Ch 13: Complex Morlet Wavelets and Extracting Power and Phase
This chapter discusses Complex Morlet Wavelets, how they extend on Morlet Wavelets from last chapter to extract phase and power information, and mechanisms of imaginary numbers to accomplish this. Also covers some practical advise on using them.
13.1 The Wavelet Complex
Complex Morlet Wavelets are needed to to extract power and phase information from EEG data. They are called complex because they involve complex numbers (IE they use imaginary numbers). This means the complex wavelet occupies 3 dimensions (real numbers, imaginary numbers, and time).
Think of this as a corkscrew going through time (with the x and y plane the corkscrew rotates on being the real and imaginary axis).
Before this point, with non-complex Morlet wavelets, we did not see the imaginary axis due simpler equations used, but also since we only showed 2D views for the data. Arguably you can do those projections of this 3D spiral onto 2D planes as well. (below are 2D plots from different sides of the above corkscrew)
13.2 Imagining the Imaginary
Imaginary numbers are denoted with i (or sometimes j ), and represent the square root of negative one. They are often scaled by a scalar (For example: 3i ). When combined with a non-imaginary number (they cannot be added directly), we end up with a 2 number expression, such as 6 + 3i . This expression is known as a complex number.
Be sure to practice and understand the concept of imaginary numbers before going further, otherwise you WILL get lost!
A good overview (that also touches on details on the next section) can be found here
13.3 Rectangular and Polar Notation and the Complex Plane
Usually when we plot graphs, we do it over perpendicular axes of X and Y. This notation is known as the Cartesian Notation. However, another notation would be Polar Notation, which instead of XY uses r (or M), being the radius of an arc/circle, and θ (theta), being the angle in radians of the segment from the starting point (typically directly to the right). Polar notation is often a lot better at describing properties of frequency band specific activity.
Converting back and forth between the 2 notations can be done trivially with basic geometry (Pythagorean theorem and tangent). In the below example, we convert a complex number from the Cartesian Notation to Polar. Can you figure out how the below numbers were calculated?
Answer
M = sqrt( (x2 + y2 ) = sqrt( (4)2 + (-8)2 ) = 8.9443
θ -> tan(θ) = Opposite / Adjacent -> θ = tan-1(-8 / 4) = -1.107
More reading here (just remember we use radians, not degrees when doing data analysis!)
13.4 Euler's Formula
Euler's Formula allows us to tie everything above together, as it allows representing complex numbers as points on a circle.
e is the natural number, which is equal to 2.718.... (continues forever...). In the context of EEG data processing, Euler's Formula allows conveying complex numbers as points on a circle.