Per-tone Equalization Research Articles

Per-Wavelet Equalization for Discrete Wavelet Transform Based Multi-Carrier Modulation Systems

The Discrete Wavelet Transform (DWT) has gained attention in the area of Multi-Carrier Modulation (MCM) because it can overcome some well known limitations of Discrete Fourier Transform (DFT) based MCM systems. Its improved spectral containment removes the need for a cyclic prefix, be it that appropriate equalization then has to be added as the cyclic convolution property no longer holds. Most DWT based MCM systems in the literature use Time-domain EQualizers (TEQs) to mitigate the channel distortion. In this paper, a Per-Wavelet EQualizer (PWEQ) is proposed which directly maximizes the Signal-to-Interference-plus-Noise Ratio (SINR) per symbol and is applicable to any wavelet family. The proposed PWEQ provides a performance upper bound for the TEQs for DWT based MCM systems. The computational complexity of the PWEQ is reduced by modifying the Filter Bank (FB) structure of the DWT. Simulations are performed to compare the PWEQ performance against the TEQs for DWT based MCM systems and the similar Per-Tone EQualizer (PTEQ) for DFT based MCM systems. The simulations are performed using measured Asymmetric Digital Subscriber Line (ADSL) and G.fast channels with Fejér-Korovkin (FK) wavelets. The proposed PWEQ increases the SINR on the received symbols compared to the TEQs at the cost of an increased computational complexity.

MIMO Per-Tone Equalizer Design for Long Reach xDSL

MIMO Per-Tone Equalizer Design for Long Reach xDSL

Signal Impropriety in Discrete Multi-Tone Systems and Widely Linear Per-Tone Equalization

In the context of digital subscriber line (DSL) systems, where the long-reach (LR) extension of G.fast has recently been proposed, interest in techniques dealing with long channel impulse responses (CIRs) without increasing the cyclic prefix (CP) length of the discrete multi-tone (DMT) modulation has recently resurfaced. The technique under consideration in this article is referred to as channel shortening, and applies FIR filters to the received signal to shorten the apparent CIR. Time-domain equalization (TEQ) as well optimal per-tone equalization (PTEQ) FIR filters will be considered. When channel shortening techniques are analyzed, it is often overlooked that the received signals after DMT demodulation are generally improper when the CP is too short. Hence, the state-of-the-art signal-to-interference-plus-noise ratio (SINR) and bit loading expressions employed to assess system performance, which implicitly assume the received signals to be proper, misrepresent the true achievable performance. New expressions for the SINR and bit loading are therefore presented that explicitly take signal impropriety into account. Based on these expressions, it is then observed that - if the received signals are improper - the PTEQ FIR filters are no longer optimal, and that the achievable bit loading depends on a particular phase shift experienced by the transmitted signals. This article therefore introduces a novel widely linear PTEQ - which is again optimal when the received signals are improper - and additionally proposes to optimally rotate signals in the complex plane prior to transmission. Finally, this article assesses the performance increase obtainable by explicitly accounting for signal impropriety.

Channel Estimation and Carrier Frequency Offset Compensation in Orthogonal Frequency Division Multiplexing System Using Adaptive Filters in Wavelet Transform Domain

In this paper, a combination of channel, receiver frequency-dependent IQ imbalance, and carrier frequency offset estimation under short cyclic prefix length is considered in orthogonal frequency division multiplexing system. An adaptive algorithm based on the set-membership filtering algorithm is used to compensate for these impairments. In short CP length, per-tone equalization structure is used to avoid inter-symbol interference. Due to CFO impairment and IQ imbalance in the receiver, we will expand the PTEQ structure to a two-branch structure. This structure has high computational complexity, so using the set-membership filtering idea with variable step size while reducing the average computation of the system can also increase the convergence speed of the estimates. On the other hand, applying wavelet transform on each branch of this structure before applying adaptive filters will increase the estimation speed. The proposed algorithm will be named SMF-WP-NLMS-PTEQ. The results of the simulations show better performance than the usual adaptive algorithms. Besides, estimation and compensation of channel effects, receiver IQ imbalance and carrier frequency offset under short CP can be easily accomplished by this algorithm.

Per-Tone Precoding and Per-Tone Equalization for OFDM and DMT Transmission Systems: Duality, Filter Optimization, and Resource Allocation

OFDM and DMT transmission systems add a cyclic prefix (CP) or zero pad (ZP) to the transmitted signal. Interference-free transmission requires this CP/ZP to be similarly long as the channel impulse response (CIR), reducing the achievable data rate in highly dispersive channels. A first strategy for dealing with long CIRs without increasing the CP/ZP overhead consists of applying a channel shortening filter to the received signal. A second strategy consists of spectral resource allocation, i.e. bit and power allocation to reduce interference. As little effort has been made towards joint channel shortening and resource allocation, a new algorithm to simultaneously optimize the channel shortening per-tone equalization (PTEQ) filters and the resource allocation is presented. In addition, transmitter-side channel shortening filters are considered, more specifically so-called per-tone precoding (PTPC) filters which apply the channel shortening filter before the IDFT modulation of the ODFM/DMT transmitter. At first glance, the FIR filter optimization for PTPC seems much more involved than the relatively straightforward FIR filter optimization for PTEQ. However, it will be demonstrated that any OFDM/DMT system with PTPC is - after time-reversing the CIR - equivalent to an OFDM/DMT system employing PTEQ. With this result in hand, systems with PTPC can take full advantage of the straightforward FIR filter optimization in systems with PTEQ, as well as of the aforementioned resource allocation algorithm. Simulation results show that the performance obtained for systems with PTPC is nearly indistinguishable from that obtained for systems with PTEQ, making PTPC an interesting alternative channel shortening strategy.

Joint Compensation of Transmitter and Receiver IQ Imbalance in OFDM Systems Based on Selective Coefficient Updating

In this paper, a selective coefficient updating (SCU) approach at each branch of the per-tone equalization (PTEQ) structure has been applied for insufficient cyclic prefix (CP) length. Because of the high number of adaptive filters and their complex adaption process in the PTEQ structure, SCU has been proposed. Using this method leads to a reduction in the computational complexity, while the performance remains almost unchanged. Moreover, the use of set-membership filtering with variable step size is proposed for a sufficient CP case to increase convergence speed and decrease the average number of calculations. Simulation results show that despite the aforementioned algorithms having similar performance in comparison with conventional algorithms, they are able to reduce the number of calculations necessary. In addition, compensation of both the channel effect and the transmitter/receiver in-phase/quadrature-phase imbalances are achievable by these algorithms.

Sparse approximation based resource allocation in DSL/DMT transceivers with per-tone equalization

Per-tone equalization has been proposed as an alternative to time domain equalization for DMT receivers in DSL modems. It optimizes the bit rate performance of the receiver as each tone can be equalized independently. It has also been shown that using variable length equalizers can significantly reduce the total number of equalizer taps and hence the run-time complexity, without compromising performance. For a given transmit power loading, it has been shown that the equalizer taps can be allocated optimally using a dual decomposition based approach with per-tone exhaustive searches over all possible equalizer lengths. However, a more general approach is needed when optimal transmit power allocation is also considered to maximize the overall bit rate, where in addition the per-tone exhaustive searches are replaced by a more efficient procedure. In this paper, a sparse approximation based resource allocation algorithm is presented to allocate equalizer taps and transmit power over tones and maximize the overall bit rate. This algorithm is shown to provide efficient allocations at a relatively low computational cost.

Time-Domain Block and Per-Tone Equalization for MIMO–OFDM in Shallow Underwater Acoustic Communication

Shallow underwater acoustic (UWA) channel exhibits rapid temporal variations, extensive multipath spreads, and severe frequency-dependent attenuations. So, high data rate communication with high spectral efficiency in this challenging medium requires efficient system design. Multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO---OFDM) is a promising solution for reliable transmission over highly dispersive channels. In this paper, we study the equalization of shallow UWA channels when a MIMO---OFDM transmission scheme is used. We address simultaneously the long multipath spread and rapid temporal variations of the channel. These features lead to interblock interference (IBI) along with intercarrier interference (ICI), thereby degrading the system performance. We describe the underwater channel using a general basis expansion model (BEM), and propose time-domain block equalization techniques to jointly eliminate the IBI and ICI. The block equalizers are derived based on minimum mean-square error and zero-forcing criteria. We also develop a novel approach to design two time-domain per-tone equalizers, which minimize bit error rate or mean-square error in each subcarrier. We simulate a typical shallow UWA channel to demonstrate the desirable performance of the proposed equalization techniques in Rayleigh and Rician fading channels.

Variable Tap-Length Mixed-Tone RLS-based Per-Tone Equalisation with Adaptive Implementation

In this paper, a methodology of mixed-tone recursive least squares algorithm development based on an orthogonal projection approach and a new variable tap-length mechanism is presented for per-tone equalisation (PTEQ) in discrete multitone systems. A mixed-tone cost function described as the sum of weight estimated errors is minimised to achieve the solutions for different per-tone equalisers simultaneously. We describe about the inverse square-root recursive least squares algorithm based upon the QRdecomposition which preserves the Hermitian symmetry of the inverse autocorrelation matrix by means of the product of square-root matrix and its Hermitian transpose. Such symmetrical property lends the benefit to the parallel implementation. In order to reduce the computational complexity, a new variable tap-length algorithm based on the sense of mean square mixted-tone errors is introduced to search a proper choice of tap-length of PTEQ. Simulation results show the improvement of achievable bit rate and signal to noise ratio performance as compared to the PTEQ exploiting conventional recursive least squares algorithm.

Bit Rate Maximising Per-Tone Equalisation with Adaptive Implementation for DMT-Based Systems

We present a bit rate maximising per-tone equalisation (BM-PTEQ) cost function that is based on an exact subchannel SNR as a function of per-tone equaliser in discrete multitone (DMT) systems. We then introduce the proposed BM-PTEQ criterion whose derivation for solution is shown to inherit from the methodology of the existing bit rate maximising time-domain equalisation (BM-TEQ). By solving a nonlinear BM-PTEQ cost function, an adaptive BM-PTEQ approach based on a recursive Levenberg-Marquardt (RLM) algorithm is presented with the adaptive inverse square-root (iQR) algorithm for DMT-based systems. Simulation results confirm that the performance of the proposed adaptive iQR RLM-based BM-PTEQ converges close to the performance of the proposed BM-PTEQ. Moreover, the performance of both these proposed BM-PTEQ algorithms is improved as compared with the BM-TEQ.

Resource Allocation in ADSL Variable Length Per-Tone Equalizers

Per-tone equalization (PTEQ) has been proposed as an alternative to time-domain equalization in asymmetric digital subscriber line (ADSL) modems. It optimizes the bit rate performance of the system as each tone can be equalized independently. In a per-tone equalization-based ADSL system a fixed-length frequency-domain equalizer is typically used for every tone. The ADSL channel has, however, a lower channel gain at high frequencies; therefore, using a longer equalizer for these frequencies does not result in a significant performance gain. With a fixed equalizer length, the length of the equalizer is determined by the required length at the lower frequencies; hence, a significant number of equalizer taps are wasted in the high-frequency range without any performance gain. Using a variable-length equalizer designed such that an overall performance criterion is optimized can significantly decrease the total number of equalizer taps without compromising performance. In this correspondence, a resource allocation technique is presented for a variable-length equalizer design using a dual problem formulation. This optimally solves the problem of equalizer length distribution over tones for given resource constraints, with a relatively low complexity.

Modified Frequency-Domain Equalization for Channel Shortening in Reduced-CP OFDMA Systems

In orthogonal frequency-division multiple access (OFDMA) systems, the cyclic prefix (CP) length needs to be no less than the longest delay spread of the channels of many users, reducing bandwidth efficiency more significantly than in single-user orthogonal frequency division multiplexing systems. In this paper, we address OFDMA downlink data transmission when a short CP is used to mitigate the inefficient bandwidth usage. Previous time-domain equalizers (TEQs) can be used to shorten the channel; however, they tend to increase noise by introducing spectral nulls. A previous work on per-tone equalization (PTEQ) structure involves a multitap frequency-domain equalizer (FEQ) for each tone, and shows better performance than TEQ-based receiver structure. We propose a novel receiver structure with only one-tap FEQ for OFDMA systems with a reduced CP, exploiting the unused subchannels for a user. We formulate an optimization problem to set the FEQ coefficients at the unused subchannels such that the channel is shortened (approximation is involved) and noise is not enhanced at the used subchannels. With the aid of computer simulations, it is demonstrated that the proposed equalization method is superior to the conventional TEQ-based receivers, and is comparable to the previous PTEQ-based receiver in terms of the achievable SNR, error performance, and bandwidth efficiency. Although the throughput curve versus synchronization delay of the PTEQ is smoother than that of the proposed receiver, the proposed method shows proper throughput over a wider range of the delay values than PTEQ receiver when the system parameters are set so that the complexities are comparable.

Joint Compensation of OFDM Frequency-Selective Transmitter and Receiver IQ Imbalance

Direct-conversion architectures are recently receiving a lot of interest in OFDM-based wireless transmission systems. However, due to component imperfections in the front-end analog processing, such systems are very sensitive to in-phase/quadrature-phase (IQ) imbalances. The IQ imbalance results in intercarrier interference (ICI) from the mirror carrier of the OFDM symbol. The resulting distortion can limit the achievable data rate and hence the performance of the system. In this paper, the joint effect of frequency-selective IQ imbalance at both the transmitter and receiver ends is studied. We consider OFDM transmission over a time-invariant frequency-selective channel. When the cyclic prefix is long enough to accommodate the channel impulse response combined with the transmitter and receiver filters, we propose a low-complexity two-tap equalizer with LMS-based adaptation to compensate for IQ imbalances along with channel distortions. When the cyclic prefix is not sufficiently long, then in addition to ICI there also exists interblock interference (IBI) between the adjacent OFDM symbols. In this case, we propose a frequency domain per-tone equalizer (PTEQ) obtained by transferring a time-domain equalizer (TEQ) to the frequency domain. The PTEQ is initialized by a training-based RLS scheme. Both algorithms provide a very efficient post-FFT adaptive equalization and their performance is shown to be close to the ideal case.

IQ-Imbalance Compensation for OFDM in the Presence of IBI and Carrier-Frequency Offset

In this paper, we propose a frequency-domain equalization technique for orthogonal frequency division multiplexing (OFDM) transmission over frequency-selective channels. We consider the case where the receiver analog front-end suffers from IQ-imbalance and the local oscillator suffers from carrier-frequency offset (CFO). While the IQ-imbalance results in a mirroring effect, the CFO induces inter-carrier interference (ICI). In addition to ICI, we consider the channel delay spread is larger than the cyclic prefix (CP). This means that inter-block interference (IBI) is present. The frequency-domain equalizer is obtained by transferring a time-domain equalizer (TEQ) to the frequency-domain resulting in a per-tone equalizer (PTEQ). Due to the presence of IQ-imbalance the conventional TEQ (where only one TEQ is applied to the received sequence) is not sufficient to cope with the mirroring effect. A sufficient TEQ consists of two time-domain filters; one applied to the received sequence and another applied to a conjugated version of the received sequence. For the case of IQ-imbalance and CFO, the TEQs are designed according the basis expansion model (BEM) which showed to be able to cope with the ICI problem. Finally, in addition to the frequency-domain PTEQ design procedure, a training-based RLS type initialization scheme for direct per-tone equalization is also proposed

Equalization for OFDM over doubly selective channels

In this paper, we propose a time-domain as well as a frequency-domain per-tone equalization for orthogonal frequency-division multiplexing (OFDM) over doubly selective channels. We consider the most general case, where the channel delay spread is larger than the cyclic prefix (CP), which results in interblock interference (IBI). IBI in conjunction with the Doppler effect destroys the orthogonality between subcarriers and, hence, results in severe intercarrier interference (ICI). In this paper, we propose a time-varying finite-impulse-response (TV-FIR) time-domain equalizer (TEQ) to restore the orthogonality between subcarriers, and hence to eliminate ICI/IBI. Due to the fact that the TEQ optimizes the performance over all subcarriers in a joint fashion, it has a poor performance. An optimal frequency-domain per-tone equalizer (PTEQ) is then obtained by transferring the TEQ operation to the frequency domain. Through computer simulations, we demonstrate the performance of the proposed equalization techniques.

Adaptive bit rate maximizing time-domain equalizer design for DMT-based systems

The classical discrete multitone receiver as used in, e.g., digital subscriber line (DSL) modems, combines a channel shortening time-domain equalizer (TEQ) with one-tap frequency-domain equalizers (FEQs). In a previous paper, the authors proposed a nonlinear bit rate maximizing (BM) TEQ design criterion and they have shown that the resulting BM-TEQ and the closely related BM per-group equalizers (PGEQs) approach the performance of the so-called per-tone equalizer (PTEQ). The PTEQ is an attractive alternative that provides a separate complex-valued equalizer for each active tone. In this paper, the authors show that the BM-TEQ and BM-PGEQ, despite their nonlinear cost criterion, can be designed adaptively, based on a recursive Levenberg-Marquardt algorithm. This adaptive BM-TEQ/BM-PGEQ makes use of the same second-order statistics as the earlier presented recursive least-squares (RLS)-based adaptive PTEQ. A complete range of adaptive BM equalizers then opens up: the RLS-based adaptive PTEQ design is computationally efficient but involves a large number of equalizer taps; the adaptive BM-TEQ has a minimal number of equalizer taps at the expense of a larger design complexity; the adaptive BM-PGEQ has a similar design complexity as the BM-TEQ and an intermediate number of equalizer taps between the BM-TEQ and the PTEQ. These adaptive equalizers allow us to track variations of transmission channel and noise, which are typical of a DSL environment.

Linear and decision-feedback per tone equalization for DMT-based transmission over IIR channels

The per-tone equalizer (PTEQ) has been presented as an attractive alternative for the classical time-domain equalizer (TEQ) in discrete multitone (DMT) based systems, such as ADSL systems. The PTEQ is based on a linear minimum mean-square-error (L-MMSE) equalizer design for each separate tone. In this paper, we reconsider DMT modulation and equalization in the ADSL context under the realistic assumption of an infinite impulse response (IIR) model for the wireline channel. First, optimum linear zero-forcing (L-ZF) block equalizers for arbitrary IIR model orders and cyclic prefix (CP) lengths are developed. It is shown that these L-ZF block equalizers can be decoupled per tone, hence they lead to an L-ZF PTEQ. Then, based on the L-ZF PTEQ, low-complexity L-MMSE PTEQ extensions are developed: the linear PTEQ extension exploits frequency-domain transmit redundancy from pilot and unused tones; alternatively, a closely related decision-feedback PTEQ extension can be applied. The PTEQ extensions then add flexibility to a DMT-based system design: the CP overhead can be reduced by exploiting frequency-domain transmit redundancy instead, so that a similar bitrate as with the original PTEQ is achieved at a lower memory and computational cost or, alternatively, a higher bitrate is achieved without a considerable cost increase. Both PTEQ extensions are also shown to improve the receiver's robustness to narrow-band interference.

Adaptive self-orthogonalizing per-tone decision feedback equalizer for single carrier modulations

Frequency-domain decision-feedback equalization (FD-DFE) for single carrier modulation avoids interblock interferences (IBIs) by using guard intervals located in between two data blocks. If the period of a guard interval is shorter than that of the channel impulse response, the IBIs cannot be removed by the FD-DFE, which significantly deteriorates its performance. To handle this IBI problem, this letter introduces the adaptive self-orthogonalizing per-tone FD-DFE without channel estimation, where time dispersiveness, one weakness of the per-tone equalization, is compensated by the self-orthogonalization coming from the lattice structure of the Levinson-Durbin algorithm, which dramatically reduces bit-error rates of the proposed FD-DFE, comparing with the existing FD-DFE.

On time-domain and frequency-domain MMSE-based TEQ design for DMT transmission

We reconsider the minimum mean square error (MMSE) time-domain equalizer (TEQ), bitrate maximizing TEQ (BM-TEQ), and per-tone equalizer design (PTEQ) for discrete multitone (DMT) transmission and cast them in a common least-squares (LS) based framework. The MMSE-TEQ design criterion can be formulated as a constrained linear least-squares (CLLS) criterion that minimizes a time-domain (TD) error energy. From this CLLS-based TD-MMSE-TEQ criterion, we derive two new least-squares (LS) based frequency-domain (FD) MMSE-TEQ design criteria: a CLLS-based FD-MMSE-TEQ criterion and a so-called separable nonlinear LS (SNLLS) based FD-MMSE-TEQ design. Finally, the original BM-TEQ design is shown to be equivalent to a so-called iteratively-reweighted (IR) version of the SNLLS-based FD-MMSE-TEQ design. This LS-based framework then results in the following contributions. The new, IR-SNLLS-based BM-TEQ design criterion gives rise to an elegant, iterative, fast converging, Gauss-Newton-based design algorithm that exploits the separability property. The intermediate FD-MMSE-TEQ designs establish a link between the TD-MMSE-TEQ on one hand and the BM-TEQ and the PTEQ on the other hand. Moreover, the considered LS-based equalizer designs-although at first sight very different in nature-exhibit remarkable correspondence when turned into equivalent generalized eigenvalue problems.

Per-Tone Equalization for Single Carrier Block Transmission with Insufficient Cyclic Prefix

This paper proposes per-tone equalization methods for single carrier block transmission with cyclic prefix (SC-CP) systems. Minimum mean-square-error (MMSE) based optimum weights of the per-tone equalizers are derived for SISO (single-input single-output), SIMO (single-input multiple-output), and MIMO (multiple-input multiple-output) SC-CP systems. Unlike conventional frequency domain equalization methods, where discrete Fourier transform (DFT) is employed, the per-tone equalizers utilize sliding DFT, which makes it possible to achieve good performance even when the length of the guard interval is shorter than the channel order. Computer simulation results show that the proposed equalizers can significantly improve the bit error rate (BER) performance of the SISO, SIMO, and MIMO SC-CP systems with the insufficient guard interval.