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Libniz

  • 2 years ago

\[\sum _{k=-\infty }^{\infty } \theta (n-k)\theta (-k-1)a^k\]

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  1. Libniz
    • 2 years ago
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    \[\theta\] is step function

  2. Libniz
    • 2 years ago
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    @phi

  3. Libniz
    • 2 years ago
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    I can show you the work , it would be great if you can explain it

  4. Libniz
    • 2 years ago
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  5. Libniz
    • 2 years ago
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    I get the n<=-1 part but not the n>-1

  6. psi9epsilon
    • 2 years ago
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    this is convultion

  7. phi
    • 2 years ago
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    I know u(-k-1) is 1 for k= -inf to -1 u(x) is 0 for x<0, and 1 for x≥0 u(n-k) is 1 for k= -inf to +n

  8. psi9epsilon
    • 2 years ago
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    use convolution property

  9. psi9epsilon
    • 2 years ago
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    you can break up the sequence from -infinity to 0 and from 0 to infinity for 0 to infinity we know what are the results of a step function

  10. phi
    • 2 years ago
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    when you multiply the 2 step functions the one that "steps down" (because we are doing u(-k) ) first sets the upper limit n wins for n< -1, and the other wins for n> -1

  11. phi
    • 2 years ago
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    |dw:1347579756548:dw|

  12. psi9epsilon
    • 2 years ago
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    you can split up the basic sequence too, its way easier to do it

  13. phi
    • 2 years ago
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    when n>-1 u(n-k) times u(-k-1) is zero for k≥ -1

  14. phi
    • 2 years ago
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    clear as mud?

  15. Libniz
    • 2 years ago
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    lol, I am reading all your posts over , will ask questions later

  16. phi
    • 2 years ago
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    The first confusion is u(k) is 0 up to k=0 , then it jumps to 1 but u(-k) is 1 and drops down to 0 at k=0 that is because when we plug in (for example) k=-2, we get u(-(-2))= u(2) =1 (2>0 means we jumped up) Both of these steps are backwards, are at 1 and drop to zero at some point

  17. Libniz
    • 2 years ago
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    I get it how u(-t) is reflected across y axis

  18. Libniz
    • 2 years ago
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    trying to parse this "when you multiply the 2 step functions the one that "steps down" (because we are doing u(-k) ) first sets the upper limit n wins for n< -1, and the other wins for n> -1"

  19. phi
    • 2 years ago
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    u(-k-1) never changes. It is 1 from -inf to -1, then drops to 0 we multiply this step times u(n-k) if n is bigger than -1 this second step drops to zero after -1 but it does not do anything, because we are multiplying the 2 steps and everything past -1 is 0 (from the first step function). if n is less than -1, it drops to zero before the first step does, and it sets the upper limit where the product of the 2 step functions is 1.

  20. phi
    • 2 years ago
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    the picture is trying to show the two cases. If it helps, think logic, both steps have to be 1

  21. Libniz
    • 2 years ago
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    oh, so u(-k-1) is one constant step function whereas u(n-k) is many different function based on differing value of n so say if n=-2 u(-2-k) is it drop to zero before first step does

  22. phi
    • 2 years ago
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    yes, u(-2-k) drops to zero when -2-k=0 or k=-2 when you multiply against the first step, you get 1 only up to k=-2 (that is, k= n)

  23. Libniz
    • 2 years ago
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    I guess my main confusion is \[\sum _{k=-\infty }^{-1} a^k\] n>-1 <---- why do we use this even though n is not used in summation itself

  24. phi
    • 2 years ago
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    n is controlling the upper limit. n bigger than -1 cannot make the limit bigger than -1 , but n< -1 can change the upper limit it n (n being smaller than -1)

  25. phi
    • 2 years ago
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    still puzzling over it?

  26. Libniz
    • 2 years ago
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    yeah, a lot of mental gymnastic, but thanks though

  27. phi
    • 2 years ago
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    maybe this helps the u(-k-1) is 1 from -inf to -1 and zero afterwards so the sum \[\sum_{k= -\infty}^{+\infty}u(-k-1)a^k= \sum_{k= -\infty}^{-1}a^k\] the step function does not "do" anything except zero out all k's bigger than -1 on the other hand, the step (n-k) affects the sum \[\sum_{k= -\infty}^{+\infty}u(n-k)a^k= \sum_{k= -\infty}^{n}a^k\] by tossing all entries where k>n so the step function is setting the upper limit. when we use both (multiply both together), both have to be true (1). you get 2 cases: n<-1 (where the upper limit is n) and n> -1 (where the upper limit is -1)

  28. phi
    • 2 years ago
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    the next trick if figuring out how to do the summations. For a>1, you can change the limits to positive values and invert a: a^-value= (1/a)^+value now it is a geometric series with a closed form solution.

  29. Libniz
    • 2 years ago
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    ^that one did it, much more clear now

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