--- title: "Fourier Analysis" author: Jed Rembold date: February 18, 2025 slideNumber: true theme: tokyo-night-light highlightjs-theme: tokyo-night-light width: 1920 height: 1080 transition: slide p5js: true --- ## Announcements - Homework 3 is coming out today! - Due in approx just under 3 weeks - I'll be scheduling short one-on-one meetings at that point to discuss with you how you are doing as a partner - Schedule Rejiggering - I am still working out details, but will likely be bumping HW deadlines back a few days to Sunday or Monday nights to give you more time after our last content lecture - Means the check-in / debrief timing might shift some as well - Quiz at the end of Thursday's class! Study up! ## Today's Plan - Why can determining periodicity be difficult? - What is a Fourier transform? - How can I use a Fourier transform to extract information from a signal? # Tricky Periodicity ## Periodicity - Identifying repetition in observations often gives important information about a source or event - There are no limits on the shape of the observation beyond that it repeats - Signal may be smooth, like a sine wave - Signal may be something non-continuous, like a square wave - Signal can be a combination of several underlying signals that all repeat differently - The _period_, often denoted with a $T$, of an observation is the elapsed time between repetitions ## The "Easy" Case - For simple, known repeating functions, you might think just fitting the data would work - It _can_, but it can also be surprisingly susceptible to initial parameter guesses ![](../images/period_sin_sensitive.png){width=80%} ## Frequency :::{style='font-size:.95em'} - There are multiple ways to think about how fast an event or observation repeats - We've already mentioned that the period is the time between repetitions - Alternatively, we could talk about how many repeats occur within a certain time - This is called the _frequency_, generally indicated with an $f$ - The period of a repeating event and the frequency of that event are easily correlated: $$ T = \frac{1}{f} $$ - Frequency is thus in units of 1 over time. If time is measured in seconds, this unit is called a _hertz_ ::: ## Complications - If the period of your signal is small compared to the frequency of your observations, it can be nearly impossible (or actually impossible) to tease out the period - Making a simple fit to the signal rapidly becomes unfeasible if the signal is a combination of several periodic sources - Even if you **know** the number of periodic sources and can model it mathematically - If fitting even a single basic sine wave is sensitive to initial guesses, it gets worse with more complexity ## Another Way - We need better ways to find and determine periodic behavior in data - One method is to construct a periodogram - A periodogram is a visual representation of the strength of differing periodic signals to a combined signal - Periodograms are inherently related and linked to Fourier Series and the Fourier Transform # For Who? ## Fourier Series :::{style='font-size:.95em'} - **Any** periodic function can be represented by summing together many differing sine waves of different amplitudes, periods, and phases - A generic sine wave would have a form of: $$ A\sin\left(\frac{2\pi t}{T} + \phi\right) $$ - The _Fourier Series_ of a periodic function is just the infinite series of all the sine wave contributions that could be added together to exactly equal that periodic function - 3Blue1Brown has several _excellent_ videos on Fourier series and the (upcoming) Fourier Transform that you should check out on YouTube - This lets us just focus on finding and determining sinusoidal periodic repetition ::: ## The Fourier Transform - The _Fourier Transform_ shifts data from looking at how it changes over time, to looking at how it changes over different frequencies $$ \hat{g}(f) = \int_{-\infty}^\infty g(t) e^{-2\pi i f t}\,dt $$ - This value will generally be a complex number - Fourier transforms are symmetric about the y-axis, since periodic motion can occur in either direction - You can generally just focus on the positive values. ## The Transform Visualized
## Getting Powerful - A larger magnitude of $\hat{g}(f)$ implies a greater contribution of that frequency $f$ to the original signal - The goal is to compute this magnitude over a wide range of frequencies and then see which frequencies contribute the most - Commonly actually computer the squared magnitude as $\left|\hat{g}(f)\right|^2$, known as the _power spectrum_ ## Examples ![](../images/python/inf_sin_and_trans.png){width=70%} ![](../images/python/inf_square_and_trans.png){width=70%} # Computational Fourier Transforms ## The Discrete Fourier Transform - When computing a Fourier Transform from data, we are technically always computing the _Discrete Fourier Transform_, since we only know the value of the function at certain points - This absolutely will have ramifications in terms of what we end of seeing, but I'd like to postpone that discussion for just a bit - So long as the data looks mostly like an infinite continuous series, the DFT will closely mimic the continuous Fourier Transform previously defined - Spacing of data points relatively close together relative to the period - Data available over many full periods - The classic algorithm for computing a DFT is the _Fast Fourier Transform_, which is what libraries will generally make available in different languages ## Python FFT - You can access functions to perform a FFT from several libraries in Python: - Numpy has it in `np.fft`, and offers several variants - Scipy has it in `scipy.fft` and also offers several variants - In both: - The `fft` function will return complex output, symmetric about 0 - The `fftfreq` function will generate a list of the corresponding frequencies to plot the power spectrum against - The `rfft` function will return complex output, only include positive frequencies - The `rfftfreq` function will generate a list of the corresponding frequencies to plot the rfft power spectrum against ## R FFT - In R, the FFT function is defined in the `stats` library - Only offers the `fft` function, which will return complex output, symmetric about 0 - No function that corresponds to `fftfreq` in Python - I wrote an R package to give you this functionality: `fftutils.r` ## Example - Suppose I wanted to try to decompose the following wave into its various parts. Data available [here](../demos/multi_wave.csv). ![](../images/python/multi_wave.png) # Determining all wave properties ## The Classic Periodogram - Related to the power spectrum, 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, and then transform that signal back and plot it atop the original noise ## HW3 Groups! - You have a bit of time 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 ::: :::: ::::::