The z transform and first principles

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In summary, the person is asking for help understanding the last two lines of an equation, but their attempt at solving it is not shown. They mention that their textbooks and lectures have given them some foundation for the problem, but they are still struggling. They also point out that there are errors and inconsistencies in the given equation and suggest using an errata sheet or finding a different text for better understanding.
  • #1
nothing909
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Homework Statement


shown in the picture is the question and answer to it, but i don't understand how they're getting it.

this is me just not understanding the maths and i know its not difficult but I've been stuck on it for a while, so can someone explain in detail how you get the last two lines of the equation.
 

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Our rules say that you must show your attempt at solution before helpers are allowed to help. Just saying, "I don't get it." is not enough. Your textbooks and lectures must have given you some foundation needed for this problem.
 
  • #3
They do show me how to get the first part, but not how to converge. I think the last line you multiple by denominator and numerator by z squared, but that’s all I got.

All I need is a quick explanation of how they are getting the last 2 lines of the equation, I understand the rest of it
 
  • #4
This text looks... sloppy. Two glaring problems:

1) The inequalities at the top of the page don't match the summation -- i.e. the top specifically states that ##a_0 = 0 ## and ##a_{N-1} = 0##, so why are they included when moving from a formal infinite series to a finite one? I'm fairly confident that they meant to write ##0 \leq n \leq N-1## and didn't pay attention to strictness of inequalities.

2) Convergence of the geometric series really has nothing to do with the result at the bottom of the page. It's just telescoping a finite geometric sum -- a technique that is frequently taught in middle school or high school. The result holds whether or not you're in the radius of convergence for a geometric series.

That's a lot of issues for a very short extract. If there's an errata sheet, get it and include relevant sections with these posts. If there isn't an errata sheet, I'd get a different text or recognize that learning from this will leave you, at best, as confused as the authors.
 

1. What is the z-transform?

The z-transform is a mathematical tool used in signal processing and control systems to convert a discrete-time signal into a complex frequency-domain representation. It is essentially the discrete-time equivalent of the Laplace transform in continuous-time systems.

2. What are the advantages of using the z-transform?

The z-transform allows us to analyze and manipulate discrete-time signals in the frequency domain, which can provide valuable insights into the behavior of a system. It also simplifies the process of solving differential equations and performing convolution operations.

3. What is the difference between the z-transform and first principles?

The z-transform is a mathematical tool used to analyze discrete-time signals, while first principles refer to the fundamental laws and equations that govern a system. The z-transform can be used to derive the transfer function of a system, which is a representation of the system's behavior based on first principles.

4. How is the z-transform related to the Fourier transform?

The z-transform is closely related to the Fourier transform, as both are used to analyze signals in the frequency domain. The main difference is that the Fourier transform is used for continuous-time signals, while the z-transform is used for discrete-time signals.

5. What are some common applications of the z-transform?

The z-transform is commonly used in digital signal processing, control systems, and communication systems. It can be used to design filters, analyze stability and performance of control systems, and process digital signals in various applications such as audio and image processing.

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