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Can I get a general function f(m,n)by Emilijo
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#1
Jun1212, 09:34 AM

P: 36

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
Jun1212, 12:03 PM

P: 295

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.)



#3
Jun1212, 12:53 PM

P: 2,251

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. 


#4
Jun1212, 01:02 PM

HW Helper
Thanks
PF Gold
P: 7,663

Can I get a general function f(m,n)
0&n=0\mod m\\ 1&\hbox{otherwise} \end{array}\right.$$ 


#5
Jun1212, 03:42 PM

P: 2,251

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


#6
Jun1212, 03:49 PM

HW Helper
Thanks
PF Gold
P: 7,663




#7
Jun1212, 04:37 PM

P: 2,251

good answer.



#8
Jun1212, 04:55 PM

P: 2,251

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 nonzero 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. 


#9
Jun1212, 05:14 PM

P: 2,251

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. 


#10
Jun1212, 05:27 PM

P: 2,251

[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}^{m1} e^{ i 2 \pi k x / m} \ + \ \frac{e^{ i 2 \pi x / m}}{2m} \sum_{k=0}^{m1} 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? 


#11
Jun1212, 05:39 PM

P: 2,251

[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] 


#12
Jun1212, 05:54 PM

P: 2,251

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. 


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