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[10] T. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution—A tool for time-frequency signal analysis—Part III: Relations with other time-frequency signal transformations,” Philips J. Res., vol. 35, no. 6, pp. 372–389, 1980. [11] A Handbook of Fourier Theorems, Cambridge Univ. Press, New York, 1987.
Reply to “Comments on ‘A High-Resolution Quadratic Time-Frequency Distribution for Multicomponent Signals Analysis’” B. Barkat and B. Boashash
I. REPLY TO COMMENT 1 The recently proposed quadratic time-frequency distribution (TFD), referred as BD, handles power signals in the same way the Wigner–Ville distribution (WVD) or any other time-frequency distribution (TFD) handles them. These TFDs, being in essence Fourier transforms (FTs) of the signal or quadratic functions of it, cannot be evaluated in an ordinary way but have to be evaluated using functionals. As an illustration, let us reconsider the signal used in the comments, namely, a sinusoid expressed as z (t) = exp(j!0 t). The WVD of this signal is given by Wz (t; f ) =
1 01
1 3 0 0:5 )] exp[0j! ] d :
[z (t + 0:5 ) z (t
(1)
Because [z (t +0:5 ) 1 z 3 (t 0 0:5 )] = exp[0j!0 ] is not absolutely integrable over the considered interval, its FT cannot be obtained by direct evaluation, and one has to resort to transforms in the limit or functionals in order to obtain the final result in (3) of the comments. The same discussion applies to the BD or any other TFD. FTs in the limit have existed for decades and are not new concepts for the community. As a consequence, Comment 1 does not provide any new information. II. REPLY TO COMMENT 2 It was clearly stated that the BD can solve some problems that the WVD or the spectrogram cannot. It was never claimed that the BD performs better than the two other distributions at all times and all situations. To be more specific, in the paper introduction, the following statement was given: “This comparison is performed with respect to some criteria detailed later in the paper.” All the criteria pertaining to the comparisons in the paper were given in detail. In addition, the paper comparisons were basically numerical and not analytical, and in a numerical implementation, it is not possible to use an infinite-length
Manuscript received November 14, 2002; revised August 25, 2003. The associate editor coordinating the review of this paper and approving it for publication was Dr. Ta-Hsin Li. B. Barkat is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail:
[email protected]). B. Boashash is with the Signal Processing Research Centre, Queensland University of Technology, Brisbane, Qld 4001, Australia (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSP.2003.820059
Fig. 1. Doppler-lag kernels of the cone-shape distribution for a small value (top plot) and a large value (bottom plot) of its parameter.
signal, and consequently, the theoretical “best possible concentration” of the WVD is not guaranteed. This point was well illustrated in the paper where Monte-Carlo analysis as well as several examples, including real-life data, were provided to support the claim.
III. REPLY TO COMMENT 3 To start with, let us observe that conditions (11) are just sufficient conditions, and consequently, there is the following. i) Other, less restrictive, conditions can also be valid conditions for expression (9). This means that conditions (10) reported in [2, p. 14] cannot be excluded without a detailed demonstration to prove it. ii) The first moment of a TFD that violates conditions (11) may still be a good estimator of the signal instantaneous frequency (IF). In addition, in the paper, expression (9) was not used to estimate the signal IF. Instead, the peak of the BD was used as an IF estimator. Therefore, whether the BD verifies or violates conditions (11) does not have any negative implications on the results of the BD paper or on the simulations therein. Furthermore, many known TFDs do not verify (11). One of them is the spectrogram, which is still today one of the most popular and widely used quadratic TFDs. Since this has not limited its application in real-life problems, the same statement can be made about the BD.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004
IV. REPLY TO COMMENT 4 In the literature, there exist many quadratic TFDs whose kernels do not have a two–dimensional (2-D) lowpass shape but are well known for their cross-terms suppression property. One particular example is the cone-shape distribution whose Doppler-lag kernel [1, p. 146] is displayed in Fig. 1 for two values of its tuning parameter , specifically, 0.01 and 100. By continuously varying the value of from small to large, the kernel evolves from an allpass filter shape (i.e., the distribution is not appropriate for cross-terms suppression) to the shape displayed in the figure (i.e., the kernel removes all cross-terms, except those on the axis = 0). There exists no value of for which this kernel has a 2-D lowpass shape. For an arbitrary signal, the user has to select an appropriate value of to decide the amount of cross-terms suppression. Similarly, by continuously varying its parameter , the BD Doppler-lag kernel shape evolves as shown in the figure of the comments. In particular, for small values of , the kernel removes all cross-terms, except those on the axis = 0, and for large values, the BD becomes inappropriate for cross-terms suppression, as explained in the original paper. Note that because of its zero value at the origin, the BD may cause some extra energy distortion for the auto-terms; however, this does not seem to adversly affect its time-frequency representation, as shown by the various examples provided in the paper. This is why the major interest of the paper is not only to define a new quadratic TFD with useful resolution properties but to also open a new direction of research in the design of quadratic TFDs with new criteria that are not limited by old thinking. In short, one can say that each kernel has its own characteristics that are, in general, different from those of the others. This situation is very normal and expected because each kernel defines a different member of the quadratic class with different properties. Further details can be found in a recent tutorial on this question [3, ch. 3]. REFERENCES [1] S. Qian and D. Chen, Joint Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1996. [2] H. Fargetton, “Frequences instantanees de signaux multicomposantes,” Ph.D. dissertation, INST, Paris, France, 1979. [3] B. Boashash, Time Frequency Signal Analysis and Processing: A Comprehensive Reference, Oxford, U.K.: Elsevier, 2003.
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A Fine Blind Frequency Offset Estimator for OFDM/OQAM Systems Philippe Ciblat and Erchin Serpedin Abstract—Like other orthogonal frequency division multiplexing (OFDM) systems, OFDM systems based on offset quadrature amplitude modulation (OFDM/OQAM) are very sensitive to carrier frequency offset. In this paper, a new blind carrier frequency offset estimator is developed for OFDM/OQAM systems by exploiting the noncircularity of the received OFDM/OQAM signal. Since the received signal exhibits conjugate cyclic frequencies at twice the carrier frequency offset, the frequency estimator is designed by maximizing a cost function expressed in terms of the sample conjugate cyclocorrelations. The theoretical asymptotic (large sample) performance analysis of the proposed estimator is established. Computer simulations are presented to illustrate the performance of the estimator. It is shown that the proposed estimator is very accurate whenever it is well initialized. Therefore, the proposed estimator appears to be very well adapted for a tracking mode rather than for an acquisition mode. Index Terms—Carrier frequency offset, conjugate cyclocorrelation, estimation, OFDM system.
I. INTRODUCTION The orthogonal frequency division multiplexing (OFDM) system, which belongs to the family of multicarrier transmission schemes, has been developed to combat efficiently the intersymbol interference (ISI) effects on frequency-selective channels. Its main advantages are the very low computational cost [the receiver consists only of a fast Fourier transform (FFT) and, if necessary, of a more general filterbank] and the simplified equalization step [1]. During the last few years, OFDM-like techniques have received increasing attention and are currently employed in the European digital radio broadcasting (DAB), digital terrestrial TV broadcasting (DVBT), indoor wireless systems (HIPERLAN), and broadband access on twisted pair (ADSL). However, it is well-known that OFDM-like techniques are more sensitive to carrier frequency offset than single carrier techniques [2]. The frequency offset (due to Doppler shifts and local oscillator drifts) gives rise to intercarrier interference (ICI), which dramatically degrades the performance. Therefore, removing the frequency offset at the front end of the receiver is a crucial task. A lot of techniques based on the OFDM principle have been proposed in the literature. The structure of a standard OFDM transmitter consists of the concatenation of an IFFT transform and a guard interval and a linear modulation shaped by means of a rectangular window. As the time-frequency localization of such shaping windows is not compact, considerable research attention has recently been allocated to developing alternative modulations such as OFDM/offset quadrature amplitude modulation (OQAM), which constitutes a combination between an offset quadrature-amplitude modulation and a square-root Nyquist pulse-shaping filter (see, e.g., [3]–[9]). For OFDM/OQAM schemes,
Manuscript received June 12, 2002; revised April 21, 2003. This work was presented in part at the IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC’2003), Roma, Italy. The associate editor coordinating the review of this paper and approving it for publication was Prof. Xiaodong Wang. P. Ciblat is with Département Communications et Electronique, Ecole Nationale Supérieure des Télécommunications, 75013 Paris, France (e-mail:
[email protected]). E. Serpedin is with Department of Electrical Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSP.2003.820062 1053-587X/04$20.00 © 2004 IEEE
