Understanding Phase

[Matt M]

I’ve been having some issues understanding phase, what are some good resources for doing so?

I haven’t (yet) gone through the Module 2 reading, is it in there?

[Catherine Lai]

There’s a bit about phase in the reading for Module 2:
https://www.ee.columbia.edu/~dpwe/pubs/Ellis10-introspeech.pdf

But it might be more practically helpful to look at some animations. There are some nice interactive ones on this website:
https://jackschaedler.github.io/circles-sines-signals/index.html

This one relates sines and cosines to rotations around the unit circle (the ‘phasor’ in the module 1 notebooks
https://jackschaedler.github.io/circles-sines-signals/sincos.html

This one shows what a change in the phase angle means:
https://jackschaedler.github.io/circles-sines-signals/trig_review.html

And here is a visualization of phase with respect to the DFT.
https://jackschaedler.github.io/circles-sines-signals/dotproduct4.html

You can also play with this by changing the params settings in the last exercise of the sp-m1-3-sampling-sinusoids.ipynb notebook and generate some new animations there.

The key point for us is that DFT produces N magnitude and phase outputs.
The phase outputs basically tell you how much you would shift a specific cosine wave (basis function) if you were to try to reconstruct the original input by adding up scaled and shifted versions of those N cosines associated with the N DFT outputs (remembering that only half of those are distinct because of aliasing!).

You can play with how changing the phase of an input component effects the DFT magnitude and phase outputs in sp-m1-5-interpreting-the-dft.ipynb by changing the values in the code under section 5.3 ‘DFT of a compound waveform’ (see the gen_sinusoid function). Basically, the phase angle changes the point on the unit circle where you start sampling your cosine waves.

In terms of visualizations, I also quite like this ‘Lead/Lag’ video from the Khan Academy Electrical Engineering course, though the teacher never actually says phase! The videos that follow in that series also have some nice visualizations of Euler’s formula.
https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-lead-lag?modal=1

[Author]

Posts