Time Frequency Analysis & Gabor Transforms

okay today well here's we're gonna do we're gonna do a little bit of MATLAB programming the objective will be to take some data we're gonna make up a little data function and yeah the nice thing about data drawing it it's a lot easier to draw than actually like make up an artificial one we have some function of time or you have to have someone who has real data to give you right but even like something like a music file is actually huge why isn't it I think if you actually take the data from that it's it can be it's a lot right so you could get it all the time resolution and so forth so we're just take some of this data and we're gonna do a couple things the morning lecture today what we're gonna do is we're going to build our own little Gabor wavelet Gabor transform we're gonna put this on here you know rock that back and forth look at time and frequency okay so mostly the idea here is to try to construct this time frequency picture we've been talking about all of last week okay we'll talk a little bit then also the connection of wavelets because all we're gonna do now is take this thing ynette and shorten it okay so that's basically a wavelet so look at how that changes what the time frequency idea looks like as well okay so we'll do that this afternoon's lecture at 4:00 I will then show you what you have available to you on MATLAB okay professional grade time frequency analysis tools in fact there's a bunch of toolboxes there is the wavelet toolbox there is the signal processing toolbox there is the filter design toolbox okay all these things kind of doing all the same thing but you got to pay three prices okay but if you have the student version it's like 29 bucks each or something like that just a little FYI for you hopefully many of you have it to be your department's most we want to do show you the kind of things that they have in them we won't be using them so much in class for homeworks partly because it's too much to expect that everybody can have access to these two boxes also the homework 3e will be up this afternoon in homework 3 is gonna allow is gonna be something that you're gonna basically construct your own little wave that things start looking very carefully at some data so you'll do some time signal processing yourself okay and I may I'll probably have you do that also on two-dimensional data until you can start looking at stuff like that alright so that's a dia for today right there sliding filters time frequency analysis everybody with that okay so let's turn on the projector I never know how long I'm gonna talk here but I don't like to turn that on because it leaves a you know you need one of those buttons that you can say you know shuts the projector without turning it off yeah oh say is it on there oh okay so that's I gotta learn how to work that okay and check it out I will we were talking one of my students he he does he takes it edge lectures and he and he takes him from a wave file and makes him mp4 you can watch him on iPod goes out Lawson he called it the 582 on the go that would be awesome right and I he was playing it on the plane and everybody was over there looking man what you got playing on there's at the coolest movie ever it's like the stars on the plane anyway I'm just saying well just an idea yeah there we go yeah alright but I'm worth it I'm sure you can convince him all right so here we are our beautiful MATLAB program and we're gonna make some cool stuff happen today whoo all right we've pumped up Kristin back there okay you get breakfast champions you should be yelling louder than everybody because you got breakfast alright so first of all here's we're gonna do start off with our very simple nice clear I'll close all could screen command so we start with a fresh slate every time we run this thing and now I want to do is take a demain let's start off with something here domain size gonna be 10 I'm just gonna take a box slice 10 now this could represent 10 seconds 10 hours and microseconds right so whatever the whatever the units you are working with that's your time frame that you've got you've got a signal of certain duration 10 and I'm gonna have I'm going to break this up because I want to use a some F of T's into 2048 points now typically this number is given by experiments these are your number of measurements time if I've got 10 seconds I'm sampling every millisecond that's gonna determine what that number is but right now we're just making up data fine and then I make up some crazy functions oh well okay first of all I define define a linear space it goes from zero to L and plus 1 points and I'm interested in the first one through n so it's gonna be since I'm using Fourier transforms right now I think about this being periodic essentially what you use the FFTs you are assuming periodicity so I'll make the time domain be the first one through endpoints standard and I define some function there's a 3 sine 2t tangent shifted over by a hyperbolic tangent shift over by 3 an exponent Gaussian another sine a cosine with an argument squared okay stuff like this now let's just see what this looks like you've already seen this function because I actually I just recycled it from my notes from lecture Wednesday lecture I think okay so let's just plot T versus s okay all right so it's not even that crazy looking a little signal alright but there it is and the important thing that this Stila straights first of all here's the kind of growth features you can see in this signal there is some kind of long scale stuff going on so low frequencies are involved right that happen over this ten seconds that's kind of clear to see there's a sort of wave that goes through there there's clearly over here or some kind of there's lack of high frequencies yeah over here very high frequencies here high frequencies again here not much going on here except flow frequency stuff so it's because we've made up some kind of crazy function most the time what you get is you get a noisy signal so there's noise on top of this too now we don't necessarily need noise on top of this right now we can but we'll just start off this right now okay and we want to analyze the time frequency domain of this signal by the way there is a very good chance you've already seen time frequency analysis you just didn't perhaps recognize it as such okay who loves whales now if you don't raise your hand you're a terrible person well bah buh I mean we all love like wheels I mean that goes without question Wow pedda who's gonna have a field day with you guys cuz like nobody raised their hand about well I'm gonna call them up after this class you guys all go into counseling I'm sure the pet of people will pay for it too so you can love whales uh the reason I bring up whales is because well not quite whales yet here let's look at this you see this figure I brought up here on the screen this is a whale doing its whale it's a killer whale talking he's saying he's wrapping up some dr.dre right here and this is what it looks like some three different things he's saying like this is nothing but a G thang baby right there okay no I don't really say here's a distant part of song but that's how it comes out looking in whale --is-- here's time here's frequency this is a time frequency analysis this is what we'll plot when you want to analyze speech signal patterns okay you don't look at this little thing that just looks all squiggly you put it on here and you start seeing what frequency is involved when the sounds are coming out how does it play out so you've probably seen these have you seen any of these before you know they always talk I don't know I just generically wail make music and I have whale CDs and you've probably accidentally seen a spectrogram this is a spectrogram you can send one to your loved one spectrum it's like a singing spectrogram okay everybody with that here's another one this is of a male human saying that tah-dah the web heyo frequency time there's your spectrogram okay so if you're in speech recognition this is the kind of stuff you look at all day okay when someone says it has a certain signature in a time frequency domain so what you do well you build something that recognizes that spectral time pattern converts it to tough right so you can do the whole alphabet you can do all these kind of words this is how you sift through things and you make speech recognition okay speech recognition doesn't just take a bunch of sound waves and say oh that's the it you put it on here you start doing statistical decision and processing of these signals and that's what converts it over to trying to say oh that I think you know when you do speech recognition it types up on your keyboard I have a very simple speech recognition right on my iPhone I push this button I say I'll see you at ten o'clock meet me at the hub but pump pops up and then I can send it as an email or a text right so even in speech recognition is quite advanced and it's all it's looking for is these kind of spectrogram patterns of speech okay many of them actually learn what's specific to you suppose if you're from the south you might have a twang and so it might have a different pattern that you would want to be generating here or Jersey Erica do you don't even have in Jersey accent you know do you need a cup of water do you need some water so that would be like water and water or you know they they probably have two different signatures all right so this is what we want to build how do you build this you throw a gabor wind it on this thing you translate it in time pull out frequencies plot it that's it that's what you're doing okay so all the stuff we talked about in lecture Wednesday is just that's it right there so we're gonna build this ourselves right now with that signal I just created okay so let's bring this back up all right do you feel motivated you see how I brought that in about whales and saving them that's awesome all right I should have worn a Greenpeace tie and then you guys be all like oh and today we saved whales in class that's right because we can know how they talk all right plot TNS we did let's go back to what we had there it is now what's important about this signal is that different parts have very different behaviors here in this region high frequencies low frequencies high frequencies and then there's some kind of overall low frequency pattern and the whole thing let's look at the Fourier transform of this thing okay first of all that's the one thing that we know how to do is Fourier transform things right we almost a little too casually can for transform anything and you get to spectrum and you just kind of say look at that there it is oh here's all the frequency components so first let's a define our frequency domain two PI over L times zero to n over two minus one minus n over two okay there's our K values so there are frequency components for the FFT and let's shift on let's call KS as a shifted version because what we want to print plot it against FFT deep shift of K so now what we can do is take this signal let me say all right Fourier transform the signal s T is just FFT of s and now what I can do let's start doing some sub plots here so figure two is gonna be a plot of KS the frequency components versus you guys probably can't see those what let me bring that up with those that better yeah a little higher up on the screen okay frequency components first is the Fourier transform which we have to take the FFT shift of this thing sorry that it wasn't even on purpose although it's pretty awesome that I messed up there we go FFT shift that absolute value that's gonna give us the Fourier components of this thing so now let's look here well oh I needed something to shift okay there we go all right there we go take that signal there's this boy a decomposition now let's zoom in a little bit here I have a lot of Fourier modes and all that's concentrated down here so that's a little bit better picture of it and by the way you'll probably recognize that from the notes that picture there I actually have that notes so you can see this isn't made up that's it you know made up with a lot of actually 40 components in fact but nothing beyond about K equals 50 this is like cosine 50 X or something like that but nice set of components there you see some interesting features like two large Peaks here okay at low frequency and this would be this thing that's sitting underneath the whole thing so you see a lot of the stuff here and in fact this is a complete characterization of all frequency components in that signal I showed you but as we talked about the whole reason you do time frequency analysis is because this thing here yes I have some frequencies like that but I have no idea where it happens in time on the signal if this thing is 10 seconds what I do care about is hey I'm looking at this high frequency spike right here for instance suppose that's of interest to me I have no idea where that happened in the 10 second for audio okay so the Fourier transform gives you all spectrum from at the price of sacrificing all time information okay so let's start working then with the Gabor transform because that's what we are trying to do here now yes say okay what am I gonna do with this Gabor transform what I want to do now is I want to basically filter this in time take little sections of it so let's define like a little Gaussian as this simplest Gabor we could start with G is the Gabor I think it's little G in the notes so let's start with that it's just a Gaussian and first let's plot what this looks thing might look like it's its width and so forth so it's t minus 5 let's Center it on this signal s and we're only gonna plot the first plot here and here's what we're gonna do figure 1 is going to be a sub subplot of T of s and that's going to be our signal in black and then we're going to put this filter on it this T of G in red and the new signal is basically s G let's call it that for the signal with the Gabor on it is going to be G times s ok so I just multiplied my signal by this good for a Gaussian and I get a new signal and now we're gonna plot that in sub plot so I got to put 3 1 1 3 1 2 here we're gonna plot T versus s G alright so now what this s G is this is my Gabor signal and I can get aboard but we just get board it so here it is you take the signal and you multiply it by that window you get to pick the width of the window you get the pic where it lives and the whole idea behind Bohr is to take this window slide it back and forth okay if I multiply these two together this is the little piece of the wave that I pull out around time equals five okay so in some sense I can think about look at time equals five I can pull this out and now what I want to do is look at the Fourier components in this part of the signal okay this is the way we're gonna construct a spectrogram because right now I can say okay at time five the Fourier components are well let's do it let's take the Fourier transform now of what's in that little Gabor window which is sgt done fourier transform that let's make that sub plot three one three plot now i'm going to plot here is KS versus the F absolute value of FFT shift of sgt is GT okay that's right and and let's close up the Fourier domain a bit here so on this one I missed a axis there was nothing really outside of negative 52 50 so let's just say negative 52 50 and let's normalize it so I can always have it between 1 & 1 divided by the absolute value of GT Oh max so I'm gonna normalize the height of this thing so between 0 & 1 alright alright so now we're getting the idea of this thing there's making sense to people I take my signal I take a small portion of the signal by choosing where to place my red filter in time okay so it's like the filtering process we learned about which is in your saved your dog with it right okay if you saved your dog what you did is you did a filter in frequency now what we're doing with the bore is a filter in time to pull out a small portion of the signal there it is and look at its Fourier components there they are this is one window of time this corresponds to one slice of my spectrogram does it make sense this is gonna be at time five okay let me draw I like drawing here we go ready here's what we're gonna try to construct today with a code is the idea of the spectrogram time and frequency right now what I just did is I filtered at time five and I have the frequency components in that window so now what I can do is okay great now let's filter it a different time a different time and the construct these few components put it on a 3d grid and I have my first spectrogram that's beautiful okay anybody with that that's what we're heading for that's the big goal all right I can see you're all on board it's great all right yeah in your homework so but yeah exactly it so you have two degrees of freedom here how well do you want to resolve across time and with what with do you want to do this with right and the problem with the Gabor is right away you can see the problem with the Gabor is that once you once you pick this red there's no way that this red for instance it's it picks out just a small time piece of the window the one thing that you throw away is all this long scale stuff because your window says doesn't exist by the way that tells you the same thing there's two big spikes right around negative 2 & 2 we're actually they're they're a little bit closer in those does get kind of washed out because the very low frequency stuff lives outside of that window and your Fourier transform doesn't handle it because you killed it off okay so when we make a little movie you ready this class is exciting today right who would have thought you'd come watch movies in class like last quarter we always hear movies downstairs today we got movies no soundtrack don't worry maybe I'll play some whale songs for you later but right now we're gonna watch a movie silent film in color first we have to build it right all right so here's what we're gonna build we're gonna make our MATLAB do the following we're gonna take this and actively keep put this in the cycle and just take our red and slide it and as we slide it were gonna watch what this thing is and what this thing is and how they evolve and then we'll construct a 3d plot of those things okay all right who's super excited at this point sweet three people four whoo I've awesome I was only expecting one who I am on a roll today all right so here we go ready figuring what I'm gonna loop this hey you know dr. Dre does the dr. pepper commercial recently and he's always talking I loop in it cuz you got a little thing and we're looping it too so it's awesome I'm just trying to make those connections for you just in case you weren't seeing him all right so let's let's do the following here is the big thing the Gabor window is what we want to slide so let's bring this into our loop all right and we're not gonna slide it anymore by five this is going to be a variable so let's make a vector of how we want to slide this thing so let's do T slide that's our sliding time it's gonna first start at zero and we're going to slide it we have to go all the way to ten steps up point to slide it's too well anyway you'll see a cool movie with lots of lots of frames okay maybe this to me let's do five point five zero point five one one point five this is where this thing is going to be centered and so I'm going to go do this for want a length of T slide and what I'm gonna do is T slide J time I come in here but I better put an end statement here so this is the way this is gonna work you've decided how what kind of increments in time you want to slide by these increments in time by the way depend very strongly on how narrow your window is or how broad it is okay we'll talk about that in a second but where we decided slide it from 0 to 10 and 1/2 scale you know so just need 12 21 slides and what we do with that it would say okay slide it so I take my Gaussian centered around that point I multiply it by my original single signal for H transform it now I plot everything and we're gonna draw it now and then it's gonna come back through do the next one to the next one so you just actively see this red thing just slide through this thing and it's great be so awesome alright okay here we go so the draw now command is a nice one because it tells it to not wait to draw it so sometimes by the way why do I do that you kind of would think oh it's obvious it's gonna do this plot it and then go to the next step no sometimes I was doing a lot of crunching in a TED MATLAB will so yeah you want stuff plotted but wait I'm still doing this so put a little figure box up there and we'll never plot it until everything else is done the draw analysis no no plot it now you are master MATLAB gives you a hard time who's the master you guys go see I am who's the master oh thank you very much look good that's good right if we this is good by week ten you guys gonna know I'm gonna be talking to your computer you're probably a little shy to talk to your computer this way but come on you're the master so you tell it you tell this guy when you want to plot how you want to plot with what kind of lines you want to plot you dominate that's how we're dominate right now we're doing a draw now we can even say pause on that plot for a tenth of a second or list it to 2/10 of a second okay so it's going to plot it see it move it move it move it so let's see if we did this right we should see a sweet little movie here there it is look at that thing walking across whoa dang wasn't that the coolest thing you ever saw in your whole life okay okay maybe not your whole life but pretty close now let's do it we can do point one and let's just have it pause just a little bit longer because that was in fast motion okay there it is there's my Magoo bore windows sliding across here's the little signal pieces it takes out look what the for you how time and frequency are changing there you go we should lock that in so amich they're looking one it has to rescale as we go across but that's giving you all this time frequency information and that's is exactly what windowed Fourier transforms do and this is the heart of analyzing your data and time frequency domain okay essentially what we just did here this plot right here was looking at these little sections right so we are looking at like making a movie of this slice this slices dices this slice so now what we want to do bring it all together with your hand no okay yeah you bring it together and you make 3d plot P color from the top and then we'll have a spectrogram okay is that kind of nice all right time frequency it tells you all the frequency components and then we'll start playing around a little bit with widths and so forth by the way one thing to notice right away let me show you something very important that like that's happening here you see this thing sliding across when I slide I actually do a lot of over sampling I slide just a little bit so all the frequency components from this window are still mostly in the next window right time and frequency are very closely matched a very good resolution if I do a lot of over sampling like this now there's my filter moves very slowly across this gives me a very accurate representation of time frequency in other words I choose how I want to slide my filter if I did my filter here one here like an under sample as well which is I missed this whole region in between right we'll do that in a minute as well so let's build some spectral grams and we'll come to that okay so we'll keep that plot going and now what I want to do is make so what we want to do now is every time we get this information this s DT which is the important thing what we're gonna do now is say oh by the way okay now that's gonna be important when we plotted this thing we always normalized the frequency components their strengths to one so now what we want to do is have an absolute scale of these frequency components at least on the if we're going to do a plot of the full spectrogram right you may have a very intense signal at time one which all weakens as I'm - so you have to make sure to see that time one is more intense in time - for instance and so you've got to make sure not you don't you got to take out that scaling there because now you're gonna look at all the frequency and all the time components on one graph okay so that's one thing but this is the quantity of interest okay and what we can simply ask about sgt is the following what is it sighs sighs sgt it's a 1 by 2048 matrix which dr. which we know right let's go it's one row 2048 columns because we've did our data in 2048 points before he transforms still 2048 points well we want to do those now right every single one of these as a row of a big matrix with how many rows well number of rows is determined by the number of slices I took okay all right so here's what we do we make the s let's call it s g t spec that's for our spectrogram okay and what this is is we're going to start off by the way I'm gonna collect all this data SG s GT spec first of all is nothing in it it's a spectrogram it's empty now when I first create my first sgt i simply add that row to what I have so it's what it was before and in the next row I want to add s GT I think I think it's right so I start with one row of data if s GT it's just as empty just begin with but what if I have a row of data women do is every row I get put it on the bottom of that so we take sgt spec is what it was before so as they start building it all I do is say take what happened before return character go to the next row place the new row in their place in your own there we go with that so so this thing here should have a total of I've done this here now with what 100 101 101 slices so this would be 101 it's it's a 101 rows by 2048 columns that's my data entry good with that at least in this construction all right provided I did that right and by the way let's save not sgt but the fft shift of sgt and it's absolute value here okay so that's we're gonna save ya Oh s PT sgz yeah thanks alright ready well okay so that's gonna be that and then at the end of the day after we get all that information we can just do a pea color on sgt spec yeah top view of our spectrogram see if we did it right now by we're gonna have it whip through here very quickly oh I got to do oh sorry sorry let me figure - okay figure - so here we go we walk it across we're starting collect all our spectrogram data look how the frequency changes quite a bit different components different times and the signal pulling everything out and and and and if we right okay and I know why it's black don't worry here's why it's black you want to know why I got the answer for you okay when you do pea color what it generically does is it draws okay I've cut it up into 2048 points this way and there's 2048 lines and there's 101 this way now if you try to fit 2048 black lines on that minor space what do you get a big thing of black okay so we need to do is shading interpret shading in Turk will take out the lines smooth anything over all right so what we do is shading a Terp so let's do this except for its this is time and that's frequency so we just need to rotate it it's still looking kinda cool so if all right everybody cool with this yeah all right so now what we do is we look perhaps edits let's try that oh I need to run it again sorry anyway you can never get tired of seeing this again though I mean seriously this is the sweetest thing you've ever saw today you guys should just go home call it a day why even buddy other classes you know it's on you're only gonna go downhill from here there is your first homemade spectrogram give it up for yourselves no okay anyway let's also work on the axis and some cool colors maybe maybe you're not ready to quite give it up yet after we do this we do some access settings on this maybe you'll be maybe you'll be quick okay with that you'll be comfortable with that so here we go what we want to do is set GCA the Y limits to negative 50 to 50 and then let's do color map hot all right oh I mess up I mean I didn't mess it up I just clearly I was just seeing if you guys saw where I've made a typo mad when you see anything come on guys help me out bail me out I'm in trouble here come on dudes come through for me all right anyway I do want to just go from negative fifty to fifty don't I just do the exilim yeah what's going on with that so if I take this out okay let's just go back to here real quick we're all good with this right okay there's that oh yeah that's right okay so but I haven't actually put on axes here so here's what we want to do is actually put our time which was tee Tee tee slide right this is a function T slide no so I actually I yeah we have T slide times right that's the times what we have actual data on and the other direction is chaos okay and then I can set sit GC a deleted it as it was what it was plotting because I didn't put the XY coordinates here it was just plotting oh there's thousand 2048 points this way and there's 101 this way so it just plots it as first versus 101 versus 2048 yes no turns out that it knows that if you do it as a vector I'm sorry I didn't need to run it again but again like I said nobody can get sick of this right all right keep going keep going we're gonna do this again Doe with a narrower and a broader little window here in a minute dang bow to the power super-sweet hardly that you built your first spectrogram and who wouldn't want to get that in the mail like you know send that off sign it love Erica here's your task for today you're gonna take this gonna print you no sign of Erica your mom dear mom I love you graduate school is fun I'm learning all kinds of things this is a whale song well you got one on four little hearts for you buddy little gold stars it's coming your way don't worry you'll be delivered in a coughs now will basement of lo alright so there you go this is what people look at that's your toll time frequency domain picture time frequency gives you all the intensity of the signal and at what time the signal occurred great perfect tool for analyzing so now we can build a speech recognition suppose that crazy signal was somehow a a P is at P and that's how it comes out well they say oh that's a characteristic for that when I get that I'm special patient I can kind of save that away somewhere and I can now start to recognize it by the way that's where we're going next after next week we're gonna end up and starting to start asking these questions how would you recognize stuff like this how would you make some statistical decisions based upon time frequency information you have so that's where we're heading we still have the whole linear algebra component to bring into our data analysis that we haven't even touched yet and when we do super powerful you think a powerful now see wait all right let's play around with filters time filters let's look at this picture house you have it here now let's get to the question of for instance let's make our filter sharper let's make it broader let's see the kind of things we see all right so there we go uh so first thing maybe we could do it make this filter narrower how's that we like that idea so let's put a something like that yeah right we'll see alright anyways kind of there it is oh no we made it broader alright still fine we're gonna do a broader first that's why all right so now we're taking huge windows we lose some time accuracy look at the spectral structure but now it's going to be harder to localize signals it changes our spectrogram so what this is really telling us we've lost essentially remember there's a Heisenberg uncertainty principle that goes along with Fourier transforms the broader the window we have we have more accurate information on frequency content at the sacrifice of less accurate localization of where the signal is in time okay so we can trust this more this way and less this way okay that's that's what that says right so that's what this thing looks like if you take a broader signal so you've gained some a frequency resolution and I think these signatures are sharper here is that right if you look I think there was a bright spot here but these were not as bright before what are these these are the low-frequency components this is what that shorter window was throwing away now you want to see these go away who wants them go away yeah well anyway we're gonna yeah we're gonna see them go away because we're gonna take a very narrow filter now in time and the narrow filter now is not going to allow for long frequencies it's gonna throw them all out so once you throw them out it's like well okay these should kind of disappear or at least be attenuated a great deal because you're gonna have to good time resolution but poor frequency resolution alright so now let's go the other way there we go and go here we are there is my little window much smaller window pulling out smaller and smaller bits of the signal and same process and this is the kind of thing that you get to play with essentially when you do time frequency analysis you have to figure out what the optimum is check this out quite a different picture excellent time localization there is a big peak you remember how this looked before kind of washed out bad time localization now we get very good time localization we threw away a lot of frequency content in this process okay so that's that's that's the trade-offs you make it still looks pretty cool right we got holes in there I didn't expect that but they are have little holes let's make it even narrower let's go here to ten so you see this is nice it's getting very good resolution the width of that thing is you know less than a unit it's about half a unit you're pulling out quite a bit of information there and so you can get the good localization in time it's just that now you start losing a lot of the frequency aspects of the signal okay this is this frequency just started getting washed out there are almost remember there's two big frequency components here completely lost so if you go to these very short short windows you throw out any of the kind of low frequency components that might exist in your signal okay and those might be very important for you right so in all the pens so you can always but you can also build histograms or these spectrograms based upon hey I want to actually filter for low frequency content let's not worry about so much localization in time low frequency content is not localized in time that's the point right it's it's long okay so you get things like this all right what else should we do with this thing that is just one example let's do one more with a very broad window now this window here pulling out almost half the signal right that thing here is half the time about half the window so you see that you got a lot of free content and when you look at this thing look at the difference we could have plots both side by side when did this thing happen here in this big window you're gonna go like there was something happening here between time I don't know about 4 to 9 so you kind of know this 5 second window range that something would have happened in now when you go to the time domain picture right this thing's squeezed down to you know what happened that right around time 6 I so right now you have so it's good here you get leta good strong signatures of the Fourier components here they totally got washed out all this middle region gets washed out essentially by the short window so you get the good short window stuff here and this is where the wavelet comes okay so no so the wavelet idea is to say can't I kind of take advantage of both can I get good time and frequency by starting with a big window getting these things out now going to a shorter window getting the next things out getting the shorter window so I can actually sort of localize this and localize this that's what wavelets are trying to do for you okay so lecture 2 today 4 o'clock super awesome here and I'm actually going to introduce you to some of the MATLAB toolboxes that do some of this for you ok so I'll go through a little 3 toolboxes we'll just walk through them a little bit and then on Monday we start image processing ok

Channel: Nathan Kutz

Views: 14,479

Rating: 4.9007092 out of 5

Keywords: Kutz, Nathan Kutz, Gabor transform, wavelets, time-frequency analysis, short-time Fourier transform, Fourier transform, spectrogram

Id: 4WWvvMkFTw0

Length: 46min 38sec (2798 seconds)

Published: Thu May 10 2018