English subtitles for clip: File:3 of 4 - Analysis - Explaining Fourier analysis with a machines.webm

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In the previous video,

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I showed how turning the crank

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on this machine generates

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twenty different frequencies

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also known as harmonics,

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which are turned into
cosines here

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multiplied by coefficients

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then summed

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magnified

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and the resulting function

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plotted here on the front.

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This process is called synthesis.

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But this machine is called a

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Harmonic Analyzer

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which means it can be

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used to solve the much more

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difficult inverse problem called analysis.

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For example

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if I want to plot a

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particular function here

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how do I set the amplitude bars

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to produce it.

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To illustrate how the machine

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does analysis let's start

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with a square wave.

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First, we use some unique properties

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of the square wave.

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Since it’s periodic

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and even

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all of the information that function

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carries is contained

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in half a period.

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So we’ll take that half a period

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and sample it at twenty

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equally spaced points.

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We use the values of these points

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as the inputs for the machine.

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When we turn the crank we produce

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a new function that reveals

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the correct coefficients.

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To see how we get these coefficients

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we’ll look at the side of the machine.

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As I turn the crank the tips

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of the rocker arms form a sinusoid.

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The indices on the rocker arms run

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from high on the left

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to low on the right.

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I’ll flip the video to

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make it a little more intuitive.

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When the rocker arm tips are

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lined up in a straight line

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we’ll call that crank number zero.

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If the horizontal axis runs from

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zero to pi

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and the vertical axis from

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minus one to plus one

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then we can describe

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the position of the rocker arms

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with a cosine.

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Every two turns of the crank

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increases the frequency

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of the cosine by one.

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At these even numbered cranks

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the values of the function

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on the platen yield the coefficients

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we’re looking for.

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Let’s rewind and watch the plot

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and the rocker arms simultaneously.

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If we pause briefly at every

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second crank a point is

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marked on the function.

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We create a total of 20 points.

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If we look at the output

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from the machine and

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compare it to a sinc

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calculated by a computer

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we see that they are very similar.

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Now, we’ll take the data points

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from this sinc

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scale them

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and use these

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values on the rocker arms.

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Remember that our goal is to

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program the analyzer so that

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it will plot a square wave.

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Now, as I turn the crank

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the pen writes a horizontal line

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then drops and writes

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another flat section

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which amazes me because

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we’re adding only cosines

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and then it rises to write

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another horizontal line.

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Of course, what we’re seeing is

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a square wave.

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What I’ve just shown

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with the machine is

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an essential feature

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of fourier methods.

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I can take a function

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perform harmonic analysis

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extract the coefficients

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and then synthesize

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that function to

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approximate the original.

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So, now that we see that

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this machine can do harmonic synthesis

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and analysis I’ll show you

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in the next video

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some details about how to

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set up the analyzer to perform

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these calculations.

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I’m Bill Hammack

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the Engineer Guy.

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Next up in the series

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is operation.

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If you haven't seen

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them already there's

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also they intro

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and synthesis videos.

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You can learn more about

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the book here.

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And if you really want to

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learn more about the book

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watch the page by page.

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If you're a fan of

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oscillatory motion

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you gotta watch the

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bonus rocker arms video.