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Winograd

Name of this serie [SOLVED]

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Hi! Is there a name for the serie below? EDIT: sum(n = 0 to N-1, cn*zn^k), where cn and zn have n as a index and k is a positive integer. And can you point to any useful site containing information about series of that form? [Edited by - Winograd on December 1, 2004 1:18:44 PM]

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Not homework, although I admit that it might seem one ;)

MathWorld didn't help.. and even if my serie is in some dark corner of mathworld I would still need to know the name first or crawl through all series and sums it has (and pretty much this is what I did :) )

This serie or sum emerges from close-form solution for recurrence equation of following kind:

y[n]=sum(k=0 to N-1, ak*y[n-k] )

Those recursive equations produce the impulse response of the IIR-filter having unity nominator in the z-plane equation. for example: H(z)=1/(1+az^-1) ==> y[n]=-a*y[n-1]

So I would REALLY appreciate if someone knows a name or place for information for the serie I described in my first post!

And even if this would be homework (which it isn't.. trust me) I don't ask you guys to solve anything for me. And what kind of home assignment goes: "Find the name for following expression."

:D

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I am very familiar with that type of expression, but I don't have a better name for it than "sum of geometric sequences". What is it that you want to know about those sequences?

EDIT: Nevermind, I read your formula wrong.

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Sounds a bit odd for a homework question, so I'll give you the benifit of the doubt.
EDIT: Nevermind. There was only 1 reply when I started my post [smile]

It's called the Power Series.

[Edited by - joanusdmentia on December 1, 2004 6:22:58 AM]

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Quote:
Original post by Winograd
This serie or sum emerges from close-form solution for recurrence equation of following kind:

y[n]=sum(k=0 to N-1, ak*y[n-k] )

Those recursive equations produce the impulse response of the IIR-filter having unity nominator in the z-plane equation. for example: H(z)=1/(1+az^-1) ==> y[n]=-a*y[n-1]


Thought it might be something along those lines :)

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Joanusdmentia, you missed one crucial point.
This is my serie:
sum(n = 0 to N-1, cn*zn^n)

This is power serie:
sum(n = 0 to N-1, cn*z^n)

Notice the index in z.

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So there is.....but that doesn't seem right. I'm assuming that z is in the z-plane (and you didn't just happen to chose z for something else [smile]), in which case having a subscript doesn't make any sense.

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Whoopsy!! Made a HUGE typo.. sorry about that! I corrected it in my first post.

So the exponent should be independent positive integer variable k and not n as in my first post it used to be.

So this is kind of family of series. One for every k :)

Here it is again.. this time correctly:

sk = sum(n = 0 to N-1, cn*zn^k), where cn and zn have n as a index and k is a positive integer

EDIT: Now the question really should be that is there name for that family of series. I'm actually intrested in solving (or aproximating the solution) following group of equations for cn and zn:

sk = sum(n=0 to N-1, cn*zn^k), for k in [0, M-1]

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Quote:
Original post by Winograd
sk = sum(n = 0 to N-1, cn*zn^k), where cn and zn have n as a index and k is a positive integer

Still doesn't make any sense. By zn^k, do you really mean (z^-n)^k? How did you get your equation?

EDIT: Forgot a minus sign.

[Edited by - joanusdmentia on December 1, 2004 7:57:18 AM]

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