Nothing Special   »   [go: up one dir, main page]
You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
3.7
1.3
Journals
Acoustics
Special Issues
Resonators in Acoustics (2nd Edition)
share announcement

Resonators in Acoustics (2nd Edition)

A special issue of Acoustics (ISSN 2624-599X).

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 15728

Special Issue Editor

Dr. Nikolay Kanev

E-Mail Website
Guest Editor
Theoretical Department, Andreyev Acoustics Institute, Moscow 117036, Russia
Interests: helmholtz resonator; low frequency
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Today, resonators are widely used in different fields of acoustics. Although the first mentionof resonant devices can be found in ancient manuscripts, their scientific study began much later with the well-known Helmholtz resonator, which was an effective absorber and scatterer of sound waves. The theory of resonant acoustic phenomena was founded by Helmholtz, Rayleigh, and other scientists, and continues to be developed intensively by researchers and engineers in various fields. Since then, resonators have been successfully applied to architectural acoustics, noise and vibration control, medical applications, measurement technologies, sound radiation, new sensors, and variety of other inventions. 

In recent years, new metamaterials and metastructures with unusual properties have been developed on the basis of using the simplest resonant elements; the theory of resonators has assisted in elaborating new techniques for active sound control. Therefore, resonators remain highly valuable to both fundamental research and engineering investigations.

Taking into account the relevance of this topic and the great interest of many researchers, we devote thisSpecial Issue of the journal Acoustics to resonators in acoustics. The first volume of this Special Issue published 10 interesting papers, with more than 20,000 views. This new volume aims to attract the latest findings concerning the application of resonators in acoustics. New ideas and approaches, theoretical studies, and the technical implementation of resonators in various acoustic fields are within the scope of this Special Issue. Submissions with original results, as well as reviews, are most welcome in this Special Issue.

Dr. Nikolay Kanev
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Acoustics is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • theory of resonators
  • modelling and simulation of acoustic resonators
  • new types of resonators
  • resonant devices in acoustics
  • resonators for noise and vibration control
  • absorption and scattering of sound waves by resonators
  • new application of resonators
  • metamaterials based on resonance phenomena
  • experimental study
  • active resonators
  • resonators in engineering

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (6 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

14 pages, 1989 KiB  
Article
Acoustic Properties of Surfaces Covered by Multipole Resonators
by Nikolay Kanev
Acoustics 2024, 6(2), 509-522; https://doi.org/10.3390/acoustics6020027 - 25 May 2024
Viewed by 1427
Abstract
Different types of resonators are used to create acoustic metamaterials and metasurfaces. Recent studies focused on the use of multiple resonators of the dipole, quadrupole, octupole, and even hexadecapole types. This paper considers the theory of an acoustic metasurface, which is a flat [...] Read more.
Different types of resonators are used to create acoustic metamaterials and metasurfaces. Recent studies focused on the use of multiple resonators of the dipole, quadrupole, octupole, and even hexadecapole types. This paper considers the theory of an acoustic metasurface, which is a flat surface with a periodic arrangement of multipole resonators. The sound field reflected by the metasurface is determined. If the distance between the resonators is less than half the wavelength of the incident plane wave, the far field can be described by a reflection coefficient that depends on the angle of incidence. This allows us to characterize the acoustic properties of the metasurface by a homogenized boundary condition, which is a high-order tangential impedance boundary condition. The tangential impedance depending on the multipole order of the resonators is introduced. In addition, we analyze the sound absorption properties of these metasurfaces, which are a critical factor in determining their performance. The paper presents a theoretical model for the subwavelength case that accounts for the multipole orders of resonators and their impact on sound absorption. The maximum absorption coefficient for a diffuse sound field, as well as the optimal value for the homogenized impedance, are calculated for arbitrary multipole orders. The examples of the multipole resonators, which can be made from a set of Helmholtz resonators or membrane resonators, are discussed as well. Full article
(This article belongs to the Special Issue Resonators in Acoustics (2nd Edition))
►▼ Show Figures

Figure 1

Figure 1
<p>Physical models of the multipole sound scatterers with the order <math display="inline"><semantics> <mi>N</mi> </semantics></math> presented as a set of <math display="inline"><semantics> <mrow> <msup> <mn>2</mn> <mi>N</mi> </msup> </mrow> </semantics></math> monopoles with the volume velocities <math display="inline"><semantics> <mrow> <mo>+</mo> <mi>q</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>−</mo> <mi>q</mi> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>An array of linear multipole scatterers near a rigid surface. The direction of the incident plane wave described by the angle <math display="inline"><semantics> <mi>θ</mi> </semantics></math> is shown by the arrow.</p> Full article ">Figure 3
<p>The reflection coefficients of the metasurface in dependence on the incidence angle for <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">˜</mo> </mover> <mi>N</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>5</mn> </mrow> </semantics></math> (<b>a</b>), <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">˜</mo> </mover> <mi>N</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>b</b>), and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">˜</mo> </mover> <mi>N</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (<b>c</b>) and orders <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mn>4</mn> <mo>,</mo> <mo> </mo> <mn>10</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The diffuse absorption coefficient of the metasurfaces formed by the resonators with the orders <math display="inline"><semantics> <mi>N</mi> </semantics></math>.</p> Full article ">Figure 5
<p>The optimal impedance (<b>a</b>) and maximum absorption coefficient (<b>b</b>) of the metasurface in dependence on the order <math display="inline"><semantics> <mi>N</mi> </semantics></math>.</p> Full article ">Figure 6
<p>Eigenmodes of two (<b>a</b>) and four (<b>b</b>) Helmholtz resonators placed at a distance <math display="inline"><semantics> <mi>a</mi> </semantics></math>. The arrows indicate the direction of air flow in the resonator necks.</p> Full article ">Figure 7
<p>Membranes embedded in a rigid baffle. The first (<b>a</b>) and second (<b>b</b>) eigenmodes of a single membrane correspond to monopole and dipole scattering, respectively. Two membranes in the second eigenmode form a quadrupole scatterer (<b>c</b>).</p> Full article ">
17 pages, 5223 KiB  
Article
Influence of the Gain–Bandwidth of the Front-End Amplifier on the Performance of a QEPAS Sensor
by Luigi Lombardi, Gianvito Matarrese and Cristoforo Marzocca
Acoustics 2024, 6(1), 240-256; https://doi.org/10.3390/acoustics6010013 - 6 Mar 2024
Viewed by 1934
Abstract
The quartz tuning fork used as an acoustic sensor in quartz-enhanced photo-acoustic spectroscopy gas detection systems is usually read out by means of a transimpedance preamplifier based on a low-noise operational amplifier closed in a feedback loop. The gain–bandwidth product of the operational [...] Read more.
The quartz tuning fork used as an acoustic sensor in quartz-enhanced photo-acoustic spectroscopy gas detection systems is usually read out by means of a transimpedance preamplifier based on a low-noise operational amplifier closed in a feedback loop. The gain–bandwidth product of the operational amplifier used in the circuit is a key parameter which must be properly chosen to guarantee that the circuit works as expected. Here, we demonstrate that if the value of this parameter is not sufficiently large, the response of the preamplifier exhibits a peak at a frequency which does not coincide with the series resonant frequency of the quartz tuning fork. If this peak frequency is selected for modulating the laser bias current and is also used as the reference frequency of the lock-in amplifier, a penalty results in terms of signal-to-noise ratio at the output of the QEPAS sensor. This worsens the performance of the gas sensing system in terms of ultimate detection limits. We show that this happens when the front-end preamplifier of the quartz tuning fork is based on some amplifier models that are typically used for such application, both when the integration time of the lock-in amplifier filter is long, to boost noise rejection, and when it is short, in order to comply with a relevant measurement rate. Full article
(This article belongs to the Special Issue Resonators in Acoustics (2nd Edition))
►▼ Show Figures

Figure 1

Figure 1
<p>Simplified block diagram of a QEPAS sensor.</p> Full article ">Figure 2
<p>QTF coupled to micro-resonator tubes.</p> Full article ">Figure 3
<p>QTF read-out by means of a transimpedance preamplifier.</p> Full article ">Figure 4
<p>Butterworth–Van Dyke model of the QTF coupled to the TIA.</p> Full article ">Figure 5
<p>SPICE simulation of the loop gain of the TIA in <a href="#acoustics-06-00013-f003" class="html-fig">Figure 3</a> with the OPAMP OP27.</p> Full article ">Figure 6
<p>Detail of the frequency behavior of |T(jω)| around the resonant frequency of the QTF, for the TIA based on the OP27 OPAMP.</p> Full article ">Figure 7
<p>Equivalent circuit of the TIA after the application of Miller’s theorem to the feedback impedance.</p> Full article ">Figure 8
<p>Simplified circuit for the evaluation of the resonant frequencies of the circuit in <a href="#acoustics-06-00013-f007" class="html-fig">Figure 7</a>.</p> Full article ">Figure 9
<p>Frequency response of the TIA realized with three different OPAMPs.</p> Full article ">Figure 10
<p>Main noise sources in the TIA.</p> Full article ">Figure 11
<p>TIA realized with the AD8067: modulus of the transfer function |H(jω)| obtained with SPICE simulations and Equation (2).</p> Full article ">Figure 12
<p>TIA realized with the AD8067: total output noise power density obtained with SPICE simulations and Equation (12).</p> Full article ">Figure 13
<p>Increasing the loop gain of the TIA based on the OP27 by inserting an ideal voltage amplifier in the feedback loop.</p> Full article ">Figure 14
<p>Comparison between the output noise power spectral densities of the TIA realized with the OP27 and with the same OPAMP, but with loop gain increased by a factor of 20 by means of an ideal voltage amplifier inserted into the feedback loop.</p> Full article ">Figure 15
<p>Comparison between the modulus of the transfer function H<sub>f</sub>(jω) of the TIA realized with the OP27 and with the same OPAMP, but with loop gain increased by a factor of 20.</p> Full article ">Figure 16
<p>SNR at the LIA output as a function of the LIA reference frequency: LIA filter bandwidth BW = 0.1 Hz, TIA realized with the OP27 and with the same OPAMP, but with loop gain increased by a factor of 20.</p> Full article ">Figure 17
<p>SNR at the LIA output as a function of the LIA reference frequency: LIA filter bandwidth BW=3 Hz, TIA achieved with OP27 and with the same OPAMP but with loop gain increased by a factor of 20.</p> Full article ">
17 pages, 14043 KiB  
Article
Investigations into the Approaches of Computational Fluid Dynamics for Flow-Excited Resonator Helmholtz Modeling within Verification on a Laboratory Benchmark
by Daniil Sergeev, Irina V’yushkina, Vladimir Eremeev, Andrei Stulenkov and Kirill Pyalov
Acoustics 2024, 6(1), 18-34; https://doi.org/10.3390/acoustics6010002 - 22 Dec 2023
Cited by 2 | Viewed by 2415
Abstract
This paper presents the results of a study of self-sustained processes excited in a Helmholtz resonator after a flow over its orifice. A comparative analysis of various approaches to the numerical modeling of this problem was carried out, taking into account both the [...] Read more.
This paper presents the results of a study of self-sustained processes excited in a Helmholtz resonator after a flow over its orifice. A comparative analysis of various approaches to the numerical modeling of this problem was carried out, taking into account both the requirements for achieving the required accuracy and taking into account the resource greediness of calculations, the results of which were verified by comparison with data obtained during a special experiment. The configuration with a spherical resonator with a natural frequency of 260 Hz and an orifice diameter (about 5 cm) in an air flow with a speed of 6 to 14 m/s was considered. A comparison of the calculation results with data obtained in experiments carried out in the wind tunnel demonstrated that the accuracy of calculations of the characteristics of the self-sustained mode using the simplest URANS class model tends to the accuracy of calculations within the large eddy simulation approach formulated in the WMLES model. At the same time, when using WMLES, it is possible to better reproduce the background level of pulsations. From the point of view of resource greediness, expressed in the number of core hours spent obtaining a solution, both models of the turbulence turned out to be almost equivalent when using the same grid models. Full article
(This article belongs to the Special Issue Resonators in Acoustics (2nd Edition))
►▼ Show Figures

Figure 1

Figure 1
<p>The resonator used in investigations: (<b>a</b>) common view, (<b>b</b>) microphone gauge installed inside resonator, (<b>c</b>) frequency response characteristic.</p> Full article ">Figure 2
<p>(<b>a</b>) Resonator installed in the working section of the wind tunnel of IAP RAS; (<b>b</b>) scheme of the location of the resonator inside the working section of the wind tunnel: top view, dimensions are given in mm.</p> Full article ">Figure 3
<p>Scheme of measurements of air flow with hot film in the inlet and hot wire close to the resonator orifice anemometers relative to a resonator in a wind tunnel: (<b>a</b>) side view, (<b>b</b>) top view.</p> Full article ">Figure 4
<p>Frequency spectra in the range up to 400 Hz for different inlet airflow speeds: (<b>a</b>) pressure power pulsations inside the resonator; (<b>b</b>) flow velocity module pulsations at a distance of 5 mm above the orifice. Line color: blue—8 m/s, red—10 m/s, green—12 m/s, black—14 m/s.</p> Full article ">Figure 5
<p>Sound pressure spectra for inlet airflow speed from 4 to 14 m/s with a step of 0.5 m/s. Red dots mark peaks corresponding to cases of excitation of the first mode of hydrodynamic instability. Black dots correspond to the second mode. For spectra, the vertical axis for amplitude is on the left. The solid line with the corresponding scale on the right shows the calculated frequency response of the resonator, for which the vertical axis is on the right.</p> Full article ">Figure 6
<p>Spectrogram of sound pressure on the inlet air flow speed at the entrance to the working section of the wind tunnel.</p> Full article ">Figure 7
<p>Dependence of the Strouhal number calculated from the peak frequency in the spectrum in <a href="#acoustics-06-00002-f005" class="html-fig">Figure 5</a> and bulk flow velocity above the orifice. The red dots show the values for the first (main) mode of hydrodynamic instability; the black dots show the values for the second one.</p> Full article ">Figure 8
<p>Illustration of numerical model of the resonator in the software SATES 2.0.</p> Full article ">Figure 9
<p>Computational domain of the air flow around a resonator. 1—inlet boundary, 2—outlet boundary, 3—side walls, 4—surface of the sphere.</p> Full article ">Figure 10
<p>Grid model (GM1) of the computational domain with selected fragments for the SST turbulence model.</p> Full article ">Figure 11
<p>The calculated vorticity field at the cross-sectional plane of symmetry of the resonator, inlet air flow 14 m/s. URANS.</p> Full article ">Figure 12
<p>The sound pressure pulsation spectra for different flow rates (experimental data and results of calculations).</p> Full article ">Figure 13
<p>Grid model (GM2) of the computational domain with selected fragments for the WMLES turbulence model.</p> Full article ">Figure 14
<p>The calculated vorticity field at the cross-sectional plane of symmetry of the resonator, inlet air flow 14 m/s. WMLES.</p> Full article ">Figure 15
<p>The sound pressure pulsation spectra for different flow rates (experimental and calculated data: SST, WMLES).</p> Full article ">Figure 16
<p>The sound pressure pulsation spectra for air flow 14 m/s (experimental and calculated data: SST, WMLES).</p> Full article ">
13 pages, 2967 KiB  
Article
Low-Frequency-Noise Attenuation through Extended-Neck Double-Degree-of-Freedom Helmholtz Resonators
by Abhishek Gautam, Alper Celik and Mahdi Azarpeyvand
Acoustics 2023, 5(4), 1123-1135; https://doi.org/10.3390/acoustics5040063 - 3 Dec 2023
Viewed by 3064
Abstract
The use of acoustic liners, based on double-degree-of-freedom Helmholtz resonators, for low-frequency-noise attenuation is limited by the volume of individual resonating cavities. This study investigates the effect of the septum neck length on the acoustic performance of double-degree-of-freedom resonators, both experimentally and numerically, [...] Read more.
The use of acoustic liners, based on double-degree-of-freedom Helmholtz resonators, for low-frequency-noise attenuation is limited by the volume of individual resonating cavities. This study investigates the effect of the septum neck length on the acoustic performance of double-degree-of-freedom resonators, both experimentally and numerically, for varying cavity volume ratios. The underlying sound attenuation mechanism is studied by analysing the acoustic pressure fields within the resonator cavities. An increase in the septum neck is shown to lower the frequencies affected by the resonator. In addition, it deteriorates and significantly improves the sound attenuation performance at the primary and secondary peak transmission-loss frequencies, respectively. Full article
(This article belongs to the Special Issue Resonators in Acoustics (2nd Edition))
►▼ Show Figures

Figure 1

Figure 1
<p>Schematics of the following components are depicted: (<b>a</b>) a grazing flow impedance tube, (<b>b</b>) a baseline resonator with two degrees of freedom (2 DoF), (<b>c</b>) a 2-DoF resonator with a two-sided septum neck extension (referred to as Case A), (<b>d</b>) a 2-DoF resonator with a one-sided septum neck extension (referred to as Case B), (<b>e</b>) comparative analyses encompassing experimental and finite element analysis results for the transmission coefficient arising from 2-DoF resonators and 2-DoF resonators with extended septum neck configurations, and (<b>f</b>) comparative analyses incorporating experimental data and finite element analysis results for the transmission loss from 2-DoF resonators and 2-DoF resonators with extended septum neck configurations. The septum neck extension <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and <span class="html-italic">m</span> = 0.5.</p> Full article ">Figure 2
<p>The numerical setup used in this study: (<b>a</b>) schematics of the steady and transient simulation; (<b>b</b>) detailed image of the mesh in the neck of a resonator sample.</p> Full article ">Figure 3
<p>Experimental and finite element analysis results for (<b>a</b>) peak transmission-loss frequencies comparison with changing volume ratio (<span class="html-italic">m</span>) at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>, (<b>b</b>) peak transmission-loss frequencies comparison with changing volume ratio (<span class="html-italic">m</span>) at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>, (<b>c</b>) change in bandwidth coefficient with changing volume ratio at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>, (<b>d</b>) change in bandwidth coefficient with changing volume ratio at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>, (<b>e</b>) normalised transmission loss comparison with changing volume ratio (<span class="html-italic">m</span>) at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>, (<b>f</b>) normalised transmission loss comparison with changing volume ratio (<span class="html-italic">m</span>) at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>Contour plots of acoustic pressure inside the 2-DoF and 2-DoF extended-neck resonator configurations at the primary peak transmission-loss frequency: (<b>a</b>) 2-DoF resonator with volume ratio <span class="html-italic">m</span> = 0.3 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>b</b>) 2-DoF resonator with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and volume ratio <span class="html-italic">m</span> = 0.3 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>c</b>) 2-DoF resonator with volume ratio <span class="html-italic">m</span> = 0.5 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>d</b>) 2-DoF resonator with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and volume ratio <span class="html-italic">m</span> = 0.5 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>e</b>) 2-DoF resonator with volume ratio <span class="html-italic">m</span> = 0.7 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>f</b>) 2-DoF resonator with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and volume ratio <span class="html-italic">m</span> = 0.7 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>.</p> Full article ">Figure 5
<p>Contour plots of acoustic pressure inside the 2-DoF and 2-DoF extended-neck resonator configurations at the secondary peak transmission-loss frequency: (<b>a</b>) 2-DoF resonator with volume ratio <span class="html-italic">m</span> = 0.3 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>b</b>) 2-DoF resonator with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and volume ratio <span class="html-italic">m</span> = 0.3 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>c</b>) 2-DoF resonator with volume ratio <span class="html-italic">m</span> = 0.5 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>d</b>) 2-DoF resonator with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and volume ratio <span class="html-italic">m</span> = 0.5 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>e</b>) 2-DoF resonator with volume ratio <span class="html-italic">m</span> = 0.7 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>f</b>) 2-DoF resonator with <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> and volume ratio <span class="html-italic">m</span> = 0.7 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 6
<p>SPL of acoustic signal for Cavity 1 (C1) and Cavity 2 (C2) captured using microphones M1 and M2, for different septum neck extension configurations of <math display="inline"><semantics> <msub> <mi>t</mi> <mn>2</mn> </msub> </semantics></math> = 0, 0.1 and 0.3: (<b>a</b>) volume ratio <span class="html-italic">m</span> = 0.3 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>b</b>) volume ratio <span class="html-italic">m</span> = 0.5 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>c</b>) volume ratio <span class="html-italic">m</span> = 0.7 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>; (<b>d</b>) volume ratio <span class="html-italic">m</span> = 0.3 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>e</b>) volume ratio <span class="html-italic">m</span> = 0.5 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>; (<b>f</b>) volume ratio <span class="html-italic">m</span> = 0.7 at <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>. Phase of the pressure signals from microphone M2 relative to M1 for different septum neck extension configurations of <math display="inline"><semantics> <msub> <mi>t</mi> <mn>2</mn> </msub> </semantics></math> = 0, 0.1, and 0.3: (<b>g</b>) volume ratio <span class="html-italic">m</span> = 0.3; (<b>h</b>) volume ratio <span class="html-italic">m</span> = 0.5; (<b>i</b>) volume ratio <span class="html-italic">m</span> = 0.7. The first peak transmission-loss frequency, <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>, is illustrated by a coloured triangle and the second peak transmission-loss frequency, <math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>, is illustrated by a coloured circle.</p> Full article ">Figure 7
<p>Phase of the pressure signals from microphones M1 and M2 relative to G8 for different septum neck extension configurations of <math display="inline"><semantics> <msub> <mi>t</mi> <mn>2</mn> </msub> </semantics></math> = 0, 0.1, and 0.3: (<b>a</b>) phase of M1 relative to G8 for a volume ratio <span class="html-italic">m</span> = 0.3; (<b>b</b>) phase of M1 relative to G8 for a volume ratio <span class="html-italic">m</span> = 0.5; (<b>c</b>) phase of M1 relative to G8 for a volume ratio <span class="html-italic">m</span> = 0.7; (<b>d</b>) phase of M2 relative to G8 for a volume ratio <span class="html-italic">m</span> = 0.3; (<b>e</b>) phase of M2 relative to G8 for a volume ratio <span class="html-italic">m</span> = 0.5; (<b>f</b>) phase of M2 relative to G8 for a volume ratio <span class="html-italic">m</span> = 0.7.</p> Full article ">
18 pages, 4773 KiB  
Article
An Acoustoelectric Approach to Neuron Function
by Jörg P. Kotthaus
Acoustics 2023, 5(3), 601-618; https://doi.org/10.3390/acoustics5030037 - 22 Jun 2023
Viewed by 3352
Abstract
An acoustoelectric approach to neuron function is proposed that combines aspects of the widely accepted electrical-circuit-based Hodgkin–Huxley model for the generation and propagation of action potentials via electric polarization with mechanical models based on propagation via capillary waves. Explaining measured velocities of action [...] Read more.
An acoustoelectric approach to neuron function is proposed that combines aspects of the widely accepted electrical-circuit-based Hodgkin–Huxley model for the generation and propagation of action potentials via electric polarization with mechanical models based on propagation via capillary waves. Explaining measured velocities of action potentials quantitatively, it also predicts the electrical tunability of highly anisotropic polarization packages that surf on the dynamic mechanical force field deforming the neuron membrane. It relies substantially on the local motion of dipoles formed by excess charges close to the inside surface of the neuron membrane, which in turn are anisotropically screened by water molecules in their hydration shell, thus modulating the strong electric field at the interface. As demonstrated on acoustic resonators of suspended nanowires fabricated out of amorphous dipolar silicon nitride, high electric fields combined with predominantly axial-strain modulation can cause transverse acoustoelectric polarization waves that propagate soliton-like with extremely low loss. In neurons, the modulation of electric polarization is confined in the nanometer-thin skin of a high electric field inside the neuron membrane and propagates phase-coherent along the axon as a lowest-order one-dimensional breathing mode, similar to transverse polarization pulses studied in nanowire resonators. Some experiments for the further manifestation of the model as well as topological protection of such breathing-mode polarization waves are discussed. Full article
(This article belongs to the Special Issue Resonators in Acoustics (2nd Edition))
►▼ Show Figures

Figure 1

Figure 1
<p>Schematic sketch of a neuron. The action potential is generated in the soma and its dendrites, propagates soliton-like along the axon, and is transduced by the synaptic terminals to other neurons or sensor cells.</p> Full article ">Figure 2
<p>An electron micrograph of a typical silicon nitride string resonator of length L, width W, and height T with its supports (green) is jointly displayed with a cross-sectional cut of the silicon substrate (grey), a SiO<sub>2</sub> insulating layer (blue), and the supported electrodes (yellow). The right-side sketches how a static electric field <span class="html-italic">E<sub>y</sub></span> polarizes the string and an oscillating electric field E<sub>yRF</sub> initiates radio frequency mechanic resonance modes at MHz frequencies.</p> Full article ">Figure 3
<p>Generation of chiral components of an action potential via injection of positively charged ions via an ion channel (green) which locally depolarizes and repolarizes the high field cylindrical Debye skin of thickness t (red) of a tubular axon of radius R. The local depolarization generates 4 chiral wave components propagating into opposing axial directions. For a given axial direction, two opposing chiral components create a breathing-mode capillary wave propagating near the electrolyte membrane interface along a spiral path (as indicated on the left of the ion channel for the right-hand polarized spiral. Here the red line visualizes the 3D nature of the spiral moving with momentum p, whereas the blue line presents a projection into the 2D axial plane) with the two opposing angular momenta M<sub>R</sub> and M<sub>L</sub> created by the ion-current pulse injected into the high electric-field region between the inside of the tubular lipid membrane of thickness d (yellow) and the Debye region in the electrolyte of thickness t (red). The capillary pulse wavelength <span class="html-italic">λ<sub>C</sub></span> is half the axial wave-length <span class="html-italic">λ<sub>E</sub></span> of the electric-field component E<sub>⊥</sub> perpendicular to the membrane capacitor and the change of the local tube radius Δ<span class="html-italic">R</span> propagating with the capillary wave causes an enhanced axial polarization by the locally redistributed ionic charge and the anisotropic oscillating water dipoles as discussed in the main text.</p> Full article ">Figure 4
<p>This sketch tries to schematically display the correlation between a typical shape of an action potential (top) reflecting the temporal voltage change at the inside of the axon membrane, initiated by the transient opening of ion channels, injecting positively charged ions and deforming the axon diameter D (bottom) through an action potential moving to the left (blue arrow) as a capillary wave pulse as viewed from a stationary outside observer. The arrows indicate the orientation and size of polarization-induced forces onto the confining axon membrane. Red arrows mark polarization-caused forces by negatively charged excess ions and their hydration shell on the inside of the membrane and green arrows are forces caused by polarization induced by positively charged excess ions.</p> Full article ">Figure 5
<p>Components of a myelinated axon, from Wikipedia, File Neuron.svg, User: Dhp1080 “Anatomy and Physiology” by the US National Cancer Institute’s Surveillance, Epidemiology and End Results Program, created: 17 March 2019 <a href="http://creativecommons.org/licenses/by-sa/3.0/" target="_blank">http://creativecommons.org/licenses/by-sa/3.0/</a> (accessed on 23 January 2023).</p> Full article ">Figure A1
<p>This photo and Video 4 show my own experiments on how a solitonic excitation can propagate through a soft chiral spring, prestressed by gravity with a weight, in an effort to imitate the propagation of a chiral action potential pulse through an axon. The spiral spring is fabricated out of a rectangular plastic wire which has a radial width of 3 mm and an axial width of 2 mm and is wound into a spiral of 9 cm diameter with right-hand turn chirality—a toy named “Slinky”. Shocking the spring locally with short radial compression, one initiates a solitonic pulse moving with low damping axially through the spring with a propagation velocity faster than the harmonic oscillation of the spring. <a href="https://cast.itunes.uni-muenchen.de/clips/3zte0uDMZo/vod/online.html" target="_blank">https://cast.itunes.uni-muenchen.de/clips/3zte0uDMZo/vod/online.html</a>, (accessed on 23 January 2023).</p> Full article ">
18 pages, 4785 KiB  
Article
IIR Cascaded-Resonator-Based Complex Filter Banks
by Miodrag D. Kušljević, Vladimir V. Vujičić, Josif J. Tomić and Predrag D. Poljak
Acoustics 2023, 5(2), 535-552; https://doi.org/10.3390/acoustics5020032 - 30 May 2023
Cited by 1 | Viewed by 2255
Abstract
The use of a filter bank of IIR filters for the spectral decomposition and analysis of signals has been popular for many years. As such, a new filter-bank resonator-based structure, representing an extremely hardware-efficient structure, has received a good deal of attention. Recently, [...] Read more.
The use of a filter bank of IIR filters for the spectral decomposition and analysis of signals has been popular for many years. As such, a new filter-bank resonator-based structure, representing an extremely hardware-efficient structure, has received a good deal of attention. Recently, multiple-resonator (MR)-based and general cascaded-resonator (CR)-based filters have been proposed. In comparison to single-resonator-based analyzers, analyzers with a higher multiplicity of resonators in the cascade provide lower side lobes and a higher attenuation in stopbands. In previous works, it was shown that the CR-based filter bank with infinite impulse response (IIR) filters, which is numerically more efficient than one with finite impulse response (FIR) filters, is suitable for dynamic harmonic analysis. This paper uses the same approach to design complex digital filter banks. In the previous case, the optimization task referred to the frequency responses of harmonic filters. In this work, the harmonic filters of the mother filter bank are reshaped so that the frequency response of the sum (or difference, depending on the parity of the number of resonators in the cascade) of two adjacent harmonic filters is optimized. This way, an online adaptive filter base can be obtained. The bandwidth of the filters in the designed filter bank can be simply changed online by adding or omitting the output signals of the corresponding harmonics of the mother filter. Full article
(This article belongs to the Special Issue Resonators in Acoustics (2nd Edition))
►▼ Show Figures

Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Block diagram of the <span class="html-italic">K</span>-type CR-based harmonic analyzer.</p> Full article ">Figure 2
<p>Frequency responses for the first up to the sixth order of resonator multiplicity (<math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>0,1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>5</mn> </mrow> </semantics></math>), with the ordinate scale in decibels for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Frequency responses for different bandwidths for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Approximation of a cycle with a square and an octagon.</p> Full article ">Figure 5
<p>The number of multiplications per one sample time instant as a function of <math display="inline"><semantics> <mrow> <mi>K</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The number of real multiplications per second as a function of <math display="inline"><semantics> <mrow> <mi>K</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>M</mi> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Frequency responses for different bandwidths, for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Frequency responses for different bandwidths, for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <mn>2</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Frequency responses for different bandwidths, for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mfenced separators="|"> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> <mo>/</mo> <mrow> <mn>2</mn> </mrow> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Frequency responses for different bandwidths, for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Pole-zero map for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Frequency responses for different bandwidths, for <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>16</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>125</mn> <mo> </mo> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mfenced separators="|"> <mrow> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>τ</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-6
Back to TopTop