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The Modified Cram´er–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 Cram´er–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— Cram´er–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 Cram´er–Rao bound (MCRB). This bound has two major features: 1) unlike the conventional Cram´er–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:
[email protected];
[email protected];
[email protected]). 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
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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
´ –RAO VECTOR BOUND II. THE MODIFIED CRAMER
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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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
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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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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
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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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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)
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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
[1] N. A. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cram´er–Rao bound and its application to synchronization problems,” IEEE Trans. Commun., vol. 42, pp. 1391–1399, Feb./Mar./Apr. 1994. [2] H. Van Trees, Detection, Estimation and Modulation Theory, Part I. New York: Wiley, 1967. [3] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [4] C. V. Kimball, P. Lewicki, and N. I. Wijeyesekera, “Error analysis of maximum likelihood estimates of physical parameters from one or more dispersive waves,” IEEE Trans. Signal Processing, vol. 43, pp. 2928–2936, Dec. 1995. [5] M. Ghosh and C. L. Weber, “Blind deconvolution using a maximum likelihood channel estimator,” Proc. IEEE Communications Workshop, 1991, pp. 448–452. [6] G. Zhou and G. B. Giannakis, “Harmonics in Gaussian multiplicative and additive noise: Cram´er–Rao bounds,” IEEE Trans. Signal Processing, vol. 43, pp. 1217–1231, May 1995. [7] K. Yao, “A representation theorem and its applications to sphericallyinvariant random processes,” IEEE Trans. Inform. Theory, vol. IT-19, Sept. 1973. ¨ [8] M. Rangaswamy, D. Weiner, and A. Oztrk, “Non-Gaussian random vector identification using spherically invariant random rocesses,” IEEE Trans. Aerosp. Electron. Syst., vol. 29, pp. 111–123, Jan. 1993. [9] E. Conte, M. Di Bisceglie, and M. Lops, “Optimum detection of fading signals in impulsive noise,” IEEE Trans. Commun., vol. 43, pp. 869–876, Feb./Mar./Apr. 1995. [10] E. Conte, M. Di Bisceglie, M. Longo, and M. Lops, “Canonical detection in spherically invariant noise,” IEEE Trans. Commun., vol. 43, pp. 347–353, Feb./Mar./Apr. 1995. [11] D. Middleton, “Statistical-physical models of urban radio-noise environment,” IEEE Trans. Electromag. Compat., vol. EMC-14, May 1972.
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[12] D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 591–598, Sept. 1974. [13] M. P. Fitz, “Planar filtered techniques for burst mode carrier synchronization,” GLOBECOM ’91, Phoenix, AZ, Dec. 2–5, 1991, pp. 365–369. [14] B. Porat, Digital Processing of Random Signals, Theory & Methods. Englewood Cliffs, NJ: Prentice-Hall, 1994. [15] M. Moeneclaey, “A fundamental lower bound to the performance of practical joint carrier and bit synchronizers,” IEEE Trans. Commun., vol. COM-32, pp. 1007–1012, Sept. 1984. [16] S. Bellini, C. Molinari, and G. Tartara, “Digital frequency estimation in burst mode QPSK transmission,” IEEE Trans. Commun., vol. 38, pp. 959–961, July 1990.
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.
