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Guzmán Abstract—In this correspondence, the use of superimposed training (ST) as a mean to estimate the finite impulse response (FIR) components of a widely linear (WL) system is proposed. The estimator here presented is based on the first-order statistics of the signal observed at the output of the system and its variance is independent of the channel components if suit- able designed training sequences are employed. The construction of such sequences having constant magnitude both in time and frequency domains is also addressed. Index Terms—Joint channel I/Q imbalance estimation, optimum channel independent sequences, superimposed training, widely linear system esti- mation. I. INTRODUCTION Different physical phenomena can be conveniently analyzed using complex random processes and widely linear (WL) systems. The pro- Manuscript received December 17, 2010; revised May 23, 2011; accepted July 11, 2011. Date of publication July 25, 2011; date of current version October 12, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Josep Vidal. The work of I. A. Arriaga-Trejo is funded by the Mexican Council for Science and Technology (Conacyt) with the graduate Grant 271258. This work has been also supported by Intel Mexico, under Grant DCIT-2006. I. A. Arriaga-Trejo and A. G. Orozco-Lugo are with the Communication Sec- tion of Cinvestav-IPN, C.P. 07360, Mexico City, Mexico (e-mail: iarriaga@cin- vestav.mx; aorozco@cinvestav.mx). A. Veloz-Guerrero and M. E. Guzmán are with Intel Labs/IPR/SIA, Intel Guadalajara Design Center, C.P. 45600 Jalisco, Mexico (e-mail: a.veloz@intel. com; m.guzman@intel.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2162834 cessing of complex valued data using WL systems has become an area of interest since improvements with respect to conventional linear pro- cessing may be achieved [1]. Since general complex random processes are improper [2], WL processing is an appropriate analysis tool to fully exploit the statistical information contained in their augmented second order statistics (covariance and complementary covariance functions). Furthermore, an additional degree of freedom is obtained with WL processing, since the conjugate of the observed process is also con- sidered. On the other hand, the use of strictly linear (SL) processing is adequate for processes with null complementary covariance. Var- ious results have been reported in the open literature where WL sys- tems are used to perform estimation, detection and prediction among other applications (see [3] and references therein). In communication theory, for instance, WL systems result from the joint modeling of the multipath propagation channel together with the in-phase and quadra- ture-phase (I/Q) imbalances at the receiver [4]. The joint estimation of these impairments is of practical interest since it allows to construct an equalizer that compensates undesired effects. Within this field of appli- cation, two approaches have been considered to perform the estimation task: blind and training based methods. Blind techniques make an ef- ficient use of the available bandwidth (see [5] and references therein), however a considerable number of samples might be required to be processed before the algorithm delivers reliable estimates. On the con- trary, with trained methods, the number of required samples is reduced since dedicated slots of time are assigned to training pilots, although this operation decreases the effective data rate. In [6]–[8] joint esti- mation of channel effects and compensation of I/Q imbalances using training sequences are documented. However, the reported design pro- cedure does not guarantee sequences possessing constant magnitude in time and frequency domains. In [9] even though the conditions to perform unbiased joint estimation of channel and I/Q imbalance are identified, the sequences there proposed do not accomplish the stated objectives. In this correspondence, we propose the use of superimposed training (ST) to perform generic WL system estimation. With ST a deterministic periodic signal is added to the information bearing sequence with the purpose of inducing cyclostationarity in the trans- mitted signal. The periodicity in the statistics of the received signal is exploited to perform channel estimation. Alike blind algorithms, with ST there is no need to reserve additional bandwidth for the training sequence since it is jointly transmitted with the information. Nonethe- less, the improved bandwidth efficiency achieved comes at the expense of obtaining suboptimal channel estimates (due to the self interfering effect of the transmitted data with the ST sequence). Besides, the use of ST is only attractive in slowly varying channels, where the channel response does not change considerably for a contiguous number of transmitted blocks. One promissory application of ST is in terrestrial digital video broadcasting, where it can outperform pilot assisted transmissions [10]. In contrast to the analysis presented in [11] and [12], where the ST sequence is used to identify a SL system, in the WL scenario the im- plicit training sequence must fulfill additional mathematical restric- tions. An important contribution of this correspondence is a systematic approach to generate families of sequences that yield unbiased channel estimators for WL systems. Optimum sequences having constant mag- nitude in time and frequency domains are obtained from the generated families using conventional optimization methods. The correspondence is organized as follows. In Section II, we in- troduce the WL model and the ST theoretical framework. The pro- posed channel estimator using ST is presented afterwards in Section III, where the restrictions required for the training sequences to obtain un- 1053-587X/$26.00 © 2011 IEEE