In this paper, we consider a transmission scheme combining precoded Faster-Than-Nyquist signaling (FTN) and Orthogonal Frequency Division Multiplexing Offset Quadrature Amplitude Modulation (OFDM/OQAM). The precoding method, named Sparse Interference Pre-Cancellation (SIPC), aims at reducing the interference introduced by FTN signaling at the transmitter side. In order to further improve the receiver performance, we propose additional enhancements to the channel encoder, notably, the bit-to-symbol mapping and symbol to time-frequency-positions mapping. Via simulations, we show that the proposed modifications improve the Bit-Error-Rate (BER) performance of the considered transceiver.

In this paper, we propose an implementation method for Orthogonal Frequency Division Multiplexing Offset Quadrature Amplitude Modulation (OFDM/OQAM) with Faster-Than-Nyquist (FTN) signaling. The proposed scheme can bring several advantages: 1) it approaches the theoretical rate gain of FTN signaling; 2) it can flexibly switch between Nyquist and FTN modes; 3) it does not cause complexity increase for the modem components while switching from Nyquist to FTN mode, and vice versa. In addition, we also present an iterative detector method which can support high constellation order transmission. With the simulations, we intend to show the FTN limits, up to a rate increase by a factor of 2, that the proposed transceiver can reach with various pulse shapes.

The Generalized Frequency Division Multiplexing (GFDM) has been proposed as a flexible modulation scheme for future cellular systems. It extends the traditional OFDM concept to a general framework by trading the orthogonality for the flexibility of employing different pulse shapes, which allows to improve the frequency localization feature. Nevertheless, considering its practical usage, the complexity issue has to be addressed. In this paper, we present a low complexity algorithm for the GFDM implementation which, under certain conditions, yields less computational complexity. Furthermore, we introduce a Span-Factor Manipulation (SFM) concept that can possibly compensate for the orthogonality loss, eventually turning the GFDM into an orthogonality improved system.

This paper considers bit-loading algorithms to maximize throughput under total power and spectral mask constraints in interference-free OFDM systems. The contribution is twofold. First, we propose a simple criterion to switch between two well-known algorithms from the literature: the conventional Greedy and Greedy-based bit-removing (with maximum allowable bit loading initialization) algorithms. Second, we present a new low-complexity loading algorithm that exploits the bit vector obtained by rounding the water-filling algorithm solution to the associated continuous-input rate maximization problem as an efficient initial bit vector of the Greedy algorithm. We theoretically prove that this bit vector has two interesting properties. The first one states that it is an efficient bit vector, i.e., there is no movement of a bit from one subcarrier to another that reduces the total used power. The second one states that the optimized throughput, starting from this initial bit vector, is achieved by adding or removing bits on each subcarrier at most once. Simulation results show the efficiency of the proposed algorithm, i.e., the achievable throughput is maximized with significant reduction of computation cost as compared to many algorithms in the literature.

Many alternative modulation schemes have been investigated to replace OFDM for radio systems. But they all have some weak points. In this paper, we present a novel modulation scheme, which minimizes the predecessors' drawbacks, while still keeping their advantages.

The aim of this paper is to study the bit-loading and power allocation problem in the presence of interference (Inter-carrier Interference (ICI) and Inter-Symbol Interference (ISI)) in Orthogonal Frequency Division Multiplexing (OFDM) systems. ISI and ICI significantly degrade the performance of OFDM systems and make the resource management optimized without the assumption of interference less efficient. To solve this problem, an initial solution based on the greedy approach is proposed in this paper. Then, several reduced complexity approaches, which yield a little degradation compared to the initial solution, have been developed. Simulation results presented in the context of Power Line Communication (PLC) show that the performance of proposed algorithms is tight with their upper bound. Moreover, these algorithms efficiently improve the system performance as compared to the constant power water-filling allocation algorithm as well as maximum power allocation algorithm.

Low complexity Multiple Input Multiple Output (MIMO) precoders for Orthogonal Frequency Division Multiplex (OFDM), such as Maximum Ratio Transmission (MRT) and Minimum Mean Square Error (MMSE) precoders enable to multiplex data streams in the spatial domain, without breaking OFDM orthogonality in the frequency domain, and therefore achieve a high spectral efficiency. OFDM using Offset Quadrature Amplitude Modulation (OFDM/OQAM) is an alternate modulation which is more spectrally efficient than OFDM. Unfortunately, when applied to OFDM/OQAM, traditional low complexity precoders introduce an inter-carrier interference (ICI) which is complex to cancel at the receiver side. This paper proposes a new family of low complexity precoders, called “Make-It-Real” (MIR) precoders. These precoders are designed so that the equivalent channel including precoding and propagation is “made real” and does not break the orthogonality of OFDM/OQAM in the frequency domain. At the receiver side, a simple extraction of the real part of the demodulated signal is necessary. Simulations show that OFDM/OQAM with MIR MMSE precoder outperforms all studied MIMO OFDM solutions, without requiring a complex ICI cancellation at the receiver.

In Power Line Networks, a link adaptation is realized by combining Multi-Carrier modulations techniques and bit-loading algorithms, resulting in the creation of tone maps. A tone map defines, for each subcarrier on a particular transmission path, the appropriate constellation that has to be applied based on a link quality criterion. Now, a frequently encountered situation happens when one station is requested to forward simultaneously multiple services to several stations of the network. Considering the fact that current PLC networks only work in a Time Division Multiplexing mode, the transmission efficiency in a point-to-multipoint scenario is not optimal. In this paper, we present a novel and practical algorithm, called TMSA (Tone Maps Splitting Algorithm), meant to split already defined point-to-point tone maps. It results in the definition of a Frequency Division Multiplexing mode to handle point-to-multipoint transmissions, efficiently exploiting the frequency diversity naturally present in the tone maps. Note that the proposed solution could work with other technologies, as long as quasi-static channels can be assumed and bit-loading algorithms are used.

Filter bank based multicarrier (FBMC) modulations have recently gained a great interest. Indeed, when coupled with an Offset QAM (OQAM) coding on each sub-carrier, FBMC/OQAM can reach the maximum Nyquist rate and provide a compact spectrum. However, as with this modulation scheme no cyclic prefix is used, equalizers more efficient than a simple 1-tap Zero Forcing may be sometimes required. In this paper, we investigate a new type of equalizers for FBMC/OQAM systems. Instead of discarding the imaginary interference component at the receiver side, as usually done in relation with the real orthogonality property of FBMC/OQAM, we exploit it. We, first, show that a strong correlation exists between the real and the appropriately shifted imaginary interference components. This correlation is used afterwards to provide a new equalizer structure which complexity is evaluated. Finally, simulation results of the FBMC/OQAM transmission over a multipath channel are provided which illustrate the improvements, in terms of bit error rate, brought by our equalizer.

In order to reduce the Peak-to-Average Power Ratio (PAPR) of multicarrier signals, a DFT precoding technique can be particularly useful. In this paper we propose to combine such a DFT precoding with the OFDM/OQAM modulation to provide a new frequency access scheme, that we call in short DFT-OQAMA. This DFT-OQAMA technique is compared to the Single-Carrier Frequency Division Multiplex Access (SC-FDMA) recently proposed for the UpLink (UL) of the 3GPP LTE system.

The Filter Bank MultiCarrier (FMBC) concept can be seen as an extension of the Orthogonal Frequency Division Multiplex (OFDM) principle. Indeed, in the case where each sub-carrier uses an Offset Quadrature Amplitude Modulation (OQAM), we get with the FBMC/OQAM system a multicarrier modulation scheme with relaxed orthogonality constraints so that, differently from OFDM, a cyclic prefix (CP) is no longer required. However, the OQAM principle, that permits to separately handle the real and imaginary parts of the complex QAM symbols, presents an implementation drawback. Indeed, compared to OFDM at the same bit rate, the Inverse Fast Fourier Transform (IFFT) has to run twice faster. In this paper, we show that for FBMC/OQAM, the IFFT outputs present properties that can be exploited by a pruned IFFT implementation. Moreover, in some specific cases, a reduction of the computational complexity can also be operated at the filtering stage.

The Filtered MultiTone (FMT) modulation can be seen as a simple extension of the OFDM modulation. Indeed, it can be efficiently realized as a Discrete Fourier Transform (DFT)-based transceiver. However, to avoid a significant increase of its computational complexity the FMT has to be implemented with short prototype filters. In this paper we propose explicit expressions to compute such short prototype filters. The corresponding closed-form expressions of the prototype filter coefficients are obtained for two different optimization criteria: out-of-band energy and time-frequency localization. In both cases it is shown that these quasi-optimal results correspond to linear-phase prototype filters.

Power Line Communications (PLC) are usually subject to severe degradations, partly due to the channel time dispersion. OFDM is a widely used multicarrier modulation scheme (MCM) for such frequency selective channels, because of the simplicity of the required equalization, and also because of its robustness toward Inter Symbol Interference (ISI), thanks to Cyclic Prefix (CP) insertion. However, other MCM schemes such as OFDM/OQAMcan sometimes be preferred to OFDM, for specific reasons, e.g. time or frequency localization. But, unlike CP-OFDM, OFDM/OQAM exhibits extra ISI and Inter-Carrier Interference (ICI) terms. This paper provides a comparison between both modulation schemes, including coding, for transmission cases where these interference terms cannot be neglected. An accurate Log-Likelihood Ratio calculation is derived for OFDM/OQAM in the case of a transmission through a PLC channel being also impaired by an impulsive noise. Simulation results are reported for a double binary turbo code considering two different block lengths.

The power line channel is characterized by a multipath behavior. A direct consequence is that multi-carrier modulation (MCM) is generally retained for power line communication (PLC). In addition to OFDM, or equivalently DMT, wavelet OFDM and Hermitian symmetry OFDM/OQAM (HS-OQAM) have been recently proposed for PLC transmission. In this paper we propose a unified and efficient modulation scheme that can generate all these different modulation schemes.

Amplitude clipping is one of the ways to fight against transmission impairments due to impulsive noise. In a recent study, in order to cope with empirical threshold determination, we used a Receiver Operating Characteristic (ROC) analysis to derive a closed form expression of the optimal threshold. In this paper we show that this ROC analysis also corresponds to a maximum likelihood (ML) criterion. Then, we evaluate, over power line channels, the efficiency of the clipping operation based on this ML decision. Furthermore, an adaptive scheme is proposed for automatic impulse mitigation over such channels. The adaptive procedure consists in estimating the impulsive noise parameters and updating the clipping threshold through ML optimization. The proposed scheme is finally evaluated over HomePlug AV (HPAV) physical layer.

The impulsive noise is one of the main reasons for bit or burst errors in power line transmission. The use of a level limiter is a simple and intuitive technique to mitigate the noxious effect of impulsive noise in such channels. However the appropriate threshold determination remains empirical. In this paper, a suitable calculation of the log-likelihood ratio (LLR) for Turbo codes (TC) in an impulsive environment is performed both for BPSK and OFDM systems. As main results of the present work, we first determine the LLR channel value when using a limiter. Furthermore, based on a receiver operating characteristic (ROC) analysis, we derive the optimal detection threshold of the limiter. Finally, the results are validated by simulation with HomePlug AV physical layer specification.

OFDM/OQAM is a special type of multi-carrier modulation that can be considered as an alternative to conventional OFDM with cyclic prefix (CP) for transmission over multi-path fading channels. Indeed, as it requires no CP, it has the advantage of a theoretically higher spectral efficiency. Furthermore, efficient pulse shaping can also be easily implemented with OFDM/OQAM. However, the classical channel estimation methods used for OFDM cannot be directly applied to OFDM/OQAM. In this paper we present an analysis of this problem and we introduce a new preamble-based channel estimation method. The performance results are obtained either by considering an IEEE802.22 channel model or regarding to the channel delay spread variation of a two-tap channel. The proposed OFDM/OQAM channel estimation method is evaluated, in both scenarios, using different pulse shaping and taking conventional CP-OFDM as reference.

OFDM/OQAM is a special type of multi-carrier modulation that can be considered as an alternative to conventional OFDM with cyclic prefix (CP) for transmission over multi- path fading channels. Indeed, as it requires no guard interval, it has the advantage of a theoretical higher spectral efficiency. In this paper, we propose a new version of OFDM/OQAM named in short HS-OQAM, where HS means that an Hermitian Symmetry property is used to directly provide a modulated real-valued signal. We analyze the links between HS-OQAM to Wavelet OFDM (WOFDM) modulation and Wilson-based modulated filter banks. The performances of HS-OQAM are compared to the ones of CP-OFDM for the transmission over Power Line Communication (PLC) channels.

Multi-carrier Modulation (MCM) and especially Orthogonal Frequency Division Multiplexing (OFDM) are currently used in many radio transmission standards. Among these modulations, OFDM/OQAM is an interesting alternative to OFDM. However, a simple one-tap Zero Forcing (ZF) equalization of OFDM/OQAM modulation may cause a performance floor problem, while the transmission meets a requirement of high order constellations. Therefore, more complex equalization needs to be considered, such as equalization with Interference Cancellation (EIC) technique, which has been presented to counteract the performance floor problem. In this paper, we propose a system design based on OFDM/OQAM modulation with cyclic prefix (CP) to also counteract the performance floor problem and keep simple equalization at receiver. The simulation results illustrate the efficiency of this system.

We address the challenges related to the development of very high speed power line communication (PLC) systems in the home environment. The development of such a technology is fundamental to realize the objective of reliably delivering new high speed services in the home environment in a network that integrates in a coordinated fashion also wireless systems. That is, an in-house network with the need of ldquonog new wiresrdquo. We start with an overview of the state of the art PLC technology, and then we describe our vision on application scenarios, and requirements. We address the problem related to transmission bandwidth, channel modeling, capacity evaluation as well as we describe novel advanced transmission technologies that enable reliable high speed transmission. We show that by enhancing current technology bandwidth up to 100 MHz, Gbps transmission is achievable.

The design problem of paraunitary filter banks has been addressed in many publications. The authors propose factorized structures that are obtained using transformations of the polyphase matrix of an analysis bank. In this letter, we focus, using a different approach, on the factorization of real-valued square paraunitary matrices. Using fundamental properties of real algebraic sets, we theoretically prove that, for the set of all paraunitary matrices of given size and order, one can get a full characterization of a complete and minimal set of mutually disjoint parameterized subsets. Thus, our analysis opens new horizons for the implementation and design of paraunitary filter banks.

The oversampled OFDM modulation is a possible alternative to conventional OFDM for the transmission of signals over multipath fading channels. Indeed with oversampled OFDM, appropriate pulse-shaping can be introduced to fight against time and frequency dispersion. In this paper, we propose a theoretical and experimental analysis of the peak-to-average power ratio (PAPR) of oversampled OFDM. Our analysis illustrates the impact of the oversampling ratio and of the pulse shape on the PAPR distribution

In this paper we focus on two multicarrier modulation schemes based either on offset-QAM (OFDM/OQAM) or on oversampled OFDM. Both modulations permit the introduction of an appropriate pulse shaping for transmission over time- varying multipath channels. As for any multicarrier modulation, these pulse-shaped systems may be also subject to large peak- to-average power ratios (PAPR). An adaptation of the selective mapping (SLM) technique, already used for OFDM, is proposed. This new version of a technique named overlapped SLM (OSLM) generalizes a previous result we got for OFDM/OQAM. It is shown that the impact on the oversampled OFDM is less strong but also that the influence of the pulse shape's length is still predominant.

OFDM/OQAM is now a recognized alternative to conventional OFDM for the transmission of signals over multipath fading channels. Indeed with OFDM/OQAM an appropriate pulse-shaping can be introduced to fight against time and frequency dispersion. In order to reduce the occurrence of large peak-to-power ratios (PAPR) that are inherent to multicarrier modulations, a PAPR reduction method is proposed. The proposed technique is an adaptation to OFDM/OQAM of the selective mapping (SLM) technique already used for OFDM. It is shown that, as with OFDM, the performance increases with the number of SLM codes, but is also dependent upon the length of the OFDM/OQAM pulse shape

Several recent publications have shown that joint source-channel decoding could be a powerful technique to take advantage of residual source redundancy for fixed- and variable-length source codes. This letter gives an in-depth analysis of a low-complexity method recently proposed by Guivarch et al., where the redundancy left by a Huffman encoder is used at a bit level in the channel decoder to improve its performance. Several simulation results are presented, showing for two first-order Markov sources of different sizes that using a priori knowledge of the source statistics yields a significant improvement, either with a Viterbi channel decoder or with a turbo decoder.

This paper describes a distributed coding system based on punctured turbo codes to compress three correlated memoryless binary and Gaussian sources. The performance bounds given by the Slepian-Wolf and Wyner-Ziv theorems are provided for three sources. The simulation results are analyzed against those provided by the same asymmetric system based on punctured turbo codes used for distributed coding of two sources.

Siohan and Roche introduced recently a new family of modulated filter banks, which was derived from a set of continuous-time orthogonal functions named extended Gaussian functions (EGFs). At first, these EGFs were obtained using the isotropic orthogonal transform algorithm (IOTA), i.e., a two-step orthogonalization procedure of the Gaussian function. As shown recently, using the Zak transform, the IOTA method yields the tight window function canonically associated with the Gaussian. In this letter, it is shown that the Zak transform can also be useful in recovering the series expansion of the EGFs. Practical guidelines are also provided in order to get accurate approximations of the orthogonal EGFs.

Modified discrete Fourier transform (MDFT) filter banks are analyzed in relation with Weyl-Heisenberg expansions over R in l/sub 2/(Z). This analysis is used to formally derive a result that could sometimes be taken for granted without any proof: the design of perfect reconstruction (PR) MDFT subband coders is equivalent to that of PR MDFT transmultiplexers. The framework of WH expansions over R in l/sub 2/(Z) is also used to prove the equivalence between the MDFT transmultiplexer and the biorthogonal frequency division multiplexing/offset quadrature amplitude modulation (BFDM/OQAM) multicarrier system. This analysis also leads to a slightly modified MDFT scheme with a reduced reconstruction delay.

A method is proposed to solve the unconstrained optimization problem related to the design of perfect reconstruction modulated filter banks. The basic idea is to take advantage of certain attractive properties of the angular parametrization of the underlying lattice structure. This approach allows the design of prototypes with thousands of coefficients. Nearly optimal results are presented for two different optimization criteria.

A new system of multi carrier modulation is described, which takes advantage of the biorthogonality property, to generalize the oversampled OFDM technique into a biorthogonal frequency division multiplex scheme named BFDM/QAM. A discrete-time analysis of this scheme leads to a discrete model of modulated transmultiplexer. Using a polyphase decomposition of this transmultiplexer, and assuming a distortion free channel, we determine the mathematical conditions on the prototype filters, allowing an exact cancellation of intersymbol and interchannel interference. Then, computationally efficient realizations of the modulators and demodulators are given. Lastly, some design examples are provided, illustrating the respective advantages of BFDM/QAM and OFDM/QAM.

Several publications have shown that joint source-channel decoding (JSCD) could be a powerful technique for taking advantage of residual source redundancy. Furthermore, it appeared that the principle could also be successfully applied to variable length codes (VLCs). This paper gives an in-depth analysis of a new method proposed by Guivarch, Carlach and Siohan (see Data Compression Conference (DCC), Snowbird, Utah, USA, Mar. 2000, p.83-92) and Guivarch, Siohan and Carlach (see International Communication Conference (ICT), Acapulco, Mexico, May 2000, p.872-76) to provide a low complexity soft decoding of VLCs with and without channel coding. Furthermore we also present new results obtained in a video coding context, using a VLC table from the MPEG4 video standard. Compared to separated decoding systems, the gain in signal to noise ratio varies from 0.3 dB up to 2 dB.

We present a new method for using a priori information at a bit level in the decoding of Huffman codes. Its basic principle is the computation and use of the bit probabilities derived from the symbol probabilities of Huffman codes. Thus, we obtain a joint source-channel decoding algorithm which takes advantage of this a priori knowledge. Even better performances are obtained when a turbo code-like iterative decoding is carried out. Compared to separate decoding systems, the gain in signal to noise ratio varies from 0.5 up to 1 dB with an added complexity which is relatively limited.

A new system of multicarrier modulation is described which takes advantage of the biorthogonality property to generalize the OFDM/OQAM technique into a biorthogonal frequency division multiplex scheme named BFDM/OQAM. A discrete-time analysis of this scheme leads to a discrete model of modulated transmultiplexer. Using a polyphase decomposition of this transmultiplexer, and assuming a distortion-free channel, we determine the mathematical conditions on the prototype filter allowing an exact cancellation of intersymbol and interchannel interference. Then, computationally efficient realizations of the modulators and demodulators are given. Lastly, some design examples are provided illustrating respective advantages of BFDM/OQAM and OFDM/OQAM.

The energy preservation property is among the most widely used properties of orthogonal transforms in image compression because the reconstruction error can be computed as the sum of the subband distortions. Thus, this is a key point in the use of efficient bit allocation techniques such as rate-distortion algorithms. Therefore, we study the nonorthogonality of biorthogonal filterbanks with reference to energy preservation from both theoretical and applicative points of view. We calculate the Riesz bounds as energy preservation bounds for filterbanks and discrete wavelet transforms, and then connect these results with the Riesz bounds of the related continuous wavelet transform. The simultaneous use of biorthogonal filterbanks and rate-distortion algorithms is then discussed as the issue of estimating the reconstruction error as an additive function of the subband distortion. We propose a weighted sum of the subband distortions as an estimate, whose accuracy is calculated by a wide range of experiments. This accuracy is shown to be correlated to the Riesz bounds of the filterbanks. We conclude that from this point of view, most of the usual biorthogonal filterbanks may be considered as nearly orthogonal.

The problem of designing two-channel perfect reconstruction filter banks with odd length FIR linear phase analysis and synthesis filters is revisited. Based on an algebraic approach we introduce a complete and canonical lattice structure. Design examples show that, for short length filters, multiplierless realizations providing an acceptable frequency behaviour were attainable.

We study the non-orthogonality of perfect-reconstruction (PR) biorthogonal filter banks by measuring the energy preservation between the spatial and transform domains. The mathematical formulation of that issue leads to the computation of the Riesz constants, and a more relevant modelization leads to a measure of near-orthogonality which is well suited for image compression systems based on filter banks. This provides a criterion for the validity of the energy preservation approximation: we can compare the latter approximation with the one that is made when estimating the perceptual quality of an image by the mean square error.

New solutions to the issue of designing transmitter and receiver filters for data transmission are proposed: We design two different linear-phase FIR filters, a shorter receiver filter and a longer transmitter filter, whose product is a Nyquist filter, with an exactly zero inter-symbol interference (ISI). Holding simultaneously exactly zero ISI and phase linearity is new for an oversampling factor equal to 2. The design algorithm takes into account various frequency specifications. Furthermore, the filters are nearly optimal with reference to signal-to-noise ratio (SNR) and the complexity of the transmission system is reduced. A design chart and several design examples are given which illustrate the capabilities of our approach.

The purpose of the paper is to show the close relations between the two types of structures, transversal and cascade form, usually proposed for the design of two-channel perfect-reconstruction filter banks with FIR linear-phase analysis and synthesis filters. Based on a linearization of the non-linear problems, related to the identification of cascade form perfect reconstruction systems, recursive algorithms are derived. Numerical results are presented which illustrate the accuracy of the identification methods. Two design examples are also given, corresponding to biorthogonal solutions for the two types of systems, i.e. with even and odd length filters; which satisfy selectivity and regularity constraints.>

Two-dimensional non-separable quadrature mirror filters (2-D QMFs) are designed. An iterative algorithm is used for the optimization of a prototype diamond-shaped filter. The variations of the reconstruction error as a function of the size and specifications of the prototype filter are tabulated.>

An unified approach to the design of optimum (in the minimax sense) finite precision 1-D and 2-D FIR filters is presented. To take into account a specific problem related to filters whose coefficient values are selected from the powers-of-two coefficient space, an improvement of a branch and bound technique is proposed. Thus some design problems, corresponding to what is defined as a critical specification, which otherwise are doomed to failure can be solved in reasonable CPU times.>

The design of 2-D finite impulse response (FIR) digital filters is presented for two-dimensional sampling structure conversion. Both the sampling structure model and the filter characteristics have been chosen to cover a large class of problems related to multidimensional linear processing for video signals. Optimal infinite and finite precision filter design methods are proposed. Experimental analysis concerning the effects of coefficient rounding on the filter frequency response was carried out, which was found to be consistent with previous theoretical results. A simulation with one test picture is presented to illustrate the problem of sampling structure conversion.>

One of the most popular technique for edge detection is based on the convolution of the image with a LoG operator, namely the Laplacian of Gaussian, characterized by a space constant

An algorithm is proposed for the design of optimal two-dimensional (2-D) finite impulse response (FIR) digital filters with finite wordlength and linear phase. This algorithm associates linear programming and a branch and bound technique for which two strategies are compared. Three examples are presented which show the efficiency of the method for the design of filters with size up to 9 × 9.

We propose some improvements to an optimal algorithm used to design finite wordlength linear phase FIR (Finite Impulse Response) digital filters. We consider also a new version, which allows the designer to introduce constraints in the discrete optimization problem. Examples are given concerning the optimal design of : a filter without multipliers, a third-band filter, a low-pass filter with a constraint on step response ripple.