analysis:course-w16:week5
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analysis:course-w16:week5 [2016/01/24 07:58] eirvine [Fourier series, transforms, power spectra] |
analysis:course-w16:week5 [2018/07/07 10:19] (current) |
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| To study oscillations we require //periodic// functions of time, such as ''sin(t)'', which repeat its values in regular intervals. | To study oscillations we require //periodic// functions of time, such as ''sin(t)'', which repeat its values in regular intervals. | ||
| - | Recall that to plot this sort of function in matlab, we first define the time axis (commonly a variable with the name ''tvec''), pass this to the function we wish to plot as an argument, and plot the result: | + | Recall that to plot this sort of function in MATLAB, we first define the time axis (commonly a variable with the name ''tvec''), pass this to the function we wish to plot as an argument, and plot the result: |
| <code matlab> | <code matlab> | ||
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| f = 2; % frequency of sine to plot | f = 2; % frequency of sine to plot | ||
| - | y = sin(2*pi*f*tvec); % note sin() expects arguments in radians, not degrees (see sind()) | + | y = sin(2*pi*f*tvec); % note sin() expects arguments in radians, not degrees (see also ''sind()'') |
| stem(tvec,y); | stem(tvec,y); | ||
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| Computing a FFT over a window of finite size is as if we are taking a finite-size window (for instance '' w(n) = 1 if 0 <= n <= N; 0 otherwise''; note that this defines a //rectangular// window) and multiplying this with a hypothetical infinite signal. | Computing a FFT over a window of finite size is as if we are taking a finite-size window (for instance '' w(n) = 1 if 0 <= n <= N; 0 otherwise''; note that this defines a //rectangular// window) and multiplying this with a hypothetical infinite signal. | ||
| - | It turns out that the spectrum of the windowed signal equals the //convolution// of the signal's spectrum and the window's spectrum; this occurs because of the [[http://en.wikipedia.org/wiki/Convolution_theorem|convolution theorem]] that states that multiplication in the time domain equals convolution in the frequency domain. | + | It turns out that the spectrum of the windowed signal equals the //convolution// of the signal's spectrum and the window's spectrum; this occurs because of the [[http://en.wikipedia.org/wiki/Convolution_theorem|convolution theorem]] that states that multiplication in the time domain equals convolution in the frequency domain. (If the idea of convolution is new to you, you can think of it as a kind of "blurring": for instance, convolving a signal with a 5-point rectangular pulse is equivalent to taking the running average with a 5-point moving window. For a graphical exploration of this idea, see [[http://pages.jh.edu/~signals/convolve/ | this website]] or look at the MATLAB doc for the ''conv()'' function.) |
| Thus, it becomes important to understand the spectral properties of the windows used for Fourier analysis. Let's plot a few: | Thus, it becomes important to understand the spectral properties of the windows used for Fourier analysis. Let's plot a few: | ||
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| <code matlab> | <code matlab> | ||
| % check if sampling is ok | % check if sampling is ok | ||
| - | plot(diff(csc_preR)); % only minimal differences | + | plot(diff(csc_pre.tvec)); % only minimal differences |
| - | Fs = 1./mean(diff(csc_preR)); | + | Fs = 1./mean(diff(csc_pre.tvec)); |
| </code> | </code> | ||
| Line 494: | Line 494: | ||
| (Note that the above is a quick hack of something that should really be implemented in a nice function, because the above commands really correspond to one conceptual operation of decimation, presenting an opportunity for making the code more readable and maintainable. But I haven't gotten around to that yet.) | (Note that the above is a quick hack of something that should really be implemented in a nice function, because the above commands really correspond to one conceptual operation of decimation, presenting an opportunity for making the code more readable and maintainable. But I haven't gotten around to that yet.) | ||
| + | |||
| + | ☛ Now that you have decimated the data, also update your sampling frequency ''Fs''. | ||
| Now we can compute the spectrum: | Now we can compute the spectrum: | ||
| Line 508: | Line 510: | ||
| {{ :analysis:course:week4_fig7.png?600 |}} | {{ :analysis:course:week4_fig7.png?600 |}} | ||
| - | Note that we are plotting not the raw power spectrum but rather the ''10*log10()'' of it; this is a convention similar to that of the definition of the decibel (dB), the unit of signal power also applied to sound waves. | + | Note that we are plotting not the raw power spectrum but rather the ''10*log10()'' of it; this is a convention similar to that of the definition of the decibel (dB), the unit of signal power also applied to sound waves. The values on your vertical axis may look different, depending on what version of ''LoadCSC()'' was used. |
| Regardless, it doesn't look very good! The estimates look very noisy, a characteristic of the periodogram method. | Regardless, it doesn't look very good! The estimates look very noisy, a characteristic of the periodogram method. | ||
analysis/course-w16/week5.1453640328.txt.gz · Last modified: 2018/07/07 10:19 (external edit)
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