and
as obtained from the MCRVB, respectively, and
comparing (40) with (35) and (36), we find
(41)
as
varies within
. From (41) we observe that the
bounds provided by (35) and (36) are tighter than their scalar
counterparts in [1], except for the case
,
where the bounds coincide due to the vanishing of the off
diagonal terms in the MFIM.
Also, the cross terms
and
become negligibly small
with respect to
,
, and
as
. Comparison of
(43)–(45) with (35)–(37) demonstrates our claim.
V. CONCLUSIONS
This paper has extended the scalar MCRB to vector parameter estimation (MCRVB). The MCRVB retains many
of the properties of the scalar MCRB. In particular, it is
generally looser than the conventional CRVB, even though
GINI et al.: MODIFIED CRAMÉR–RAO BOUND
57
there are some situations of practical interest where it strictly
approaches the CRVB. A specific application of the MCRVB
has been discussed wherein frequency, phase, and timing
epoch of a linearly modulated waveform are to be jointly
estimated in the presence of correlated impulsive noise. In
this example, data symbols and noise power are assumed
as unknown random nuisance parameters. The above is a
typical situation in which the CRVB cannot be analytically
evaluated in closed form, whereas calculation of the MCRVB
leads to explicit bounds with moderate analytical effort. These
bounds are found to be tighter than their scalar counterparts
as calculated by the MCRB. To the authors’ knowledge,
no other bounds for this application have been proposed so
far in the literature. The MCRVB approach seems to be
applicable in many other different situations where the search
for conventional bounds is analytically infeasible.
The equality sign holds in (A6), and therefore in (A1), if
and only if
(A7)
vector that does not depend on . Note that
where is a
if (9) is satisfied then (A7) is also satisfied, but the reverse
is not true.
APPENDIX B
CALCULATION OF THE MCRVB FOR THE
JOINT ESTIMATION PROBLEM OF SECTION III
The observed data can be written in vector form as
where
APPENDIX A
PROOF OF (4)
To prove inequality (4), we demonstrate that the matrix
is positive semidefinite, i.e.,
(B1)
(A1)
where is an arbitrary
vector and
are the
FIM and MFIM, defined in (5) and (3), respectively. We start
observing that
(A2)
Substituting into (5) produces
(A3)
Next, we premultiply by
and postmultiply by
to get
where
Also, the noise vector can be written as
is a random variable with a priori known pdf and
is an -dimensional complex-valued Gaussian circular
vector. In particular, we assume that the in-phase
and
quadrature
components of
have zero mean unit
variance and covariance matrix
.
As a first step toward the derivation of the MCRVB,
we calculate the FIM
defined in (20), relative to
the estimation of the unknown vector when the nuisance
parameters
are known. Afterwards, the elements of the
MFIM will be derived from (19). To compute
we need
the conditional pdf
, which is easily recognized
to be Gaussian
(A4)
Differentiating
is not a function of
with respect to
yields
and recalling that
(B2)
(A5)
Also, application of the Cauchy–Schwartz inequality produces,
after standard manipulations
(A6)
and bearing in mind (3) we conclude that
.
where the dependence of the signal on and the data
has been explicitly indicated. The calculation of the FIM for
a vector of correlated Gaussian observations is carried out in
[3, Ch. 3] and produces
(B3)
. Inserting (B2) into (B3) and (B3) into (19)
where
we finally obtain the elements of the modified FIM.
In the following, we discuss the important case of exponentially correlated noise, i.e.,
is modeled as a first-order
autoregressive process. This entails that the elements of
are
where denotes the one-lag correlation
coefficient
.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998
Inserting into (B9) gives, after some algebra,
A) Calculation of
Averaging (B3) with respect to
yields
(B4)
while the second is over
where the first expectation is over
. Now, recalling the well-known result for quadratic forms
is a generic complex-valued
random vector,
is a known matrix, and
denotes the
trace of a matrix), produces
(B5)
On the other hand, using (B1) produces
(B11)
(B6)
denote the one-lag correlation coefficient of
we
Letting
have
. Also, assuming that the observation
interval is much longer than the symbol duration
,
(B11) can be approximated by keeping only the cubic terms
in . Correspondingly, (B9) becomes
so that
(B12)
B) Calculation of
This element may be written as
(B7)
where
Let
denote an auxiliary matrix whose elements are
defined as
(B13)
From (B1) we get
(B8)
(B14)
Collecting (B5), (B7), and (B8) yields
Hence
(B9)
where
and
. Recalling the
assumption of exponentially correlated noise, the elements of
the inverse of
are found to be
(B15)
Define the matrix
with elements
(B16)
or
Collecting (B13) and (B15) yields
and
(B17)
(B10)
GINI et al.: MODIFIED CRAMÉR–RAO BOUND
59
algebra we arrive at
and after some manipulations (assuming
(B27)
(B18)
which becomes (under the usual assumption
and
C) Calculation of
Consider the off-diagonal elements
(B28)
is the one-lag correlation of
where
(B19)
Reasoning as above, from (B6) and (B14) we get
,
E) Calculation of
and
(B20)
Letting
such that
.
With the same reasoning it can be shown that
. The previous results
, , and , as defined in
can be expressed in terms of
(23), (24), and (25), respectively. Collecting the above results
produces the modified FIM
(B21)
from (B19) we obtain
(B29)
(B22)
which becomes (under the assumption
whose inverse gives (22).
REFERENCES
(B23)
D) Calculation of
Consider the element
(B24)
From (B1) we have
(B25)
where
. Define
(B26)
where
the autocorrelation function of
. After some
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Fulvio Gini (M’93) received the Doctor Engineer
(cum laude) and the Research Doctor degrees in
electronic engineering from the University of Pisa,
Pisa, Italy, in 1990 and 1995, respectively.
During his military service from 1991–1992,
he joined the Istituto per le Telecomunicazioni e
l’Elettronica of the Italian Navy, assigned to the
radar division. In 1993 he joined the Department of
Information Engineering of the University of Pisa,
where he is now a Research Scientist. From July
1996 through January 1997, he was a Visiting
Researcher at the Department of Electrical Engineering, University of
Virginia, Charlottesville. His general interests are in the areas of statistical
signal processing, estimation, and detection theory. In particular, his research
interests include non-Gaussian signal detection and estimation using higher
order statistics, cyclostationary signal analysis, and estimation of nonstationary
signals, with applications to communication and radar processing.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998
Ruggero Reggiannini received the Dr.Ing degree in
electronic engineering from the University of Pisa,
Pisa, Italy, in 1978.
From 1978 to 1983 he was with USEA S.P.A.,
where he was engaged in the design and development of underwater acoustic systems. Since 1984,
he has been with the Department of Information
Engineering, University of Pisa, Pisa, Italy, where
he is currently an Associate Professor of radio
communications. His research interests are in the
field of digital satellite and mobile communication
systems.
Umberto Mengali (M’69–SM’85–F’90) received
the degree in electrical engineering from the University of Pisa, Pisa, Italy. In 1963, he obtained
the Libera Docenza in telecommunications from the
Italian Education Ministry.
Since 1963, he has been with the Department
of Information Engineering at the University of
Pisa, where he is a Professor of telecommunications.
His research interests include digital communication
theory, with emphasis on synchronization methods
and modulation techniques. He has served for six
years as Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS. He is now
an Editor of the European Transactions on Telecommunications. He has been
a consultant to industry in the area of communications.
Prof. Mengali is a member of the Communication Theory Committee. He
is listed in American Men and Women in Science.