What does the frequency representation of a function show?

[Tosh5457]

I don't understand what electrical engineers mean by the frequency of a signal... Frequency is the inverse of the period, but they speak of the frequency of non-periodic signals.

I know that I can derive the function using the Fourier transform, I just don't understand what frequency means in this context... For example, why is the Fourier transform of f(x) = 1 is $$\hat{f}(\xi )=\delta (\xi)$$ (dirac delta)? The only frequency that gives a non-zero value for f is 0, why is that?

 

[mathman]

f(x) = 1 contains no variable components, so its spectrum is simply the delta function.

 

[Tosh5457]

f(x) = 1 contains no variable components, so its spectrum is simply the delta function.

I see. But if I'm not mistaken, for f(x) = x, the function is

fˆ(ξ)=δ(ξ)

as well, but the function f(x) has variable components in this case.

The frequency measures the number of occurrences of an event. What's the occurrence in these examples?

 

[Mute]

I see. But if I'm not mistaken, for f(x) = x, the function is

fˆ(ξ)=δ(ξ)

as well, but the function f(x) has variable components in this case.

The frequency measures the number of occurrences of an event. What's the occurrence in these examples?

No, that's not correct. For the function f(x) = x, the Fourier transform is the derivative of the delta function. (The Fourier transform in this case must be interpreted as a generalised function aka a distribution).

To understand what the frequency of a non-periodic function is: For periodic functions you can write down a Fourier series; that is, the function can be written as the sum of infinitely many sinusoids. However, the frequencies of the sinusoids are restricted to multiples of the fundamental frequency of that function. For a non-periodic function, the Fourier transform is the same idea, except that now any frequency is possible because there is no fundamental frequency. The Fourier transform is essentially the coefficient in the Fourier series, and it tells you the weight of a given frequency in the composition of the non-periodic function.

 

[Tosh5457]

No, that's not correct. For the function f(x) = x, the Fourier transform is the derivative of the delta function. (The Fourier transform in this case must be interpreted as a generalised function aka a distribution).

To understand what the frequency of a non-periodic function is: For periodic functions you can write down a Fourier series; that is, the function can be written as the sum of infinitely many sinusoids. However, the frequencies of the sinusoids are restricted to multiples of the fundamental frequency of that function. For a non-periodic function, the Fourier transform is the same idea, except that now any frequency is possible because there is no fundamental frequency. The Fourier transform is essentially the coefficient in the Fourier series, and it tells you the weight of a given frequency in the composition of the non-periodic function.

Oh yes that's right, it's the derivative of delta, my mistake.

Ok I understand. Just one thing: what coefficient of the Fourier series are you talking about? an, bn or $$\frac{nπ}{L}$$?

 

[dodo]

I have a possible misconception that relates to the OP, and that maybe someone can clarify. I thought that, when making a Fourier approximation of an arbitrary, non-periodic function, you would make the approximation on a specific region of the curve (for example, with x in the interval [-2,2], or whatever)... and imagine this curve segment as "one period" of a (nonexistent) periodic function that would repeat over and over this curve segment. Similarly, when analyzing an incoming, "data-stream-like" signal, you would take a fixed-sized "window" on that data, and treat it again like "one period". Is this how Fourier analysis is supposed to work, or am I too far off?