This video series is very informative - Thanks! At [7:15] "... H(z) only converges at points where the magnitude of z is less than the magnitude of the largest pole location". I think "less" should be "greater".
my question to you is : where did you learn all this amazing stuff? no school I know of teaches this kind stuff as professors come in and read out the slides never add value to teaching. also there are no texts i know of that can explain this stuff like you . one thing i can tell you is that whoever learns from you- are the most fortunate ones- just to have someone like you - who knows this stuff so well- is a blessing. I wish I had a just single professor (dumb headed and enjoying the tenured jobs for life) in school who can do the job so well like you do
mainly working with it and a lot of thinking about how to present it in a simple way. Most of my understanding comes from trying to visualise what the equations are doing and then putting that understanding into practical situations. The text books I found most useful when I started out in this area are "The Scientists and Engineers Guide to DSP" by Smith and "Understanding Digital Signal Processing" by Lyons.
Hats off to Dave. Salute you Sir! I understand the variable z. It is essentially a complex frequency comprising a decay factor r and a regular angular frequency w. H(z) is the system response for a value of z. In case of FIR, clearly z is an attribute of input x[n]. In case of IIR, there is both input and feedback. So what is really the source of z for IIR?
Hi David, wonderful videos. Since the stability of the system can be determined solely from pole location (inside the unit circle), then why bother calculating the ROC? What extra information does the ROC provide? Why does it matter if H(z) converges? Don't we only care if h[n] converges? Thank you!
There isn't any extra information gained from the region of convergence that I can think of. The main reason I put this video together is to explain what it is, as it appears in most text books and can be a tricky concept to appreciate. In fact I find the z surface plots easier to interpret if the region of non convergence (the undetermined region) isn't shown 'correctly' and is filled in as shown in the earlier videos.
I have an intuitive question. In the beginning of the series, you mentioned that Z transform is testing the presence of a signal (z^-n) in another and the higher the number the greater the presence of one signal in another. Applying that to the impulse response signal, I don't get why the system would be stable if the poles are inside the unit circle. Values of z where |z| is lower than one correspond to exp increasing sinusoids. A pole represents a high value of |H(z)| which means high presence of these signals in h[n]. So, it hints to instability if we were to consider the analogy in the beginning of the series.
Yes - I understand the issue you're raising. My comment about a larger correlation value indicating stronger presence has some caveats, but can provide a nice intuition for how correlation works in general. Correlation results will be impacted by the 'energy' of signals (see ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-ngEC3sXeUb4.html) and the energy of the signal z^-n is dependent on the magnitude of z, so really direct correlation comparisons should only be drawn between values of z which have the same magnitude values. I absolutely appreciate why my comment in the earlier video would cause someone to expect poles of unstable system to lie inside the unit circle - a small part of me regrets saying it because of the confusion it can cause. However, a larger part likes the analogy of correlation being an indication of how strongly present one signal is in another - it seems to sit well with a lot of people. I guess if you're at the stage where you're asking this type of question you're well on the way to internalising your own understanding of how the z-transform works. Best of luck with your studies!
@@ddorran Thanks for the reply. I was just trying to tie up your intuitive explanation in the beginning with stability criteria. I know that poles outside the unit circle lead to an unbounded impulse response, but I was trying to relate it to the correlation concept. What I come up with though, which might be somehow an explanation, is from Laplace transform. If I consider e^(-st) the signal I am trying to check my impulse response against, here again stability is at the left half of the plane which corresponds to an overall e to a positive power for e^(-st) meaning an ever increasing signal. But when you have a pole in the left half plane ( e^(-st) = e^(positive number) ) that means there is an e^(negative number) in your impulse response and they cancelled out which leads to a high integration number. Hence, the system is stable.
@@RaedMohsen Yes - I see where you're going with that. Not sure if this helps but this video attempts to show the relationship between the Laplace and Z domans ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-acQecd6dmxw.html
Hey, there is a smale mistake in the vid, around 4:20 you say that the values in the series are decreasing and the series there for converges. This however is not always the case, the best example of this is the harmonic series that diverges to infinity. In most cases however you are right and they wil converge but it is not the case that by definition when the values in the series are decreasing that the series converges. (excuse my bad english pls)
could you please show a practical use of z-trasform ? I really not able to visualize its use in practical life, like we use PID controller in quadcopter or balancing robot
You can use it to determine if a system is stable. Also, you can use it to design a system to have a particular frequency response by locating the poles and zeros at particular locations.
David Dorran I know this but I want some practical example like someone said that he used a transformer in ball balancing bot but he didn't tell details ..I have also seen use of bode plot in amplifier design...or if we use z trasform in moving averager that you showed previously?
I use the z-transform to give me insight into the behavior of systems in the same way as I use the fourier transform to give me insight into signals. The practical applications are quite broad. You can use the z-transform to design an amplifier, model the behavior of a car suspension, model a vocal tract, design a PID control to control a quadcopted in a particular manner, etc.
Maybe a little late, but Z transform is the digital counterpart to the Laplace transform. So anytime you want to implement a digital controller of a continuous system, chances are you will have to transfer the controller from Laplace Domain to Z Domain. Essentially Z Transform is just Digital Laplace Transform
I understand where you're coming from but mathematically speaking a series that doesn't converge (diverges) is undefined in the same way that 1/0 is undefined - although a lot of the time people think that 1/0 is infinity. Even some programming languages like matlab give a result of 1/0 as being equal to infinity. I have to admit that I find thinking of 1/0 as being infinity quite useful conceptually. Maybe it's conceptually useful to think of a divergent series summing to infinity as well - but mathematically speaking its not defined.
I think dspguide.com (Steven Smith) and " understanding digital signal processing" (Richard Lyons) are two excellent resources. I've starting compiling some notes at eleceng.tudublin.ie/dsp/docs that might also be of help.
