@@iain_explains Profesor Iain. I have a curiosity, can i use the series expansion obtained from Dft in fuzzy logic? Generally i use Z transform to do it.
Amazing explanation. Ties everything very nicely. This video should be shown in all students of control system and signal processing. Thank you for creating this.
Sir Can You Please now make videos on digital filters?By the way your playlist really helped me to grow interest in Digital Signal processing. Thank You so much.😊
I just completed a course of signals and system to understand all the things but not quite connect and remember now rather than practically solving I'm tryna solve some online
Thank you for the explanation! I have a question: why the magnitude of the DTFT of square function has negative values, while the CTFT counterpart has only positive values?
Let's call it a 'minor typo'. Although it's not really a typo, just an inconsistency. Actually that particular DTFT function is real valued (there is no complex component), so it's a plot of the actual function (rather than the magnitude - which the other plots are).
In discrete time, all the discrete values are spaced apart by 1 sample time. More explanation can be found here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-7-4uEHoY1m4.html
Thank you so much for this series of videos. This is the first time i've seen such a comprehensive explanation of the purpose behind Z and Laplace transform and the region of convergences !
Thanks for your nice comment. I'm glad you have found the videos helpful. I'm planning to set up a Patreon page, so people can support what I'm doing if they wish, and also to potentially run interactive sessions, but I don't have anything set up yet. For now, it's just great to know that you have found the channel useful. Thanks.
I'm so glad you liked it! It's great to know that it's connecting all around the globe. Unfortunately I've never been to Norway, although one of my good friends during my PhD was a Norwegian student who spent a year here in Australia. I don't think I'll ever forget the "rotten fish delicacy" his mother used to send out to remind him of home! 😁
The discrete time basis functions repeat every 2pi. So that means 0 frequency is the same as 2pi, 4pi, ... and also the same as -2pi, -4pi, ... See this video for more explanation: "Discrete Time Basis Functions" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-P7q2YMQiat8.html
Thank you a lot for your videos! They are very helpful. However, I'm a bit confused about one point. In your other video ( ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-lLq3D-v4kPU.html ) you said that the CTFT is periodically replicated/has repeats on the sampling frequency, but here it is not. When is it and when is it not?
When a continuous-time signal is sampled with a sequence of (ideal continuous-time) delta functions, the resultant "continuous-time sampled signal" has a (continuous time) Fourier transform that repeats at the sample rate. For all other (non-sampled) continuous-time signals, there is no frequency repetition. In other words, the repetition is because of the sampling.
I'm confused isn't the discrete time Signal is a group of impulse delta functions ? And the fourier transform of delta function is 1 meaning it's got continuous frequency components? How we are getting discrete frequency components in for example DTFS instead of 1
Good question. Hopefully these points will help to explain it: 1) In general, the DTFT is a continuous valued function. See this video for more explanation: "Fourier Transform of Discrete Time Signals are not Discrete" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AOQAlrtGUzo.html 2) The FT of a delta function has a magnitude of 1 (as you point out), but it also has a phase which is a function of frequency (depending on its time offset in the time-domain). This phase is most often not plotted, and is sometimes overlooked. The phases from all the different time-domain delta functions add up to give an overall function (in the frequency domain) that is not a constant magnitude. 3)The plots that show discrete frequency components for the DTFS and DFT (the 3rd and 4th plots on the far right hand side) both correspond to sinusoids in the time-domain. For time-domain signals that are periodic, the Fourier transform will consist of discrete impulses. This video explains this more: "Why do Periodic Signals have Discrete Frequency Spectra?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-wA3VXyl9xVg.html 4) and finally, note that there is a slight error in the plot for the DFT. I explain this in the notes below the video, and I've fixed it in the Summary Sheet on my website: drive.google.com/file/d/1fh7TzeT4HCeoRECnHiQYmjFRSeWLYnDI/view
Thank you for a good lecture. This lecture makes me understand region of convergence. However, I am a bit confused with transforming from time domain to frequency domain and frequency domain to image domain. I don't get this relationship. some lectures talk about only one part time domain to freq. domain or only image to freq domain. In reality, it seems the process includes all of this steps, time domain >> frequency domain (k space)>> image (object) domain.
Great question. The Laplace transform is a generalisation of the Fourier transform that allows for functions (eg. signals, system responses, ...) that have infinite energy (eg. the impulse response of an unstable system). In Communications, we're mostly dealing with communication channels that are inherently stable (if the input has finite energy, then the output will have finite energy) and we're interested in frequency domain aspects (eg. inter-channel interference, bandwidth efficiency, ...), so the Fourier transform is appropriate. In Control Systems, there's feedback (for controlling in the "plant") and this can lead to instabilities if not designed appropriately (eg. positive feedback in guitar amplifiers), so the Laplace transform is needed, in order to investigate aspects of stability in cases where a function potentially has infinite energy.
Yes, that's right. As I said, in Communications, we're mostly dealing with communication channels that are inherently stable (if the input has finite energy, then the output will have finite energy) and we're interested in frequency domain aspects (eg. inter-channel interference, bandwidth efficiency, ...), so the Fourier transform is appropriate.
Hi, little remark, just to dot the i's and bar the t's... when you say that the spectrum in DT (the basisfunctions) repeats around 2pi, on the omega-axis, do you actually mean that they repeat around 2pi*fs, fs being the sample rate? I was just wondering because w is in radians times Hz. So, any point on it should be too, no? So, basisfunction repeats around 2pi*fs and -2pi*fs etc...? Is that correct?
No, w is _not_ radians times Hz. It is just radians. In discrete-time, the "time" samples are just numbers stored in a vector. They are just indexed by integers.
I don't tend to pay too much attention to the Fourier Series, because there aren't really any periodic signals in the real world that go for an infinite amount of time. I prefer to think in terms of the Fourier Transform. However I do have one video on the FS, and I also have some on the DFT/FFT. Have you checked out my webpage? iaincollings.com Here's the link to the video on FS: "Fourier Series and Eigen Functions of LTI Systems" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-gRq3K4ZQKi8.html
@@iain_explains yeah I have checked your website after I finish the entire playlist of signals and systems I will replay your videos again and make notes or downloads your summery sheets depending on the time I have. thank you very much
Dear professor, Do you have a video explaining the Hilbert transform when it used to extract the instantaneous amplitude and phase, and calculate the phase locking value? Thank you for uploading great content. Very much appreciated. Eitan
Caveat: My memory is not what it was! However, if we start from the Fourier Transform, we end up with positive and negative frequencies. If you think of the spectrum of a cosine wave it has two impulses: one at the positive frequency and one at its negative counterpart. If you now think in 3D, you can imagine those impulses rotating around the frequency axis where the positive frequency rotates towards you "out of the paper" and the negative frequency also rotates but "into the paper." If you now make a phasor sum of both those components you get your cosine wave back if you plot the resultant amplitude against time. (If you turn the picture round so you are looking straight down the frequency axis you would see two phasors rotating in opposite directions which you can then sum to give a purely real wave.) There is another way of doing this by not having negative frequencies. You just keep the positive frequency, double its amplitude and rotate that about the frequency axis. The result, when plotted against time, is the same as the Fourier approach. What we now have is a rotating phasor that is drawing out a helix in 3D space. (Mathematically, that is what exp(jωt) looks like.) When looking at the projection on the real plane we see a cosine wave but what does it look like on the imaginary plane? That is what the Hilbert Transform tells us. In this case, the imaginary projection would be a sine wave. I hope that helps.
Dear Professor I have seen your CP-OFDM (Cyclic Prefix ) explanation in another video and really enjoyed it , If you have some spare time I appreciate you also to explain about W-OFDM (Wide band), F-OFDM (Filtered) and other types? Thank you a lot.
First of all, thank you for uploading great content. Secondly, I have a question, what is the difference between Fast Fourier Transform (FFT) and Fractional Fourier Transform (FrFT) ? and what is its applications? I searched on RU-vid on it but I don't find videos explain it.
hello lain, as you have mentioned that a sinusoidal which is having infinite energy is defined by the delta function in the Fourier domain so my question is how this can be done as impulse function is itself an unpractical signal which exists only theoretically is there any point I am missing means can you give an insight into this? means i want to say that is there any boundation or condition apllied in defining sineij terms of the delta function Thanks !
Well, when you think about it, the sinusoidal signal sin(wt) is also an impractical signal which only exists theoretically (because it starts at negative infinite time, and goes until positive infinite time.) If you think about multiplying sin(wt) by a "window function" (eg. rect(t) ) to limit its duration to a finite range of time, then in the frequency domain you would be convolving the delta function with the Fourier transform of the rect function, which is a sinc function. These videos might help: "How to Understand the Delta Impulse Function" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-xxGcI9WVoCY.html and "Fourier Transform Duality Rect and Sinc Functions" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-rUgBhEpeqxo.html
Hi Iain! Thanks for the great video. I notice that the magnitude of the DTFT example has some negative regions. Is that actually just a plot of the real part of the DTFT rather than the magnitude which should always be positive?
Yes, that's right. Let's call it a 'minor typo'. Actually that particular function is real valued (there is no complex component), so it's a plot of the actual function.
I'm glad you liked the video. If you'd like to see more like this, check out iaincollings.com where you'll find a categorised listing of all the videos on the channel, as well as summary sheets.
Thank you. I'm not sure I get the explanation on why a discrete signal is both aperiodic and periodic. I thought that a discrete signal is a sampled signal already. I was of the thought that discrete signals were aperiodic.
I think you're talking about the DFT, right? If you've sampled a signal for a finite period of time, you will have a vector of a certain length (depending on the sampling rate you used). The Fourier transform is defined as an integral over all time - not just over the time period that you sampled over. So the question is, what to do? One approach would be to assume that the signal is in fact zero outside the period of time that you sampled for. Another approach is to assume that the signal keeps repeating itself outside the period of time that you sampled for. The DFT takes the second approach.
@@iain_explains Thanks. Yes DFT/FFT since I'm considering seismic signals. And I believe you mentioned the FT is for processing finite signals? I read a material that finite signals were aperiodic so is the seismic signals aperiodic or assumed to be periodic in the DFT. Your videos are great by the way. I work with transform software processes but I'm still trying to understand HOW f(t) transforms to F(w). Like how 2π/T actually works. Your videos are helping me though.
@@iain_explains Thanks for replying :). I meant that If you have a vector of values, lets say x = [1, 2, 3], and you perform the DFT (for example, with MATLAB: fft(x) ), the result is a vector of 3 finite values. If I undertood well, the result shown in 15:00 is made up of impulses, with infinite values.
Ah yes, I remember now. Unfortunately I wasn't as accurate as I should have been in that diagram. I drew "continuous" impulses (delta functions), when I should have drawn "finite/discrete" impulses only over a finite range of frequencies. I fixed it on the summary sheet on my website: drive.google.com/file/d/1fh7TzeT4HCeoRECnHiQYmjFRSeWLYnDI/view
Instead of a book to explain this in signals and systems theory the universities should have a special oscilloscope that allows periodic and non periodic signals to be represented and explained in terms of all these various transforms and relationships...a specific machine to hone in on these concepts for education.
