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03-03-02-ACA-Input-TF-CQT.tex
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03-03-02-ACA-Input-TF-CQT.tex
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% move all configuration stuff into includes file so we can focus on the content
\input{include}
\subtitle{module 3.3.2: time-frequency representations~---~constant Q transform}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
% generate title page
\input{include/titlepage}
\section[overview]{lecture overview}
\begin{frame}{introduction}{overview}
\begin{block}{corresponding textbook section}
%\href{http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6331119&}{Chapter 2~---~Fundamentals}: pp.~24--26
section~3.3.2
\end{block}
\begin{itemize}
\item \textbf{lecture content}
\begin{itemize}
\item constant-Q transform (CQT)
\end{itemize}
\bigskip
\item<2-> \textbf{learning objectives}
\begin{itemize}
\item discussing advantages and disadvantages of different time-frequency transforms
\item explaining the principles of the CQT and auditory filterbanks
\end{itemize}
\end{itemize}
\inserticon{directions}
\end{frame}
\section[intro]{introduction}
\begin{frame}{non-FT time frequency transforms}{introduction}
\begin{itemize}
\item Fourier transform continues to be much-used tool in audio signal processing and MIR
\bigskip
\item but there are disadvantages, e.g.
\begin{itemize}
\item frequency axis does not directly map to (perceptual) pitch axis
\item frequency and time resolution inversely related
\smallskip
\item<2->[$\Rightarrow$] \textbf{alternative transforms} can be used
\end{itemize}
\end{itemize}
\end{frame}
\section[CQT]{constant-Q transform}
\begin{frame}{constant-Q transform}{introduction}
\begin{itemize}
\item<1-> DFT has a \textit{linear} frequency axis:
\begin{itemize}
\item not perceptually meaningful: \textit{logarithmic} is better match
\item low pitch resolution at low frequencies
\end{itemize}
\bigskip
\item<2->[$\Rightarrow$] compute DFT-like transform {\color{highlight}{at specific frequencies}}
\begin{itemize}
\item space frequencies logarithmically (constant $\mathcal{Q}$)
\item resulting abscissa resolution is pitch-related
\item parameter $c$ adjusts number of bins per octave
\end{itemize}
\end{itemize}
\bigskip
\visible<2->{\begin{equation*}
\mathcal{Q} = \frac{f}{\Delta f} = \frac{1}{2^{\nicefrac{1}{c}}-1}
\end{equation*}
}
\end{frame}
\begin{frame}{constant Q transform}{implementation 1/2}
\begin{columns}
\column{0.55\textwidth}
\begin{eqnarray*}
X_\mathrm{CQ}(k,n) &=& \frac{1}{\mathcal{K}(k)}\sum\limits_{i=i_{\mathrm{s}}(n)}^{i_{\mathrm{e}}(n)}{w_k(i-i_{\mathrm{s}})\cdot x(i) \e^{\mathrm{j}2\pi \frac{\mathcal{Q}\cdot(i-i_{\mathrm{s}})}{\mathcal{K}(k)}}}\\
\mathcal{K}(k) &=& \frac{f_{\mathrm{S}}}{f(k)} \mathcal{Q}
\end{eqnarray*}
\column{0.45\textwidth}
\begin{itemize}
\item $f(k)$: frequency of bin index $k$
\item $\mathcal{K}(k)$: blocklength for bin index $k$
\item $\mathcal{Q}$: measure of pitch res.
\item $w_k$: window function
\item $i_\mathrm{s},i_\mathrm{e}$: start and stop time indices of block
\item $f_\mathrm{S}$: sample rate
\end{itemize}
\end{columns}
\bigskip
\begin{itemize}
\item long window for low frequencies (high freq res, low time res)
\item short window for high frequencies (low freq res, high time res)
\end{itemize}
\end{frame}
\begin{frame}{constant Q transform}{implementation 2/2}
\vspace{-5mm}
\begin{columns}
\column{.3\linewidth}
\begin{center}\textbf{non-overlapping}\end{center}
\vspace{-8mm}
\begin{figure}
\centering
\scalebox{1.2}{\input{pict/fundamentals_CQT}}
\end{figure}
\column{.3\linewidth}
\begin{center}\textbf{overlapping}\end{center}
\vspace{10mm}
\vspace{15mm}
\column{.3\linewidth}
\begin{center}\textbf{differences}\end{center}
\begin{itemize}
\item outputs at multiple vs.\ one time resolution
\item multiple different FFT lengths vs.\ one FFT length (zero-padded)
\item dependent vs.\ independent definition of block and hop length
\end{itemize}
\vspace{15mm}
\end{columns}
\end{frame}
\begin{frame}{constant Q transform}{CQT vs.\ DFT}
\textbf{CQT}:
\bigskip
\begin{itemize}
\item<1->[+] perceptually/musically adapted frequency resolution
\smallskip
\item<2->[--] time resolution depends on frequency
\smallskip
\item<3->[--] not invertible
\smallskip
\item<4->[--] no optimized implementation (compare FFT)
\end{itemize}
\end{frame}
%\section{Auditory Filterbanks}
%\begin{frame}{auditory filterbanks}{introduction}
%FT and related transforms bad models of physiological properties of the human ear:
%\begin{itemize}
%\item frequency resolution (critical bands)
%\item frequency scale (pitch resolution)
%\item loudness \& masking
%\item event perception \& time integration
%\end{itemize}
%
%\smallskip
%$\Rightarrow$ \textbf{auditory filterbanks}
%
%\pause
%\bigskip
%not as widely used as one might think because
%
%\begin{enumerate}
%\item<3-> computationally inefficient
%\item<3-> analysis only: no invertibility (mostly)
%\item<3-> not proven to be superior
%\end{enumerate}
%\end{frame}
%
%\begin{frame}{auditory filterbanks}{gammatone filterbank}
%\vspace{-6mm}
%\begin{equation*}
%h(i) = \frac{ a \cdot\left(\nicefrac{i}{f_{\mathrm{S}}}\right)^{\mathcal{O}-1}\cdot \cos\left( 2\pi \cdot f_{\mathrm{c}} \frac{ i}{f_{\mathrm{S}}} \right) }{\e^{\nicefrac{2\pi i \Delta f}{f_{\mathrm{S}}}}}
%\end{equation*}
%\vspace{-4mm}
%\figwithmatlab{Gammatone}
%\end{frame}
\section{summary}
\begin{frame}{summary}{lecture content}
\begin{itemize}
\item \textbf{DFT has disadvantages}
\begin{itemize}
\item low frequency resolution for low pitches
\item non-logarithmic/perceptually relevant pitch resolution
\end{itemize}
\bigskip
\item \textbf{CQT}
\begin{itemize}
\item similar to Fourier Transform but logarithmically spaced frequency bins
\item not invertible and inefficient
\end{itemize}
%\bigskip
%\item \textbf{filterbanks}
%\begin{itemize}
%\item good model of human physiology
%\item not invertible and inefficient
%\item not proven to be superior
%\end{itemize}
\end{itemize}
\inserticon{summary}
\end{frame}
\end{document}