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So I have the question

$$\frac{2z+1}{z^2-z+0.5}$$

do I solve the problem of power series by long division like the following?

what I've done

I know the power series is infinite but I'm just a little confused on the terminology for the question I'm doing I'm supposed to go to the 4th coefficient but I'm not sure if I'm calculating correctly.

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The exact wording of the question is as follows

A causal system is defined by transfer function

$$\frac{2z+1}{z^2-z+0.5}$$

Determine the first four non-zero coefficients of the system’s impulse response, ℎ(), using the power series expansion by long division method.

when I asked the TA who was teaching this section where I could find more info on this he said look at google so I don't know what to do because every example I get from google is talking about doing the inverse of a transfer function.

Livingstone
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  • OP, can you please clarify whether you want a power series (with positive powers) or a long divison (laurent series)? – Gareth Ma Dec 10 '20 at 07:11
  • The question says using the power series by long division method, I've never heard of this and what I have accomplished so far is what I've seen on the internet that I suspect is the correct thing to do – Livingstone Dec 10 '20 at 07:32
  • Ah. Then what I have replied with should be what you want. Feel free to ask for clarifications if you don't understand, and accept if it helps!~ – Gareth Ma Dec 10 '20 at 07:33
  • Is there a better name for it that I can look it up by? – Livingstone Dec 10 '20 at 07:33
  • Polynomial long divsion – Gareth Ma Dec 10 '20 at 07:34
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    Please clarify, Livingstone, whether you are looking for an expansion in positive powers of $z$, or an expansion in negative powers of $z$. – Gerry Myerson Dec 10 '20 at 12:04
  • I don't know the question is as stated above, the work I did was what I did guessing off what I saw on the internet – Livingstone Dec 10 '20 at 23:26
  • OK. Do you know anything relating "impulse response" to "transfer function"? – Gerry Myerson Dec 11 '20 at 23:27
  • @Gareth, I found, "If the transfer function of a system is given by $H(s)$, then the impulse response of a system is given by $h(t)$ where $h(t)$ is the inverse Laplace Transform of $H(s)$." So I think what we're really supposed to find here is the inverse Laplace transform of $(2z+1)/(z^2-z+0.5)$. I think we've all been barking up wrong trees. – Gerry Myerson Dec 12 '20 at 12:13
  • Have you been able to get any clarification, Livingstone? If your TA isn't helpful, is there someone else in charge of the course who can help? – Gerry Myerson Dec 14 '20 at 01:34
  • The TA has been left in charge so no – Livingstone Dec 15 '20 at 04:17
  • How about the head of the department? – Gerry Myerson Dec 15 '20 at 06:41
  • head of department would just tell me to do what I was told – Livingstone Dec 17 '20 at 02:43

1 Answers1

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Try to think of it as $$\frac{1}{z^4}\cdot\frac{2z^5+z^4}{z^2-z+0.5}=\frac{1}{z^4}(2z^3+3z^2+2z+\cdots)$$

Gareth Ma
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