Finished Lab 2
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Sam Perry
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-16
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{\multicitedelim}
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{}
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\newcommand{\code}[1]{\texttt{#1}}
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% MATLAB Code block stuff...
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\usepackage{color}
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\usepackage{listings}
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@@ -54,25 +56,28 @@
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\definecolor{gray}{rgb}{0.5,0.5,0.5}
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\lstset{language=Matlab,
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keywords={break,case,catch,continue,else,elseif,end,for,function,
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global,if,otherwise,persistent,return,switch,try,while},
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basicstyle=\ttfamily,
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keywordstyle=\color{blue},
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commentstyle=\color{gray},
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stringstyle=\color{dkgreen},
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numbers=left,
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numberstyle=\tiny\color{gray},
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stepnumber=1,
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numbersep=10pt,
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backgroundcolor=\color{white},
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tabsize=4,
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showspaces=false,
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showstringspaces=false}
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keywords={break,case,catch,continue,else,elseif,end,for,function,
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global,if,otherwise,persistent,return,switch,try,while},
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basicstyle=\ttfamily,
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keywordstyle=\color{blue},
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commentstyle=\color{gray},
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stringstyle=\color{dkgreen},
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numbers=left,
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numberstyle=\tiny\color{gray},
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stepnumber=1,
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numbersep=10pt,
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backgroundcolor=\color{white},
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tabsize=4,
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showspaces=false,
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showstringspaces=false
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frame=single,
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breaklines=true,
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%postbreak=\raisebox{0ex}[0ex][0ex]{\ensuremath{\color{red}\hookrightarrow\space}}
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}
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\begin{document}
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\title{ECS707P - DSP}
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\subtitle{\LARGE{DSP Lab 2 Writeup}}
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\author{Sam Perry - ec16039}
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\author{Sam Perry - EC16039}
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\maketitle
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@@ -98,4 +103,191 @@ function main()
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stem(x)
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\end{lstlisting}
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\begin{figure}[H]
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\caption{\code{stem} Output Plot}
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\makebox[\textwidth]{\includegraphics[width=\textwidth]{SineStem}}
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\end{figure}
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\section{Generating DFT using \code{fft}}
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\begin{lstlisting}
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% Main function for running exercises
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function main()
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% Declare the number of samples in the sine wave
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N = 32;
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% Declare the sample rate
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fs = 2048;
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% Create indexes 1 - N
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time = [0:N-1];
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% Declare the frequency as 128Hz
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frequency = 128;
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% Declare the amplitude
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amplitude = 1.0;
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% Generate signal x from parameters
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x = amplitude * sin(2*pi*frequency*(time/2048));
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% Compute the DFT of the sine wave
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X = fft(x);
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% Create a new figure window
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figure
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% Take only the values < the nyquist frequency (1024)
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X = abs(X)'
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X = X(1:N/2)
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% Plot the absolute magnitude of the DFT to a graph
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stem(X);
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% Turn on grid
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grid on;
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% Add a title to the graph
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title('DFT of x')
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% Create labels for x ticks
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labels = 0:(fs/2)/(N/2):fs/2
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% Setup x ticks
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set(gca, 'XTick', 0:length(labels)); % Change x-axis ticks
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set(gca, 'XTickLabel', labels); % Change x-axis ticks labels.
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% Label the x axis as 'Frequency'
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xlabel('Frequency (Hz)')
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% Label the y axis as 'Magnitude'
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ylabel('Magnitude |X|')
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\end{lstlisting}
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\section{What is the phase at the frequency 128 Hz?}
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The angle returned is -1.5708 ($\frac{-\pi}{2}$). This is because the
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difference between the sinusoid generated and the cosine of the same frequency
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at this bin is $\frac{-\pi}{2}$.
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\section{sample a sinusoid at 220 Hz at the same sampling rate}
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\begin{figure}[H]
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\caption{\code{stem} Plot of a Sine Wave at 220Hz}
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\makebox[\textwidth]{\includegraphics[width=\textwidth]{Sine220Stem}}
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\end{figure}
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\section{Draw a picture of what you hypothesize will be the magnitude DFT of
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this signal}
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It is expected that two frequency bins closest will have magnitudes $> 0$. This
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is because there is not frequency bin that represents 220Hz exactly and the
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bins either side will correlate a certain amount with the sinusoid.
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\section{Find the DFT of this signal and plot the magnitude of the result versus frequency}
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\begin{figure}[H]
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\caption{32 Bin DFT of a Sine Wave at 220Hz}
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\makebox[\textwidth]{\includegraphics[width=\textwidth]{DFT220}}
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\end{figure}
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\subsection{How does this plot compare to that in 1.2?}
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There is significant spectral content across all frequency bins, with a peak
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at bins around the frequency of the sinusoid generated. This contrasts the
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single peak of magnitude $\frac{N}{2}$ at the exact frequency of the sinusoid
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generated in section 1.2.
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\subsection{Was your hypothesized plot, created in 1.5, correct?}
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The hypothesized plot drawn in section 1.5 was partially correct. It predicted
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the dispersion of magnitude across the nearest frequency bins due to the lack
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of an exact bin for that frequency, however it did not anticipate the
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distribution of magnitudes across all frequency bins due to the discontinuity
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in signal as stated in the lab document.
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\section{Create a 512-length sequence of the sinusoid in equation (3) with a
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frequency of 220 Hz, amplitude of 1, and a sampling rate of 2,048 Hz}
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\begin{figure}[H]
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\caption{\code{512 Bin DFT of a Sine Wave at 220Hz}}
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\makebox[\textwidth]{\includegraphics[width=\textwidth]{DFT220_512}}
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\end{figure}
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\section{Comment the program, line-by- line, to describe what it is doing}
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\begin{lstlisting}
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% Main function for running exercises
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function main()
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% Clear all previously defined variables
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clear all
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% Initialize string with name of sound file
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sndfile = 'speech_female.wav';
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% Read the audio samples of the sound file into matrix x and save the
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% integer samplerate in variable Fs
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[x,Fs] = audioread(sndfile);
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% Set the variable N to 512
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N = 512;
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% Returns a matrix S, Frequency range F and Timestemp T for 1.4second of
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% the input vector x. Argument 2 is the window size used for windows.
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% Argument 3 is the fft size for each window. Larger values will results in
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% a higher frequency resolution, lower values results in a better temporal
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% resolution. Argument 4 is the sample rate of the signal x.
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[S,F,T] = spectrogram(x(1:Fs*1.4),N,3*N/4,N*4,Fs);
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% Generate a figure with set position and other display related parameters.
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f = figure('Position',[500 300 700 500],'MenuBar','none', 'Units','Normalized');
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% More display related settings...
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set(f,'PaperPosition',[0.25 1.5 8 5]);
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% same as above...
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axes('FontSize',14);
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% same as above...
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colormap('jet');
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% Display spectrogram over time T, with frequency in KHz. Matrix of values
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% are converted to absolute magnitudes and scaled to decibels.
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imagesc(T,F./1000,20*log10(abs(S)));
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% Set labels of spectrogram image
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axis xy; set(gca,'YTick',[0:2000:Fs/2]./1000,'YTickLabel',[0:2000:Fs/2]./1000);
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ylabel('Frequency (kHz)'); xlabel('Time (s)');
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print(gcf,'-depsc2','p2i1.eps');
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\end{lstlisting}
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\section{Subsection}
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\subsection{In what frequency band is most of the energy located?}
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For the majority of the sample, frequency of 5-6KHz and below are most
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prominent. The exception is during noisy phonemes where the energy is spread
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across higher frequencies.
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\subsection{Knowing the Nyquist-Shannon Sampling Theorem, at what minimum rate
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would you sample this signal such that it could be reconstructed well—but
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not exactly?}
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From visual inspection, it may be possible to reconstruct the sound to an
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intelligible degree at a sample rate of around 11KHz. It may be possible to
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lower this further.
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\section{Which sounds of which words correspond to the energy between 4 and 18
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kHz?} These frequency ranges contain the most energy during sibilant phonemes.
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This is clearly demonstrated in the word ``administer'' at the `s'.
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\section{What is the highest decimation factor you think this signal can
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withstand and you can still understand the speech? What would the sampling
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rate be at that factor?}
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A decimation factor 6 was found to be the highest factor that was still
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intelligible. This results in a sample rate of 7350Hz.
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\section{Do it. Include your code. Include your observations.}
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\begin{lstlisting}
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% Main function for running exercises
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function main()
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% Clear all previously defined variables
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clear all
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% Initialize string with name of sound file
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sndfile = 'speech_female.wav';
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% Read the audio samples of the sound file into matrix x and save the
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% integer samplerate in variable Fs
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[x,Fs] = audioread(sndfile);
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% Set the variable N to 512
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% Set decimation factor
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decimation_factor = 16;
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% Pick every $decimation_factor samples
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x = x(1:decimation_factor:length(x));
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% Set sample rate accordingly
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Fs = Fs / decimation_factor;
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% Play sound.
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sound(x(1:Fs*1.4), Fs);
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return;
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\end{lstlisting}
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Decimation is applied by simply taking every nth sample. A more sophisticated
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approach would be to apply a low pass filter to the audio before decimation in
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order to avoid aliasing.
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\section{Say you have an animal that needs medicine. Knowing what you know now,
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how would you administer said medicine?}
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Although it is a very difficult matter, the simplest method is to mix the
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medicine with butter, or some other grease, and smear it on the nose of the
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animal from time to time...
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\end{document}
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