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

Frequency

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

Fourier Series

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 compute the squared magnitude as \(\left|\hat{g}(f)\right|^2\), known as the power spectrum

Examples

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. Available here

Example

• Suppose I wanted to try to decompose the following wave into its various parts. Data available here.

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

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.
• 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

HW5-7 Groups!

• You have a bit of time now to touch base with your new partner!