Which of the signals is not the result of fourier series expansion?

[dexterdev]

Homework Statement

Which of the signals is not the result of Fourier series expansion?
options :
(a) 2cos(t) + 3 cos(3t)
(b) 2cos($\pi$t) + 7cos(t)
(c) cos(t) + 0.5

Homework Equations

Dirichlet conditions

The Attempt at a Solution

From observation, I thought all are periodic and so must be Fourier series expansions. But since this was a question in an objective exam, I went for option (b). Although the answer is right, I am not satisfied, by the explanation of Dirichlet conditions. And the plot of the function 2cos($\pi$t) + 7cos(t) seems periodic too and no discontinuities etc.

My question is actually is 2cos($\pi$t) + 7cos(t) a function which cannot be a result of Fourier expansion or not?

 

[Simon Bridge]

What are the "Directlet Conditions" conditions for exactly?
If each of the possible answers are the Fourier series of something - what are they each a Fourier series of?

 

[dexterdev]

f(x) must be absolutely integrable over a period.
f(x) must have a finite number of extrema in any given interval, i.e. there must be a finite number of maxima and minima in the interval.
f(x) must have a finite number of discontinuities in any given interval, however the discontinuity cannot be infinite.
f(x) must be bounded.

These dirichlet conditions were told not obeying in the case of option (b). But I am confused.

 

[AlephZero]

From observation, I thought all are periodic and so must be Fourier series expansions.

Two questions:

What is the mathematical definition of a periodic function?

If you think function (b) is periodic, what is the period?

 

[Simon Bridge]

Please answer the questions in post #2.

f(x) must be absolutely integrable over a period.
f(x) must have a finite number of extrema in any given interval, i.e. there must be a finite number of maxima and minima in the interval.
f(x) must have a finite number of discontinuities in any given interval, however the discontinuity cannot be infinite.
f(x) must be bounded.

Please answer the questions in post #2 - thank you.
The above merely says what the Direchtlet conditions are. I asked what they were conditions for.
If an equations satisfied the Directlet conditions - what does it mean?

These dirichlet conditions were told not obeying in the case of option (b). But I am confused.

Consider - you have a statement that each of the equations are exact Fourier transforms of some other function. But for one of them, this statement is false.

Like AlephZero suggests - you should use the conditions as a test.

 

[dexterdev]

If a periodic signal meets dirichlet conditions , that mean theoretically I can expand it using FOURIER SERIES EXPANSION THEORY. And a periodic function means self repeating function. Theoretically it expands from -infinity to infinity. For periodic function f(t) = f(t + n T) , n - integer and T time period.

 

[Simon Bridge]

If a periodic signal meets dirichlet conditions , that mean theoretically I can expand it using FOURIER SERIES EXPANSION THEORY.

What does that mean?
Surely all functions have a Fourier series expansion.
What is special about the Fourier expansion of those functions that meet the Directlet conditions?

And a periodic function means self repeating function. Theoretically it expands from -infinity to infinity. For periodic function f(t) = f(t + n T) , n - integer and T time period.

Great - now, what AlephZero is suggesting is this: for the function in (b), find T.

 

[LCKurtz]

Another approach to answering the OP's question. FS have sums of terms $\cos\frac{n\pi t}p$ and $\sin\frac{n\pi t}p$. What about the ratio of the angular frequencies $\frac{n\pi} p$ for two different values of $n$?

 

[Ray Vickson]

Homework Statement

Which of the signals is not the result of Fourier series expansion?
options :
(a) 2cos(t) + 3 cos(3t)
(b) 2cos($\pi$t) + 7cos(t)
(c) cos(t) + 0.5

Homework Equations

Dirichlet conditions

The Attempt at a Solution

From observation, I thought all are periodic and so must be Fourier series expansions. But since this was a question in an objective exam, I went for option (b). Although the answer is right, I am not satisfied, by the explanation of Dirichlet conditions. And the plot of the function 2cos($\pi$t) + 7cos(t) seems periodic too and no discontinuities etc.

My question is actually is 2cos($\pi$t) + 7cos(t) a function which cannot be a result of Fourier expansion or not?

I hope you realize that the Dirichlet conditions are irrelevant to this problem. They give (sufficient) conditions for the Fourier series of a function f(x) to converge to f(x) itself (except at jump discontinituies). In the current problem you need to work backwards from this: you are told a trigonometric series, and you want to know if it could be the Fourier series of some unknown f(x).

 

[dexterdev]

me too thought the same only. One online test was saying that 2cos(πt) + 7cos(t) cannot be a Fourier series outcome since it is not obeying the dirichlet conditions. But actually now I think that the question has a mistake. Thanks every one who took effort for this.

 

[vanhees71]

The solution is correct. Think about the periodicity of the function!

 

[dexterdev]

All functions here are periodic right, since it has only sinusoids. Am I wrong somewhere? If so please correct me.

 

[Dick]

All functions here are periodic right, since it has only sinusoids. Am I wrong somewhere? If so please correct me.

You are wrong. 2cos(πt) is periodic. 7cos(t) is periodic. 2cos(πt)+7cos(t) is NOT periodic.

 

[Ray Vickson]

All functions here are periodic right, since it has only sinusoids. Am I wrong somewhere? If so please correct me.

To expand on what Dick said: the term cos(t) is periodic, with period 2π, while cos(πt) is periodic, with period 2. Since π is an irrational number, the two periods are not "commensurate" and that means that when you add the two terms together the result is non-periodic.

Added note: the above statement seems clear enough, but I will admit it needs proof, and I don't see at the moment how that can be tackled. It might not be easy!

 

[LCKurtz]

Another approach to answering the OP's question. FS have sums of terms $\cos\frac{n\pi t}p$ and $\sin\frac{n\pi t}p$. What about the ratio of the angular frequencies $\frac{n\pi} p$ for two different values of $n$?

At the risk of beating a dead horse: You don't have to analyze the behavior of the original function nor get into a discussion of periodicity or Dirichlet conditions. The ratio of any two angular frequencies in any FS is$$
\frac{\frac{n\pi}{p}}{\frac{m\pi} p}=\frac n m$$which is rational, and which isn't in the given example.

 

[dexterdev]

Thank you friends. Now that is a point I have to dwell into. Let me try proving it.
Finally let me ask one more thing. Fourier series representation is only for periodic functions. 2cos(πt) is periodic. 7cos(t) is periodic. 2cos(πt)+7cos(t) is NOT periodic. So 2cos(πt)+7cos(t) must be represented using FOURIER TRANSFORM right?

 

[Simon Bridge]

Thank you friends. Now that is a point I have to dwell into.

That would be interesting [1]

Let me try proving it.
Finally let me ask one more thing. Fourier series representation is only for periodic functions.

Nope.

2cos(πt) is periodic. 7cos(t) is periodic. 2cos(πt)+7cos(t) is NOT periodic. So 2cos(πt)+7cos(t) must be represented using FOURIER TRANSFORM right?

Nope.

Remember - each of the functions, except one, is supposed to be the complete Fourier transform for another function ... that is what I was trying to get you to realize about the conditions.
When you get the Fourier transform of a function, it has a characteristic look - especially if the transform is finite. The question is testing that you understand this.