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'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’m not going into detail in this integral, but the important part is that an <strong>FT transforms the function <em>f(x)</em> to another function g<em>(ω)</em> in the frequency space. </strong>Hold that information for later, it’s going to be important. (To better understand the FT I highly recommend the video from <a href="https://www.youtube.com/watch?v=spUNpyF58BY" rel="noopener ugc nofollow" target="_blank">3Blue1Brow</a>.)</p>
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