<story><title>Demystifying Fourier analysis</title><url>https://dsego.github.io/demystifying-fourier/</url></story><parent_chain><item><author>femto</author><text>A funky thing is that you can replace the &amp;#x27;N&amp;#x27; sin&amp;#x2F;cos functions used in Fourier analysis with any set of &amp;#x27;N&amp;#x27; orthogonal functions (aka basis functions).  In this context, a Fourier transform is just a bunch of correlations of the function to be analysed with each basis function.  Two functions are orthogonal if their correlation is zero.&lt;p&gt;PCM is &amp;quot;Fourier analysis&amp;quot; with impulse functions in place of sin&amp;#x2F;cos.  CDMA replaces sin&amp;#x2F;cos with a set of (almost) orthogonal functions with a wide bandwidth.&lt;p&gt;From an information perspective, orthogonality of the basis functions means each component&amp;#x2F;coefficient is giving you new information about the signal.  In real life you can only compute a finite number of coefficients, meaning you must have a finite amount of information, so there is an inherent uncertainty in the relationship between the signal being analysed and the output of Fourier analysis.  (Physical quantities that have an uncertainty principle, such as position and momentum, are related to each other by a Fourier transform.)</text></item></parent_chain><comment><author>lupire</author><text>No! This is a huge misconception that misses the essence of quantum mechanics.
The Uncertainty is not due to finite information. It is an inherent property!&lt;p&gt;Uncertainty is due to  (or better, &amp;quot;modeled by&amp;quot;, because FT is math, not physics, and you could use wavelets or something instead), Fourier (and similar transforms) is a (bidirectional) function that maps (local) points to (dispersed) waves, and vice versa. This holds true even with infinite precision&amp;#x2F;information.&lt;p&gt;&lt;a href=&quot;http:&amp;#x2F;&amp;#x2F;math.uchicago.edu&amp;#x2F;~may&amp;#x2F;REU2021&amp;#x2F;REUPapers&amp;#x2F;Dubey.pdf&quot; rel=&quot;nofollow&quot;&gt;http:&amp;#x2F;&amp;#x2F;math.uchicago.edu&amp;#x2F;~may&amp;#x2F;REU2021&amp;#x2F;REUPapers&amp;#x2F;Dubey.pdf&lt;/a&gt;&lt;p&gt;&amp;quot;Qualitatively, this means a narrow function has a wide Fourier transform, and a wide function has a narrow Fourier transform. In either domain, a wider function means there is literally a wide distribution of data, so there always exists uncertainty in one domain.&amp;quot;</text></comment>
<story><title>Demystifying Fourier analysis</title><url>https://dsego.github.io/demystifying-fourier/</url></story><parent_chain><item><author>femto</author><text>A funky thing is that you can replace the &amp;#x27;N&amp;#x27; sin&amp;#x2F;cos functions used in Fourier analysis with any set of &amp;#x27;N&amp;#x27; orthogonal functions (aka basis functions).  In this context, a Fourier transform is just a bunch of correlations of the function to be analysed with each basis function.  Two functions are orthogonal if their correlation is zero.&lt;p&gt;PCM is &amp;quot;Fourier analysis&amp;quot; with impulse functions in place of sin&amp;#x2F;cos.  CDMA replaces sin&amp;#x2F;cos with a set of (almost) orthogonal functions with a wide bandwidth.&lt;p&gt;From an information perspective, orthogonality of the basis functions means each component&amp;#x2F;coefficient is giving you new information about the signal.  In real life you can only compute a finite number of coefficients, meaning you must have a finite amount of information, so there is an inherent uncertainty in the relationship between the signal being analysed and the output of Fourier analysis.  (Physical quantities that have an uncertainty principle, such as position and momentum, are related to each other by a Fourier transform.)</text></item></parent_chain><comment><author>djmips</author><text>From this perspective you can look at it as comparable to the dot product concept of &amp;#x27;how much&amp;#x27; of this thing is in that thing. You can also think of it as a &amp;#x27;rotation&amp;#x27; operation - a change of basis into the frequency domain. That insight I garnered from graphics pioneer James Blinn.&lt;p&gt;A related funky thing is that if you choose the right functions you can engineer the analysis to require only shifts and adds which was a boon for making bespoke compression schemes on limited hardware. An example is the Slant transform used in an early Intel video codec. IIRC the Hadamard transform was also similarly useful where the coefficients were all 1s and -1s. I also seem to recall that some modern codecs have modes that use transforms that avoid the traditional DCT.&lt;p&gt;One last thing, I might have dreamt this but I recall reading a NASA tech report from the 70s or 80s outlining a video compression scheme that used a non DCT approach that could be calculated with simpler hardware without multiplication. I remember thinking, wow they were out front but never commercialized that.</text></comment>