Ch 11: The Discrete Time Fourier Transform, the FFT, and the Convolution Theorem
This chapter directly expands on the previous by talking about how the Fourier Transform works and what it is used for, as well as the Convolution Theorem and its implicationsimplications. This is an important chapter as the the Fourier Transform is the backbone of many analyses!
11.1 Making Waves
The Fourier transform works by computing the convolution using various sine waves as the kernels. The general equation of a sine wave is:
Asin(2πf t + θ)
- A is the amplitude
- Power is amplitude squared, and is a term you may see used interchanged
- f is the frequency
- The number of cycles per seconds, expressed as Hz
- t is the time
- The time you are calculating the wave in seconds. This is the dependent variable of the equation
- θ is the phase
- Measured in radians or in degrees (be careful here), essentially is the offset of the whole wave in regards to time
Use online graphing calculates and play with the sine equation until you feel familiar with how changing these properties changes the resultant wave!
On a 2D representation, we see sine waves cycling up and down. However, as discussed in the future, sine waves can be better described as a 3D spiral going through space (phasors), and the 2D plot is just a projection of this onto a plane.
You can take several sine waves, and add them together to get a more complicated signal.
Given the complicated signal, is it possible to go back? It can be taken that every periodic signal can be depicted as a sum of various sine waves, and this is what the Fourier Transform lets us do - break down a complicated signal back into the component sine waves.
This is done by doing the convolution of a signal using various sine kernels of different amplitudes, frequencies, and phase. This is the Fourier transform, which is a 3D representation of the time-series data with the dimensions being frequency, power, and phase (not time!).
11.2 Finding Waves in EEG Data with the Fourier Transform
11.3 Th Discrete Time Fourier Transform
11.4 Visualizing the Results of a Fourier Transform
11.5 Complex Results and Negative Frequencies
11.6 Inverse Fourier Transform
11.7 The Fast Fourier Transform
11.8 Stationarity and the Fourier Transform
11.9 Extracting More or Fewer Frequencies than Data Points
11.10 The Convolution Theorem