Z Transform Example 

Only "almost" all of them? 😁

I'm so glad the videos have been helping you. It's great to hear.

Hi Kirsty, great to hear from you. It certainly is a small world. It's good to hear that my videos are popping up when people are searching for biomed topics.

​@@iain_explains Sir, what does this all mean physically? and why can't we take as |az^-1| > 1 instead of

You might like to watch this video: "Z Transform Region of Convergence Explained" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-uq_qv3Spzbs.html

Not quite. I plotted the case for values of the parameter 'a' between 0 and 1. In that case, yes, it is stable, and you could just use the Fourier Transform. For values of 'a' bigger than 1, or less than -1, the function is increasing with time 'n', and the Fourier Transform is not defined. Even if a function is stable though, the z-transform is still generally used, because it gives you extra insights. You can see how close your system is to being unstable, by observing the relative locations of the poles and zeros. You can't tell this just from the Fourier Transform.

@@iain_explains thank you so much

Thanks. You are welcome!

If a>1 then the region of convergence would not include the unit circle, and therefore the signal x[n] will grow to infinity over time. If your Z-transform is the transform of a system's impulse response, then a>1 means the system is unstable.

Maybe this video might help: "Z Transform Region of Convergence Explained" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-uq_qv3Spzbs.html

@@iain_explains Thank you for the incredibly speedy response! While a great video, I don't think it answers the question I had; at least not directly. The question was what, mathematically, determines whether the ROC is on the inside or outside? The answer I ended up finding has to do with flipping the summation terms when your signal is left sided.

Thanks for the topic suggestion. Are there any properties in particular that you'd like to see? Most of the properties are essentially the same as the continuous time Laplace transform properties, and I've already got quite a few videos on my channel about them.

sir there is liniarty property and time swfting @@iain_explains

OK, I'll add those to my list.

thanks for accept my request@@iain_explains

The "time reversal" property states that x[-n] has Z-transform X(z^(-1)), with an inverted region of convergence. So therefore, u[-n] has Z-transform 1/(1-z) for |z|

@@iain_explains Is it that the graph of a^n should be increasing Because that one is similar to a^-n

Glad you found it useful.