How Does the Uncertainty Principle Limit Time Series Analysis?

<h1>1. Introduction</h1> <p>The connection between the Fourier Transform, Uncertainty Principle, and time series analysis unveils a fascinating interplay that shapes the extraction of simultaneous temporal and frequency information. To understand this relationship, it&#39;s important to first briefly understand what a Fourier Transform (FT) and the uncertainty principle are as a consequence. Then, we explore Wavelet Transform (WT) as a promising tool for this limitation, revealing specific frequency temporal occurrences with sufficient clarity.</p> <h2>1.1 Fourier Transform</h2> <p>The Fourier Transform (FT) serves as a mathematical bridge between the time and frequency domains of a function. An FT can be described as:</p> <p><img alt="" src="https://miro.medium.com/v2/resize:fit:676/1*cZnGbs4fK7wMXkfWqgbH2g.png" style="height:126px; width:676px" /></p> <p>I&rsquo;m not going into detail in this integral, but the important part is that an&nbsp;<strong>FT transforms the function&nbsp;<em>f(x)</em>&nbsp;to another function g<em>(&omega;)</em>&nbsp;in the frequency space.&nbsp;</strong>Hold that information for later, it&rsquo;s going to be important. (To better understand the FT I highly recommend the video from&nbsp;<a href="https://www.youtube.com/watch?v=spUNpyF58BY" rel="noopener ugc nofollow" target="_blank">3Blue1Brow</a>.)</p> <p><a href="https://towardsdatascience.com/how-does-the-uncertainty-principle-limit-time-series-analysis-c94c442ba953"><strong>Visit Now</strong></a></p>