Fourier Transform

We just learned a bit about fourier transform. The theory of fourier transform is that any signal can be represented as a sum of sinusoids. Then we saw how this theory can be put into action by recreating a real accelerometer signal using only the addition of sinusoids. The frequency of the specific sinusoids that make up a signal can tell us important information that we can use to build algorithms to process that signal.

Summary

We demonstrate the addition of sinusoids and demonstrate how the Fourier transform allows us to describe any signal as a summation of sinusoids. The frequencies of the sinusoids that comprise a signal represent the signal’s frequency components. The range of frequency components for a signal is called its bandwidth.

We then discuss the Nyquist frequency and the limits this imposes on the sampling rate and the bandwidth of the signals that we sample.

If we try to sample a signal that has higher frequency components than the Nyquist frequency, we will see aliasing, which means those high-frequency components will show up at mirrored lower frequencies.

Notebook Review

If you wanted to interact with the notebook in the video, you can access it here in the repo /intro-to-dsp/walkthroughs/fourier-transform-I/ or in the workspace below.

Disclaimer
The sections below are not fully functional in the following workspace as they are interactive plots:

• Fourier Transform Demo
• Nyquist Frequency
  I'd suggest you either watch that part of the video again or view it on your local machine. You can follow the instructions found in Introduction to Wearable Data's Concept Developer Workflow for how to set up your local environment to interact with the 2 sections.

New Vocabulary

• Frequency component: The Fourier transform explains a signal as a sum of sinusoids. Each of these sinusoids is a frequency component of the signal.
• Nyquist frequency: Half of the sampling frequency. Signal components above this frequency will get aliased in the sampled signal.
• Bandwidth: A range of frequencies within a band.
• Aliasing: The effect that causes frequency components greater than the Nyquist frequency to become indistinguishable from frequencies below the Nyquist frequency.