Why is the Fourier transform so important?

This is quite a broad question and it indeed is quite hard to pinpoint why exactly Fourier transforms are important in signal processing. The simplest, hand waving answer one can provide is that it is an extremely powerful mathematical tool that allows you to view your signals in a different domain, inside which several difficult problems become very simple to analyze.

Its ubiquity in nearly every field of engineering and physical sciences, all for different reasons, makes it all the more harder to narrow down a reason. I hope that looking at some of its properties which led to its widespread adoption along with some practical examples and a dash of history might help one to understand its importance.

History:

To understand the importance of the Fourier transform, it is important to step back a little and appreciate the power of the Fourier series put forth by Joseph Fourier. In a nut-shell, any periodic function g(x)g(x) integrable on the domain D=[−π,π]D=[−π,π] can be written as an infinite sum of sines and cosines as

Lorem Ipsum's great answer misses one thing: The Fourier transform decomposes signals into constituent complex exponentials:

and complex exponentials are the eigenfunctions for linear, time invariant systems.

Put simply, if a system, HH is linear and time-invariant, then its response to a complex exponential will be a complex exponential of the same frequency but (possibly) different phase, ϕϕ, and amplitude, AA, --- and the amplitude may be zero:

So the Fourier transform is a useful tool for analyzing linear, time-invariant systems.

Additional to Peter's answer, there is another reason which is also related to the eigenfunction. That's, ekxekx is the eigenfunction of the differential operator dndxndndxn. That's why Fourier transform (corresponding to pure imaginary kk) and Laplace transform (corresponding to complex kk) can be used to solve differential equations.

Since ekxekx is the eigenfunction of both convolution and differential operator, maybe that's one the of the reasons why LSIV system can be represented by differential equations.

EDIT: As a matter of fact, differential (and integral) operators are LSIV operators, see here.

Another reason:

It's fast (e.g. useful for convolution), due to its linearithmic time complexity (specifically, that of the FFT).
I would argue that, if this were not the case, we would probably be doing a lot more in the time domain, and a lot less in the Fourier domain.

Edit: Since people asked me to write why the FFT is fast...

It's because it cleverly avoids doing extra work.

To give an concrete example of how it works, suppose you were multiplying two polynomials, a0x0+a1x1+…+anxna0x0+a1x1+…+anxn and b0x0+b1x1+…+bnxnb0x0+b1x1+…+bnxn.

If you were to do this naively (using the FOIL method), you would need approximately n2n2 arithmetic operations (give or take a constant factor).

However, we can make a seemingly mundane observation: in order to multiply two polynomials, we don't need to FOIL the coefficients. Instead, we can simply evaluate the polynomials at a (sufficient) number of points, do a pointwise multiplication of the evaluated values, and then interpolate to get back the result.

Why is that useful? After all, each polynomial has nn terms, and if we were to evaluate each one at 2n2n points, that would still result in ≈n2≈n2 operations, so it doesn't seem to help.

But it does, if we do it correctly! Evaluating a single polynomial at many points at once is faster than evaluating it at those points individually, if we evaluate at the "right" points. What are the "right" points?

It turns out those are the roots of unity (i.e. all complex numbers zz such that zn=1zn=1). If we choose to evaluate the polynomial at the roots of unity, then a lot of expressions will turn out the same (because a lot of monomials will turn out to be the same). This means we can do their arithmetic once, and re-use it thereafter for evaluating the polynomial at all the other points.

We can do a very similar process for interpolating through the points to get back the polynomial coefficients of the result, just by using the inverse roots of unity.

There's obviously a lot of math I'm skipping here, but effectively, the FFT is basically the algorithm I just described, to evaluate and interpolate the polynomials.
One of its uses, as I showed, was to multiply polynomials in a lot less time than normal. It turns out that this saves a tremendous amount of work, bringing down the running time to being proportional to nlognnlog⁡n (i.e. linearithmic) instead of n2n2 (quadratic).

Thus the ability to use the FFT to perform a typical operation (such as polynomial multiplication) much faster is what makes it useful, and that is also why people are now excited by MIT's new discovery of the Sparse FFT algorithm.

The other people have given great, useful answers. Just think about some signal: you only care what frequencies are in it (and their phase), not about the time domain. I do not this that this is a final or complete answer, but just another reason why the Fourier transform is useful.

When you have some signal, it could be composed of an infinite (or close to) number of frequencies, depending on your sampling rate. But, that's not the case: we know that most signals have the fewest number of frequencies possible, or that we're sampling at a high enough rate.

If we know that, why can't we use it? That's what the field of compressed sensing does. They know that the most likely signal is one that has the least error and has the fewest frequencies. So, they minimize the overall error relative to our measurements as well as the magnitude of the Fourier transform.

A signal of a few frequencies often has a minimal Fourier transform, or mostly zeros (aka "sparse," as they say in compressed sensing). A signal of one frequency just has a delta function as the transform, for example.

We can use the formal mathematical definition too.

x¯=arg min ||y−Ax||+λ||F(x)||x¯=arg min ||y−Ax||+λ||F(x)||

Here, all we're doing is minimizing the error (first set of ||⋅||||⋅||) and minimizing the Fourier transform (second set of ||⋅||||⋅||). Here, we have

• x¯x¯ as our reconstructed signal (most likely close to the original)
• yy, our measurements
• AA, a selection matrix
• xx, our signal
• λλ some constant
• F(x)F(x) the Fourier transform.

You may recall that Nyquist said that you have to measure at twice the highest frequency to get a good representation. Well, that was assuming you had infinite frequencies in your signal. We can get past that!

The field of compressed sensing can reconstruct any signal that's mostly zeros (or sparse) in some domain. Well, that's the case for the Fourier transform.

The main importance of the Fourier transform lies with system analysis. The main constituent of our universe is vacuum, and vacuum is a fundamentally linear and time-invariant carrier of fields: different fields superimpose by adding their respective vectors, and regardless of when you repeat the application of certain fields, the outcome will be the same.

As a consequence, a lot of systems also involving physical matter are to a good approximation behaving as linear, time-invariant systems.

Such LTI systems can be described by their "impulse response", and the response to any time-distributed signal is described by convolving the signal with the impulse response.

Convolution is a commutative and associative operation, but it is also quite computationally and conceptually expensive. However, the convolution of functions is mapped by Fourier transform into piecewise multiplication.

That means that the properties of linear time invariant systems and their combinations are much better described and manipulated after Fourier transformation.

As a result, things like "frequency response" are quite characteristic for describing the behavior of a lot of systems and become useful for characterizing them.

Fast Fourier transforms are in the "almost, but not quite, entirely unlike Fourier transforms" class as their results are not really sensibly interpretable as Fourier transforms though firmly routed in their theory. They correspond to Fourier transforms completely only when talking about a sampled signal with the periodicity of the transform interval. In particular the "periodicity" criterion is almost always not met.

There are several techniques for working around that, like the use of overlapping windowing functions.

However the FFT can be employed for doing discrete-time convolution when doing things right, and is it is an efficient algorithm, that makes it useful for a lot of things.

One can employ the basic FFT algorithm also for number theoretic transforms (which work in discrete number fields rather than complex "reals") in order to do fast convolution, like when multiplying humongous numbers or polynomials. In this case, the "frequency domain" is indistinguishable from white noise for basically any input and has no useful interpretation before you do the inverse transform again.

Some of the other answers in this thread have excellent mathematical discussions of the definition and properties of the Fourier transform; as an audio programmer, I merely want to provide my own personal intuition as to why it's important to me.

The Fourier transform permits me to answer questions about a sound that are difficult or impossible to answer with other methods. It makes hard problems easy.

A recording contains a set of three musical notes. What are the notes? If you leave the recording as a set of amplitudes over time, this is not an easy problem. If you convert the recording to a set of frequencies over time, it's really easy.

I want to change the pitch of a recording without changing its duration. How do I do this? It's possible, but not easy to do, by just manipulating the amplitude of an input signal. But it's easy if you know the frequencies that comprise the signal.

Does this recording contain speech or does it contain music? Super hard to do using only amplitude-based methods. But there are good solutions that guess the right answer nearly all of the time based on the Fourier transform and its family.

Almost every question you'd like to ask about a digital audio recording is made easier by transforming the recording using a discrete version of the Fourier transform.

In practice, every modern digital audio device relies heavily on functions very similar to the Fourier transform.

Again, forgive the highly informal description; this is merely my personal intuition as to why the Fourier transform is important.

the physics relevance of fourier transform is that it tells the relative amplitude of frequencies present in the signal . it can be defined for both discrete time and continuous time signal. Any signal can be represented as mixture of many harmonic frequencies. Fourier transform help in filter applications , where we need only certain range of frequencies then we first need to know what are the amplitudes of frequencies contains in the signal.