Introduction to the Fourier Transform (Part 1)

welcome back to control system lectures this is the first video covering the fundamentals of the Fourier transform and in this video I'll focus on what we use Fourier transforms for then I'll give you an intuitive way of thinking of the transform then we'll back up that thinking with a little bit of math and in the process hopefully simplify the topic for you so let's get started I'm going to write two equations right off the bat and now if you're not already familiar with them they're probably going to look menacing and a bit difficult to comprehend it first however by the end of this lecture I think you'll come to realize that they're just as intuitive as multiplication or addition this first equation is called the Fourier transform or the forward Fourier transform the second equation is called the inverse Fourier transform and as you can see the first equation turns a function of time into a function of frequency and the second equation does the reverse it turns a function of frequency into a function of time and this process of turning one into another is called transformation and so these equations are transforms so what exactly is a transform it transform is a mapping between two different sets of data or domains in this case it changes information in the time domain into information in the frequency domain here I'm using the variable nu to represent frequency in units of Hertz or cycles per second the data in these two domains look different and can even vary in length of the data but it still represents the exact same information for example if I asked you to find me at the White House assuming you knew where that was that title would be all you would need to know to find my location however I could have also said find me at 1600 Pennsylvania Avenue Washington DC or find me at these specific GPS locations each of these descriptions represent the exact same location they're just stated a little bit differently and depending on your situation one way of presenting the information might be better than another and a transform would get you from one representation to another and this is exactly what the Fourier transform does it jumps between two sets of information but instead of converting between two addresses it converts between the time domain F of T and the frequency domain F of Nu now John Fourier and others figured out in the early 18th century that any continuous signal in the time domain could be represented by a sum of carefully chosen sinusoids and since sinusoids can in turn be completely described by their amplitude their frequency and their phase those three parameters are all the information we need to know to describe the signal at that point we can plot the amplitude and phase at every frequency or across the entire spectrum and call it the frequency domain representation of the signal and this information is exceedingly useful in control systems and digital signal processing of course most signals aren't just comprised of a single sinusoid but multiple ones at different amplitudes and phases across the entire spectrum but why decomposes signal into sinusoids why not some other repeating signal like a square wave or sawtooth or why not in two different a periodic signals altogether well what we're trying to do is transform the signal into something that's easier to work with at least for our particular purpose and sinusoids are great because they're the only waveform that doesn't change shape when subjected to a linear time-invariant system or LTI system there's a wing at the bottom of the screen to a video that describes why this is true all right so the benefit of having an input waveform that doesn't change shape is that it allows us to think in terms of change in amplitude or gain and change in phase through a system and any given frequency and that lets us make use of a whole arsenal of tools for designing and analyzing control systems so we know that representing a signal in the time domain is good because that's the way that is going to behave in real life but also representing it in the frequency domain is good in other situations when we're designing or analyzing a control system and the Fourier transform gives us the ability to move from one domain to the other which is awesome also the other reason that I like Fourier transforms is that it's a stepping stone to the Laplace transform which is even a more important tool in classical control theory but I'll cover that in a future video for now let's finally dive into some of the math if I was to give you these two points in the frequency domain and asked you to construct the time domain signal how would you do it this first point has a frequency of 1 Hertz with amplitude of 1 and the second point has a frequency of 2 Hertz with amplitude of 0.5 but neither signals are phase shifted well since you know that these correspond to sinusoidal equation like this you'll see why I chose cosine instead of sines in a moment but what you have are two separate sinusoids and when you add them together you'd get a result that looks something like this in the time domain no I actually didn't plot this out ahead of time I've just kind of drawn this from what I expect it to look like so it might not be exact now if I added a third point you'd have three cosines and so on however it'll start to get really cumbersome if we use more than three points so instead I can reduce all of these summations into a discrete summation and out of all of the frequencies between negative infinity and positive infinity but don't worry too much right now about negative frequencies just know that they're there so I've rewritten the above equation in a more compact form and this first term a of Nu is all of the amplitudes at each of those given frequencies and then each one would be multiplied by a cosine of that specific frequency but what if I have a continuous spectrum of frequencies then you can just replace that discrete summation with an integral and you can have an equation that looks something more like this and now you should start to see that this is looking a lot similar to that inverse Fourier transform from above you can actually get a good sense of what the inverse Fourier transform is doing just by studying this equation and you could stop right here but we'll continue it's taking parameters like amplitude and multiplying them by sinusoids and adding them all up to get the signal in the time domain and that's exactly what you did with the two or three cosines from earlier but the equation still needs some work and that's because we chose cosine waves that had zero degrees of phase shift but in general signals can have sinusoids of any phase shift remember that phase and amplitude can be described by a single complex number which is made up of real and imaginary components and if we plot that complex number on a real imaginary plane then the amplitude is the distance from the point to the origin and phase is the angle of that line from the positive real line when we do this we see that a sinusoidal Atun of 1 and no phase shift results in a point that lies on the real line in this case 1 which is a real number and a real number is exactly what we used in our example above so it all works out but for this next example we're going to say that the sinusoidal amplitude of 1 and finding where this red dot is on the real imaginary plane is pretty straightforward the amplitude is the hypotenuse of the right triangle and the tangent of the phase is the imaginary part over the real part and solving for these two equations gives us the square root of 2 over 2 for both the real and imaginary components so now we have complex signal information that captures both phase and amplitude not just amplitude like we had with the real number and I'll change my a of new function which only had amplitude information into a function of new which contains both amplitude and phase additionally we can describe sines and cosines in the complex plane using Euler's formula which can be written like this e to the I T equals the cosine of T plus I times the sine of T so now with the added phase information and writing things in the complex form amplitude times the cosine becomes the function of nu times Euler's formula and this new equation can handle complex inputs all right so let's continue our example of a cosine signal with amplitude of 1 and phase shift of 45 degrees remember that corresponds to the square root of 2 for both the real and imaginary component for F of Nu which is written at the top of the screen so if I multiply out this equation I'm left with four separate terms the first two are real numbers and the last two are imaginary so wait what does imaginary time even mean well it really doesn't mean anything in our case and that's where negative frequencies come in to help us the imaginary time signals are going to cancel out when you consider the negative frequencies remember we're summing all frequencies between negative infinity and positive infinity and so far we've only looked at the positive one Hertz so we need to add this to the results of the negative one Hertz now at this point I have a small side note the function f of Nu is complex as I've stated earlier but if you were to plot the real component of F of Nu and the imaginary component separately you would find that the following is always true in dealing with real time problems the real component would be an even function and the imaginary component would be an odd function and what I mean by that is that the real value between a positive frequency and it's negative counterpart would have the same value and the same sign and the imaginary value between the two would have the same value but opposite signs also the values of each is split between the positive and negative values so they would each be half of the original so in our case we would have frequency content that looked like this with square root of two over four once we realize that only half of the signal is captured into positive frequencies and the other half and the negative so we already have the positive frequency content although the amplitude is actually half of what I wrote earlier so let's generate the negative frequency values and when we sum these two results together you're going to see that the imaginary parts cancel out and the real parts add together I'll let you work that out and make sure that it's correct but if you have any problems with it leave a question in the comment section below and I'll try to address it so what you should be left with though is a signal that contains both a sine wave and a cosine wave and this further reduces to a single cosine wave with a phase shift and that phase shift is 45 degrees which is exactly what we intended from the beginning so it all worked out but that really seemed like it took a lot of effort to get to this point right well really it isn't because using the Fourier transform as an integral you don't have to work through all of those steps individually like we just did it takes care of all of that for you now in most cases this integral might be difficult to actually perform but I'll leave that up to the math department to explain how to go through all of that what I really wanted to do is just give you an understanding of how the Fourier transform works so just to sum it up we can write the transformation from the frequency domain to the time domain like this you take a continuous set of frequency information and this is complex information since it has both amplitude and phase we call that information F of Nu then you multiply that by Oilers formula or by a bunch of phase-shifted cosine waves with frequencies ranging from negative infinity to infinity then finally you just sum up each of those waves at all of those frequencies and what you're left with is the time domain signal so now you can see that the inverse Fourier transform is really as intuitive as multiplication and summing since that's all its really doing now I had intended to cover both the inverse Fourier transform and the forward for u transform in the same video unfortunately I got writing too much and now I've run out of time therefore this is only part one of the video and in part two I'll cover the forward direction of the Fourier transform along with a simple way to remember it don't forget to subscribe so you don't miss any future videos and as always thanks for watching

Channel: Brian Douglas

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Keywords: Feedback, Control Systems Lectures, Theory, Flight Controls, Education, Lecture, Lesson, Brian Douglas, Automatic Control, Control Theory, Control System Tutorial, Laplace Transform, Linear Control, Fourier Transform, Fourier Series, Time domain, frequency domain

Id: 1JnayXHhjlg

Length: 13min 2sec (782 seconds)

Published: Thu Jan 10 2013