Time Frequency Analysis & Wavelets

all right so follow-up from lecture from this morning but let's pretend the last lecture was Wednesday and this is Friday Eve had two days to forget what I talked about all right so now when I continue on though with this idea and introduce the concept of what are called wavelets okay so that's going to be the basic thrust of today is to understand these objects how to use them we're already talking about time frequency and what we want to talk about is begin this concept of when I take a signal I want frequency information I want time information I want it all right Fourier transform doesn't give it all to me time series doesn't give it all to me Gabor doesn't give it all to me but wavelets they're getting closer okay so that's what we're gonna go after and try to figure out how it does that okay so here's the problem I have some stigma on time that I measure there it is beautiful signal in time okay and I want to understand both frequency content of that signal I want to understand the time content I want to know where the frequencies actually occur right I mean if you look at this signal look here during this little interval there's a high frequency stuff but here there might be some low frequency stuff right so you're gonna ultimately what we learn to say the one idea that Gabor hat is to say if I want to try to localize in time in space as I could potentially then put like a filter on this thing something like this and it has some width a and what I do I slide this filter back and forth not back in I just I slide it once I don't have to do it back and forth start here filter portion move it move it move it move it computationally so theoretically I continuously move it all across all of time but computationally right I would take discrete jumps you know move it delta T to the future delta T to the future and I would actually say what is the frequency content in every single little window of time that I've got okay what is the problem with doing this it's a nice concept right but the fact is this window what are the longest wavelengths it can handle it's determined by that window size right I mean if you're gonna do a Fourier mode expansion in here ultimately which is what we're still doing we're still using Fourier modes to represent the spectrum and whatever fits in here that's your box size that's like your longest wavelength piece that you can have in there so what if this signal I mean I've drawn it here what if there's signal that has some really low frequency components in it you just threw them all away with this Gabor window so you do a very poor job in getting let's say frequency resolution you retain a lot of frequencies okay but the cost of throwing that away is that now you actually have the frequency content of this in a specific time so you've you've you've traded throwing some of this away for now I have some time information right so this is what you're trying to do at a time I get time information I get frequency information both but I've thrown something away okay but the window size that you pick is actually the problem because the window size you pick is actually determining if you're throwing away a lot of frequency information or very little frequency information if I make this window narrower and narrower and narrower then I actually pick out as a part of the signal very nicely where this piece of the signal is in time but oh there's all the frequency content however if I make a big window I got lots of frequency content but no time localization so that's the problem with this with this issue however it straight Stookey things okay this is the part that Gabor really pointed out that in this Gabor piece there's two key concepts first there is this short time window controlled by a so the parameter a tells me the width of the window so in some sense I have a scaling parameter I can scale my time that I want to pick out it's gonna be an important concept the second thing that Gabor illustrates is this idea of translation in time right parameter tau it's a slide and I can scale those are your two options but with Gabor at the beginning you fix a slide tau so right away this is where the resolution problem comes because you fix the window and you slide it around okay but we don't have to do that and it's actually not that hard to see that well what I could do is instead of doing this why do I instead decide to say a slide tau and why don't I so slide a around I can do that right so for instance if I come back to this picture I can say why don't I first make a window captures the whole thing what is my objective in doing that my objective in making a big window over the whole thing is to pull out low frequency content low frequency content exists over that whole time so I'll use a big window pull out low frequencies and I use smaller and smaller and smaller windows to pull out high frequencies right simple low frequencies are not localized in time so I can deal with the big window high frequencies are localized in time so I can use very small windows Gabor just does the whole thing with one window and now what wavelets are going to do is basically take this concept these two fundamental ideas and now slide it both in tau and in a buddy with that not a really trivial extension of or a way to think about it are you ready now we introduce what we call the mother wavelet not the mother of all wavelets just a mother wavelet okay ready I don't think there is a mother of all wavelets or they'd probably be in a movie somewhere okay right for the mother wavelet or is it too late in the afternoon to sell jokes I'm tired I'm super tired that's probably why I tell more than I normally would okay here we go the mother wavelet or a sorry a mother wavelet by the way the term wavelet come from idea of a little wave like if I was little you might call me Nathan lit or something right so but I'm not huge so you wouldn't but if you had a little wave you call the wavelet and this is the idea you're gonna take the little waves or at first you get to take a big wave but then you're gonna make a smaller smaller smaller cloud features and these little pieces of wave you're gonna pull those out with little wavelets and that cute it's not so cute when you start doing the math on it but right now it sounds cute and here's the idea we're gonna reduce the function call it sy a B T to our mother wavelet it's gonna kind of have this scaling function here I'm not going to give you a specific way that wavelet more of a functional relationship between the parameters a and B where a and B are real and a is not zero what are these two parameters do I'm gonna give you some sigh it's gonna be a localized function essentially this is just saw here by the way there's all kinds of wavelets so when I talked about some different wavelets that are out there but even when we talk about doing wavelet decompositions it's not like the Fourier transform there's kind of one Fourier transform when you do a wavelet transform you could do a hard wavelet you could you a dovish she's wavelet you could do a moral a wavelet there's all kinds of wavelets okay people make up different wavelets for different purposes so part of your job is if you're gonna do wavelets is you better understand your field well enough to know what people use in your field there's probably a reason they picked a wavelet in your field maybe you work on seismology and there's a very reason specific reason you pick out certain wavelets because have some very nice properties related to seismology okay so when you think about doing generic wavelets it's not as easy as just saying oh I'll just apply a wavelet transform and which wavelet transform okay so we'll come to talk about that a little bit more key parameters here are a scaling and B translation these are the two key things introduced by Gabor slide that thing around but here's the deal here's our degree of freedom we're introducing now that Gabor didn't he made this you know it picked a to be a fixed and I do it now we're not gonna do that we're gonna actually slide around and use a to pull out frequency and time content across a huge range okay and get better time frequency resolution than any of the current methods that we have available okay so let me introduce you to your first wavelet you guys ready the mother wavelet of this what's called a by the way wavelets are kind of old the first one was the harlot 1910 okay so here's the idea I want to localize in time for instance this is what the idea was between the hard way flit it's like well you could expand in Fourier modes but what do forty modes do when you have a signal you expand let's let's take a simple signal suppose have a signal here this is my time domain suppose I wanted to do this with Fourier transforms what do I do what I do before he transforms well you expand in sines and cosines that live on this huge interval but the thing is just sitting right there that seems kind of dumb right so when I introduced the basis functions to expand in that are kind of by construction localized okay so hard did this hard introduced this is a hard mother wavelet as the following this sigh of tea mother wavelet looks like this exciting there it is the haar wavelet okay but in theory you could use this wavelet and as the mother wavelet and expand a function in terms of these things instead of sines and cosines what's nice about it what's really nice about this head's compact support it's localized oh yeah this between 0 & 1 the fact is I can slide it around right I'm gonna take this wavelet slide it compress it expand it so you give me a signal and I can say I'll construct a bunch of these and by sliding them and contracting and compressing or compress whoa sorry what's the opposite of compression expansion yeah yeah anyway some combination of these things I'm going to construct a signal out of these kind of functions okay excellent localization in time right there it is this is the in time it's localized right so it's just this function where it's it's 1 or negative 1 or 0 so it's 1 between T between 1/2 and 0 and it's happening and negative 1 between ok so that's your mother wavelet by the way very important is what its Fourier transform looks like because again when we introduce a wavelet by the way we never actually get away from the ideas of for a for yeah right Fourier tells us expansion of frequencies it's still it still has that interpretation right so whenever we introduce it say a wavelet we're always gonna have to introduce it's or think about its Fourier transform even though we're going to expand now in these functions and not in the Fourier basis you'll see how the Fourier transform plays a role here is the Fourier transform of this thing it's like a sinc function okay so if you take a step function that's Tory transform of a step function indistinct function now we basically have two steps so it's kind of a double type sinc function here yeah what by the way one important thing this envelope decays like 1 over Omega so even the horror basis function is it's the first wavelet it was introduced not in the context of what we're doing here but it was introduced tennis trying to do local is it local expansions of functions the one bad thing about it is it decays very slowly in frequency so it has great localization here in time but very bad localization and frequency so ultimately again when we go to this idea of looking at our signal time and frequency this function would give us very nice time bad frequency right and what we're trying to do is get the best of both if we can so what I'm going to do with this hard wavelet well there's lots of things to do with it and some properties we should note about it so first of all some of the properties are of this haar wavelet that are kind of obvious if you integrate over this thing you get 0 the area under the square of the curve of the absolute value is 1 and it has we call compact support so compact support all that means right is that it lives on a finite interval to zero outside of that okay any function that goes to zero outside of some range we call it has say that's compact support okay so those are some properties about this thing and let me tell you what you're gonna do with us something like this with a hard wavelet you're gonna construct signals and the idea is now is with this haar wavelet basis function you're gonna turn this thing into a function of a b t a is going to be essentially our scaling b is our translation okay so for instance i could do the following with this function I could take the mother wavelet is 1 or negative 1 between so it goes from 0 to half its 1/2 to 1 negative 1 I could for instance translate it over here somewhere by B and then i could compress it by saying hey how about if I make a is equal to 1/2 instead of 1 so in other words I'm gonna say is the mother wavelet that I just plotted over there is that means over on that board when I plotted that's obvious when I draw an arrow right it means what I drew him that board over there two boards over when I plotted over there was this one zero it was one and it was sitting started at the origin so that's the mother wavelet that would start with so for instance what if I wanted to plot something like this use a is 1/2 and B is 2 well serve first I'd slide it over 2 that's the translation piece a is 1/2 it's actually going to scale it so that now this thing this is one two three so now this thing looks like the following my heart wavelet and now it goes to height two negative two and you know this is two and a half right there so I compressed it so the area of the curve still preserved and I've slid it over by two okay so that is Sai 1/2 2t right so for instance suppose I have the mother wavelet to start with and I'm looking at a signal and I want to stay okay I need to resolve the signal well and I can start scaling and translating so for instance that's my mother wavelet and let's take something like here's my signal there's nothing and then all the sudden they're stuffed right over here around five or something like this well what I'd want to do is you know first of all probably use like V equals five get my heart waveless over here and then start scaling it so it's around the width of this thing and keep adding more and more of them higher and higher resolution so that I actually can with very few of these car wavelets define that signal does that make sense to everybody it's just another way to do it all right this is no different than before yay it's just it's a little different looking basis instead of basis functions essentially okay all right so this is just like the idea of Gabor in some sense because essentially you're you're using a window but you're sort of using a dynamic scaling window just go look for the signal over here and then you get Bohr says hey is fixed but you say well not have a fixed so just kind of keep resolving it finer and finer using more and more refined scalings and then I can actually get this thing out pretty quickly with very few wavelets right part of the thing you want to do computationally is use as few wavelets as you have to write you want to take a signal and you want to represent that signal with as few wavelets or for emos as you can write that's that's the ideas of extracting good signal okay alright so this is kind of the idea by the way what happens to the Fourier domain of things like this okay so I get good localization of time but when I compress this signal the spectrum doubles its width okay so it has bad localization in frequency okay all right everybody okay with that the haar wavelet that kind of make some intuitive sense hopefully a little bit hopefully it does so this is the concept then signal and what I want to do I'll give you a mother wavelet the mother way that can go two ways and this is important I can scale it fatter I can scale it thinner whatever I want to do with it right I have a and B to work with and of initially I have call it Cylons ero is the mother sitting in the center of my domain and all I'm gonna do is say okay first thing I want to do is why don't I make a wavelet just goes across that whole domain my whole time signal so I want to take this scale and pick a so that I get a wavelet that compromises the whole domain why am i doing that I'm just getting the low frequency content okay once I get it I'm done I don't worry about well essentially it in some sense I pick it out okay I know what the low frequency content is now I'll scale it so that I do have the domain so I can just change this a cut it in half look at this half of the domain pull out the frequency content there right cut it in half again pull off the frequency content there if I were to take this window of course I'm losing all kinds of frequencies but already I already got those frequencies by doing the bigger scale right through the big scale cut it in half get the next intermediate long scales get the finer scale you keep resolving higher and higher and higher does that make sense to everybody you just multi it's called multi resolution okay we'll talk a lot more about that on Monday you just keep cutting it in half there's your window there's your window you shrinking it down every time you shrink it you get really nice time localization properties you lose frequency content but you already picked it up that was your first step big wavelet you got the frequency content on the big scale you keep shrinking that is the algorithm of the wavelet okay all right in fact I think it's it's time it's time for a picture everybody everything everybody ready for a picture me too okay ready let's draw a picture well you don't know what to say next you just draw pictures right I actually kind of think I know what I'm going to say next but I'm gonna draw a picture and Stansted here we go bhaktas you've already seen some of this this morning but this could cartoon like I said you have to get this cartoon in your head if you don't understand this cartoon you don't understand wavelets you don't understand time frequency analysis this cartoon is critical I understand this cartoon so I'm telling you to do I guess four boxes let's start off with the idea of time series analysis what you do in time-series analysis you don't look at any spectral information you just look at the time series but you chop it up the time very finely you know exactly what happens at every instant of time where the signal is so you have a lot of information at every single point in time no frequency information okay actually looks like this remember this picture like I said this is where it helps like if I had the election Wednesday because now I'd be two days later and you were forgotten I drew that picture but since I drew it this morning you're like here he did that but you just but you have to pretend like it was to get days ago okay there it is that time freaks analysis all right what about the Fourier transform Fourier transform method boy transform you take the whole time signal when you throw it into the frequency domain you have no more time information everything's just frequency content and this is a great thing to do if you have a stationary signal but you don't so you have all the frequency content you have no idea when in the signal there was this for instance this nice little high frequency ringing right you lose track of any of that information but you really have great resolution in the frequency and you have no idea about where it isn't time okay so this is again that picture that we drew this morning then there was the idea of Gabor which is I'm gonna split it I'm gonna do a little time window slide it across and when I do the sliding across my data wherever my window width is that gives me time resolution and for every time window I have I have a certain frequency resolution so here you're kind of saying I'll trade some of my time information for frequency and some of my frequency for time you get patches which you can say at this time how much frequency content do I have localized around this spot you can actually calculate this okay grant it you throw away some information in this right cuz whatever your window size you chopped off or you filter perhaps all the low frequencies and things like this okay how much information you lose depends a lot on how you pick you're filtering in time to do this here at the wavelet picture what I'm first gonna do in the wavelet picture is an algorithm I'm gonna start off saying hey spectral transform the whole dang thing FFT it essentially to some extent right I'm just gonna take the whole time signal and look at its Fourier components so I don't have any spectral resolution but now what I'm doing to do the experiment over again and now I'm gonna take my mother wavelet and I'm gonna scale it be half the size or some fraction of the box and I must slide it around so here's the canonical picture representing this process there it is you start off here great frequency resolution you have no idea where these frequencies live but you have all the frequency you took the whole signal you for a trip you know you put it in the frequency domain say I know every single frequency piece of this signal have no idea where it is in time but I have it all that's okay let's scale let's use this a parameter that I have to do the scaling of the wavelets now let's cut this let's say for instance in half sample two boxes half the size well now I get some information right I actually get some tine information oh is it in the front half or the back half of the signal I can get that okay you get some you get less frequency information okay so you're starting to chop up frequency then you say okay let's sample it again with half the size now I get more time resolutions less frequency and again then I keep chopping up in time less frequency but I already have a lot of frequency from here right it's not only go through this way know all this already so I'm just improving my time frequency resolution by doing this so when I come up here on the scaling I have great time race resolution very bad frequency resolution but that's okay I already got the good frequency resolution everybody good so you're just going level by level and pulling away frequency content you take the same signal and you say alright pull out the low frequencies next step my filter that I'm applying my wavelet mother my mother wavelet there's no difference right my wavelet mother you go there and now you've lost a bunch of frequency information but you already had it so you don't need it now you pull out what's there go one level down keep going one level down when you stop whatever you want right this is a process that ultimately you control how much resolution do you need in time how much resolution do you need in frequency that's when you stop okay so when you when you actually do this in a discreet fashion a compute computational fashion you get to decide how many levels do I want to go down okay and every level I go down I get finer and finer time scale resolution higher frequency content at the expense of throwing away all my low frequencies but I already had them anyway okay all right so that is the picture you take the same signal and you just keep processing seeing that same signal over and over again by using the scaling and translation it's the whole process of wavelets questions on that is that that cartoon that's that's that's the key the rest of all math right but it's hard to understand what you're gonna try to do mathematically if you don't understand that picture okay so contemplate that over the weekend I would suggest a nice pose relaxed legs crossed I'm very uncomfortable that way so I've contemplated for two seconds but some of you are really comfortable and uh you know pose like that and you could just think about it for a while I'm like pictured in your head draw it over and maybe in the sand it guess I'm sand at your house draw it in the sand no I don't also don't like Seattle so what can we say about Edwin alright everybody so far so now we're going to talk about the mathematics of this what is the machinery that actually gets you the information in these cells okay frequency content along these cells so we're going to introduce the continuous wave a continuous wavelet transform okay it's like any other transform its you plot Colonel transform it take your data put it in another domain no different no different than a Fourier transform okay the nice thing about the boy transform is we have a very specific understanding of you know what this what it means in this other domain right we're taking something in time and translating to frequency okay so at least in that case we know what that transform does sort of intuitively but really in the end you're just asking about let's just take some transform some function and generically you know here's what the transform looks like transform some signal with this kernel here and if you'll notice Fourier transform had a very simple kernel of e to the I the Gabor transform had e to the I plus some time filter and there's all kinds of filters you could put in there and we're just gonna do something a little bit differently all we're gonna do now is put in some other wavelets as our transform kernel okay so here's the deal let me introduce a couple things first there's first gonna be a concept about admissibility okay when can actually do a wavelet transform can I just make up any mother wavelet and do this no there's a couple conditions that have to hold first let me define the following take a wavelet and again take the mother wavelet for a transform it okay what's that gonna be take the mother wavelet this thing has a four-day transform that looks like the following let me write it in this form here I can write it all out but you can always rescale it to something like this so this is the Fourier transform in the month the wavelet why this is why is this important because we're gonna define a constant C here's what's important some constant C 2 depends upon a mother wavelet you pick which is defined as the integral from negative infinity to infinity if you take the Fourier transform of your method wavelet absolute value squared divided by the frequency absolute value and integrate over all frequencies this thing has to be bounded a finite value book value okay this is very important for basically not having things blow up okay you have to be able to integrate this stuff and this is a condition so it miscibility is this condition right here so whenever I define a wavelet or if you make up a wavelet that condition has to be satisfied once that satisfied then the definition of the continuous for a trance wavelet transform holds I like that one I don't want to race that yet okay so let's go and define the transform often called the c WT for obvious reasons and here it is just simply something very simple the wavelet transform of some function f which is a function of M P there's the important thing now a function of a and B so I can understand how this function behaves as a function of my scaling and translation is equal it's just an inner product with F times the mother wavelet simple the mother wavelet plays the role of the colonel that's it okay that is the definition of a wavelet transform again I leave it in this form here or the thing to keep in mind remember is that there's lots and lots of different types of wavelets so we leave it generically like this the haar wavelet the senate goes really easy to do right because the harlot is one minus one zero everywhere else so you know for instance it's kind of nice right computationally it's very nice okay so it's like a Fourier transform except just you just have a different kernel all right by the way let's check that the haar wavelet yeah we could do the highway look okay the haar wavelet we define so let's go to that case if you look at the haar wavelets fourier transform it's it's a sinc function like I promised okay well something like a sinc it's actually has a sine squared up top not a sign but this is the Fourier transform of it so what we want to ask is the following is the haar wavelet an admissible class of wavelets right there's only one condition I've stated making this an admissible class and the immiscible class says that what I got to do is integrate this thing here it's absolute value squared over Omega T Omega and I got to show that that's less than infinity it turns out you can calculate it for this wavelet and you get 16 1 over Omega cubed sine absolute value of Omega over 4 to the fourth DW mega and this is in fact less than infinity why this is bounded between 1 and minus 1 the kernel dies like 1 over you know Omega cubed right you the problematic case would be 1 over Omega right do you remember doing in calculus so if I have a paint can that goes like one over Omega can I paint the outside or how's it go does it take an infinite amount of paint paint the outside or an enfant amount of paint to fill it in one case there's a finite amount of paint maybe to paint an infinite amount of paint to fill I can't remember what it was does any member yeah so you can fill it just can it goes like 1 over Omega with a finite amount of paint but if you try to paint it you have an instrument of pain crazy so don't try to paint cans like that all right so that's the kind of thing it's the one over make a case but this goes like 1 over mega cubed it's clearly gonna be bounded finite this is in fact an invisible class so that's the kind of games you play with things like this other cool things this this is one of the more remarkable properties I have to say and this is it's partly because of this property that people get carried away ok this is why we have so many different wavelets to some extent supposed to have a wavelet a tried and trued wavelet just like something I define there it satisfies admissibility now let's just consider another function feet or a sigh Phi Phi that one alright right consider Phi and suppose this Phi of T is just bounded integrable function okay then convolution of cyan fee is another wavelet now of course you could say wow that means I have a lot of potential for creating wavelets in fact you do right I mean I get to start with the horror I can take any integral function and just keep making any many as many ways I want right and people have done this they've played around but what are you trying to do in doing that you're trying to get the best time and frequency resolution jointly that you possibly can okay I'm gonna show you some wavelets on Monday that when you look at them go like that makes no sense at all like it's completely non-intuitive but they have great properties okay and you construct these things in chronic crazy ways the dobashi wavelets if you've ever seen these things they're like that doesn't even make sense it's like just crazy looking function but who cares what it looks like in the end you just gonna use a piece of software to create it expand your basis in there and go on right and the product it's it's more important what the properties of localization and time and space are okay I have some examples in the notes of how to construct some other ones but we've got things to do we got other things to move on - all right so row how you doing back there yeah you ready for the weekend who's got the best plans for the weekend yeah it's Friday at 4:30 we can take a minute to talk about this okay who was going clubbing tonight everybody people home I hope you have better plans than this sad come on nobody who at least might go to the bar for a drink one two three three there should really be more of you dang okay exactly who's gonna hug the pillow and we suavely violently we violently into the night all right all right great I'll stop making fun maybe it's uh thanks well that was a good one I should have been more courageous with that with that said you ready here we go this is just getting ready for the party what do you think the first word complete the sentence hat wavelet party anybody guessing come on I heard it maybe MMX it Mexican Hat party I mean Mexican hat wavelet okay okay all right you guys are not long with me yet Mexican hat wavelets here's what they look like it's like they're pretty important especially seismology this guy morally kind of played around with these things it turns out they have really great resolution properties for multiple scales for the seismology problems and essentially how they're defined five t equal to 1 minus G squared and if I were to plot that did it I did it it uh no and it was on there Mexican hat all right everyone Matt no you guys should go to the bar these are my students and they my homework to you is you go to the bar give me a report by Saturday morning not early though cuz I'm really didn't go then okay there is the Mexican hat wavelet localized beautifully it has some width and we just translate it and use the width before you know basically define everything on this thing by the way so this is sy its Fourier transform so I have a mega go to PI Omega squared either minus Omega squared over 2 before a transform is a double bump thing looking like this okay so what I could do with this is basically say look I know how to construct this thing and I know how to change its width and its center position right change the center position you just changed T to t minus B change its width by just putting a scaling primer under the Gaussian an a so I can make these thinner fatter whatever I want to do with them great localization satisfies all the properties of the wavelet and this thing here then becomes a really nice wavelet basis and by the way if I've any integral function I can just convolute it with it and get another way for it right so this is one of the more famous of the wavelets okay this is quite old and so I just want you to be educated about Mexican hat parties that you're all gonna go to after this right okay all right all right I don't think you guys are getting prepared for the final yet because the final is gonna be on Friday last day of class so we're gonna go out to the big turn not big time college in I'm gonna get the back room there we all just have a final which means we're all there and we the final is this to you know have fun with me with each other okay put that on your calendar start getting prepared for it now because you guys looks like you need some help with that all right now here's what's important about these wavelets time frequency localization you can calculate time frequency localization so Sigma T we did this by the way for the Gabor for the Gabor we said if I have a wavelet window how localizing time space is it same thing here all this is is you calculate the second moment around some mean so you say I look around some mean time T T bracket I want to go look there and see how much energy is localized around that spot and here it is I want to look at how much energy is around some frequency there it is these are important quantities because it tells you how widely spread the stuff is in time and frequency and these relationships are closed off to calculating energy in each one of these cells how much energy is at some time frequency component well you can do a little integral around that point - okay those are some good properties to know another property knows how to invert it right that's kind of always important here's how you get the wavelet transform of a function to get it back to invert this thing's get back F of T this is why you need that C coefficient to be bounded okay because conversion formula has it in it one over C so you want this thing to be finite not zero obviously it's not gonna be zero and you do a double integral from negative infinity to infinity of you take this wavelet transform here of a and B and you just multiply it by DB be the a over a squared there it is this is the inversion formula you integrate over all of your translations all of your scalings pipe signal reconstruct it okay there's the process wavelet transform inverse wavelet transform and there's a discrete version of this ultimately what you're going to work on is a lattice you're going to break up into main to frequency components and time components and once you've broken it into this part of there's always as pictured that they basically paint for you paint draw whatever you want to do it paint nobody paints the picture in a book anymore they plot it there that's what I was looking for plotting a picture they plot this foot picture here are you see if I'm going to want five one two three four five one two five these are the B values how you slide things left-right and you do it in discrete fashion so for instance on your first pass you would slide your window and every one of these points why because here on this scale I'm plotting the log base two of a so if a is one you're here it's the whole window and now I'm gonna get better resolution or it's sorry now so if I go up this way I'm making fatter and fatter wavelets okay so here now because I got a fatter wavelet I only need to do every other point to cover that range that makes sense I go up one more level I've doubled my fatness to the wavelet again covers more of a so then I only need the sample these so there's this hierarchical way of thinking right you do big sampling you cut it you more points more this is the same idea this plot in some sense which you will see a lot in wavelet books I want you just I'm doing it just for kind of completeness this plot here is in some sense just to take on that plot right there big domain cut in half but what I did is I drew it from this side up so I turned it upside down but that's typically once those people draw it that way so I just want you to if you see that in a book you won't get all confused you'll say oh that's that and this guy's all right with that we will end I do hope you have an exciting weekend better than you're looking like you might have right now because right now it looks like you might have a pathetic weekend but there's always hope that maybe in an hour you'll be feeling better maybe you'll have a Chipotle burrito and life will look so rosy after that one can only hope Huggle weekend guys

Channel: Nathan Kutz

Views: 59,271

Rating: 4.9764705 out of 5

Keywords: Kutz, Nathan Kutz, wavelets, time-frequency analysis

Id: ViZYXxuxUKA

Length: 51min 46sec (3106 seconds)

Published: Thu May 10 2018