--- title: "Discrete Fourier" author: Jed Rembold date: February 20, 2025 slideNumber: true theme: tokyo-night-light highlightjs-theme: tokyo-night-light width: 1920 height: 1080 transition: slide --- ## Announcements - I'll be working on HW2 feedback over the next week - HW3 is out - Don't forget the check-in form this weekend! - 5 min for partner meet-n-greet in just a moment - `fftutils.r` library posted [here](../libs/fftutils.r), though I did include in starting repo ## HW3 Groups! - Since I forgot on Tuesday, you have 5 min now to touch base with your new partner! ::::::{.cols style='align-items: flex-start'} ::::col :::{.block name='Left side'} - Conor & Clay - M & Ema - Mamadou & Gabby - Aurora & Elliott - Maddie & Sadie - Jared & Luca - Salem & Izzy ::: :::: ::::col :::{.block name='Right side'} - Tegan & Sawyer - Luna & Sage - Lucca & Sergio - Pearson & Greg - Evyn & Evan - Oscar & Felicity ::: :::: :::::: ## Recall - Measuring periodicity for noisy data can be very sensitive to initial guesses - Any periodic signal can be represented as a combination of sine waves - The Fourier transform wraps a signal at different periods and measures the "overlap" of that signal at each period - Moves a signal from "time-space" to "frequency-space" - The signal's power spectrum is the square of the amplitude of its Fourier transform - Can use `fft` or `rfft` to compute the power-spectrum, and `fftfreq` (or `rfftfreq`) to compute the corresponding frequencies ## Today's Plan - Understanding how the Fourier Transform can compute other wave parameters - Going from frequency-space to time-space - Understanding discrete data's effect on the Fourier Transform - Quiz # Wave Properties ## The Classic Periodogram - Though we've touched on it, the classic, or Schuster, periodogram is defined as: $$ P_S(f) = \frac{1}{N}\left|\mathcal{F}(f)\right|^2 $$ where $N$ is the number of discrete measurements in the observing window - Differs from the power spectrum by a factor of $\tfrac{1}{N}$, which accounts for the fact that otherwise longer signals will have higher power spectrum values - Technically, the periodogram is our observational statistic, which serves as an _estimator_ for the underlying power spectrum ## Amplitudes and Phases - Periodograms can help identify the prominent frequencies, but what if you want the other sinusoidal parameters? - Amplitude: determined from just the magnitude of the FFT scaled by the number of observations $$ A = \frac{1}{N} \left| \mathcal{F}(f) \right| \times 2 $$ - The $\times 2$ comes from the symmetric nature of the FFT - Phase: determined from the angle formed by the real and imaginary parts of the FFT $$ \phi = \arctan\left(\frac{\operatorname{Im}(\mathcal{F}(f))}{\operatorname{Re}(\mathcal{F}(f))}\right) $$ - The returned phase is in radians # Back it up ## The Inverse FFT - You can also go backwards! - The _Inverse Fourier Transform_ moves you back from the frequency-domain to the time-domain - In Python, this is given by `ifft` - In R, use the `inverse = TRUE` flag inside `fft` - Make it possible to filter out certain frequencies, and then transform back to a clean signal ## Activity! - I've generated noisy data of a single oscillation [here](../demos/dopplerdata.csv). - Your task is to determine the period/frequency, filter out everything else by setting it to 0, and then transform that signal back and plot it atop the original noise # Discrete Effects ## Common Transforms ![Important Fourier Transform Pairs](../images/python/important_fts.png) ## Convolutions - Mathematically, a convolution is defined as: $$ [f * g](t) = \int_{-\infty}^{\infty} f(t)g(t-\tau)\,d\tau $$ - Conceptually, this is the same as: - Taking the second function and flipping it about the y-axis - Then "sliding" that function across the other, from left to right - Each step, summing the area beneath both functions ## Visual Convolutions ![https://commons.wikimedia.org/wiki/File:Convolution_of_box_signal_with_itself2.gif](https://upload.wikimedia.org/wikipedia/commons/6/6a/Convolution_of_box_signal_with_itself2.gif){width=100%} ## Why do we care? - Fourier Transforms have a particular attribute: $$ \mathcal{F}\{f * g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\} $$ or $$ \mathcal{F}\{f \cdot g\} = \mathcal{F}\{f\} * \mathcal{F}\{g\} $$ - In other words: convolutions in one space are the same as just multiplying the function point-wise in the other space ## Example 1: Window ![](../images/python/conv1_annotated.svg) ## Example 2: Discrete Measurements ![](../images/python/conv2_annotated.svg) ## Our Powers Combined... - In practice, most real world data consists of both effects: - Data was measured over some duration: the window - Will cause broadening of our peaks. The narrower the window, the greater the broadening. - Data was collected at some frequency: the discrete measurements - Will cause _aliases_ of the signal, spaced according to the sampling rate - The slower the sampling rate, the more densely packed the aliases ## The Nyquist Limit - Note that if the window of observations gets too small, or the time between observations too large, our Fourier Transform peaks will begin to overlap! - In this case not all of the frequency information can be recovered - This is called the Nyquist Limit, and occurs at a frequency of half the sampling frequency - The FFT algorithm generally measures frequencies _up to_ but not beyond this point, so you shouldn't see aliases in your results, but your results might not capture what you were hoping to see. ## Nyquist Visual ![](../images/python/conv3_annotated.svg) # Quiz 1 ## Quiz Time! - Put your notes away and have just a writing implement and a calculator out! - Show as much work or your thought process as you can on all problems for the potential of partial credit - When you are finished you can leave