1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Change of variables in summations

  1. Jun 27, 2011 #1
    Hi,

    although this may sound trivial, I stumbled upon this problem while studying decimation process in digitial signal processing. I can't find anything on the web about some definition for the change of variables in sumations (as there is one for integrations), so maybe someone here could help me.

    Consider the sum
    1j19K.png
    (the summation range is [-inf, +inf])

    If I just do the substitution [itex]m=2n+5[/itex] and get this
    BbapZ.png
    it wouldn't be right.

    Decimation leads to irreversible changes, aka, I should still have the sum of elements [itex]x[1], x[3], x[5],...[/itex], but, somehow, I now have the sum of all elements of [itex]x[n][/itex] with this simple substitution.

    So, I think the right answer would be
    wjmCf.png

    I did this ad-hoc, using logic. I was wondering if there is a proper definition for the change of variables in summations, which takes into the account the effects of decimation (which do not exist in continuous case of course)?

    Thanks in advance :)
     
  2. jcsd
  3. Jun 29, 2011 #2
    The problem is that, if n goes from minus to plus infinity, then m = 2n + 5 also goes from minus to plus infinity, but taking only odd values. So you could write

    [itex]\sum_{n=-\infty}^{+\infty}x_{2n+5}=
    \sum_{m=-\infty}^{+\infty}x_{m}\frac{1-(-1)^{m+1}}{2}[/itex]

    This because the extra term I introduced is 0 for even m and 1 for odd m.
     
  4. Jul 3, 2011 #3
    Yeah, that does the same thing as the sequence of deltas, but I kinda like yours more :D

    I guess there isn't a rule for this 'substitution' because it's a trivial matter, but I wanted to check it still.

    Thank you for your response!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Change of variables in summations
  1. Variable Change (Replies: 69)

Loading...