From a transfer function to a difference equation

In summary, the conversation is about finding the difference equation for a digital system based on a given transfer function. The speaker has attached their work and is asking for feedback on whether they are on the right track. They also mention a typo that was later corrected. The other person replies with feedback on the equation and wishes a happy new year.
  • #1
SMOF
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Hello.

I have been given a transfer function for a digital system, and I need to find the difference equation for that system.

I have attached a pdf with my work. I would be grateful if someone could take a look at it, and see if I am along the right lines.

Thanks in advance, and happy new year to you all.

Seán
 

Attachments

  • Untitled 8.pdf
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  • #2
I thought I had responded to this before.

You have a typo in your 6th equation, 3rd term on RHS.

Otherwise, I see nothing wrong with your derivation of the differencce equation.
 
  • #3
Hello,

Thank you very much for the reply!

Yea, I spotted that one a bit after I posted.

Thanks for the feedback! :)

Seán
 

1. What is a transfer function and how is it related to a difference equation?

A transfer function is a mathematical representation of the relationship between the input and output of a system. It describes how the input signal is transformed into the output signal. A difference equation, on the other hand, is a discrete-time representation of a system that relates the current output to previous inputs and outputs. The transfer function can be converted into a difference equation and vice versa, as they both represent the same system.

2. How do you convert a transfer function to a difference equation?

To convert a transfer function to a difference equation, the transfer function is first transformed into a z-domain representation using the z-transform. Then, the inverse z-transform is applied to the z-domain representation to obtain the difference equation. This process involves using algebraic manipulations and partial fraction decomposition.

3. What are the advantages of using a difference equation over a transfer function?

One advantage of using a difference equation is that it is a discrete-time representation, which is more suitable for digital signal processing systems. This allows for easier implementation on digital systems. Additionally, difference equations are easier to analyze and manipulate compared to transfer functions.

4. Can a difference equation be used to model continuous-time systems?

No, a difference equation is only applicable for discrete-time systems. Continuous-time systems are described using differential equations. However, a continuous-time system can be converted into a discrete-time system using techniques such as sampling, which involves taking discrete samples of the continuous-time signal at specific time intervals.

5. What is the importance of understanding the relationship between transfer functions and difference equations?

Understanding the relationship between transfer functions and difference equations allows for a better understanding and analysis of systems in both the continuous-time and discrete-time domains. It also allows for easier implementation and design of digital signal processing systems, which are essential in various fields such as telecommunications, control systems, and image and audio processing.

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