Finished first draft

writeup.tex

@@ -78,42 +78,42 @@
 
     \section*{Section 2.1 Buoy data parsing function}
     \begin{lstlisting}
-    % Declare a function that takes a signle argument (the
-    % file path) and returns a 'data' object containing the
-    % data parsed from the file object.  A count integer is
-    % also returned with the number of entries read from
-    % file.
-    function [data, count] = readbuoydata(datafile)
+% Declare a function that takes a signle argument (the
+% file path) and returns a 'data' object containing the
+% data parsed from the file object.  A count integer is
+% also returned with the number of entries read from
+% file.
+function [data, count] = readbuoydata(datafile)
 
-        % Create a file object with the path provided by the
-        % 'datafile' variable
-        fid = fopen(datafile,'r');
-        % Read first two header lines of file so that they
-        % are ignored in processing below.
-        tline = fgetl(fid);
-        tline = fgetl(fid);
+    % Create a file object with the path provided by the
+    % 'datafile' variable
+    fid = fopen(datafile,'r');
+    % Read first two header lines of file so that they
+    % are ignored in processing below.
+    tline = fgetl(fid);
+    tline = fgetl(fid);
 
-        % Parse each line of the file from line 3 onwards
-        % and store in variable 'A' first argument specifies
-        % the file object.
-        % second arguments specifies the data type to expect
-        % for each of the 10 elements per line.
-        % the third argument specifies to read 10 elements
-        % per line, and to read until the end of the file.
-        [A,count] = fscanf(fid, ...
-        '%d %d %d %d %d %f %f %d %f %f',[10 inf]);
+    % Parse each line of the file from line 3 onwards
+    % and store in variable 'A' first argument specifies
+    % the file object.
+    % second arguments specifies the data type to expect
+    % for each of the 10 elements per line.
+    % the third argument specifies to read 10 elements
+    % per line, and to read until the end of the file.
+    [A,count] = fscanf(fid, ...
+    '%d %d %d %d %d %f %f %d %f %f',[10 inf]);
 
-        % Creates a member of the data object that contains
-        % dates in the 'serial date number' format
-        data.date = datenum([A(1:5,:); zeros(1,size(A,2))]')';
-        data.Hs = A(6,:); % significant wave height
-        data.Tp = A(7,:); % peak period
-        data.Dp = A(8,:); % peak period direction
-        data.Ta = A(9,:); % average period
-        data.SST = A(10,:); % sea surface temperature
+    % Creates a member of the data object that contains
+    % dates in the 'serial date number' format
+    data.date = datenum([A(1:5,:); zeros(1,size(A,2))]')';
+    data.Hs = A(6,:); % significant wave height
+    data.Tp = A(7,:); % peak period
+    data.Dp = A(8,:); % peak period direction
+    data.Ta = A(9,:); % average period
+    data.SST = A(10,:); % sea surface temperature
 
-        % Close the file 
-        fclose(fid);
+    % Close the file 
+    fclose(fid);
     \end{lstlisting}
 
     \section*{Section 2.2 Peak Period and Wave Height Plots}
@@ -124,46 +124,67 @@
 
     \section*{Section 2.3 Moving Average Filter Function}
     \begin{lstlisting}
-    % Function takes an 1-dimensional input signal and
-    % an M value (>0) 
-    % A filtered signal of the same size is returned
-    function [outputSignal] = movingAverage(inputSignal, M)
-        % Initialize output array with zeros.
-        outputSignal = zeros(1, length(inputSignal));
-        % Pad input with zeros at the begining.
-        inputSignal = [zeros(1, M-1), inputSignal];
+% Function takes an 1-dimensional input signal and
+% an M value (>0) 
+% A filtered signal of the same size is returned
+function [outputSignal] = movingAverage(inputSignal, M)
+    % Initialize output array with zeros.
+    outputSignal = zeros(1, length(inputSignal));
+    % Pad input with zeros at the begining.
+    inputSignal = [zeros(1, M-1), inputSignal];
 
-        % For each sample in input...
-        for n = M:length(inputSignal)
-            % Take the last M samples and save their
-            % mean value as the output sample.
-            outputSignal(n-M+1) = mean(inputSignal(n-M+1:n));
-        end
+    % For each sample in input...
+    for n = M:length(inputSignal)
+        % Take the last M samples and save their
+        % mean value as the output sample.
+        outputSignal(n-M+1) = mean(inputSignal(n-M+1:n));
+    end
     \end{lstlisting}
 
     \section*{Section 2.4 Filtered Peak Period Data Plots}
-    FIX Plot to show "(s)"
     \begin{figure}[H]
         \makebox[\textwidth]{\includegraphics[width=1.3\textwidth]{MovingAveragePlot}}
     \end{figure}
 
     \section*{Section 2.5 Questions}
     \subsection*{Q1. What do you observe as $M$ increases?}
-    The output signal appears to be smoothed at increasing
-    amounts as values for $M$ increase.
+    The output signal appears to be smoothed at increasing amounts as values
+    for $M$ increase.
+
     \subsection*{Q2. Why do you think you observe this ``thing''?}
-    Increasing $M$ increases the size of the ``slice'' of
-    signal averaged to form each samples in the output
-    signal. By averaging larger slices of signal to form the
-    output, output samples will deviate less from the
-    previous sample due to the increased similarity between
-    the consecutive slices. This causes the smoothing effect
-    observed.
+    Increasing $M$ increases the size of the window of signal averaged to
+    form each samples in the output signal. By averaging larger windows of
+    signal to form the output, output samples will deviate less from the
+    previous sample due to the increased similarity between the consecutive
+    windows. This causes the smoothing effect observed.
+
     \subsection*{Q3. What is happening at the beginning of the averaged dataset, and why does it happen?}
+    There is a ramp up from 0 that is increasingly prominent with increasing
+    values of $M$. This is due to the $M-1$ zeros padding the start of the
+    input signal to allow initial samples to be processed. As these are
+    included in the averaging of input samples, they decrease the value of
+    output samples. As $n$ increases, less zero-padding is included in the
+    average, causing a ramp up. As there are more zeros with larger values of
+    $M$, this effect is more prominent with higher values of $M$ as it will
+    affect $M-1$ samples at the beginning of the output signal.
 
     \subsection*{Q4. What happens to the running average when the peak period suddenly drops?}
+    If the peak period suddenly drops and remains stable at the new value, then
+    the running average will drop to this value in $M$ samples. Essentially the
+    running average will reach any new value in $M$ samples if that new value
+    remains constant, if it doesn't then the running average will vary based on
+    the average of samples in the consecutive windows. For this reason, large
+    windows filter out quick drops or spikes in the input, as the mean of the
+    window is still relatively close to the majority of samples.
+
     \subsection*{Q5. Are these drops preserved?}
+    Sudden drops will be smoothed increasingly into ramps with larger values of
+    $M$ for the reasons stated above.
+
     \subsection*{Q6. Are the wave trains more clear?}
+    Wave trains are clearer with higher values of $M$.  Due to the removal of
+    small fluctuations in the signal, the overall contour of the averaged
+    signal shows the increase and gradual decrease more clearly.
 
     \section*{Section 2.6 Filtered Wave Height Plots}
     \begin{figure}[H]
@@ -172,8 +193,60 @@
 
     \section*{Section 2.7 Questions}
     \subsection*{Q1. Do you observe anything different in the plots from 2.6 to those you saw in 2.4?}
+    Aside from the time shift mentioned in the lab instructions, a significant
+    observation is that for wave height, lower values of $M$ resemble the
+    original signal more closely than the equivalent plots of peak period. As
+    there are fewer abrupt fluctuations in the original signal, the output at
+    $M=5$ is very similar to the input. This is not true of the peak period as
+    the original signal has many small variations that are smoothed even with
+    small values.
+
     \subsection*{Q2. How does the size of this shift relate to M?}
-    \subsection*{Q3. Explain why the peaks move?}
+    The samples will lag by $M\div2$ samples. (Explained below...)
+
+    \subsection*{Q3. Explain why the peaks move? (in complete sentences)}
+    This lag is caused by the window comprising entirely of samples on one side
+    of $n$. As the window centres around samples before $n$, the output sample
+    will be an average of samples around the centre of the window rather than
+    centring on $n$. This causes the lag of half a window size.
+
+    \section*{2.8 Non-causal moving average filter}
+
+    %\subsection*{}
+    \begin{figure}[H]
+        \caption{Peak Period Plots}
+        \makebox[\textwidth]{\includegraphics[width=1.3\textwidth]{NCMovingAveragePeakPlot}}
+    \end{figure}
+    
+    %\subsection*{}
+    \begin{figure}[H]
+        \caption{Wave Height Plots}
+        \makebox[\textwidth]{\includegraphics[width=1.3\textwidth]{NCMovingAverageWavePlot}}
+    \end{figure}
+
+    \subsection*{Non-causal moving average filter code}
+    \begin{lstlisting}
+% Function takes an 1-dimensional input signal and
+% an even M value (>0) 
+% A filtered signal of the same size is returned
+function [outputSignal] = nonCausalMovingAverage(inputSignal, M)
+    % Check that M is odd
+    validateattributes(1, {'numeric'}, {'odd'});
+
+    % Initialize output array with zeros.
+    outputSignal = zeros(1, length(inputSignal));
+    % Pad input with zeros at beginning and end
+    inputSignal = [zeros(1, (M-1)/2), inputSignal, ...
+        zeros(1, (M-1)/2)];
+
+    % For each sample in input...
+    for n = M:length(inputSignal)
+        % Take the last M samples and save their
+        % mean value as the output sample.
+        outputSignal(n-M+1) = mean(inputSignal(n-M+1:n));
+    end
+    \end{lstlisting}
+
 
     % \printbibliography
 \end{document}