How are the Fourier Series, Fourier Transform, DTFT, DFT, FFT, LT and ZT Related?

so how are the fourier series the fourier transform the discrete time fourier transform the discrete fourier transform the fast fourier transform the laplace transform and the z transform all related to each other so let's start with the fourier series we're going to start in continuous time so this is where we have a t for continuous time and the fourier series applies to signals which are periodic so i'm going to draw a signal here which is periodic that means it repeats with a period and the period is important and is has a fundamental frequency omega naught so i'm going to write the equation here i won't write the equation for all of the transforms but i will for this one just to make a point here so this is from k equals minus infinity to infinity of a k times e to the j k omega naught t and so here omega naught is the fundamental frequency so what is the fourier series well again i'm not going to give all of the details there's plenty of videos on the channel you can find in the notes below this video but for this continuous time fourier series case with a periodic signal then the fourier series is written in terms of k so this axis here is k so it's not a frequency axis this is an index of these a's so we're plotting a k as a function of k so it's an index of the case and the index is related to multiples of the fundamental frequency okay and in this case it is discrete because these are discrete it's a discrete summation of multiples of the fundamental frequency components okay so it's discrete and it's a periodic so this is not a periodic uh function of k it's a periodic okay so this is one of the properties of the fourier series so when does it apply again the important thing is it applies for periodic signals and where the period here capital t equals 2 pi divided by that fundamental frequency omega naught okay so that's the fourier series in continuous time what about the fourier transform well the fourier transform is a generalization of the fourier series for signals which are not periodic and the most common signal that we or a very common one that we consider here is a square function it's very important for digital communications where you switch something on and switch something off and all sorts of other electronics and other situations so in this case what is the fourier transform okay it's not periodic and that's when the fourier transform applies it's more general than the fourier series and in this case you're probably familiar with the sink function i'm going to be drawing here the magnitude and this is important to remember that there is also a phase component but in this case some differences from the fourier series is that this is not this this is the actual frequency in plotted in radians when you plot with omega so this is the actual frequency and they're all different frequency components that are possible whereas for fourier series it was only the multiples of the fundamental frequency so that's a that's an important difference between the fourier series and the fourier transform so this is all frequencies are possible and they all can contribute to this non-periodic or a periodic signals so this is what we call the continuous time fourier transform so the ct ft and this one up here was the continuous time fourier series often ctfs okay so this here also it's important that i note here i'll put a magnitude with a dot there that there's also a phase component but we often only plot the magnitude of the fourier transform so it's just something important to remember and of course also this is continuous as i said with all possible frequencies and it's a periodic the signal is aperiodic in time and it's a periodic in frequency as well okay now uh one thing though we can define the fourier transform for periodic signals and so one common one that we do is that is one of the fundamental basis functions i'm just going to draw it here so this is the fourier transform for the periodic signal so typically one thing i should say for the fourier transform it's defined for signals which have finite energy now these periodic signals do not have finite energy but we can define them using the delta function we can define them in a particular way under the fourier transform so there is a definition for periodic signals under the fourier transform and in this case for this cos wave you might be familiar and this is where i mentioned with the delta function uh that it the fourier transform the magnitude of the fourier transform in the frequency domain is now actually continuous and it is impulsive so when you have a periodic signal if you are doing the fourier transform then you get an impulsive fourier transform we call it impulsive uh where we have these delta functions and this is omega naught and minus omega naught and again for more information on exactly what this is check out the videos in the link below okay so this is for the continuous time what about for the discrete time well discrete time we have things in we we use little typically use little n to index the discrete time and we're going to start with the fourier series in discrete time so again the fourier series relates to periodic signals so here's a signal that is periodic this signal here is periodic i'll just try and sketch one here that's periodic okay so this is a discrete time so it only exists for for uh integer values of n in the time domain so this is why it's called discrete time and it's periodic so we're looking at the fourier series this is the discrete time fourier series dtfs in this case it is a discrete similar to in continuous time this time it is discrete but it is periodic so this is turns out this is periodic so why is it periodic uh let's let me just draw something here that is uh periodic um it it's just a sketch representation uh it's not exactly the i'm not exactly trying to replicate the fourier trans fourier series for this function over here but i'm just trying to indicate that it is periodic so this is a periodic function here and the period of this function is n so this is capital n is the period of that function uh where you have the fourier uh the the period here relates to the uh the fundamental frequency sorry relates to 2 pi divided by capital n okay so this was the period over here also as n so this is the period n and this is in the frequency domain also has a period n actually i've i haven't sketched enough points in here to be exactly precise but i think you'll get the message so this is in the time domain the discrete time domain and the fourier series is discrete and periodic with period n so this is so one point to note as i mentioned the the basis functions repeat when you have discrete time signals and so for more information on why they repeat it's uh there's plenty of links in the description below uh where you'll be able to find information on that okay so now let's look at the equivalent to this one over here so now we've got the fourier transform in discrete time so what do we look at here and a common one equivalent to this one over here i'm just drawing it with an offset here because i've got to squeeze one more in on the discrete side so hopefully you recognize that this one is going to be equivalent and matching up to this one over here um in terms of that this is the discrete uh sorry the the um discrete time fourier transform so this is the transform line and this is the transform line on discrete time so here we've got it's important not to forget these zeros off here so in this case here this function here is a periodic therefore we need the fourier transform just like we had in continuous time and it's zero for all the different times other than these times here i'm just using this square function as an example in this case again the basis functions repeat what does this mean uh well i'll draw that just in a minute but i'll make the point first of all that the fourier transform of a discrete time signal that's a periodic is a continuous function uh and so here's uh it's continuous and exactly the same reason for over here this one was discrete in in terms of the fourier series and then the fourier transform was continuous it had the possibility can to contain all different basis functions at all different frequencies and this is exactly the same in the discrete time case and once again if that's something that you would like to know more about there's a video in the link below in the description below and so this is because the basis functions repeat this is periodic and the period is 2 pi so this is minus 2 pi and this is 2 pi and this is a zero here and again we're plotting the magnitude so this is the fourier series has discrete is discrete in the fourier series uh the fourier transform even though it's discrete time the fourier transform is continuous okay so we've got a periodic signals now what about the a periodic signal in discrete time to match with this one over here so we said the fourier transform was for a periodic signals signals that are not periodic but you can define it as the same in continuous time you can define the fourier transform in discrete time for a periodic signal so let's pick one here for an example uh let's pick matching this one over here so let's piece cos omega naught n so this function here cos omega naught n looks like this function here where it goes sort of follows a cos shape so i'm just drawing the that curve there is an indicator of where the points are i'm not trying to fill it in just these are the actual discrete points here of a cos function in discrete time okay so this is uh just trying to sketch that in there so this is in discrete time a periodic function can have a fourier transform so let's remind ourselves it's in this case because it is a periodic function just like in this case over here the fourier transform is going to be impulsive so over here in discrete time the fourier transform of a discrete time signal is still continuous so it has potential to have all the values of omega but because it is periodic this is going to be impulsive and so here for example we're going to have matching with the continuous time version we're going to have these two impulse functions at omega naught and minus omega naught but again because the basis functions repeat we're also going to have those repeating our round centered at 2 pi and also repeating centered around minus 2 pi so this is minus 2 pi this is 2 pi and this is the fourier transform of a periodic discrete time signal so it matches you can look at the the similarities over here in continuous time the continuous time cos waveform has a fourier transform that just has two impulses at minus omega norton omega naught in discrete time it also has two impulses but because it's in discrete time the basis functions repeat at 2 pi and multiples 4 pi 6 pi 8 pi and so on as as well as minus 2 pi minus 4 pi and so on so this is the uh the equivalence across from continuous time to discrete time okay so now we've covered fourier series and fourier transform and the discrete time fourier transform so what's this other thing called the discrete fourier transform well the discrete fourier transform is mostly what you implement in a computer and this is because a discrete fourier transform is just an equivalent of the discrete time fourier transform but where you only have a finite number of samples so in this case here because this is what you've stored and you've measured and you've stored on a computer of course in reality you don't have an infinite number of samples so you only ever have a finite number in reality and so if you've sampled so what does the discrete fourier transform so this is the dft what what's the difference between this and the discrete time fourier transform well as i say it only knows about the samples between two limits so it doesn't know anything about the samples out here it's not that they're zero like they were out here they just don't exist all you've done is sampled for a finite amount of time so you've got a finite number of samples so there's no arrow on here it's not in n index that goes on to infinity or anything is just over the finite sampling period of capital n and what does the dft do well it assumes that what you have sampled is periodic so it assumes that these samples repeat as if they were like in the example above here as if they were a full periodic waveform that does go from minus infinity to infinity all you've done is sampled a finite number of samples which is all you can ever do in practice but the dft assumes implicitly assumes that you have sampled a full period from something which is periodic and does go on for an infinite amount of time and so the fourier the discrete fourier transform will give you exactly the same result as the discrete time fourier transform assuming for the periodic signal so if this signal here were to be if this was a cos that you have sampled then the dft process assumes it goes forever so you'll get it will give you and you can do this in matlab or or other python and so on programming languages and you'll see that you given back an exact version of the discrete time fourier transform when you do the discrete fourier transform now you can then see what happens this is the case if you have sampled over an exact period if i had taken less samples than a full period then you wouldn't have a waveform which would be exactly the same as cos because you would have a shorter waveform which then would not repeat periodically to form a cause because it would have discontinuities at the ends and then you would not get exactly matching into in the frequency domain and again there's a video on that on the channel about the dft okay so what is the fft well the fft is exactly the same as the dft it's just a clever way of improv implementing the dft in a fast way of fast with using less calculations than the full dft formula so the fft or the fast fourier transform is the same as the discrete fourier transform it's just a more efficient implementation so you get the same relationship there okay in the last two transforms of the laplace transform and the z transform and in the laplace transform how does that relate well the laplace transform is in continuous time the z transform is in discrete time so let's look at those two now uh so we said before that the fourier transform applies for signals which have finite energy so what do you do if you had a signal like this which is continue oscillating and continuing to increase so this is like for example one example is positive feedback in a microphone system where the signal keeps increasing in its amplitude as time goes on well this signal if it keeps going like that forever it will not have finite energy so in this case uh instead of doing the fourier transform in this case we generalize and we instead of having j omega in the formulas which i haven't shown the formulas here but you can look them up instead of having j omega in the formula we replace that by another variable s which simply equals sigma plus j omega so it's almost the same as the fourier transform it's just that there's an extra component a real component in which you're replacing it by and again there's a video on this that explains the region of convergence and so mostly here we draw the region of convergence instead of drawing the transform because now we've got an extra variable so not only do we have amplitude and phase but we also have a real and imaginary component of the variable of the transform and so normally we say this is the real and imaginary and you're you've probably seen things where there's a vertical there's poles and zeros and a vertical line from the poles and you might have seen a cross hashed area which is the region of convergence what that means is it's the the error the region of s values which you can see sigma values are bigger than this number here for example would give you that whole area to the right so it's it's values of sigma for which the transform converges for which you can evaluate the integral and it turns out that if you if it includes if this region of convergence includes the vertical axis when sigma equals zero then you do in fact have the fourier transform so you can see here if sigma equals zero then you would exactly have exactly the same as the fourier transform and that would be looking along this line here so if we took that line this is where this gets a greater here and this gets left if we took this line and we rotated it around before sigma equals zero then we would have the fourier transform so i think of the fourier transform that we've drawn in this line here if i flip it up so it's vertical and then shift it around like this then it lives above that line there and the other values for other values of sigma you'll have other shapes and there'll be a smooth function and again for more insights on this there's a video on the channel and so the final one is so this is the laplace transform so the final one is the z transform okay and the z transform is equivalent to the laplace transform but just for discrete time signals so if we had a signal that was increasing as time went on or as the samples went on it's not a finite signal it's a signal that goes forever so like this discrete time signal up here then if this signal here this is this does not have finite energy so again as with laplace but in discrete time we can replace and again you can see more details on this but we can replace the e to the j omega by an r times e to the j omega so again we're putting a real value into or we're putting an amplitude here into our pot this has been polar coordinates uh where we put a real value in over here and i should just go back and say the real value here dampens this expansion so it ends up by multiplying this the effect is to multiply this function here by an exponential of the reverse slope so that in the overall signal uh resulting from doing that doesn't grow and is finite and again you can see more on the video about the region of convergence for the laplace transform in the case of the z transform it's in polar coordinates and so you'll see that there are circles there's a unit circle uh in and and you you you get other another circle let's say inside the unit circle so let's say that was one and so again you get a region of convergence uh which goes from the uh where the poles are uh and outside that and so again for all these values that are in the shaded region you would get a convergence of your z transform and so again when you see this picture you should be thinking of a transform sitting above this above this plane where again if you take this flip it up put it above the plane and you can see now it wraps around a circle in this complex plane so this this and this is another way of visualizing the fact that these basis functions repeat because if we took this function here and we flipped it up vertically and wrapped it around the unit circle then once you've got round to two pi you'd be back at the start again and so you'll be going around and around this unit circle here with the transform coming up out of the page and you can see once you go around two pi you get back to the start and so that is another way of visualizing the repetition of the basis functions in discrete time so hopefully this has given you more insights into the relationship between all of these different transforms if you've found it useful give it a thumbs up it helps others to find the video subscribe to the channel for more videos and check out as i've said a number of times and check out the links in the description below the video where you'll find links to all the other videos with more details on each of these transforms and there's a web page there with a complete list of all videos on the channel in categorised order

Channel: Iain Explains Signals, Systems, and Digital Comms

Views: 6,691

Rating: 4.9643917 out of 5

Keywords: Signals and Systems, Communications, Fourier series, Fourier transform, Discrete Time Fourier transform, CTFS, CTFT, DTFS, DTFT, DFT, Discrete Fourier transform, Fast Fourier transform, FFT, Laplace Transform, Z Transform

Id: 2kMSLqAbLj4

Length: 22min 46sec (1366 seconds)

Published: Mon Mar 29 2021