Thanks man, i needed this before signals and systems unit

Tremendous video .. nice work 🎉

Just to further simplify for those that don't understand it that well. The maximum of two "values" when multiplied is when they are the same. a*b is a maximum when a = b when then gives a^2 = b^2. Hence when you pointwise multiply two functions you are just multiplying their values and their maximum is always when they are the same. E.g., suppose you have a = 3 and b = 5 then you have 3*5 = 15 < 5*5. Because you are using the same signal and shifting it over time f(x-t) at some point it must match with itself f(x)*f(x-t) for some t. In fact, it matches exactly when t = 0 and so one has f(x)^2 as an output which is the maximum possible at all points. But if the function is periodic or approximately then it will repeat itself for some non-zero t so you can find that t by looking at other maximum other than t = 0. Hence the period is simply the smallest nonzero t that. But the idea is simply that of squaring always gives the largest value in the sense that min(a,b)^2 = min(a,b)*min(a,b) <= a*b <= max(a,b)*max(a,b) = (max(a,b))^2 and when have a continuous function then we can ask about cross(e.g., at different points in time) values. You can, of course, use any binary function g(x,y) that has a maximum when x = y and the effect will be the same. E.g., x^n will also enable doing this and it can also make finding periods easier since it will, at least for normalized functions push the "noise" down to the floor. Also, the reason the shape decays is because the signal is being zero extended. If one makes them periodic then there is no decay. The problem with this though is then the signal has a natural period of the window size and this might cause some problems determining the correct period. Just to be clear, when one does f(x)*f(t - x) one is computing a pointwise product for at every x. One can then add up all the values to get the total relative "correlation". We know that at t = 0 this total is maximum because the product is f(x)*f(x) = f(x)^2 and this is the maximum value we could ever hope to get for each x. We can add up over the entire range of x to get the total "energy". At other values of t we would get other values we get other values but none can have more total power than at t = 0 or some period. All this comes out of the basic idea that a*b <= max(a,b)^2 and the only time a*b = max(a,b) is when a = b. This extends to entire functions by simply doing it pointwise and adding up to get an overall value. g(t) = sum(f(x)*f(t - x)). g(0) is maximum at t = k*period. This also extends to two different functions for the exact same reason so one can simply compare functions using this method knowing that it gives the maximum "comparison" at the maximum value of t. The reason for f(t - x) vs f(x - t) is just so our periods are positive rather than negative but it obviously makes no difference ultimately. Another way to think about it is that multiplying f(x) by a shifted copy will typically introduce a lot of destructive interference because the shifted copy will not match up and will introduce another set of zeros. E.g., f(x)*f(t - x) will have twice as many zeros, typically than f(x). If it is shifted by the period, if it has one, then those periods exactly align and one not have any excess. Generally if there are more zeros this means more crossings which means higher frequencies. Also note that when squaring one has no negative values. So when f(x) and f(t - x) align perfectly it must all be positive while there will otherwise be negative values. Hence when summing over the additional negative values will lower the overall sum value.

This is a really good approach for understanding Z transform!

This is a really good approach for understanding Z transform!

This is a really good approach for understanding Z transform!

This is a really good approach for understanding Z transform!

This is a really good approach for understanding Z transform

This is a really good approach for understanding Z transform!

JUST AMAIZING

Wonderful details. The x-axis of the transform is easy to forget

Excellent

thank you David. Great demonstration. It is extremely helpful for me.

Thks for explaining the crux of correlation

unipd? qualcuno?

10 years later, this video still helps people out!

You are a beautiful man for uploading this treasure of a video.

Sir, about the linear phase spectrum, is it really just a random straight line that you chose, or is it based on something rigorous? and i suppose the steeper the line, the smoother the resulting magnitude of the IFFT is, right? and is it ever possible to achieve a perfect "blue line" IFFT like one you mentioned in 1:25? completely disregarding how messy the phase spectrum would be, is that ever possible?

The Best

Simple and efficient Thanks !

Why is the region of stability within the unit circle for the Z Tranform? Isnt that where the signal grows exponentially i.e. instability?

Thank you so much, why do people overcomplicate the explenation of this topic when you can explain it as simply as this

super nice demo thank you so much

is there a less laggy version that I can download?

Really appreciate your video and the detailed documentation.. Thanks a lot

Could you pls. tell me what is this 3D GUI tool?

Very nice comparison

σε αγαπώ!

nice, thank you so much!

what about the effect function in Audacity? One can also apply church effect, isn't it?

your accent is wild, but your explanation on the topic was very helpful review for my understanding for my FE review

I understand more now than I did in 3 months of control lectures, thank you for putting all the pieces together

I binged this whole series overnight studying for my DSP midterm. Dude none of this connected until i watched YOUR videos. I had no idea what the Z transform was even supposed to represent in terms of a system or what the magnitude meant or anything. This all cleared it up so much!

I'm about to graduate with my degree in Electrical Engineering at Oregon State University. I've had to watch hundreds(yes, literally hundreds) of videos to understand that concepts i've had to learn since i'm a slow learner, and i can say with complete confidence that this is the absolute best and most intuitive description of the Z transform i've seen.

Good video ,thank You Sir !

What about the frequency dependent phase shift response for V-in vs. V-out?

finally , I found useful video to understand cross correlation , thank you David

Thank you.

Our teacher taught us this and I was so confused what the purpose was. I was also confused why you were solving for h at first. But after you said I could get the output signal just by the impulse response of the system. Everything clicked right there.

🐐

Funny😊, thank you. Please let me have the link for other applications, if there is. Thank you

If I have this: H(z) = (-z^(-2)-2*z^(-1)+2*z+z^2)/8T, Which are the coefficiant a and b?

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)

What a wonderful explanation!,much appreciated!

Amazing explanation, thanks a lot

Best explanation on this I have ever seen. Thank you for the colors!!

truly great material! no better video to explain the z transform than this one !

Amazingly explained in simple words, thank you

Your videos truly have helped me a ton as a newbie. I'm a little slow on the uptake... so this alaising catches signals from the waveform at different points rather than changing the sample rate directly (i.e frequency)? I also suppose each of these points has many different signals/sounds within them. This must be where the distortion comes from.