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Complex periodic functions in a vector space

  1. Feb 25, 2017 #1
    1. The problem statement, all variables and given/known data
    Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V.

    Show that if f(t) and g(t) are members of V then so is f(t) + g(t)


    2. Relevant equations


    3. The attempt at a solution

    f(t) = e(i*w0*t))

    g(t) =e(i*w0*t +φ))

    Where W0 = 2*π


    Using 'Euler's' I can write these as;

    f(t) =cos(w0*t) + i*sin(w0*t)

    g(t) =cos(w0*t +φ) + i*sin(w0*t +φ)


    So for part a) I would plot these functions separating the real and imaginary parts and choosing a value for φ to illustrate the phase shift?


    Partb) f(t) + g(t) = e(i*w0*t)) + e(i*w0*t +φ))

    = (1 + eφ) * ei*w0*t

    The signal remains a periodic complex function of t with a period of 1 and is therefore a member of V.


    Thanks
     
    Last edited by a moderator: Feb 25, 2017
  2. jcsd
  3. Feb 25, 2017 #2

    PeroK

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    What makes you think the functions are complex valued? Is that in the question? I ask because that make them hard to draw.

    Are all complex functions of the form ##e^{iwt}##?
     
  4. Feb 25, 2017 #3

    pasmith

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    Part (2) is asking you to prove the result for all possible choices of [itex]f[/itex] and [itex]g[/itex], not just your two example functions.
     
  5. Feb 25, 2017 #4
    Sorry, I missed the complex part out of the question, I have edited my post now
     
  6. Feb 25, 2017 #5

    PeroK

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    Okay. What's the definition of a periodic function?
     
  7. Feb 25, 2017 #6

    I thought all periodic ones were, or could be represented by the exponential equivalent?
     
  8. Feb 25, 2017 #7
    A function that repeats itself over a set period
     
  9. Feb 25, 2017 #8

    PeroK

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    Not at all!

    Can you express that mathematically? In this case for a function of period ##1##.
     
  10. Feb 25, 2017 #9

    Sin(wt) ≡ Sin(wt+t) ?
     
  11. Feb 25, 2017 #10

    PeroK

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    Come on, that's not a definition of anything! A definition would have to be something like:

    ##f## is periodic with period ##1## if ...
     
  12. Feb 25, 2017 #11
    f is periodic with a period of 1 if ... Sin(2π*t) ≡Sin(2πt+T)
     
  13. Feb 25, 2017 #12

    PeroK

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    To be honest, it's difficult to know what to say to that. It suggests that you lack some basic understanding of mathematics. ##\sin## is an example of a periodic function, but in no way is it the only periodic function or the definition of a periodic function.

    https://en.wikipedia.org/wiki/Periodic_function
     
  14. Feb 25, 2017 #13

    Sorry, I was using Sin as an example, I appreciate it is not the only periodic function. Therefore, a function f(x) is said to be periodic if f(x) = f(x+P)
     
  15. Feb 26, 2017 #14
    h(t+1) = f(t+1) + g(t+1) =f(t)+g(t)=h(t)

    Would I use a general form for a complex periodic function? And prove the above using that?


    Thanks
     
  16. Feb 26, 2017 #15

    PeroK

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    Yes, exactly.
     
  17. Feb 26, 2017 #16
    Would the general form be related to the complex fourier series?


    Thanks
     
  18. Feb 26, 2017 #17

    PeroK

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    Absolutely nothing to do with Fourier series. This is an algebraic result and requires no analytical structure whatsoever.
     
  19. Feb 26, 2017 #18
    Ah, I though as fourier series was for representing a periodic signal I could use the Cke(-iwt) as a model for a complex periodic signal


    Would I instead set

    f(t) =A*ei(wot +θ1)

    g(t) =B*ei(wot +θ2)


    And set about proving h(t+1) = f(t+1) + g(t+1) =f(t)+g(t)=h(t) ?



    Thanks
     
  20. Feb 26, 2017 #19

    PeroK

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    You still have a fundamental misunderstandinhg of what is a periodic function. It has absolutely nothing to do with sines, cosines, exponentials, Fourier Series or whatever else. Periodicity is a simple algebraic property:

    A function, ##f## is periodic with period ##P## if ##\forall x \ f(x+P) = f(x)##. That is it. And that is all you can use.

    Here's an example:

    ##f(x) = 1## when ##x## is an integer, and ##f(x) = 0## otherwise.

    ##f## is periodic with period ##1##, but is clearly not a sine, cosine or exponential.

    So, if you tried to prove that all periodic functions are of the form ##exp(iwt)## then you would be wrong, as the function above demonstrates.

    I misunderstood your post #14, which I thought was a proof for general periodic functions, ##f## and ##g##. Post #14 is essentially a valid proof of the result. So, your misunderstanding extends to not recognising a proof even when you've done it!

    Can you see why post #14 is a proof? And why there is nothing more to do, other to to say more formally what you doing?
     
  21. Feb 26, 2017 #20
    It's definitely becoming clear now, the complex part was confusing me though, is it possible that while h(t+1) = f(t+1) + g(t+1) =f(t)+g(t)=h(t) holds true, is it possible that h(t) could no longer be a complex function and thus not a part of V?

    Say if f(t) =-g(t) then h(t) would still = h(t+T) but would it still be considered a complex function?


    Thanks
     
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