Can I get a general function f(m,n)

  1. How can I get a general function f(m,n) that represents a series of 1 and 0, for example :
    1,0,1,0,1,0,1,0...; but also
    1,1,0,1,1,0,1,1,0...;
    1,1,1,0,1,1,1,0,1...

    where m is period and n nth number in certain period. In example two:
    m=3 (...1,1,0...) f(3,4)=1

    The function must be composed only by elementary function
     
  2. jcsd
  3. Re: 1010

    This might be a little desperate, but Fourier series should do the trick. I do not know if there is any other elementary function that would do the same, and I do not know if you consider Fourier series to be elementary (despite it involving only sine and cosine, which are both considered elementary.)
     
  4. Re: 1010

    oh c'mon.

    consider

    [tex] g(m) = \frac{1}{2} \left( 1 + (-1)^m \right) [/tex]

    for integer [itex]m[/itex]

    sure, find a Fourier series that, when sampled at integer values, gives you the values you want. but instead of sines and cosines, use the exponential version derived from Euler's formula:

    [tex] e^{i \theta} \ = \ \cos(\theta) \ + \ i \sin(\theta) [/tex]

    and

    [tex] -1 = e^{i \pi} [/tex]

    and

    [tex] g(x) = \sum_{n=-\infty}^{+\infty} c_n \ e^{i \ n (2 \pi/P) x } [/tex]

    where [itex] P [/itex] is the period of the periodic function

    [tex] g(x + P) = g(x) [/tex]

    for all [itex]x[/itex], and [itex]c_n[/itex] are the Fourier coefficients.

    but you can start with the simple equation with [itex](-1)^m[/itex] and imagine how you might put it together to get the gating functions you're looking for.
     
    Last edited: Jun 12, 2012
  5. LCKurtz

    LCKurtz 8,336
    Homework Helper
    Gold Member

    Re: 1010

    $$f(m,n)=\left\{ \begin{array}{rl}
    0&n=0\mod m\\
    1&\hbox{otherwise}
    \end{array}\right.$$
     
  6. Re: 1010


    does that count as an "elementary function"? do the mod or floor operators count as elementary functions?
     
  7. LCKurtz

    LCKurtz 8,336
    Homework Helper
    Gold Member

    Re: 1010

    You will have to check with the Elementary Function Arbitration Committee to get a definitive answer.
     
  8. Re: 1010

    good answer.
     
  9. Re: 1010

    i think i figured out a possible answer, but it involves the dirac delta function (or more precisely, the dirac comb). the dirac comb is a periodic sequence of equally space delta functions and can be represented as an infinite series. maybe we can construction this thing with the sinc() function:

    [tex] \ \operatorname{sinc}(x) \ = \ \frac{\sin(\pi x)}{\pi x} [/tex]

    and it has a removable singularity at zero so that

    [tex] \ \operatorname{sinc}(0) \ = \ \lim_{x \rightarrow 0} \frac{\sin(\pi x)}{\pi x} \ = \ 1[/tex]

    it's also true that the sinc() function is 0 for all non-zero integers. does that count as an "elementary function"? does an infinite sum of these sinc() functions count as an "elementary function"?

    if yes, i can assemble a general function. actually, this periodic sinc() function (the infinite sum) can be represented as a Fourier series with a finite number of terms, so i think that's where the answer is.
     
  10. Re: 1010

    okay, so first we define

    [tex] \ \operatorname{sinc}(x) \ = \ \frac{\sin(\pi x)}{\pi x} [/tex]

    and then we define this periodic function:

    [tex] g_m(x) = \sum_{k=-\infty}^{+\infty} \operatorname{sinc}(x - mk) [/tex]

    where m is an integer (and so is k and so is mk). we know that for integer n that

    [tex] g_m(n) =
    \begin{cases}
    1 & \ \ \text{if }n = \text{ any multiple of }m \\
    0 & \ \ \text{if }n = \text{ is any other integer}
    \end{cases} [/tex]

    now, i am pretty sure that this is the case:

    [tex] g_m(x) = \frac{1}{m} \sum_{k=1}^{m} \cos\left( 2 \pi \frac{k x}{m} \right) [/tex]

    even if it isn't, i think that this function has the property we need for when x is an integer n. it is zero for any integer n except when n is a multiple of m. since this is a harmonic series, you can get a closed form expression for it (someone else want to do it)? then subtract this from 1.
     
  11. Re: 1010

    [tex] \begin{align}
    g_m(x) \ &= \ \frac{1}{m} \sum_{k=1}^{m} \cos\left( 2 \pi \frac{k x}{m} \right) \\
    &= \ \frac{1}{2m} \sum_{k=1}^{m} e^{ i 2 \pi k x / m} \ + \ \frac{1}{2m} \sum_{k=1}^{m} e^{ -i 2 \pi k x / m} \\
    &= \ \frac{e^{ i 2 \pi x / m}}{2m} \sum_{k=0}^{m-1} e^{ i 2 \pi k x / m} \ + \ \frac{e^{ -i 2 \pi x / m}}{2m} \sum_{k=0}^{m-1} e^{ -i 2 \pi k x / m} \\
    &= \ \frac{e^{ i 2 \pi x / m}}{2m} \frac{ e^{ i 2 \pi m x / m} - 1}{e^{ i 2 \pi x / m} - 1} \ + \frac{e^{ -i 2 \pi x / m}}{2m} \frac{ e^{ -i 2 \pi m x / m} - 1}{e^{ -i 2 \pi x / m} - 1} \\
    &= \ \frac{1}{2m} \frac{ e^{ i 2 \pi x} - 1}{1 - e^{ -i 2 \pi x / m}} \ + \frac{1}{2m} \frac{ e^{ -i 2 \pi x } - 1}{1 - e^{ i 2 \pi x / m}} \\
    \end{align} [/tex]

    i'm getting tired, can someone else finish this?
     
  12. Re: 1010

    [tex] \begin{align}
    g_m(x) \ &= \ \frac{1}{2m} \frac{ e^{ i 2 \pi x} - 1}{1 - e^{ -i 2 \pi x / m}} \ + \ \frac{1}{2m} \frac{ e^{ -i 2 \pi x } - 1}{1 - e^{ i 2 \pi x / m}} \\
    &= \ \frac{e^{ i \pi x} \ e^{ i \pi x / m}}{2m} \frac{ e^{ i \pi x} - e^{ -i \pi x}}{e^{ i \pi x / m} - e^{ -i \pi x / m}} \ + \ \frac{e^{ -i \pi x} \ e^{ -i \pi x / m}}{2m} \frac{ e^{ -i \pi x} - e^{ i \pi x}}{e^{ -i \pi x / m} - e^{ i \pi x / m}} \\
    &= \ \frac{e^{ i \pi (m+1) x / m}}{2m} \frac{ e^{ i \pi x} - e^{ -i \pi x}}{e^{ i \pi x / m} - e^{ -i \pi x / m}} \ + \ \frac{ e^{ -i \pi x (m+1) / m}}{2m} \frac{ e^{ i \pi x} - e^{ -i \pi x}}{e^{ i \pi x / m} - e^{ -i \pi x / m}} \\
    &= \ \frac{ e^{ i \pi x} - e^{ -i \pi x}}{e^{ i \pi x / m} - e^{ -i \pi x / m}} \ \left( \frac{e^{ i \pi (m+1) x / m}}{2m} + \ \frac{ e^{ -i \pi x (m+1) / m}}{2m} \right) \\
    &= \ \frac{ \sin(\pi x) }{m \sin(\pi x / m)} \ \cos\left(\pi (m+1) x / m \right) \\
    &= \ \frac{ \sin(\pi x) }{m \sin(\pi x / m)} \ \cos(\pi x + \pi x / m) \\
    &= \ \frac{ \sin(\pi x) }{m \sin(\pi x / m)} \ \left( \cos(\pi x) \cos(\pi x / m) - \sin(\pi x) \sin(\pi x / m) \right) \\
    &= \ \frac{ \sin(\pi x) }{m} \ \left( \frac{\cos(\pi x)}{\tan(\pi x / m)} - \sin(\pi x) \right) \\
    \end{align} [/tex]
     
    Last edited: Jun 12, 2012
  13. Re: 1010

    i see it's that Dirichlet kernel on the left. can someone check out to see that this is 1 for n being a multiple of m and 0 for all other integers?

    maybe the bottom two or three equalities does not help us.
     
    Last edited: Jun 12, 2012
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