commit a22f988b8a148e014f8c89507893a90fc3d6d0d5 Author: J.A. de Jong - Redu-Sone B.V., ASCEE V.O.F Date: Mon Jun 21 15:28:51 2021 +0200 Initial commit diff --git a/ASCEE_lasp.lyx b/ASCEE_lasp.lyx new file mode 100644 index 0000000..b34ed14 --- /dev/null +++ b/ASCEE_lasp.lyx @@ -0,0 +1,3882 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\begin_preamble +\input{tex/preamble_article.tex} +\end_preamble +\options twoside,final +\use_default_options true +\maintain_unincluded_children false +\language american +\language_package babel +\inputencoding utf8 +\fontencoding global +\font_roman "libertine" "Linux Libertine O" +\font_sans "default" "Courier New" +\font_typewriter "default" "default" +\font_math "libertine-ntxm" "auto" +\font_default_family default +\use_non_tex_fonts true +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures false +\graphics default +\default_output_format default +\output_sync 1 +\output_sync_macro "\synctex=1" +\bibtex_command biber +\index_command default +\paperfontsize 10 +\spacing single +\use_hyperref true +\pdf_author "Dr.ir. J.A. de Jong - ASCEE" +\pdf_bookmarks true +\pdf_bookmarksnumbered false +\pdf_bookmarksopen false +\pdf_bookmarksopenlevel 1 +\pdf_breaklinks true +\pdf_pdfborder true +\pdf_colorlinks true +\pdf_backref false +\pdf_pdfusetitle true +\papersize a4paper +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine biblatex +\cite_engine_type authoryear +\biblio_style plain +\biblatex_bibstyle authoryear +\biblatex_citestyle numeric +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 0 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 3cm +\topmargin 3cm +\rightmargin 2.5cm +\bottommargin 3.5cm +\headsep 1cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation skip +\defskip smallskip +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +date{ +\backslash +today} +\end_layout + +\begin_layout Plain Layout + +% +\backslash +begin{center} +\end_layout + +\begin_layout Plain Layout + +% +\backslash +includegraphics{/home/anne/nextcloud/template_huisstijl/lyx/%ascee_beeldmerk_wit +hacr.eps} +\end_layout + +\begin_layout Plain Layout + +% +\backslash +end{center} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Title +LASP +\begin_inset Newline newline +\end_inset + + +\size normal +Library for Acoustic Signal Processing +\end_layout + +\begin_layout Author + +\series bold +J.A. + de Jong +\begin_inset Formula $^{1}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset VSpace medskip +\end_inset + + +\end_layout + +\begin_layout Standard +\align center +\begin_inset Graphics + filename /home/anne/nextcloud/templates_huisstijl/lyx/ascee_beeldmerk_withacr.eps + width 45text% + +\end_inset + + +\end_layout + +\begin_layout Standard +\align center + +\size small +\begin_inset Formula $^{1}$ +\end_inset + +ASCEE, Máximastraat 1, 7442 NW Nijverdal, info@ascee.nl +\end_layout + +\begin_layout Standard +\begin_inset VSpace medskip +\end_inset + + +\end_layout + +\begin_layout Standard +\align left +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +arrayrulecolor{asceelightblue} +\end_layout + +\begin_layout Plain Layout + + +\backslash +midrule[2pt] +\end_layout + +\end_inset + +Internal document ID: +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +31416 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +External document ID: +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Document status: +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Draft +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Standard +\align center + +\size small +\begin_inset VSpace medskip +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\align left +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +arrayrulecolor{asceelightblue} +\end_layout + +\begin_layout Plain Layout + + +\backslash +midrule[2pt] +\end_layout + +\end_inset + +Internal document ID: +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +External document ID: +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Document status: +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Draft +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Note Note +status open + +\begin_layout Plain Layout + +\lang dutch +Revisiehistorie: +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Note Note +status open + +\begin_layout Plain Layout + +\lang dutch +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +today +\end_layout + +\end_inset + +: rev. + 1 +\begin_inset Newline newline +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +fourteendaysahead +\end_layout + +\end_inset + +: rev. + 2 +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + +\backslash +bottomrule[2pt]% +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset CommandInset toc +LatexCommand tableofcontents + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +thispagestyle{empty} +\end_layout + +\begin_layout Plain Layout + +% Optionally: set this document to confidential +\end_layout + +\begin_layout Plain Layout + +% +\backslash +confidential +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Fourier transform vs discrete Fourier transform +\end_layout + +\begin_layout Standard +Our definition of the Fourier transform for the real-valued function +\begin_inset Formula $x(t)$ +\end_inset + + with unit +\begin_inset Formula $U$ +\end_inset + + is: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +X(f)=\int\limits _{t=-\infty}^{\infty}x(t)\exp\left(-i2\pi ft\right)\mathrm{d}t, +\end{equation} + +\end_inset + +such that: +\begin_inset Formula +\begin{equation} +x(t)=\int\limits _{f=-\infty}^{\infty}X(f)\exp\left(2\pi ift\right)\mathrm{d}f +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Using the radian frequency +\begin_inset Formula $\omega$ +\end_inset + +, a scaling factor should be used in front: +\begin_inset Formula +\begin{equation} +x(t)=\frac{1}{2\pi}\int\limits _{\omega=-\infty}^{\infty}X(\omega)\exp\left(i\omega t\right)\mathrm{d}\omega, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +X(\omega)=\int\limits _{\omega=-\infty}^{\infty}x(t)\exp\left(-i\omega t\right)\mathrm{d}t, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Our definition of the +\bar under +power spectral density +\bar default + is: +\begin_inset Formula +\begin{equation} +P_{x}=\underbrace{\lim_{T\to\infty}\frac{1}{T}\int\limits _{t=-T}^{T}x^{2}(t)\mathrm{d}t}_{\mathrm{Signal\,power}}=E\left[x^{2}(t)\right]\equiv\int\limits _{f=-\infty}^{\infty}S_{xx}(f)\mathrm{d}f +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +From Parseval's theorem, we know that +\begin_inset Formula +\begin{equation} +S_{xx}(f)=\lim_{T\to\infty}\frac{1}{T}X(f)X^{*}(f). +\end{equation} + +\end_inset + +Hence for signals for which the Fourier transform formally exist, the power + spectral density is zero. + In practice, signal +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Filling in: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\lim_{T\to\infty}\frac{1}{T}\int\limits _{t=-T}^{T}\left[\int\limits _{\omega=-\infty}^{\infty}X(\omega)\exp\left(i\omega t\right)\mathrm{d}\omega\int\limits _{\omega=-\infty}^{\infty}X^{*}(\omega)\exp\left(-i\omega t\right)\mathrm{d}\omega\right]\mathrm{d}t=\int\limits _{\omega=-\infty}^{\infty}S_{xx}(\omega)\mathrm{d}\omega +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\lim_{T\to\infty}\frac{1}{T}\int\limits _{t=-T}^{T}\left[\int\limits _{\omega=-\infty}^{\infty}X(\omega)\exp\left(i\omega t\right)\mathrm{d}\omega\int\limits _{\omega=-\infty}^{\infty}X^{*}(\omega)\exp\left(-i\omega t\right)\mathrm{d}\omega\right]\mathrm{d}t=\int\limits _{\omega=-\infty}^{\infty}S_{xx}(\omega)\mathrm{d}\omega +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Results in: +\end_layout + +\begin_layout Standard +Plancheler theorem: +\begin_inset Formula +\begin{equation} +\int\limits _{t=-\infty}^{\infty}x(t)^{2}\mathrm{d}t=\int\limits _{\omega=-\infty}^{\infty}X(\omega)X^{*}(\omega)\mathrm{d}\omega +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Estimation of power spectra +\end_layout + +\begin_layout Itemize +Sample frequency: +\begin_inset Formula $f_{s}=1/\Delta t$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $N_{\mathrm{DFT}}$ +\end_inset + + The number of samples taking into the FFT (nfft) +\end_layout + +\begin_layout Standard +Frequency resolution +\begin_inset Formula +\begin{equation} +\Delta f=\frac{f_{s}}{N_{\mathrm{DFT}}}, +\end{equation} + +\end_inset + +i.e. + the smallest frequency that fits into the measured time +\begin_inset Formula $T=N_{\mathrm{DFT}}\Delta t$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Definition of the DFT: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +X[k]=\mathrm{DFT}\left(x_{n}\right)=\sum_{n=0}^{N_{\mathrm{DFT}}-1}x_{n}e^{-2\pi ikn/N_{\mathrm{DFT}}},\label{eq:dft_definition} +\end{equation} + +\end_inset + +such that +\begin_inset Formula +\begin{equation} +x[n]=\mathrm{iDFT}\left(X[k]\right)=\frac{1}{N_{\mathrm{DFT}}}\sum_{k=0}^{N_{\mathrm{DFT}}-1}X[k]e^{2\pi ikn/N_{\mathrm{DFT}}}. +\end{equation} + +\end_inset + +The advantage of this definition is that it preserves the duality between + transfer function and impulse response. + However, for proper scaling of signal (power) spectra, it requires more + inspection. +\end_layout + +\begin_layout Standard +Using the definition of Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:dft_definition" + +\end_inset + +, using the real-valued DFT (DFTR), we directly obtain the positive frequency + half-spectrum. +\end_layout + +\begin_layout Standard +Parseval's theorem states: +\begin_inset Formula +\begin{equation} +\sum_{n=0}^{N_{\mathrm{DFT}}-1}x[n]y[n]^{*}=\frac{1}{N}\sum_{k=0}^{N_{\mathrm{DFT}}-1}X[k]Y[k]^{*} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The average signal power is defined as +\begin_inset Formula +\begin{equation} +P(x[n])=\frac{1}{N_{\mathrm{DFT}}}\sum_{n=0}^{N_{\mathrm{DFT}}-1}x[n]^{2} +\end{equation} + +\end_inset + +which can be written in frequency domain, using Parseval's theorem as: +\begin_inset Formula +\begin{equation} +P(x[n])=\frac{1}{N_{\mathrm{DFT}}^{2}}\sum_{k=0}^{N_{\mathrm{DFT}}-1}X[k]X[k]^{*}. +\end{equation} + +\end_inset + +Hence the signal power as a function of frequency is +\begin_inset Formula +\begin{equation} +P_{k}=\frac{1}{N_{\mathrm{DFT}}^{2}}\left\Vert X[k]\right\Vert ^{2}, +\end{equation} + +\end_inset + +such that +\begin_inset Formula $P_{k}$ +\end_inset + + is the signal power in frequency bin +\begin_inset Formula $k$ +\end_inset + +. + Now, the power spectral density is defined as the signal power per unit + frequency, for which the signal power needs to be divided by the frequency + resolution. + Hence +\begin_inset Formula +\begin{equation} +S_{x}=\frac{1}{N_{\mathrm{DFT}}f_{s}}\left\Vert X[k]\right\Vert ^{2} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Single sided amplitude spectra +\end_layout + +\begin_layout Standard +If +\begin_inset Formula $X_{s}$ +\end_inset + + is defined such that +\begin_inset Formula +\begin{equation} +x[n]=\frac{1}{\sqrt{2}}X_{s}[0]+\sum_{k=1}^{N_{\mathrm{DFT}}/2-1}X_{s}[k]e^{2\pi ikn/N_{\mathrm{DFT}}}+\frac{1}{\sqrt{2}}X_{s}[N_{\mathrm{DFT}}/2]e^{i\pi n}, +\end{equation} + +\end_inset + +then the value of +\begin_inset Formula $X_{s}$ +\end_inset + + directly corresponds to the amplitude of the sinusoid, for all frequencies + not equal to the DC and Nyquist rate. + +\begin_inset Formula $N_{\mathrm{DFT}}/2$ +\end_inset + + is an integer division which rounds down to the nearest integer. + From this definition, we can conclude that +\begin_inset Formula +\begin{eqnarray} +X_{s}[0] & =\frac{\sqrt{2}}{N_{\mathrm{DFT}}}X[0] & \mathrm{for}\,\,\,k=0\\ +X_{s}[k] & =\frac{2}{N_{\mathrm{DFT}}}X[k] & \mathrm{for}\,\,\,0 + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Dimension +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Value +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Lower frequency +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Upper frequency +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s_{1}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +430 +\begin_inset space ~ +\end_inset + +mm +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +40 +\begin_inset space ~ +\end_inset + +Hz +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +320 +\begin_inset space ~ +\end_inset + +Hz +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s_{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +190 +\begin_inset space ~ +\end_inset + +mm +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +90 +\begin_inset space ~ +\end_inset + +Hz +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +724 +\begin_inset space ~ +\end_inset + +Hz +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s_{3}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +34 +\begin_inset space ~ +\end_inset + +mm +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +491 +\begin_inset space ~ +\end_inset + +Hz +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +4047 +\begin_inset space ~ +\end_inset + +Hz +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Subsection +Microphone switching technique for estimating the transfer function corrections +\end_layout + +\begin_layout Itemize +Normal measurement, mic 0 at position A, mic 1 at position B +\end_layout + +\begin_layout Itemize +Switched measurement, mic 0 at position B, mic 1 at position A +\end_layout + +\begin_layout Standard +Definitions: +\end_layout + +\begin_layout Itemize +\begin_inset Formula $K_{0}$ +\end_inset + +: Microphone calibration correction factor for mic 0. + Such that +\begin_inset Formula $p_{0}=K_{0}\tilde{p}_{0}$ +\end_inset + +, where +\begin_inset Formula $\hat{p}_{0}$ +\end_inset + + is the measured microphone pressure, and +\begin_inset Formula $p_{0}$ +\end_inset + + the actual pressure at the measurement position. +\end_layout + +\begin_layout Itemize +\begin_inset Formula $K_{1}$ +\end_inset + +: Microphone calibration correction factor for mic 1. + Such that +\begin_inset Formula $p_{1}=K_{1}\tilde{p}_{1}$ +\end_inset + +, where +\begin_inset Formula $\hat{p}_{1}$ +\end_inset + + is the measured microphone pressure, and +\begin_inset Formula $p_{0}$ +\end_inset + + the actual pressure at the measurement position. +\end_layout + +\begin_layout Standard +We are only able to measure and estimate cross-spectrum (or equivalently + the cross-spectral density): +\begin_inset Formula +\begin{equation} +\tilde{C}_{ij}=\tilde{p}_{i}\tilde{p}_{j}^{*}, +\end{equation} + +\end_inset + +from which the transfer functions can be estimated. + We require to correct these transfer functions for the relative microphone + calibration. + The final quantity of interest is often the acoustic pressure transfer + function for the two mic's in the impedance tube: +\begin_inset Formula +\begin{equation} +G_{AB}=\frac{p_{B}}{p_{A}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +he microphone switching method is used to estimate the calibration constant + that should be used to estimate the transfer function from the measured + CPS's. + In measurement 1: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +G_{AB}^{(1)}=\frac{p_{1}^{(1)}}{p_{0}^{(1)}}\approx\frac{K_{1}}{K_{0}}\frac{\tilde{C}_{10}^{(1)}}{\tilde{C}_{00}^{(1)}}, +\end{equation} + +\end_inset + +From the second measurement in switched configuration, the estimation of + +\begin_inset Formula $G_{AB}$ +\end_inset + + yields: +\begin_inset Formula +\begin{equation} +G_{AB}^{(2)}=\frac{p_{0}^{(2)}}{p_{1}^{(2)}}\approx\frac{K_{0}}{K_{1}}\frac{\tilde{C}_{01}^{(2)}}{\tilde{C}_{11}^{(2)}}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Equating both expressions yields an exact expression for the calibration + correction factor: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $G_{AB}^{(1)}=G_{AB}^{(2)}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(\frac{K_{1}}{K_{0}}\right)^{2}=\frac{\tilde{C}_{00}^{(1)}}{\tilde{C}_{10}^{(1)}}\frac{\tilde{C}_{01}^{(2)}}{\tilde{C}_{11}^{(2)}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{K_{1}}{K_{0}}=\sqrt{\frac{\tilde{C}_{01}^{(2)}}{\tilde{C}_{11}^{(2)}}\frac{\tilde{C}_{00}^{(1)}}{\tilde{C}_{10}^{(1)}}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +K\equiv\frac{K_{1}}{K_{0}}=\sqrt{\frac{\tilde{C}_{01}^{(2)}}{\tilde{C}_{11}^{(2)}}\frac{\tilde{C}_{00}^{(1)}}{\tilde{C}_{10}^{(1)}}}. +\end{equation} + +\end_inset + +Then, for all measurements beyond the calibration, we can write for +\begin_inset Formula $G_{AB}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +G_{AB}=\frac{p_{B}}{p_{A}}=KG_{01}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $G_{01}$ +\end_inset + + is the measured transfer function from mic 0 to mic 1 ( +\begin_inset Formula $G_{01}=p_{1}/p_{0}$ +\end_inset + +). +\end_layout + +\begin_layout Subsection +Method for computing the sample impedance, absorption and reflection coefficient +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\noindent +\align center +\begin_inset Graphics + filename img/imptube_meas_setups.pdf + width 100text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Configurations for acoustic sample testing +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:sampletesting_configs" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Fig. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:sampletesting_configs" + +\end_inset + + shows two configurations that can be used for acoustic sample testing. + The top configuration is used for relatively open samples. +\end_layout + +\begin_layout Subsubsection +Open samples - configuration 1 +\end_layout + +\begin_layout Standard +For both microphones on the same side of the sample, the reflection coefficient + at the position of microphone +\begin_inset Formula $A$ +\end_inset + + can be evaluated as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{A}\equiv R(x=0)=\frac{G_{AB}-e^{-iks}}{e^{iks}-G_{AB}}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $k$ +\end_inset + + is the wave number, and +\begin_inset Formula $s$ +\end_inset + + is the microphone spacing. + The reflection coefficient rotates in phase going to the position of the + sample: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R=R_{A}\exp\left(2ik\left(s+d_{1}\right)\right). +\end{equation} + +\end_inset + +The absorption coefficient +\begin_inset Formula $\alpha$ +\end_inset + + is: +\begin_inset Formula +\begin{equation} +\alpha=1-\left|R\right|^{2} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The impedance at the sample location ( +\begin_inset Formula $x=s+d_{1}$ +\end_inset + +) can be computed as: +\begin_inset Formula +\begin{equation} +z=z_{0}\frac{1+R}{1-R} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For an arbitrary back side impedance, this impedance is due to the sample, + and the back side impedance. + +\end_layout + +\begin_layout Standard +For a thin sample (w.r.t. + the wavelength) with back cavity, the sample impedance can be computed, + by knowing the distance behind the sample ( +\begin_inset Formula $d_{2}$ +\end_inset + +). + We know for a fixed back cavity that, we set: +\begin_inset Formula +\begin{equation} +\Delta p_{s}=Z_{s}U, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The impedance on the back side of the sample, for a closed cavity can be + computed by assuming 100% reflection at the end of the back cavity: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $p=\cos\left(k\left(L-x\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $i\omega\rho_{0}U=-\frac{\partial p}{\partial x}\Rightarrow U=\frac{ik}{\omega\rho_{0}}\sin\left(k\left(L-x\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +Z|_{x=0}=\frac{p}{U}|_{x=0}=\frac{\cos\left(kL\right)}{\frac{ik}{\omega\rho_{0}}\sin\left(kL\right)}=-iz_{0}\cot\left(kL\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $R(x)=R|_{x=0}\exp\left(2ikx\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $R(-L)=R_{0}\exp\left(-2ikL\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $R_{0}=1$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $z=z_{0}\frac{1+R}{1-R}=z_{0}\frac{1+\exp\left(-2ikL\right)}{1-\exp\left(-2ikL\right)}=z_{0}\frac{\exp\left(ikL\right)+\exp\left(-ikL\right)}{\exp\left(ikL\right)-\exp\left(-ikL\right)}=-z_{0}i\frac{\cos\left(kL\right)}{\sin\left(kL\right)}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +z_{c}=-iz_{0}\cot\left(kd_{2}\right), +\end{equation} + +\end_inset + +Using that, we are able to compute the sample impedance (jump impedance) + as: +\begin_inset Formula +\begin{equation} +z_{s}=2z_{0}\frac{1-Re^{2ikd_{2}}}{\left(R-1\right)\left(e^{2ikd_{2}}-1\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Error analysis +\end_layout + +\begin_layout Standard +Types of error: +\end_layout + +\begin_layout Itemize +Positioning errors, both sample and microphones +\end_layout + +\begin_deeper +\begin_layout Itemize +Bias and random +\end_layout + +\end_deeper +\begin_layout Itemize +Model errors (speed of sound, transport parameters) +\end_layout + +\begin_layout Itemize +Sensor errors +\end_layout + +\begin_layout Itemize +Nonlinearities +\end_layout + +\begin_layout Itemize +Noise +\end_layout + +\begin_layout Section +Filter bank design +\end_layout + +\begin_layout Subsection +Overlap-save method +\end_layout + +\begin_layout Standard +Limitations: +\end_layout + +\begin_layout Itemize +Time sample block size +\begin_inset Formula $L$ +\end_inset + + should be larger than the filter order. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/overlap_save.pdf + width 90text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The yellow blocks are saved for the next +\end_layout + +\begin_layout Section +Digital filters +\end_layout + +\begin_layout Subsection +\begin_inset Formula $z$ +\end_inset + +-transform +\end_layout + +\begin_layout Standard +\begin_inset Formula $z$ +\end_inset + +-transform of a sequence +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +Z\left(h[n]\right)=\sum_{n=-\infty}^{\infty}h[n]z^{-n} +\end{equation} + +\end_inset + + +\begin_inset Formula $z$ +\end_inset + +-transform of the discrete impulse +\begin_inset Formula +\begin{equation} +Z\left(\delta[n]\right)=z^{0}=1 +\end{equation} + +\end_inset + + +\begin_inset Formula $z$ +\end_inset + +-transform of the unit step function +\begin_inset Formula +\begin{equation} +Z\left(u[z]\right)=\frac{z}{z-1} +\end{equation} + +\end_inset + + +\begin_inset Formula $z$ +\end_inset + +-transform of the exponentially decaying function +\begin_inset Formula +\begin{equation} +f(t)=e^{-at}, +\end{equation} + +\end_inset + +is discrete +\begin_inset Formula +\begin{equation} +f[n]=e^{-anT}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $T=f_{s}^{-1}$ +\end_inset + +. + +\begin_inset Formula +\begin{equation} +Z\left(f[n]\right)=\frac{z}{z-e^{-aT}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +DSP Equation +\end_layout + +\begin_layout Standard +Sample time is +\begin_inset Formula $nT$ +\end_inset + +, where +\begin_inset Formula $T$ +\end_inset + + is the sampling period (inverse sampling frequency). +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +y[n]=\sum_{m=1}^{M}b_{m}y[n-m]+\sum_{p=0}^{P}a_{p}x[n-p] +\end{equation} + +\end_inset + +In the +\begin_inset Formula $z$ +\end_inset + +-domain, this equation can be written as +\begin_inset Formula +\begin{equation} +Y[z]=\frac{\sum\limits _{p=0}^{P}a_{p}z^{-p}}{1-\sum\limits _{m=1}^{M}b_{m}z^{-m}}X[z]=H[z]\cdot X[z] +\end{equation} + +\end_inset + +Analog input signal frequency: +\begin_inset Formula $\omega$ +\end_inset + +. + Scaled frequency: +\begin_inset Formula $\Omega=\omega T$ +\end_inset + +. + Frequency response of the scaled frequency goes from 0 to +\begin_inset Formula $\pi$ +\end_inset + +. + Each DSP system has a frequency response that repeats at the sampling frequency + ( +\begin_inset Formula $\Omega=2\pi$ +\end_inset + +). + The frequency response can be found from the +\begin_inset Formula $z$ +\end_inset + +-transform by filling in +\begin_inset Formula $z=e^{sT}$ +\end_inset + +, where +\begin_inset Formula $s$ +\end_inset + + is the Laplace variable. + And filling in for +\begin_inset Formula $s=i\omega$ +\end_inset + +, to find: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +H\left(\omega\right)=H[z=e^{i\omega T}], +\end{equation} + +\end_inset + +For the DSP equation: +\begin_inset Formula +\begin{equation} +H(\omega)=\frac{\sum\limits _{p=0}^{P}a_{p}e^{-ip\Omega}}{1-\sum\limits _{m=1}^{M}b_{m}e^{-im\Omega}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +FIR Filter design +\end_layout + +\begin_layout Standard +Transfer function of a FIR filter: +\begin_inset Formula +\begin{equation} +T(e^{i\Omega})=\sum_{k=-N}^{N}a_{k}z^{-ik\Omega}\label{eq:fir_freq_response} +\end{equation} + +\end_inset + +The left side of Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:fir_freq_response" + +\end_inset + + is the desired frequency response, The right side are the corresponding + filter coefficients. + Multiplying the LHS and RHS with +\begin_inset Formula $e^{in\Omega}$ +\end_inset + + and integrating from 0 to +\begin_inset Formula $2\pi$ +\end_inset + +, we find the following equation for the filter coefficient +\begin_inset Formula $a_{n}$ +\end_inset + +: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $T(e^{i\Omega})=\sum_{k=-N}^{N}a_{k}z^{-i\Omega}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{0}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega=\int_{0}^{2\pi}\sum_{k=-N}^{N}a_{k}z^{-ik\Omega}e^{in\Omega}\mathrm{d}\Omega$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{0}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega=\int_{0}^{2\pi}\sum_{k=-N}^{N}a_{k}z^{-ik\Omega}e^{in\Omega}\mathrm{d}\Omega$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{0}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega=\int_{0}^{2\pi}\sum_{k=-N}^{N}a_{k}z^{i\Omega\left(n-k\right)}\mathrm{d}\Omega$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{0}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega=\int_{0}^{2\pi}a_{n}\mathrm{d}\Omega$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\int_{0}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega=2\pi a_{n}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +a_{n}=\frac{1}{2\pi}\int\limits _{0}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega +\end{equation} + +\end_inset + +This relation gives the coefficients of the +\emph on +noncausal +\emph default + form of the filter coefficients. + As the coefficients are symmetrical, only for positive +\begin_inset Formula $n$ +\end_inset + +, the +\begin_inset Formula $a_{n}$ +\end_inset + +'s need to be computed. + As the frequency spectrum is repeated and symmetrical around the Nyquist + frequency, we can write this as: +\begin_inset Formula +\begin{equation} +a_{n}=\frac{1}{2\pi}\left[\int\limits _{0}^{\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega+\int\limits _{\pi}^{2\pi}T(e^{i\Omega})e^{in\Omega}\mathrm{d}\Omega\right]. +\end{equation} + +\end_inset + +If we specify a certain frequency response below the Nyquist frequency, + and let the part above the Nyquist frequency be its mirror image, this + can be written as +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $T\left(e^{i\left(\Omega+\pi\right)}\right)=T^{*}\left(e^{i\Omega}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using +\begin_inset Formula $k=\Omega-2\pi\Rightarrow\Omega=k+2\pi$ +\end_inset + +, we can write the second integral as +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $a_{n}=\frac{1}{2\pi}\left[\int\limits _{0}^{\pi}Te^{in\Omega}\mathrm{d}\Omega+\int\limits _{k=-\pi}^{0}T(e^{i\left(k+2\pi\right)})e^{in\left(k+2\pi\right)}\mathrm{d}k\right].$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $a_{n}=\frac{1}{2\pi}\left[\int\limits _{0}^{\pi}Te^{in\Omega}\mathrm{d}\Omega+\int\limits _{\Omega=-\pi}^{0}T(e^{i\left(\Omega+2\pi\right)})e^{in\left(\Omega+2\pi\right)}\mathrm{d}\Omega\right].$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using the fact that +\begin_inset Formula $e^{2in\pi}=1$ +\end_inset + + for all integer +\begin_inset Formula $n$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $a_{n}=\frac{1}{2\pi}\left[\int\limits _{-\pi}^{\pi}Te^{in\Omega}\mathrm{d}\Omega\right].$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +But we know that +\begin_inset Formula $T\left(e^{-i\Omega}\right)=T^{*}\left(e^{i\Omega}\right)$ +\end_inset + +, such that: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $a_{n}=\frac{1}{2\pi}\left[\int\limits _{0}^{\pi}T\left(e^{i\Omega}\right)e^{in\Omega}+T^{*}\left(e^{i\Omega}\right)e^{-in\Omega}\mathrm{d}\Omega\right]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +a_{n}=\frac{1}{2\pi}\left[\int\limits _{0}^{\pi}T\left(e^{i\Omega}\right)e^{in\Omega}+T^{*}\left(e^{i\Omega}\right)e^{-in\Omega}\mathrm{d}\Omega\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Ideal pass-band filter +\end_layout + +\begin_layout Standard +For an ideal band filter, this results in +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $a_{n}=\frac{1}{2\pi}\int\limits _{\Omega_{l}}^{\Omega_{u}}e^{in\Omega}+e^{-in\Omega}\mathrm{d}\Omega=$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $a_{n}=\frac{1}{\pi}\int\limits _{\Omega_{l}}^{\Omega_{u}}\cos\left(n\Omega\right)\mathrm{d}\Omega=\frac{\sin\left(n\Omega\right)}{n\pi}|_{\Omega_{l}}^{\Omega_{u}}=\frac{\sin\left(n\Omega_{u}\right)-\sin\left(n\Omega_{l}\right)}{n\pi}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +a_{n}=\frac{1}{2\pi}\int\limits _{\Omega_{l}}^{\Omega_{u}}e^{in\Omega}+e^{-in\Omega}\mathrm{d}\Omega=\frac{\sin\left(n\Omega_{u}\right)-\sin\left(n\Omega_{l}\right)}{n\pi} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Ideal low-pass filter +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +a_{n}=\frac{\sin\left(n\omega_{p}T\right)}{n\pi}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\omega_{p}$ +\end_inset + + is the pass-band to stop-band transition frequency. +\end_layout + +\begin_layout Subsubsection +Gibbs phenomenon +\end_layout + +\begin_layout Standard +The abrupt change in filter gain from pass band to stop band results is + pass-band ripple and . + Leftover ripple at an abrupt transition in filter coefficients is about + 9% of the gain amplitude. + Windowing the filter coefficients: +\end_layout + +\begin_layout Itemize +Magnitude of the ripple decreases +\end_layout + +\begin_layout Itemize +Width of the transition band increases +\end_layout + +\begin_layout Standard +If the unwindowed filter coefficients are +\begin_inset Formula $f[n]$ +\end_inset + +, the windowed filter coefficients are +\begin_inset Formula +\begin{equation} +a_{n}=w[n]f[n] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Hamming window is a bit better than Hann window. + Hann window is an improvement over the Bartlett window. + In terms of the transition band width. + However, stop-band gain increases. +\end_layout + +\begin_layout Section +Band-pass digital filter design +\end_layout + +\begin_layout Standard +Low pass filter +\end_layout + +\begin_layout Itemize +\begin_inset Formula $g_{s,\mathrm{max}}$ +\end_inset + +: maximum allowed gain in the stop-band +\end_layout + +\begin_layout Itemize +\begin_inset Formula $g_{p,\mathrm{max}}$ +\end_inset + +: maximum allowed gain in the pass-band +\end_layout + +\begin_layout Itemize +\begin_inset Formula $g_{p,\mathrm{min}}$ +\end_inset + +: minimum allowed gain in the pass-band +\end_layout + +\begin_layout Subsection +Stereo band pass filter display +\end_layout + +\begin_layout Itemize +Digital filter should be designed in frequency domain +\end_layout + +\begin_layout Itemize +The RMS of the output of each filter is computed over a period on once the + center frequency of the band. +\end_layout + +\begin_layout Standard +Example: 1000 Hz octave band: pass-band is from 750 Hz to 1500 Hz. + Use a Hamming window to reduce the pass and stop-band ripple. +\end_layout + +\begin_layout Section +Sound level meter implementation +\end_layout + +\begin_layout Subsection +Time-weighted sound level +\end_layout + +\begin_layout Standard +Fast time-weighted sound level,of the A-weighted pressure signal +\begin_inset Formula $p_{A}(t)$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +L_{AF}=10\log_{10}\left(\frac{1}{\tau_{F}}\int\limits _{-\infty}^{t}p_{A}^{2}\left(\xi\right)e^{-\left(t-\xi\right)/\tau_{F}}\right)-10\log_{10}\left(p_{\mathrm{ref}}^{2}\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula $\tau_{F}$ +\end_inset + + is the exponential time constant in seconds for the fast time weighting. + Implementation suggestion: square the frequency-weighted input signal, + and apply a single pole low-pass filter with one pole at +\begin_inset Formula $-\tau_{F}^{-1}$ +\end_inset + +. + +\end_layout + +\begin_layout Itemize +Fast time weighting: +\begin_inset Formula $\tau_{F}=0,125$ +\end_inset + + s +\end_layout + +\begin_layout Itemize +Slow time weighting: +\begin_inset Formula $\tau_{s}=1$ +\end_inset + + s. +\end_layout + +\begin_layout Itemize +Impulse time weighting: +\begin_inset Formula $\tau_{i}=35$ +\end_inset + + +\begin_inset space ~ +\end_inset + +ms +\end_layout + +\begin_layout Subsection +Implementation of single pole low pass filter +\end_layout + +\begin_layout Standard +A single pole low pass filter has a frequency response of +\begin_inset Formula +\begin{equation} +G_{\mathrm{splp}}=\frac{1}{1+\tau s}, +\end{equation} + +\end_inset + +we create a digital filter from this one using the bilinear transform: +\begin_inset Formula +\begin{equation} +s\to2f_{s}\frac{z-1}{z+1}, +\end{equation} + +\end_inset + +which yields the digital filter: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $G_{\mathrm{splp},d}=\frac{1}{1+\tau2f_{s}\frac{z-1}{z+1}}=\frac{z+1}{z+1+\tau2f_{s}\left(z-1\right)}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $G_{\mathrm{splp},d}=\frac{z+1}{\left(1+\tau2f_{s}\right)z+1-\tau2f_{s}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Divide by +\begin_inset Formula $z$ +\end_inset + +, both numerator and denominator: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $G_{\mathrm{splp},d}=\frac{1+z^{-1}}{\left(1+\tau2f_{s}\right)+\left(1-\tau2f_{s}\right)z^{-1}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Normalizing such that +\begin_inset Formula $a_{0}=1$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $G_{\mathrm{splp},d}=\frac{\left(1+2\tau f_{s}\right)^{-1}\left(1+z^{-1}\right)}{1+\frac{\left(1-\tau2f_{s}\right)}{\left(1+2\tau f_{s}\right)}z^{-1}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +G_{\mathrm{splp},d}=\frac{1+z^{-1}}{\left(\frac{1}{\tau}+2f_{s}\right)+\left(\frac{1}{\tau}-2f_{s}\right)z^{-1}}, +\end{equation} + +\end_inset + +So its digital filter coefficients are: +\begin_inset Formula +\begin{align} +\boldsymbol{b} & =\left[\begin{array}{ccc} +\left(1+2\tau f_{s}\right)^{-1} & \left(1+2\tau f_{s}\right)^{-1} & 0\end{array}\right]^{\mathrm{T}}\\ +\boldsymbol{a} & =\left[\begin{array}{ccc} +1 & \frac{\left(1-\tau2f_{s}\right)}{\left(1+2\tau f_{s}\right)} & 0\end{array}\right]^{\mathrm{T}} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +No correction for frequency warping has been done, as for all cases +\begin_inset Formula $\tau f_{s}\gg1$ +\end_inset + +. + The output frequency of the sound level meter will be decimated to a sampling + frequency, where the single pole low pass filter has a -20 dB point: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1-0.01}{0.01}=\left|\tau s\right|\Rightarrow\left|s\right|=\frac{1}{\tau}\frac{1-0.01}{0.01}\Rightarrow f_{s,\mathrm{slm}}=$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\left|G_{\mathrm{splp}}\right|=\left|\frac{1}{1+\tau s}\right|=-20\,\mathrm{dB}=0.01\Rightarrow s=\frac{1}{\tau}\frac{1-0.01}{0.01} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Hence the minus 20 +\begin_inset space ~ +\end_inset + +dB point lies at a sampling frequency of: +\begin_inset Formula +\begin{equation} +f_{s,\mathrm{slm}}=\frac{1}{2\pi\tau}\frac{1-0.01}{0.01} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Then, the downsampling factor is set at: +\begin_inset Formula +\begin{equation} +d=\left\lfloor \frac{f_{s}}{f_{s,\mathrm{slm}}}\right\rfloor , +\end{equation} + +\end_inset + +where +\begin_inset Formula $\left\lfloor \dots\right\rfloor $ +\end_inset + + denotes the floor operation. +\end_layout + +\begin_layout Standard +Each subsample corresponds to +\begin_inset Formula +\begin{equation} +n=o+id +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Find the highest +\begin_inset Formula $i$ +\end_inset + +, that does not fit into +\begin_inset Formula $N$ +\end_inset + + anymore: +\begin_inset Formula +\[ +id>N-o +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Time-averaged sound level +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +L_{A\mathrm{eq},T}=10\log_{10}\left(\frac{1}{p_{\mathrm{ref}}^{2}T}\int\limits _{t-T}^{t}p_{A}^{2}(\xi)\mathrm{d}\xi\right), +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +C-weighting filter +\end_layout + +\begin_layout Standard +Linear amplitude scaling for the C-weighted frequency response: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +C(f)=C_{1000}^{-1}\left(\frac{f_{4}^{2}f^{2}}{\left(f^{2}+f_{1}^{2}\right)\left(f^{2}+f_{4}^{2}\right)}\right)^{2},\label{eq:C_weighting_freqresponse} +\end{equation} + +\end_inset + +where +\begin_inset Formula $C_{1000}$ +\end_inset + + is the numerator of Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:C_weighting_freqresponse" + +\end_inset + + evaluated at 1 kHz. + In this Eq. + +\begin_inset Formula +\begin{equation} +f_{1}=\sqrt{\left(\frac{-b-\sqrt{b^{2}-4c}}{2}\right)}\label{eq:f_1} +\end{equation} + +\end_inset + +and +\begin_inset Formula +\begin{equation} +f_{4}=\sqrt{\left(\frac{-b+\sqrt{b^{2}-4c}}{2}\right)}\label{eq:f_4} +\end{equation} + +\end_inset + +, where +\begin_inset Formula $c=f_{L}^{2}f_{H}^{2}$ +\end_inset + + and +\begin_inset Formula +\begin{equation} +b=\frac{1}{1-D}\left[f_{r}^{2}+\frac{f_{L}^{2}f_{H}^{2}}{f_{r}^{2}}-D\left(f_{L}^{2}+f_{H}^{2}\right)\right] +\end{equation} + +\end_inset + +, with +\begin_inset Formula $D=+\frac{1}{2}\sqrt{2}$ +\end_inset + +. + +\begin_inset Formula $f_{r}=1$ +\end_inset + + kHz, +\begin_inset Formula $f_{L}=10^{1.5}$ +\end_inset + + Hz and +\begin_inset Formula $f_{H}=10^{3.9}$ +\end_inset + + Hz. +\end_layout + +\begin_layout Subsection +A-weighting filter +\end_layout + +\begin_layout Standard +Linear amplitude scaling for the A-weighted frequency response: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +A(f)=A_{1000}^{-1}\frac{f_{4}^{2}f^{4}}{\left(f^{2}+f_{1}^{2}\right)\sqrt{\left(f^{2}+f_{2}^{2}\right)\left(f^{2}+f_{3}^{2}\right)}\left(f^{2}+f_{4}^{2}\right)},\label{eq:A_freq_norm} +\end{equation} + +\end_inset + +in which +\begin_inset Formula $f_{1}$ +\end_inset + + and +\begin_inset Formula $f_{4}$ +\end_inset + + are defined in Eqs. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:f_1" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:f_4" + +\end_inset + +, respectively. + And +\begin_inset Formula +\begin{equation} +f_{2}=\frac{3-\sqrt{5}}{2}f_{A}, +\end{equation} + +\end_inset + +and +\begin_inset Formula +\begin{equation} +f_{3}=\frac{3+\sqrt{5}}{2}f_{A}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $f_{A}=10^{2.45}$ +\end_inset + + Hz. + The transfer function of +\begin_inset Formula $A$ +\end_inset + + can be written as Equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:A_freq_norm" + +\end_inset + + can be rewritten to the following equivalent form +\begin_inset CommandInset citation +LatexCommand cite +key "rimell_design_2015" +literal "false" + +\end_inset + +: +\begin_inset Formula +\begin{equation} +A(s)=K_{A}\frac{\omega_{4}^{2}s^{4}}{\left(s^{2}+\omega_{1}^{2}\right)\left(s+\omega_{2}\right)\left(s+\omega_{3}\right)\left(s^{2}+\omega_{4}^{2}\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Exact midband frequencies +\end_layout + +\begin_layout Standard +The midband frequencies of a (fractional) octave band filter are defined + as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +f_{x}=f_{r}\left[10^{\left(\frac{3}{10}\right)\left(\frac{x}{b}\right)}\right], +\end{equation} + +\end_inset + +where +\begin_inset Formula $f_{r}$ +\end_inset + + is the reference frequency of 1000 Hz, +\begin_inset Formula $10^{\left(3/10\right)}$ +\end_inset + + is the nominal octave ratio for a base-10 system. + +\begin_inset Formula $b$ +\end_inset + + is the step-width designator, +\begin_inset Formula $b=3$ +\end_inset + + for one-third-octave intervals. +\end_layout + +\begin_layout Standard +Octave ratio: Nominal frequency ratio of 2:1. + Base ten system is preferred, where +\begin_inset Formula +\begin{equation} +G_{10}=10^{3/10}\approx1.995 +\end{equation} + +\end_inset + +, in base 2: +\begin_inset Formula +\begin{equation} +G_{2}=2, +\end{equation} + +\end_inset + + Bandwidth designator ( +\begin_inset Formula $b$ +\end_inset + +). + Exact midband frequencies: +\begin_inset Formula +\begin{align} +f_{m} & =\left(G^{x/b}\right)f_{r},;\,b\,\mathrm{even}\\ +f_{m} & =\left(G^{\left(2x+1\right)/(2b)}\right)f_{r},;\,b\,\mathrm{odd} +\end{align} + +\end_inset + +where +\begin_inset Formula $f_{r}$ +\end_inset + + is the reference frequency (1 kHz). + Bandedge frequencies: frequencies of the lower and upper edges of the passband + of a bandpass filter suchthat the exact midband frequency is the geometric + mean of the lower and upper bandedge frequencies. +\begin_inset Formula +\begin{align} +f_{\ell} & =\left(G^{-1/\left(2b\right)}\right)f_{m}\\ +f_{u} & =\left(G^{+1/\left(2b\right)}\right)f_{m} +\end{align} + +\end_inset + +where +\begin_inset Formula $f_{m}$ +\end_inset + + is the exact midband frequency. +\end_layout + +\begin_layout Subsection +Nominal midband frequencies +\end_layout + +\begin_layout Standard +- See standard +\end_layout + +\begin_layout Section +Reverberation time +\end_layout + +\begin_layout Standard +Reverberation time is the time that is required to let the instantaneous + sound pressure level drop with 60 db. + Model: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +P^{2}(t)=P_{0}^{2}10^{-\alpha t} +\end{equation} + +\end_inset + +Then at +\begin_inset Formula $T_{60}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\frac{P^{2}}{P_{0}^{2}}=10^{\frac{-60}{20}}=10^{-3} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Such that at +\begin_inset Formula $T_{60}$ +\end_inset + +, +\begin_inset Formula $\alpha T_{60}=3$ +\end_inset + +, hence +\begin_inset Formula +\begin{equation} +T_{60}=\frac{3}{\alpha}. +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +P^{2}(t)=P_{0}^{2}10^{-\frac{3}{T_{60}}t} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Suppose we have the data +\begin_inset Formula $\boldsymbol{P}^{2}$ +\end_inset + + for given time instances +\begin_inset Formula $\boldsymbol{t}$ +\end_inset + +. + Then we determine +\begin_inset Formula $\alpha$ +\end_inset + + and +\begin_inset Formula $P_{0}^{2}$ +\end_inset + + by minimizing: +\begin_inset Formula +\begin{equation} +\left\Vert \boldsymbol{P}^{2}-P_{0}^{2}10^{-\alpha\boldsymbol{t}}\right\Vert ^{2}, +\end{equation} + +\end_inset + +which is a nonlinear least squares problem +\end_layout + +\begin_layout Standard +Look at Schroeder-back integration! +\end_layout + +\begin_layout Section +Sweep signals +\end_layout + +\begin_layout Standard +A sweep signal is a signal which instantaneous frequency changes over time + with a specific profile, the signal follows a sine function with a specific + phase: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +c(t)=\sin\left(\phi(t)\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula $\phi(t)$ +\end_inset + + is the phase. + For a sine wave with frequency +\begin_inset Formula $f$ +\end_inset + +, +\begin_inset Formula $\phi(t)=2\pi ft$ +\end_inset + +. + The instantaneous frequency +\begin_inset Formula $f(t)$ +\end_inset + + is defined as +\begin_inset Formula +\begin{equation} +f(t)=\frac{\mathrm{d}\phi}{\mathrm{d}t} +\end{equation} + +\end_inset + + A linear sweep exhibits a profile where +\begin_inset Formula $\frac{\mathrm{d}^{2}\phi}{\mathrm{d}t^{2}}=\mathrm{constant}$ +\end_inset + +. + For all the sweep implementations, we specify: +\end_layout + +\begin_layout Itemize +The lower frequency +\begin_inset Formula $f_{L}$ +\end_inset + +, +\end_layout + +\begin_layout Itemize +The upper frequency +\begin_inset Formula $f_{U}$ +\end_inset + +, +\end_layout + +\begin_layout Itemize +The sweep time of a single period +\begin_inset Formula $T_{c}$ +\end_inset + +, +\end_layout + +\begin_layout Itemize +The sweep profile, i.e. + forward, backward or continuous +\end_layout + +\begin_layout Itemize +The sweep type (linear, exponential, hyperbolic) +\end_layout + +\begin_layout Standard +For the numerical implementation, the phase updating is done based on a + forward Euler estimate of the derivative of the phase: +\begin_inset Formula +\begin{equation} +\phi_{n+1}=\phi_{n}+2\pi f_{n}\Delta t,\qquad n=0...N-1 +\end{equation} + +\end_inset + +where +\begin_inset Formula $N=\left\lfloor T_{c}f_{s}\right\rfloor $ +\end_inset + +, and +\begin_inset Formula $\left\lfloor \dots\right\rfloor $ +\end_inset + + denotes rounding down to the nearest integer. + The input parameters are slightly adjusted, such that the sweep becomes + periodic in the discrete time domain, with with +\begin_inset Formula $C_{0}$ +\end_inset + +-continuity, and a sign of the derivative pointing in the same direction. + Note that for this type of +\begin_inset Formula $C_{0}$ +\end_inset + +-continuity, we require that +\begin_inset Formula $\phi_{N}=2\pi K$ +\end_inset + +, where +\begin_inset Formula $K\in\mathbb{N}_{0}$ +\end_inset + + (all natural numbers including 0). + The boundary conditions are thus: +\begin_inset Formula +\begin{align} +\phi_{0} & =0\\ +\phi_{N} & =0\\ +\frac{\mathrm{d}\phi_{0}}{\mathrm{d}n} & =\frac{\mathrm{d}\phi_{N}}{\mathrm{d}n} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Forward linear sweep +\end_layout + +\begin_layout Standard +For a forward linear sweep with period +\begin_inset Formula $T_{c}$ +\end_inset + +, the discrete sweep phase in a single period is defined as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +f_{n}=f_{L}+\frac{n}{N}\left(f_{U}+\varepsilon-f_{L}\right).\label{eq:forward_linear} +\end{equation} + +\end_inset + +This results in the following equation for the phase: +\begin_inset Formula +\begin{align} +\phi_{n+1}=\phi_{n}+2\pi f_{L}\Delta t+2\pi\frac{n}{N}\Delta t\left(f_{U}+\varepsilon-f_{L}\right) +\end{align} + +\end_inset + +Solving this recurrence relation yields: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\phi_{n}=2\pi\Delta t\left[f_{L}n+\frac{1}{2N}\left(n^{2}-n\right)\left(f_{U}+\varepsilon-f_{L}\right)\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +We first make a proper estimate of +\begin_inset Formula $K$ +\end_inset + +, by setting +\begin_inset Formula $\phi_{N}=2\pi K$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +K=\left\lfloor \Delta t\left[f_{L}N+\frac{1}{2}\left(N-1\right)\left(f_{U}-f_{L}\right)\right]\right\rfloor , +\end{equation} + +\end_inset + +which is used to set the correction +\begin_inset Formula $\varepsilon$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\varepsilon=\frac{\frac{K}{\Delta t}-f_{L}N-\frac{1}{2}\left(N-1\right)\left(f_{U}-f_{L}\right)}{\frac{1}{2}\left(N-1\right)} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Backward linear sweep +\end_layout + +\begin_layout Standard +The procedure for creating a backward linear sweep is similar to a forward + linear sweep, only we replace +\begin_inset Formula $f_{U}$ +\end_inset + + with +\begin_inset Formula $f_{L}$ +\end_inset + + and vice versa. +\end_layout + +\begin_layout Subsection +Continuous linear sweep +\end_layout + +\begin_layout Standard +We define +\begin_inset Formula $N_{f}$ +\end_inset + + as +\begin_inset Formula $\left\lfloor N/2\right\rfloor $ +\end_inset + +, and +\begin_inset Formula $N_{b}=N-N_{f}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\phi_{n+1}=\begin{cases} +\phi_{n}+2\pi\Delta t\left(f_{L}+\frac{n}{N_{f}}\left(f_{U}-f_{L}\right)\right) & 0\leq n\leq N_{f}\\ 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{https://www.jstage.jst.go.jp/article/indhealth/53/1/53_2013-0003/_article}, + doi = {10.2486/indhealth.2013-0003}, + abstract = {Many workers are exposed to noise in their industrial environment. Excessive noise exposure can cause health problems and therefore it is important that the worker’s noise exposure is assessed. This may require measurement by an equipment manufacturer or the employer. Human exposure to noise may be measured using microphones; however, weighting filters are required to correlate the physical noise sound pressure level measurements to the human’s response to an auditory stimulus. {IEC} 61672-1 and {ANSI} S1.43 describe suitable weighting filters, but do not explain how to implement them for digitally recorded sound pressure level data. By using the bilinear transform, it is possible to transform the analogue equations given in the standards into digital filters. This paper describes the implementation of the weighting filters as digital {IIR} (Infinite Impulse Response) filters and provides all the necessary formulae to directly calculate the filter coefficients for any sampling frequency. Thus, the filters in the standards can be implemented in any numerical processing software (such as a spreadsheet or programming language running on a {PC}, mobile device or embedded system).}, + pages = {21--27}, + number = {1}, + journaltitle = {{INDUSTRIAL} {HEALTH}}, + shortjournal = {{INDUSTRIAL} {HEALTH}}, + author = {Rimell, Andrew N. and Mansfield, Neil J. and Paddan, Gurmail S.}, + urldate = {2020-01-18}, + date = {2015}, + langid = {english}, + file = {Rimell et al. - 2015 - Design of digital filters for frequency weightings.pdf:/home/anne/.literature/storage/XKC3AE4F/Rimell et al. - 2015 - Design of digital filters for frequency weightings.pdf:application/pdf} +} \ No newline at end of file diff --git a/tex/preamble.tex b/tex/preamble.tex new file mode 100644 index 0000000..d1ffa91 --- /dev/null +++ b/tex/preamble.tex @@ -0,0 +1,264 @@ + +% The format should be: +% height: 240mm +% width: 170 mm +% \showtrimsoff + + + +% This code is for stock size a4 and small crop signs +% a4 stock size +% \stockaiv +% a4 is 297 hoog bij 210 breed. 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