# Complex periodic functions in a vector space

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1. Feb 25, 2017

### jendrix

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. Feb 25, 2017

### PeroK

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 $e^{iwt}$?

3. Feb 25, 2017

### pasmith

Part (2) is asking you to prove the result for all possible choices of $f$ and $g$, not just your two example functions.

4. Feb 25, 2017

### jendrix

Sorry, I missed the complex part out of the question, I have edited my post now

5. Feb 25, 2017

### PeroK

Okay. What's the definition of a periodic function?

6. Feb 25, 2017

### jendrix

I thought all periodic ones were, or could be represented by the exponential equivalent?

7. Feb 25, 2017

### jendrix

A function that repeats itself over a set period

8. Feb 25, 2017

### PeroK

Not at all!

Can you express that mathematically? In this case for a function of period $1$.

9. Feb 25, 2017

### jendrix

Sin(wt) ≡ Sin(wt+t) ?

10. Feb 25, 2017

### PeroK

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

$f$ is periodic with period $1$ if ...

11. Feb 25, 2017

### jendrix

f is periodic with a period of 1 if ... Sin(2π*t) ≡Sin(2πt+T)

12. Feb 25, 2017

### PeroK

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

13. Feb 25, 2017

### jendrix

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)

14. Feb 26, 2017

### jendrix

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

15. Feb 26, 2017

### PeroK

Yes, exactly.

16. Feb 26, 2017

### jendrix

Would the general form be related to the complex fourier series?

Thanks

17. Feb 26, 2017

### PeroK

Absolutely nothing to do with Fourier series. This is an algebraic result and requires no analytical structure whatsoever.

18. Feb 26, 2017

### jendrix

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

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

19. Feb 26, 2017

### PeroK

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?

20. Feb 26, 2017

### jendrix

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