Rozprawy doktorskie na temat „Recursive digital filters”

Filters constitute an essential tool for manipulating the spectral content of a signal. While there is a plethora of filtering tools, both in the hardware and software domain, the majority of them are geared towards engineers and scientists, rather than sound designers and electroacoustic composers. The "common-practice" approach is to consider filters as post-production tools. This can be restrictive if filters are to be used as artistic tools, dynamically involved in the shaping of the sound. This thesis was written with this approach in mind its aim is (a) to provide a survey of the various digital recursive filters, enabling a filter designer to choose the one that suits his needs, (b) to teach filter designers, such as electroacoustic composers and sound designers how to calculate digital filter coefficients, and (c) implement filter algorithms using the familiar syntax of computer music languages such as Csound and SuperCollider .

Neste trabalho é desenvolvido um método de análise de ciclo limite devido à quantização, à entrada zero, para redes operando com aritmética em ponto flutuante. Condições de inexistência de ciclo limite são facilmente obtidas via cálculo computacional. O método de análise se aplica a redes genéricas de qualquer ordem. É desenvolvido, em seguida, um método de síntese de redes operando com aritmética em ponto fixo, que são imunes a ciclo limite devido à quantização, à entrada zero, utilizando para isso o conceito de redes estruturalmente passivas. As redes assim sintetizadas apresentam sub-redes estruturalmente LBR ou BR na sua malha de realimentação. São as redes de segunda ordem, sintetizadas pelo método proposto. É provado que algumas dessas redes são também imunes a ciclo limite devido a overflow, à entrada zero e a resposta forçada.
This thesis presents a method for analysis of zero-input limit cycles due to quantization, in digital filters realized with floating point arithmetic. Conditions for absence of limit cycles are easily derived by computational calculus. The method of analysis is applicable to generic structures of any order. Following this, a method is presented a method for the synthesis of digital filters realized with fixed point arithmetic, that are free from zero-input limit cycles due to quantization, using the concept of structurally passive networks. The structures synthetized present sub-filters structurally LBR or BR in the feedback loop. Second order structures are synthetized and studied. It is proved that some of these stuctures are also free from zero-input limit cycles due to overflow and stable to forced response.

In this thesis two versions of a single filter frequency masking narrow-band high-speed recursive digital filter structure, proposed in [1], have been implemented and evaluated considering the maximal clock frequency, the maximal sample frequency and the power consumption. The structures were compared to a conventional filter structure, that was also implemented. The aim was to see if the proposed structure had some benefits when implemented and synthesized, not only in theory. For the synthesis standard cells from AMS csx 0.35 mm CMOS technology were used.

Muitos pesquisadores têm se interessado em explorar o vasto poder computacional das recentes unidades de processamento gráfico (GPUs) em aplicações fora do domínio gráfico. Essa tendência ao desenvolvimento de propósitos gerais com a GPU (GPGPU) foi intensificada com a produção de APIs não-gráficas, tais como a Compute Unified Device Architecture (CUDA), da NVIDIA. Com elas, estudou-se a solução na GPU de muitos problemas de processamento de sinal 2D e 3D envolvendo álgebra linear e equações diferenciais parciais, mas pouca atenção tem sido dada ao processamento de sinais 1D, que também podem exigir recursos computacionais significativos. Já havia sido demonstrado que a GPU pode ser usada para processamento de sinais em tempo-real, mas alguns processos não se adequavam bem à arquitetura da GPU. Neste trabalho, apresento uma nova técnica para implementar um filtro digital linear recursivo usando a GPU. Até onde eu sei, a solução aqui apresentada é a primeira na literatura. Uma comparação entre esta abordagem e uma implementação equivalente baseada na CPU demonstra que, quando usada em um sistema de processamento de áudio em temporeal, esta técnica permite o processamento de duas a quatro vezes mais coeficientes do que era possível anteriormente. A técnica também elimina a necessidade de processar o filtro na CPU - evitando transferências de memória adicionais entre CPU e GPU - quando se deseja usar o filtro junto a outros processos, tais como síntese de som. A recursividade estabelecida pela equação do filtro torna difícil obter uma implementação eficiente em uma arquitetura paralela como a da GPU. Já que cada amostra de saída é computada em paralelo, os valores necessários de amostras de saída anteriores não estão disponíveis no momento do cômputo. Poder-se-ia forçar a GPU a executar o filtro sequencialmente usando sincronização, mas isso seria um uso ineficiente da GPU. Este problema foi resolvido desdobrando-se a equação e "trocando-se" as dependências de amostras próximas à saída atual por outras precedentes, assim exigindo apenas o armazenamento de um certo número de amostras de saída. A equação resultante contém convoluções que então são eficientemente computadas usando a FFT. A implementação da técnica é geral e funciona para qualquer filtro recursivo linear invariante no tempo. Para demonstrar sua relevância, construímos um filtro LPC para sintetizar em tempo-real sons realísticos de colisões de objetos feitos de diferentes materiais, tais como vidro, plástico e madeira. Os sons podem ser parametrizados por material dos objetos, velocidade e ângulo das colisões. Apesar de flexível, esta abordagem usa pouca memória, exigindo apenas alguns coeficientes para representar a resposta ao impulso do filtro para cada material. Isso torna esta abordagem uma alternativa atraente frente às técnicas tradicionais baseadas em CPU que apenas realizam a reprodução de sons gravados.
Many researchers have been interested in exploring the vast computational power of recent graphics processing units (GPUs) in applications outside the graphics domain. This trend towards General-Purpose GPU (GPGPU) development has been intensified with the release of non-graphics APIs for GPU programming, such as NVIDIA's Compute Unified Device Architecture (CUDA). With them, the GPU has been widely studied for solving many 2D and 3D signal processing problems involving linear algebra and partial differential equations, but little attention has been given to 1D signal processing, which may demand significant computational resources likewise. It has been previously demonstrated that the GPU can be used for real-time signal processing, but several processes did not fit the GPU architecture well. In this work, a new technique for implementing a digital recursive linear filter using the GPU is presented. To the best of my knowledge, the solution presented here is the first in the literature. A comparison between this approach and an equivalent CPU-based implementation demonstrates that, when used in a real-time audio processing system, this technique supports processing of two to four times more coefficients than it was possible previously. The technique also eliminates the necessity of processing the filter on the CPU - avoiding additional memory transfers between CPU and GPU - when one wishes to use the filter in conjunction with other processes, such as sound synthesis. The recursivity established by the filter equation makes it difficult to obtain an efficient implementation on a parallel architecture like the GPU. Since every output sample is computed in parallel, the necessary values of previous output samples are unavailable at the time the computation takes place. One could force the GPU to execute the filter sequentially using synchronization, but this would be a very inefficient use of GPU resources. This problem is solved by unrolling the equation and "trading" dependences on samples close to the current output by other preceding ones, thus requiring only the storage of a limited number of previous output samples. The resulting equation contains convolutions which are then efficiently computed using the FFT. The proposed technique's implementation is general and works for any time-invariant recursive linear filter. To demonstrate its relevance, an LPC filter is designed to synthesize in real-time realistic sounds of collisions between objects made of different materials, such as glass, plastic, and wood. The synthesized sounds can be parameterized by the objects' materials, velocities and collision angles. Despite its flexibility, this approach uses very little memory, requiring only a few coefficients to represent the impulse response for the filter of each material. This turns this approach into an attractive alternative to traditional CPU-based techniques that use playback of pre-recorded sounds.

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A pipeline technique by Loomis and Sinha has been applied to the design of recursive digital filters. Recursive digital filters operating at hitherto impossibly high rates can be designed by this technique. An alternate technique by R. Gnanasekaran allows high speed implementation using the state-space structure directly. High throughput is also achieved by use of pipelined multiply-add modules. The actual hardware complexity will depend upon the number of pipeline stages. These techniques are used for the design of the I IR notch filter and finally, a comparison of the performance and complexity of these two techniques is presented.
http://archive.org/details/infiniteimpulser00jang
Lieutenant, Royal Thai Navy

Orientador: Amauri Lopes
Dissertação (mestrado) - universidade Estadual de Campinas, Faculdade de Engenharia Eletrica
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Resumo: Este trabalho descreve os aspectos teóricos e práticos envolvidos na elaboração do programa de computador FOREST, que sintetiza e analisa filtros digitais recursivos. São sintetizados filtros passa-baixa, passa-alta, passa faixa e corta-faixa, utilizando-se as aproximações de Butterworth, Chebychev e elíptica. Os efeitos não lineares advindos da utilização de registros de comprimento finito para a representação de coeficientes e variáveis são analisados em detalhe. Apresenta-se a teoria de otimização dos filtros digitais com relação ao ruído de quantização do sinal, baseada na descrição por variáveis de estado, e o programa incorpora essa teoria gerando filtros descritos por variáveis de estado com reduzidos efeitos de quantização
Abstract: Not informed.
Mestrado
Mestre em Engenharia Elétrica

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Capes
Um novo método de obtenção de parâmetros de filtros digitais recursivos (FDR) é apresentado para reduzir o impacto da resposta transitória dos Transformadores de Potencial Capacitivos (TPC) sobre o desempenho dos sistemas de medição, proteção e controle. Assumindo uma topologia predefinida, os parâmetros dos filtros são obtidos a partir da resposta em frequência do TPC de interesse. Diferentemente das técnicas reportadas na literatura, o método se aplica com facilidade a TPC de diferentes classes de tensão e independe das características operacionais do sistema. Para tanto, faz-se necessário conhecer a resposta em frequência do TPC em questão, no espectro de frequência de interesse. A validação do método é realizada através de simulações digitais em tempo real via simulador RTDSTM (Real Time Digital Simulator). As análises são baseadas em dados de sistemas elétricos reais e no funcionamento dinâmico dos filtros através da estimação dos fasores das tensões e estudos de localização de falta. A partir dos resultados obtidos, verifica-se que a presença dos FDR reduz significativamente os erros de medição causados pelos TPC quando submetidos a condições transitórias. Portanto, os FDR surgem como uma forma simples e de baixo custo para melhorar o desempenho e a confiabilidade dos sistemas de medição, proteção e controle.
A new method for obtaining recursive digital filter (FDR) parameters is presented in order to reduce the impact of Coupling Capacitor Voltage Transformer (CCVT) transient response on the performance of the measurement, protection and control systems. Assuming a pre-defined topology, the filter parameters are obtained from the CCVT frequency response of interest. Unlike the techniques reported in the literature, the method applies easily to CCVT of different voltage classes and it does not depend on the operating characteristics of the system, therefore, it is necessary to know the frequency response of the CCVT on the frequency spectrum of interest. The method is validated is through digital simulation using the RTDSTM (Real Time Digital Simulator). The analyzes are based on data obtained from electrical systems in service and on the dynamic performance of the filters by estimating the phasors of voltages and fault location studies. It is shown that the presence of FDR significantly reduces measurement errors caused by CCVT when subjected to transient conditions, therefore, the FDR arises as a simple and low cost alternative to improve the performance and reliability of measurement systems, protection and control.

博士
國立臺灣大學
電信工程學研究所
104
A two-dimensional (2-D) digital allpass filter (DAF) has a property of varying only phase with constant magnitude and it has mainly been used as a phase compensator for distorted signals. It is a structure that has some desirable attributes such as low complexity and low coefficient quantization error. It also can be used to design a wide range of filtering functions. In this doctoral dissertation, we present the monotone phase-response property of a two-dimensional (2-D) causal digital allpass filter (DAF) with real coefficients or complex coefficients in the quarter-plane (QP) support region. Regarding the circumstance of real coefficients, we also prove that the previously proposed bounded-input bounded-output (BIBO) stability criterion on the viewpoint of unwrapped phase is necessary and sufficient for 2-D separable DAFs, but is only sufficient for QP DAFs. The resultant property possesses the advantage of increasing the freedom of phase design over the previously proposed one. A remarkable application of the presented property is choosing an appropriate specification for the desired phase response of a 2-D QP DAF design. A 2-D nonsymmetric half-plane (NSHP) recursive DAF possesses more general causality and performs better than a 2-D quarter-plane (QP) recursive DAF. Hence, we also present the phase-response property for the BIBO stability of a 2-D causal recursive DAF with NSHP support region. Both cases of filters with real coefficients and complex coefficients are explored. Moreover, the effect of the numerator polynomial of a 2-D NSHP DAF on stability is also considered. The presented phase-response property has several applications. A remarkable application is that it can be utilized to enforce stability for a 2-D NSHP DAF design by choosing an appropriate phase specification. The eigenfilter design of 2-D NSHP DAFs for this application is also presented. The 1-D lattice filter structure exhibits the attractive advantages of low passband sensitivity and robustness to quantization error. The modularity of this structure makes industrial application. Additionally, 1-D digital lattice filter structure requires lower computational cost than 1-D direct form digital filter. The filter coefficients of 1-D direct-form allpass filter and the reflection coefficients of 1-D lattice allpass filter have a one-to-one mapping relationship. However, 2-D lattice allpass structures always do not have this relationship. Hence, we present a lattice structure for the realization of 2-D recursive DAFs with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a NSHP support region. Two variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Then, we present a parallel-combination structure composed of the 2-D lattice DAFs for the design of 2-D recursive filters. The novelty of the 2-D recursive filter is that it not only inherits the desirable attributes of lattice filters but also possesses the advantage of better performance over the 2-D recursive NSHP filters.

博士
臺灣大學
電信工程學研究所
96
Abstract The purpose of this dissertation is to devise novel and efficient techniques for optimally designing two-dimensional (2-D) recursive digital filters and 2-D recursive multirate filter banks by employing allpass sections. First, we review the well-known trust-region method that can efficiently solve the nonlinear optimization problem of designing the proposed 2-D recursive digital filter structure composed of allpass subfilters. Secondly, we develop the efficient optimization algorithms based on the primal affine-scaling variant of Karmarkar''s algorithm (PAS algorithm) to iteratively solve the design problems in L1 and L_infinite senses, respectively, when we consider the phase approximation problem. The essences and central ideas of these algorithms are employed throughout. A novel structure composed of 2-D non-symmetric half-plane (NSHP) digital allpass filters (DAFs) is utilized to design general 2-D recursive digital filters. An appropriate nonlinear objective function is formulated by considering the magnitude, group delay, and stability errors, simultaneously. It is worthy noting that the proposed structure is recursive computable and can be used to design some filters that cannot be accomplished by the existing quarter-plane (QP) allpass-based structures. According to the results obtained by the novel structure mentioned above, we present the design of 2-D recursive doubly complementary (DC) filters by parallel interconnecting two 2-D allpass sections. The design problem is appropriately formulated to result in a simple linear optimization problem that minimizes the phase error. Thus, the design problem can be efficiently solved by using the PAS algorithm in L1 and L_infinite criteria. It is worthy noting that the 2-D DC filter exhibits very attractive DC symmetric characteristics when the passband and stopband of the 2-D DC filter are symmetric with respect to certain frequency point. Owing to this DC symmetric characteristic, the 2-D DC filter can be designed and implemented very efficiently. Besides, we find that the design of the widely used diamond-shaped filters can be efficiently realized by our proposed DC structure because the diamond-shaped filters possess quadrantal symmetry. This result shows the more general design capability of our design than the design based on 2-D QP allpass filters. With regard to the 2-D filter bank systems, the application of 2-D DC filter for designing 2-D QMF banks is given. The 2-D recursive DAFs are the fundamental building blocks and we only need to focus on the phase approximation of them. The allpass-based structure will not induce any magnitude distortion. Besides, the phase distortion of the overall QMF system can be compensated by a suitable DAF that plays a role as a phase equalizer. It is shown that the quincunx QMF bank and the parallelogram QMF bank can be easily designed by applying the proposed linear approximation techniques. Additionally, we deal with the widely considered design example of 2-D recursive circularly symmetric lowpass filter by proposing a novel structure composed of 1-D and 2-D recursive DAFs. The simulation results show very satisfactory performance in comparison with the existing researches. The minimal realization of digital filters is widely interested because it needs the least hardware requirement and less computational complexity. However, it is not an easy task to develop a minimal realization of a 2-D filter as in the 1-D cases. We consider the realization of a generalized 2-D digital lattice filter by employing the corresponding matrix representation. In addition, the minimal realization of the proposed structure is verified by utilizing the Roesser 2-D state space model. The corresponding lattice structure of the direct-form 2-D DAF with symmetric-half plane support (SHP) is presented. By solving the backward recursive equations, the reflection coefficient functions of the lattice-form 2-D SHP DAF are obtained. Besides, we present the technique based on the trust-region method to directly calculate the reflection coefficient functions. Thus, the filter structures composed of direct-form 2-D SHP DAFs can be implemented by the this lattice structure. The stability problem of designing 2-D SHP DAF can be easily guaranteed by evaluating the absolute values of the reflection coefficient functions.

The design of nearly linear-phase recursive digital filters using constrained optimization is investigated. The design technique proposed is expected to be useful in applications where both magnitude and phase response specifications need to be satisfied. The overall constrained optimization method is formulated as a quadratic programming problem based on Newton’s method. The objective function, its gradient vector and Hessian matrix as well as a set of linear constraints are derived. In this analysis, the independent variables are assumed to be the transfer function coefficients. The filter stability issue and convergence efficiency, as well as a ‘real axis attraction’ problem are solved by integrating the corresponding bounds into the linear constraints of the optimization method. Also, two initialization techniques for providing efficient starting points for the optimization are investigated and the relation between the zero and pole positions and the group delay are examined. Based on these ideas, a new objective function is formulated in terms of the zeros and poles of the transfer function expressed in polar form and integrated into the optimization process. The coefficient-based and polar-based objective functions are tested and compared and it is shown that designs using the polar-based objective function produce improved results. Finally, several other modern methods for the design of nearly linear-phase recursive filters are compared with the proposed method. These include an elliptic design combined with an optimal equalization technique that uses a prescribed group delay, an optimal design method with robust stability using conic-quadratic-programming updates, and an unconstrained optimization technique that uses parameterization to guarantee filter stability. It was found that the proposed method generates similar or improved results in all comparative examples suggesting that the new method is an attractive alternative for linear-phase recursive filters of orders up to about 30.

碩士
國立臺灣大學
電信工程學研究所
102
Using allpass filters as basic building block to construct digital filters has the advantages of passband magnitude low sensitivity and doubly complementary property. Therefore, it has been widely used in digital filter design. Lattice form allpass filter has its entire frequency response less sensitive to round-off error. So, using lattice allpass filter as basic building block can achieve low sensitivity not only in the passband magnitude, but also in the stopband magnitude and the entire phase response. Additionally, the stability of lattice allpass filter can be easily determined by its reflection coefficients, and the structure of lattice allpass filter is highly modular. These advantages bring a lot of convenience to filter design process and hardware implementation. So far, lattice form is mostly used in one-dimensional filters. Since direct form and lattice form have one-to-one mapping in 1-D case, the two structures can theoretically achieve the same performance. But two-dimensional polynomials are not factorizable in general. Mapping from direct form to lattice form no longer exists in 2-D case. Some 2-D lattice structures have been proposed, but most of them have the performance much worse than direct form. Only the one with symmetric half plane support region using finite impulse response polynomials as its reflection coefficients can achieve the performance close to direct form. In fact, 2-D lattice form has several different patterns. We propose a generalized 2-D lattice structure, which formats and parameterizes all possible variations of 2-D lattice forms. By adjusting the structural parameters, we can arbitrarily generate new lattice structures. For a specific filter design problem, we try several kinds of lattice structures. Computer simulations show that on the design of general 2-D filters, 2-D doubly complementary filter pairs, and 2-D quadrature mirror filters, there exists some lattice structures which can provide the design performance very close to or even better than direct form. We also improve the optimization algorithm and objective function used in our experiment. Computer simulations show that the improved algorithm and objective function really enhance the design performance and reduce the required computation time.

Two-dimensional variable recursive digital Band-pass and Band-stop filters are applied in signal processing and pro-imagining process, as well as communication systems where the frequency domain characteristics of digital filters are required to be adjustable. The main objective of this thesis has been to propose a new method of designing 2-D recursive Band-Pass and Band-Stop digital filters with variable characteristics. From the identical analog 1-D second order Butterworth Low-Pass analog ladder network, 2-D Band-Pass and Band-Stop digital filters can be obtained through the application of Low-Pass to Band-Pass/Band-Stop transformation and double generalized bilinear transformations. The denominators of these filters transfer functions are verified for VSHP. Sensitivity analyses are performed by varying the coefficients of the double generalized bilinear transformation such as k 1 , k 2 , a 1 , a 2 , b 1 and b 2 with respect to center frequency '} o ' and Bandwidth 'B' on the resulted 2-D Band-Pass and Band-Stop filter, which are obtained by varying the coefficients of the double generalized bilinear transformation in the specific ranges in order to maintain the stability of the filter of the Band-Pass and Band-Stop transfer function respectively.

Two-dimensional variable recursive digital filters are applied in signal processing and communication systems where the frequency-domain characteristics of digital filters are required to be adjustable. The main objective of this thesis is to propose a new technique of designing 2-D recursive digital filters with variable characteristics. From a 1-D second order Butterworth low-pass analog ladder structure, 2-D low-pass and high-pass digital filters can be obtained through the application of double generalized bilinear transformations when the coefficients of the transformations are chosen in their specified ranges. And when one or more these coefficients are changing, the resulting 2-D low-pass and high-pass filters possess variable magnitude responses. Another two important types of 2-D digital filters, 2-D band-pass and band-elimination filters, can also be obtained by properly combining a 2-D low-pass filter and a 2-D high-pass filter. When the coefficients used to obtain the 2-D low-pass and high-pass filters are changeable, the resulting 2-D band-pass and band-elimination filters also possess variable magnitude characteristics. The manner how each coefficient of generalized bilinear transformation affects each type of desiring 2-D recursive digital filters is investigated in detail. Stability is always an important issue in 2-D recursive digital filter design. The stability conditions of generalized bilinear transformation and the stability conditions of the 2-D digital filters having a denominator with single degree of each variable are discussed in detail here.

Two-dimensional recursive digital filters are widely used in signal processing and image processing, as well as divers communication systems. The main objective of this thesis has been to propose a new technique of designing 2-D recursive digital filters from an analog Bridged-T network. Starting from transfer function of a Bridged-T network in the analog domain which is VSHP, 2-D recursive digital filters can be obtained through the application of the double generalized bilinear transformations with the coefficients in their specified ranges. The impedance values of the transfer function of the Bridged-T network are obtained with compare to the fourth order Butterworth polynomial. For different impedance values of the Bridged-T network we get different types of filter output--all pass filter, band pass filter, band stop filter and low pass filter. The manner how each coefficient of generalized bilinear transformation affects each kind of 2-D recursive digital filter is investigated in details.

В кваліфікаційній роботі магістра розглянуто питання фільтрації зачумлених сигналів, зокрема мовних та аудіо сигналів. Проаналізовано принципи аналогової та цифрової фільтрації і встановлено, що в плані якості фільтрації, коли необхідно використати фільтри із складною передавальною характеристикрою, оптимальним є використання методів цифрової фільтрації. Розглянуто принципи роботи та проектування рекурсивних і не рекурсивних цифрових фільтрів. Проведено моделювання не рекурсивних фільтрів в середовищі Matlab та проведено фільтрацію створеного сестового сигналу, що являє собою адитивну суміш мовного сигналу та білого шуму. Оцінено результат фільтрації такого сигналу цифровим фільтром низької частоти та вінерівським фільтром. Встановлено, що в другому випадку при збільшенні порядку фільтра значно покращується словесна розбірливість мовного сигналу
In the qualification work of the master the questions of filtering of plastered signals, in particular speech and audio signals are considered. The principles of analog and digital filtering are analyzed and it is established that in terms of filtration quality, when it is necessary to use filters with complex transmission characteristics, the use of digital filtering methods is optimal. The principles of operation and design of recursive and non-recursive digital filters are considered. Simulations of non-recursive filters in the Matlab environment were performed and the generated sest signal, which is an additive mixture of speech signal and white noise, was filtered. The result of filtering such a signal by a low-pass digital filter and a Wiener filter is evaluated. It is established that in the second case, increasing the order of the filter significantly improves the verbal intelligibility of the speech signal