# 13.1 An introduction to the analysis of brain waves  (Page 8/8)

 Page 8 / 8

Another approach would be to find the center of mass of the data. This decision rule is more comprehensive because it takes into account all data from the Fourier transform, not just the maximum value. In order to find the average weight of all amplitudes, we change the inner part of code to the following (starting with "%find the maximum one"):

```... %find the frequency corresponding to the "average" amplitudeavg_freq = sum(freq.*amps)/(sum(amps)*df);%decided which way the car should move based on the max frequency if avg_freq<freq_criterion; ...```

One can imagine myriad other ways to approach this problem. Many strategies have been developed, but the question is open-ended. A natural next step is for the reader to think of new ways to interpret spectrogram data. The most effective characterizations probably have yet to be discovered!

## Conclusion

In this module we developed the tools to decompose an arbitrary signal, such as an EEG, into is component frequencies. We began with sine waves, established the trapezoid scheme, and finally introduced Fourier analysis. This same flavor of analysis is used in many other settings, too–see the related documents.

## Code for `mytrapz.m`

```function curve_area = mytrapz(x, y, fast) % function curve_area = mytrapz(x, y, fast)% % mytrapz.m performs the trapezoid rule on the vector given by x and y.% % Input:% x - a vector containing the domain of the function % y - a vector containing values of the function corresponding to the% values in 'x' if nargin<3 curve_area = 0;%loop through and add up trapezoids for as many points as we are givenfor n = 2 : numel(x) height = (y(n) + y(n-1))/2; %average height of function across intervalbase = x(n) - x(n-1); %length of interval trap_area = base * height; %area of trapezoidcurve_area = curve_area + trap_area; %add to continuing sum endelseif fast%alternate (fast) implementation xvals = x(3:end) - x(1:end-2);yvals = y(2:end-1); curve_area = yvals(:)'*xvals(:);curve_area = curve_area + y(1)*(x(2) - x(1)) + y(end)*(x(end) - x(end-1)); curve_area = curve_area/2;end```

## Code for `myfreq.m`

```% myfreq.m %% find the frequencies and amplitudes at which a wave is "vibrating" %% Contrast simple (but laborious) trapezoid computations to the fast % and flexible built-in fft command (fft stands for fast Fourier% transform). % To make full sense of this we will need to think about complex% numbers and the complex exponential function. %T = 5;% duration of signal dt = 0.001; % time between signal samplest = 0:dt:T; N = length(t);y = 2.5*sin(3*2*pi*t) - 4.2*sin(4*2*pi*t); % a 2-piece wave plot(t,y)xlabel('time (seconds)') ylabel('signal')for f = 1:5, % compute the amplitudes as ratios of areas a(f) = trapz(t,y.*sin(f*2*pi*t))/trapz(t,sin(f*2*pi*t).^2);end figureplot(1:5,a,'ko') % plot the amplitudes vs frequency hold onplot(1:5, [0 0 2.5 -4.2 0], 'b*')figure(34) f = (0:N-1)/T;% fft frequenciessc = N*trapz(t,sin(2*pi*t).^2)/T; % fft scale factor A = fft(y);newa = -imag(A)/sc; plot(f,newa,'r+')y = y + 3*cos(6*2*pi*t); % add a cosine piece figure(1)hold on plot(t,y,'g') % plot ithold off legend('2 sines','2 sines and 1 cosine')figure(2) A = fft(y); % take the fft of the new signalnewa = -imag(A)/sc; plot(f,newa,'gx')b = real(A)/sc; plot(f,b,'gx')xlim([0 7]) % focus in on the low frequencieshold off xlabel('frequency (Hz)')ylabel('amplitude') legend('by hand','by fft','with cosine')```

## Code for `myfourier.m`

```% function [mag freq] = myfourier(y, dt, use_fft)% % myfourier.m decomposes the signal 'y', taken with sample interval dt,% into its component frequencies. %% Input: %% y -- signal vection % dt -- sample interval (s/sample) of y% use_fft -- if designated, use matlab's fft instead of trapezoid method %% Output: %% freq -- frequency domain % mag -- magnitude of frequency components of y corresponding to 'freq'function [freq mag] = myfourier(y, dt, use_fft)y = y(:); N = numel(y); %number of samplesT = N*dt; %total time t = linspace(0,T,N)'; %reconstruct time vectorhalf_N = floor(N/2); %ensures that N/2 is an integer if mod(N,2) %treat differently if f odd or evenfreq = (-half_N:half_N)'/T; %fft frequencies elsefreq = (-half_N:half_N-1)'/T; %fft frequencies endif nargin<3 %perform explicit Fourier transform sinmag = zeros(size(freq)); %vector for component magnitudescosmag = zeros(size(freq)); %vector for component magnitudes%loop through each frequency we will test for n = 1 : numel(freq)%obtain coefficient for freqency 'freq(n)' sinmag(n) = mytrapz(t, y.*sin(freq(n)*2*pi*t), 1);cosmag(n) = mytrapz(t, y.*cos(freq(n)*2*pi*t), 1); end%scale to account for sample lengthscale_factor = mytrapz(t, sin(2*pi*t).^2); sinmag = sinmag / scale_factor;cosmag = cosmag / scale_factor; mag = [sinmag(:) cosmag(:)];elseif use_fft %use built-in MATLAB fft() for speed fft_scale_factor = mytrapz(t, sin(2*pi*t).^2) * N / T;A = fft(y); mag(:,1) = -imag(A)/fft_scale_factor;mag(:,2) = real(A)/fft_scale_factor; mag = circshift(mag, half_N);end```

## Code for `mysgram.m`

```% % function [stft_plot freq tm]= my_stft(y, dt, Nwindow) %% my_stft splits the signa 'y' into time windows, the breaks each % segment into its component frequencies. See "Short-time Fourier Transform"% %% Input: % y -- signal% dt -- sample interval % Nwindow -- number of time intervals to analyze% % Output:% stft_plot -- values plotted in the spectrogram % freq -- frequency domain% tm -- time domain function [stft_plot freq tm hh]= mysgram(y, dt, Nwindow) %count the number of windowsN = numel(y); win_len = floor(N/Nwindow);sm = zeros(win_len, Nwindow); cm = zeros(win_len, Nwindow);tm = linspace(0, numel(y) * dt, Nwindow); %for each windowfor n = 1:Nwindow %isolate the part of the signal we want to deal withsig_win = y((n-1)*win_len + 1 : n*win_len); %perform the fourier transform[freq mg] = myfourier(sig_win, dt, 1);sm(:,n) = mg(1:win_len,1); cm(:,n) = mg(1:win_len,2);end stft_plot = abs(sm + cm);stft_plot = stft_plot(end/2:end, :); %plot the fourier transform over timehh = imagesc(tm, freq(round(end/2):end), stft_plot); title('Spectrogram', 'FontSize', 20)xlabel('time', 'FontSize', 16) ylabel('frequency', 'FontSize', 16)set(gca, 'ydir', 'normal') %just look at lower frequenciesylim([0-win_len/2 50+win_len/2])```

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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