What is Convolution? And Two Examples where it arises

so what is convolution well fundamentally it's an operation that takes two functions or two signals and produces a third one and it does it in a particular way so let's look into it here let's take these two signals here's a signal which is zero before time equals zero and then ramps up and then dies down and let's to call that time one for example and here's another which is zero for all time except a spike at time equals zero and we're going to see what the output is going to be from a convolution of these two signals so it often helps to think of this as the what we call the impulse response of a linear system and this is the input now think of the think of a system imagine that this is the response of that system the output of that system if you put a spike of energy a short spike of energy into that system maybe it ramps up its output and then its output dies down think of it in that way for this example well if that was the case then when we put this input into this system you're going to get this output exactly the impulse response now what happens if we put a second impulse let's say we put a second impulse into our system well if this is the case and we're going to put it in at time equals one so if this is a linear system against then the uh what happens is and if it's time invariant so we call this an linear time invariant system lti system then when we put a second impulse into our system this response is going to happen but it's going to start after the impulse so here it is here and because it's time invariant the response at a later time is the same as the response from the earlier time so we're going to get this if we had another impulse that we put into our system at time equals 2 then we would get another response and so on so this is the output yt of a linear time invariant system if these are the inputs and this is the impulse response i think the that that's intuitive that this would be the output okay what if we decided to put the input at a different times instead of not one and two maybe we put them closer together so if we put our inputs now at let me draw it under here let me say that we had our inputs now so this is a pair here i'm going to draw them say that those two are a pair this is the input into our system that gives this as the output so if we had now an input at zero a half and one instead of one zero one and two zero a half and one then what would the output be well the output now would be because it's linear they're going to add and it would be the same effect as here except they'd now be closer together so now we'd have one output coming at this time i'm just going to draw the components here a second component here and a third component here and so now because they're linear so if we're looking at a linear system they're going to add up so the overall effect will be the addition of these three components and we'd be getting this and then a straight line across here because as that goes down this goes up a straight line across here because that goes down this goes up and then we'll be going down again so the overall function output of our system if we had this system with this impulse response then the output of our system here if this was the input would be this function here okay now this let me write down the components of this system here so what is this waveform here this waveform here is x naught times h of t that's what that triangle there is and then this triangle we're going to be adding because they're adding its linear time invariant plus the second triangle which is x of 1 so it's the height of x at time 1 multiplied by this impulse response so that's h of it's it's this impulse response but shifted in time by one so it's t minus one plus this one here which is x at time two times h of t minus two okay so these are the three components from three delta function inputs gives us this result which is this in mathematics okay now here we can see we've got exactly the same except they're closer together so they're overlapping in the output but it's still the same aspect you've still got three components here that are adding it's just that instead of it being at naught one and two they're now at naught half and one and so this would be t minus a half and this would be t minus 1. so this gives us a general function if we had our delta functions if we had an infinite number of them and i'm going to now think of that so let's think of general convolution not just from delta function inputs but any two signals so let me look at an input signal so this is still we're still going to consider this for our input function here for our sorry for our impulse response here but now let's look at a signal which might look like this for example okay so if this was our input signal so let's uh let's just say we're going to x and y let's call this z t so let's say we're going to put zt into our system now so this is our system and we're going to put a signal into that system if it's a linear time invariant system one way to think about this is to think about this as being made up of an infinite number of delta functions infinitely close together and that's the way i think about all of the functions actually if i'm thinking about linear time invariant functions so let's think there's an infinite number of delta functions infinitely close together so we are going to get a response which is going to be one of these triangles for each of these delta functions just exactly as we had one for each of the delta functions here and one for each of the delta functions here in our output so now we've got an infinite number of them because there's an infinite we're going to have to scale the amplitude let's not worry about that too much now but just in concept they're going to be infinitely close together and then for each one of them we're going to get this output response and they're going to change in height because as this changes in height these output response triangles will change in height and i'm trying to match them with the height above uh they're responding from that height is going to be responsive to the triangle that comes next so they're starting to get bigger now here and they're all adding together to give the overall output which is the addition of all of these components here when you add up those triangles there you get a value up here add up all those triangles you get a value here okay so this is going to be let's call that wt so this is wt is the output of our system when we have an input zt convolved with the impulse response h t and the general equation here we can just see a generalization of this equation here now there's an infinite number here so instead of a pluses of individual terms we're going to have an integral which is also an addition but it's a of things finitely close together of x of tor h of t minus tau and integrated over tor so that's the generalization of this expression when you have them infinitely close together so this is convolution as applied to signals and systems for a linear time invariant system if you want more information on this equation you can check out the video in the link below which explains this equation in more detail convolution doesn't just happen for linear time invariant systems though as i said it's a general operation that takes two waveforms or two functions and produces a third according to this equation here okay so where i want to give you one other example of where it comes up and this is a common example in digital communication systems so let's consider as i say it's not just linear time in variant systems so let's look at another example and in this example we're going to consider digital communications so let's consider digital communication system where the output signal of a digital data bit let's just i'm just going to use a capitals here these is that these are random numbers now equals the input plus noise so this is what happens in a digital communication system okay so the input x is either plus or minus one in a binary digital communication system and noise is typically gaussian noise and again there's a link to a video below which talks about gaussian noise and white gaussian noise in communication systems but let's look at these two x and n and i want to look at the probability density function so in the case of x uh because it's random plus and minus ones if this is the probability density function pdf for x then for values of x the only takes this is the values of x here and this is the probability of getting that value then x can only take the values in a binary system let's say for example that it's either plus or minus one okay so this is the way we draw the pdf for this is it's it's a delt it's two delta functions one at plus one and one at minus one and there's no there's zero in between these delta functions because the only possibilities for the value of x in our digital system is plus one or minus one to represent a digital one or a zero so this is the input to our system a one or a zero in binary digital communications and then noise is added let's draw the pdf for the noise the probability density function for the noise so this one for the noise so this is the pdf of noise so again that's a half because there's a half probability half chance of wanting to send a one and half chance of wanting to send a zero you never know what the data is you're going to be wanting to send until you've measured the signal or recorded my voice and digitized it or whatever so there's a half chance and okay so let's look at the noise and typically as explained in those other videos the noise has a gaussian probability density function what that means is there's the it's quite this is the probability of getting this value of noise so what it says is there's a small probability that you'll get a big value of noise either positive or negative a small probability but there's a bigger probability of getting a small value of noise so the noise typically isn't too big sometimes very rarely it's very big either positive or negative mostly it's around zero and it has this gaussian shape okay so where does why am i telling about this for convolution well it turns out the the pdf of y is the convolution of these two pdfs so if you have two random variables and you add them together if they're independent random variables then if you add them together so i'm going to write independent up here so if they're independent random variables and you add them together then the probability density function of the result is the convolution of the two probability density functions so this is another place where convolution comes up so let's look at that function here what have we what's that going to look like well we're going to have the values of y that we can take and the pdf of y here of y capital y the random variable capital y well when you do the convolution of a function with a delta function it takes that function and it puts the zero of that function over the top of the delta function we actually saw that over here see this function over here when we did a convolution of this triangle with the delta function it took the zero from this function which was here the start of the triangle and placed it at the location of the delta function so this triangle moved so that the zero part was placed at the location of that as this one and then again at the location of that the zero was placed here that's that triangle and so it's a general property of convolution and so here we take the convolution of these two we're going to take this this gaussian shape is going to be located over this delta function and this delta function and of course they're added together just as these were added together and they were added together down here because it's linear and added together here the same thing happens now in the pdf space so this is minus one and plus one and so the overall probability density function for the output of a digital communication system when you have binary inputs looks like this so there's an equal chance of getting a plus one and an equal chance of getting a minus one and the area under this curve is the probability so the area under this is the probability of getting a plus one and the area on this side is the probability of getting a minus one uh because you're adding up all the probability for values which you would detect to be plus one and all the probability for those would take to be minus one this is a symmetric function it's come about because of the convolution of the two and i think you can see you would expect to get plus ones but never exactly because of noise so you get them close to there very rarely so low probability you're going to get a very high value a very low probability you'll get a very large negative value small probability you'll get something on the border of zero where you couldn't really tell if it's a plus one or a minus one but mostly you're going to get values around the plus one and the minus one because of the noise so this is another example where convolution arises it's not just in linear time invariant systems for the inputs and outputs of a system it also arises when you add random variables and you see the output of the probability density function as a is a convolution of the two inputs so if this video helped you don't forget to like the video it helps others to find it subscribe to the channel for more videos and check out the webpage on the link

Channel: Iain Explains Signals, Systems, and Digital Comms

Views: 54,761

Keywords: Signals and Systems, Communications, Convolution, LTI, PDF, Linear Time Invariant, Probability Density Function

Id: X2cJ8vAc0MU

Length: 15min 47sec (947 seconds)

Published: Tue Oct 20 2020