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I am a beginner of Fourier analysis, but my major is economics, I have not much of idea about frequency domain. My question is, since we have whole set of theory to work on time domain, such as time series analysis and stochastic process, why we need frequency domain.

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The frequency domain's meaning is closely related to the subject of your model. – user864940 Aug 3 '11 at 12:56
A generic answer for questions of the form: "I'm a beginner in ... but why do I need to study it?" The answer is: Keep going a few weeks to find out, because that topic was put into the course for a reason. – GEdgar Aug 3 '11 at 13:15
up vote 4 down vote accepted

I don't know much about its utility in economics, but Fourier analysis is invaluable in physics.

The main reason for that is that it transforms linear differential equations into simple algebraic equations. For instance, consider a solution $f(t)$ to the equation

$$ \frac{d^2}{dt^2} f(t) + v\frac{d}{dt}f(t) + \omega_0^2 f(t) = h(t)$$

where $\omega_0,v$ are given numbers and $h(t)$ is a given function. (This describes the motion of a damped spring subject to some external driving force $h(t)$)

The magic of Fourier transformation is that it turns derivatives $\frac d{dt}$ into multiplication with the frequency $-i\omega$. For example, the fourier transform of the equation above is

$$ -\omega^2\hat f(\omega) -iv\omega\hat f(\omega) + \omega_0^2\hat f(\omega) = \hat h(\omega)$$

which is readily solved as

$$ f(\omega) = \frac{h(\omega)}{\omega_0^2 - \omega^2 + vi\omega} .$$

This is the familiar Lorentz-peak.

This kind of calculation is pervasive in many fields of physics, see also: linear response, dispersion, electrical signal processing.

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The "frequency domain" is simply a different way to look at the data that you have in the time domain. When you listen to a symphony, the time domain description tells you what sound you hear in every given instant, while the frequency domain description tells you, roughly, what instruments are involved and the ways they are played.

To have a possible application at hand: If your stocks change on a weekly basis only, that fact is most easily seen via the frequency domain description: Higher frequencies than weekly ones won't turn up in the power spectrum of your stock value time series. If, on the other hand, very high frequencies turn up in the power spectrum, you know that you'll better keep your eyes on your news ticker.

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In short: there's just stuff that's easier to do in the frequency domain than in the time domain. To use an analogy: I could do stuff in rectangular coordinates, but sometimes the problem becomes much simpler (or even trivial!) if I switch to polar coordinates... – J. M. Aug 4 '11 at 7:10

We have a whole left eye, so why do we need a right eye? The answer is perspective. Being able to view a problem in both the time and frequency domains is a powerful tool and often reveals structure to a problem that isn't obvious in just the time domain.

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While what you said is true, I suspect to the OP your answer basically boils down to "because I say so." – Willie Wong Aug 3 '11 at 20:26

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