Michael@0: \documentclass[handout]{beamer}
Michael@0: 
Michael@0: \usepackage{../C4DMlecturetheme}
Michael@0: 
Michael@0: \usecolortheme{beaver}
Michael@0: 
Michael@0: % Set up the lecture details here
Michael@0: \setlecturedetails
Michael@0: %COURSE DETAILS:
Michael@0: 	{Sound Recording and Production} % COURSE TITLE
Michael@0: 	{ECS614U/ECS749P} % COURSE CODE
Michael@0: %
Michael@0: %LECTURER DETAILS:
Michael@0: 	{Michael Terrell} % LECTURER NAME(s)
Michael@0: 	{michael.terrell@eecs.qmul.ac.uk} % LECTURER EMAIL(s)
Michael@0: 	{http://qmplus.qmul.ac.uk/course/view.php?id=3243} % COURSE WEBPAGE URL
Michael@0: %
Michael@0: %LECTURE DETAILS:
Michael@0: 	{1} % LECTURE NUMBER
Michael@0: 	{The Physics of Sound} % LECTURE TITLE
Michael@0: 
Michael@0: 
Michael@0: \institute[C4DM]%
Michael@0: {Centre for Digital Music\\
Michael@0: School of Electronic Engineering and Computer Science\\
Michael@0: Queen Mary University of London}
Michael@0: 
Michael@0: \date[Semester 1, 2013--14]{Semester 1, 2013--14}
Michael@0: 
Michael@0: %\pgfdeclareimage[height=5mm]{theLogo}{QMULlogo}
Michael@0: %\logo{\pgfuseimage{theLogo}}
Michael@0: 
Michael@0: \beamerdefaultoverlayspecification{<+->}
Michael@0: 
Michael@0: 
Michael@0: \begin{document}
Michael@0: \section{Lecture \lecturenumber}
Michael@0: \subsection{ \lecturetitle}
Michael@0: \maketitle
Michael@0: 
Michael@0: 
Michael@0: \separator{Course Overview}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Lectures}
Michael@0: 	{\small
Michael@0: 	\begin{columns}
Michael@0: 	
Michael@0: 	\column{4.5cm}
Michael@0: 	\begin{enumerate}
Michael@0: 		\item The Physics of Sound.
Michael@0: 		\item Microphones.
Michael@0: 		\item The Audio Chain.
Michael@0: 		\item MIDI.
Michael@0: 		\item Sound Design.
Michael@0: 		\item Mixing: Gain.
Michael@0: 	\end{enumerate}
Michael@0: 		
Michael@0: 		
Michael@0: 	\column{4.5cm}
Michael@0: 	\begin{enumerate}\setcounter{enumi}{6}
Michael@0: 		\item Mixing: Delay.
Michael@0: 		\item Mixing: Dynamics.
Michael@0: 		\item Sound Reproduction.
Michael@0: 		\item Psychoacoustics.
Michael@0: 		\item Mastering.
Michael@0: 		\item 
Michael@0: 	\end{enumerate}
Michael@0: 	
Michael@0: 	\end{columns}
Michael@0: 	}
Michael@0: 	
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Coursework}
Michael@0: 	\begin{enumerate}\itemsep12pt
Michael@0: 		\item Microphone project: {\bf 5\%} (11/10/2013).
Michael@4: 		\item Apple loops project: {\bf 10\%} (1/11/2013).
Michael@0: 		\item Soundscape concept document: {\bf 30\%} (22/11/2013).
Michael@0: 		\item Soundscape audio and technical document: {\bf 55\%} (13/12/2013).
Michael@0: 	\end{enumerate}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \separator{The Physics of Sound}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{What is a sound?}
Michael@0: 	\begin{itemize}\itemsep8pt
Michael@0: 		\item A sound is a pressure wave.
Michael@0: 		\item The pressure wave travels through an acoustic medium, i.e. air.
Michael@0: 		\item The pressure wave consisting of compression and rarefaction.
Michael@0: 		\item In the compression and rarefaction parts of the wave, the particles which form the acoustic medium are respectively squashed together and pulled apart.
Michael@0: 		\item \href{http://illuminations.nctm.org/ActivityDetail.aspx?id=37}{\textit{Vibrating string animation}.}
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{The waveform}
Michael@0: 	\begin{itemize}\itemsep12pt
Michael@0: 		\item A waveform is a graphical representation of a sound wave.
Michael@0: 	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width = 0.9  \textwidth]{Figures/waveform.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{The waveform}
Michael@0: 	\begin{itemize}
Michael@0: 		\item A waveform plot can represent one of two things:
Michael@0: 		\setlength{\parskip}{0.25cm}
Michael@0: 		\begin{enumerate}\itemsep8pt
Michael@0: 			\item The waveform at a given point in space as it changes with time.
Michael@0: 			\item The waveform at a given moment in time as it changes in space.
Michael@0: 		\end{enumerate}
Michael@0: 		\item \href{http://www.kettering.edu/physics/drussell/Demos.html}{\textit{Waves in space and time}.}
Michael@0: 		\item When listening to a sound we are sensing the changes in pressure with time.
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Sound wave properties}
Michael@0: 	\begin{itemize}\itemsep12pt
Michael@0: 		\item Amplitude, \textbf{A} (Pa).
Michael@0: 		\item Frequency, \textbf{f} (Hz): number of cycles per second.
Michael@0: 		\item Time period, \textbf{T} (s): the time for one cycle.
Michael@0: 		\item Wavelength, $\mathbf{\lambda}$ (m): the distance taken up by one cycle.
Michael@0: 		\item Speed \textbf{c} (m/s): the speed at which the wave travels.
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{A waveform versus time}
Michael@0: 	\setlength{\parskip}{0.5cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/100HzVsTime.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{A waveform versus distance}
Michael@0: 	\setlength{\parskip}{0.5cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/100HzVsDistance.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{The relationship between time and space}
Michael@0: 	\vspace{-0.75cm}
Michael@0: 	\begin{center}
Michael@0: 		\begin{equation*}
Michael@0: 		\mathbf {TIME = \frac{1}{FREQUENCY}  \: \: \: \: \: \: \: \: \: \: \Longrightarrow } \: \: \: \: \mathbf {T = \frac{1}{f}}	
Michael@0: 		\end{equation*}
Michael@0: 		
Michael@0: 		\vspace{-0.25cm}
Michael@0: 		
Michael@0: 		\begin{equation*}
Michael@0: 		\mathbf {WAVELENGTH \times FREQUENCY = SPEED  \: \: \: \:  \Longrightarrow}  \: \: \: \: \mathbf {\lambda \times f = c}
Michael@0: 		\end{equation*}
Michael@0: 		
Michael@0: 		\vspace{-0.25cm}
Michael@0: 			
Michael@0: 		\begin{equation*}
Michael@0: 		\mathbf {DISTANCE = SPEED\times TIME  \: \: \: \:  \Longrightarrow}  \: \: \: \: \mathbf {d = c \times t}
Michael@0: 		\end{equation*}	
Michael@0: 			
Michael@0: 		\vspace{0.5cm}
Michael@0: 		
Michael@0: 		(The speed of sound in air (\textbf{c}) is 343 m/s)
Michael@0: 		
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Complex waveforms}
Michael@0: 	\begin{itemize}\itemsep16pt
Michael@0: 		\item Real musical sounds are more complex than the sine waves shown so far.
Michael@0: 		\item But...we can think of a complex waveform as a summation of many different sine waves of different amplitude, frequency (and phase).
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Complex waveforms}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	This complex waveform...
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/SummedWave.eps}
Michael@0: 	\end{center}
Michael@0: 	
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Complex waveforms}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	...is made by summing these six simple waveforems.
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/IndividualWaves.eps}
Michael@0: 	\end{center}
Michael@0: 	
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Sound Features}
Michael@0: 	\begin{itemize}\itemsep10pt
Michael@0: 		\item There are many different features that we can use to describe a sound.
Michael@0: 		\item Today we will consider two types of sound feature:
Michael@0: 		\vspace{0.25cm}
Michael@0: 		\begin{itemize}\itemsep8pt
Michael@0: 			\item Level features.
Michael@0: 			\item Spectral features.
Michael@0: 		\end{itemize}
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	There are two key level features: {\bf RMS} and {\bf Peak} level.
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/RMSandPeak.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\setlength{\parskip}{0.5cm}
Michael@0: 	\begin{itemize}\itemsep10pt
Michael@0: 			\item The term {\bf dynamics} is used to describe how much a sound varies over time.
Michael@0: 	\setlength{\parskip}{0.25cm}
Michael@0: 		\begin{itemize}\itemsep6pt
Michael@0: 			\item \textbf{Transient sounds} - large fluctuations in amplitude, e.g. percussion.
Michael@0: 			\item \textbf{Steady-state sounds} - minimal fluctuations in amplitude, e.g. constant sine-wave.
Michael@0: 		\end{itemize}
Michael@0: 		\item The {\bf dynamics} are quantified using the {\bf Crest Factor}, which is the logarithmic ratio of {\bf Peak} and {\bf RMS} levels:		
Michael@0: 	\end{itemize}
Michael@0: 	\begin{equation}
Michael@0: 		\mathbf{Crest \ \ Factor} = 20\log_{10}\left(\frac{\mathbf{Peak}}{\mathbf{RMS}}\right)
Michael@0: 	\end{equation}		
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/CF_SineWave.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/CF_Piano.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 	
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/CF_Drum.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/CF_Clarinet.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/CF_Voice.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Level Features: dynamics}
Michael@0: 	\setlength{\parskip}{0.5cm}
Michael@0: 	\begin{itemize}\itemsep10pt
Michael@0: 		\item The {\bf dynamics} are quantified using the {\bf Crest Factor}, which is the logarithmic ratio of {\bf Peak} and {\bf RMS} levels:		
Michael@0: 	\begin{equation}
Michael@0: 		\mathbf{Crest \ \ Factor} = 20\log_{10}\left(\frac{\mathbf{Peak}}{\mathbf{RMS}}\right)
Michael@0: 	\end{equation}		
Michael@0: 		\item High Crest Factor $\rightarrow$ Transient.
Michael@0: 		\item Low Crest Factor $\rightarrow$ Steady-state.
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The frequency spectrum of a sound tell us how the energy within the sound is divided into different frequencies.
Michael@0: 	\vspace{-0.15cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/Spectrum.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMulti.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMultiP1.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMultiP2.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMultiP3.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMultiP4.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMultiP5.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Spectral Features}
Michael@0: 	\vspace{0.2cm}
Michael@0: 	The spikes on the spectrum relate to the individual sine waves from which the sound was composed:	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/FrequencyMultiP6.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@1: \separator{Music Production}
Michael@1: 
Michael@1: \begin{frame}
Michael@1: 	\frametitle{Music Production}
Michael@1: 	\begin{center}
Michael@1: 		\includegraphics[width = \textwidth]{Figures/mixingCartoon.pdf}
Michael@1: 	\end{center}
Michael@1: \end{frame}
Michael@1: 		
Michael@1: 
Michael@1: \separator{Wave Phase}
Michael@1: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Wave phase}
Michael@0:  	\begin{itemize}
Michael@0: 		\item The position within a cycle of a wave is called the phase and it is defined as a fraction of the wavelength.
Michael@0: 	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width = 0.85  \textwidth]{Figures/phasefigure.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Wave phase}
Michael@0:  	\begin{itemize}
Michael@0: 		\item The positions are repeated at subsequent cycles of the wave.
Michael@0: 	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width = 0.85  \textwidth]{Figures/phasefigure2.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Wave phase}
Michael@0: 	\begin{itemize}
Michael@0: 		\item The wave phase can be represented on a circle, as an angle.
Michael@0: 	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[height = 0.8 \textheight]{Figures/phaseAngle.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Wave phase}
Michael@0:  	\begin{itemize}\itemsep24pt
Michael@0: 		\item Why do we care about wave phase as audio people?
Michael@0: 		\item We care, because the \textbf{difference} in phase is critical when we are adding waves together, and this is something we do \textbf{A LOT} in audio!	
Michael@0: 		\item Adding waves: 1 + 1 = ...?
Michael@0: 	\end{itemize}
Michael@0: 
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - in phase}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves which are in phase (no difference in phase).
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd1.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - 1/8 cycle}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{1}{8}\lambda$.
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd2.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - 1/4 cycle}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{1}{4}\lambda$.
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd3.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - 3/8 cycle}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{3}{8}\lambda$.
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd4.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - out of phase}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{1}{2}\lambda$ (completely out of phase).
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd5.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - 5/8 cycle}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{5}{8}\lambda$.
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd6.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - 3/4 cycle}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{3}{4}\lambda$.
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd7.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - 7/8 cycle}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\frac{7}{8}\lambda$.
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd8.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves - back in phase}
Michael@0: % 	\begin{itemize}
Michael@0: %		\item Adding waves with a phase difference of $\lambda$ (back in phase).
Michael@0: %	\end{itemize}
Michael@0: 	\begin{center}
Michael@0: 	\includegraphics[width = 0.85  \textwidth]{Figures/phaseadd9.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width = 0.85  \textwidth]{Figures/phaseAngleOnePlusOne.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Adding waves}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width = 0.85  \textwidth]{Figures/phaseAngleOnePlusOne-180.pdf}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Inverting Phase}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	Phase is inverted when we `flip' the signal across the time axis.	
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/WaveInPhase.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Inverting Phase}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	Phase is inverted when we `flip'  the signal across the time axis.
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/WavePhaseInvert.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Inverting Phase}
Michael@0: 	\vspace{0.3cm}
Michael@0: 	Adding inverted and non-inverted signals causes cancellation!
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/InvertCancellation.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase change with frequency}
Michael@0: 	\begin{itemize}\itemsep16pt
Michael@0: 		\item Phase differences between two sounds can vary as a function of frequency.
Michael@0: 		\item You cannot hear the difference in phase when the signal is played in isolation, but you will hear it when two signals are added together!
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase change with frequency}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	Sound A:
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/SoundA.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase change with frequency}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	Sound B:
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width =  \textwidth]{Figures/SoundB.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase changes due to time delay}
Michael@0: 	\begin{itemize}\itemsep16pt
Michael@0: 		\item If two sounds are added with a time offset there will be a frequency dependent phase difference.
Michael@0: 		\item A time delay of $\mathbf{\tau}$ ms is added and can be expressed as a percentage of the time period, $\mathbf{T}$, to give a phase shift.
Michael@0: 		\begin{equation*}
Michael@0: 			 \theta = \frac{\tau} {\mathbf{T}} \ \ \times \ \ 360 .
Michael@0: 		\end{equation*}
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase changes due to time delay}
Michael@0: 	\begin{itemize}\itemsep10pt
Michael@0: 		\item What happens if we add a delayed copy of Sound A to the original?
Michael@0: 		\item Sound A has frequency components: $\mathbf{F_1} = 100$ Hz, $\mathbf{F_2}=500$ Hz and $\mathbf{F_3}=8000$ Hz.
Michael@0: 		\item These relate to time periods: $\mathbf{T_1} = 10$ ms, $\mathbf{T_2}=2$ ms Hz and $\mathbf{T_3}=0.125$ ms.
Michael@0: 		\end{itemize}
Michael@0: \end{frame}
Michael@0: 
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase changes due to time delay}
Michael@0: 	\begin{itemize}\itemsep10pt
Michael@0: 		\item If $\tau=1$ ms:
Michael@0: 		\begin{eqnarray}
Michael@0: 			\theta_1 = \frac{1}{10} \times 360 = 36^o. \\			
Michael@0: 			\theta_2 = \frac{1}{2} \times 360 = 180^o. \\
Michael@0: 			\theta_3 = \frac{1}{0.125} \times 360 = 2880^o = 0^o.
Michael@0: 		\end{eqnarray}
Michael@0: 			
Michael@0: 	\end{itemize}
Michael@0: \end{frame}
Michael@0: 		
Michael@0: \begin{frame}
Michael@0: 	\frametitle{Phase changes due to time delay}
Michael@0: 	\vspace{0.5cm}
Michael@0: 	The effect of $\tau = 1$ ms plotted against frequency: referred to as a comb filter.
Michael@0: 	\vspace{-0.2cm}
Michael@0: 	\begin{center}
Michael@0: 		\includegraphics[width = 0.95  \textwidth]{Figures/CombFilter.eps}
Michael@0: 	\end{center}
Michael@0: \end{frame}
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Michael@0: \end{document}
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