1 Two-stage interference-resistant adaptive periodically time-varying CMA blind equalization Giacinto Gelli and Francesco Verde Abstract—In this paper, we consider the problem of blindly equalizing a digital communication signal distorted by a linear time-invariant channel, and contaminated by severe co-channel or adjacent-channel digital inter- ference, under the assumption that the latter exhibits a different symbol rate from the desired signal. The proposed equalizer is composed of two stages, both periodically time-varying (PTV), in order to better match the periodical statistics of the received signal. The first stage employs linear PTV filtering to mitigate interference, allowing thus the second stage, based on the constant modulus algorithm (CMA), to reliably recover the transmit- ted information symbols. Computer simulations confirm the effectiveness of the new approach, and comparisons with existing blind methods show that a significant performance gain can be attained. Keywords— Blind equalization, time-varying filters, cyclostationary sig- nals, constrained optimization, constant-modulus algorithm. I. I NTRODUCTION I N high-speed digital communication systems, the transmit- ted signal is subjected to time dispersion due to non-ideal transmission media, which gives rise to intersymbol interfer- ence (ISI), and is affected by thermal noise introduced at the receiver input. Channel equalization is then aimed at coun- teracting the harmful effects of ISI and noise, in order to re- liably recover the transmitted symbol sequence. Most equal- ization techniques belong to the class of non-blind or trained algorithms, since they make use of training sequences in or- der to solve for the equalizer parameters, usually under a zero- forcing, minimum mean-square error (MMSE), or maximum- likelihood criterion. To overcome the waste of resources as- sociated to transmission of training symbols, the blind equal- ization approach has been receiving a great deal of attention in the last years [1], [2], [3]. Usually, blind techniques perform channel equalization by exploiting some known property of the desired signal (e.g., second-order or higher-order statistics, cy- clostationarity, constant modulus, or finite alphabet). A major breakthrough in blind channel equalization was the pioneering work of Tong et al. [3], where it was recognized that possi- bly nonminimum-phase channels can be blindly identified us- ing only the second-order statistics (SOS) of the received signal, by exploiting temporal diversity (induced by fractionally-spaced sampling) and/or spatial diversity (associated to the use of mul- tiple sensors). In both cases, the problem can be described by a single-input/multiple-output (SIMO) or multi-channel model [1], [2], [3], whose outputs are corrupted by wide-sense station- ary (WSS) thermal noise; equalization is then performed by a multiple-input/single-output (MISO) equalizer, composed by a bank of linear time-invariant (LTI) filters. The authors are with Dipartimento di Ingegneria Elettronica e delle Teleco- municazioni, Universit` a degli Studi Federico II di Napoli, via Claudio 21, I- 80125 Napoli, Italy. Tel.: +39-0817683121, Fax : +39-0817683149, E-mail: {gelli,f.verde}@unina.it However, in both wired and wireless systems, one might be faced with the more challenging problem of counteracting not only the ubiquitous presence of ISI, but also the deleterious ef- fects of co-channel interference (CCI) and adjacent-channel in- terference (ACI). Recently, the problem of channel equalization in the presence of CCI and/or ACI has been considered in [4], [5], where it is assumed that the desired and the interfering sig- nals exhibit the same symbol rate. Thus, since the disturbance is WSS and the received signal is sampled at the symbol rate, the equalizers proposed in [4], [5] turn out to be LTI as well. However, a more general situation arises when the desired and interference signals exhibit different symbol rates. This hap- pens, for example, in wired networks, due to crosstalk between adjacent pairs transmitting at different rates; or in wireless over- lay networks, where the same geographical area is covered by different systems sharing the same spectrum band; or finally in wireless multi-rate networks, where different services with different rates need to be accommodated efficiently in an uni- fied bandwidth-on-demand manner. If the received signal is fractionally-sampled with respect to the symbol period of the de- sired signal, or multiple sensors are employed at the receiver and their outputs are sampled at the symbol rate, the problem can be conveniently described by a linear multiple-input/multiple- output system model, which, however, due to the different sym- bol rates of the desired and interfering signals, turns out to be linear periodically time-varying (LPTV) or linear almost period- ically time-varying (LAPTV). An equivalent LTI SIMO model can be obtained regarding as input only the desired symbol se- quence, but now the additive disturbances affecting the outputs of the SIMO system turn out to be the sum of WSS thermal noise plus cyclostationary or almost cyclostationary interference. In both cases, optimal equalization and interference suppression is achieved by a MISO time-varying equalizer, composed by a bank of LPTV or LAPTV filters [6]. Preliminary work on this topic has been carried out in [7], where a non-blind PTV MMSE equalization technique has been proposed. In particular, in [7] it has been shown that, by exploiting the time-varying properties of the additive disturbances at the outputs of the SIMO channel, it is possible to derive the equalizer structure without explicit knowledge or estimation of the interfering channel. A blind PTV approach has also been proposed in [8], where estimation of the desired-signal channel is based on the different circularity and/or cyclostationarity properties of the desired and interfer- ing signals. Simulation results in [7], [8] confirm that the PTV equalizers achieve a significant performance gain over their LTI counterparts. In this paper, we propose a new approach to blind PTV equal-