Can an entire signal with transients be reconstructed from an inverse Fourier transform?

Question

If we have a long signal (say some music) that contains a lot of transients at different frequencies (i.e. the relative "volume" of each different frequency changes over time) and we take a Fourier transform of that signal (the whole thing in one transform,) will the inverse Fourier transform give us the exact original signal back? If so, how?

I understand that the Fourier transform produces a "plot" with one axis the frequency and on the y axis a complex amplitude that is phase + magnitude, but this phase+magnitude information would be a kind of average of the entire signal (for each frequency.) How would it be possible to get back information about how the volume of each frequency changes as the signal/music progresses?

Assume we are talking about the Fourier transform of the whole signal and not a bunch of different windows.

Answer 1

First, mathematically, the Fourier transform contains just as much information as the original time-domain signal. Since each step of the transform is both linear and reversible, it is possible to re-generate the time-domain signal from the frequency-domain data.

If you really want to get an intuitive feel for how all this works, I would recommend getting familiar with software that is designed to handle and plot large amounts of data. Free examples would include numpy and GNU octave; there's also commercial software like MathCAD and MATLAB. There are also websites that allow you to play around with this kind of data. Learn how to create time-domain signals and plot them along with their Fourier transforms.

Try working with signals that consist of a single pulse placed in different positions in the time window. Observe what happens in the frequency domain — especially the phase information — as the time position varies. Note that when you do the inverse transform, you do indeed get the pulse back in the correct position.

A second key concept is how amplitude modulation of a tone results in sidebands in the frequency domain. A musical note is basically a tone that is amplitude modulated (multiplied) by a pulse of some shape. Try forward and inverse transforms of notes of various shapes and positions. A key fact about the Fourier transform is that multiplication in the time domain becomes convolution in the frequency domain, and vice-versa. Note how the transform of the pulse becomes convolved with the transform of the tone.

Finally, a piece of music is a collection of notes. Since the Fourier transform is linear, the transform of the sum of the notes (i.e., the transform of the entire piece) is simply the sum of the transforms of the individual notes. It's all there — just not in an easily-recognizable form!

Answer 2

The Fourier transform is an orthogonal transform. If the function is transformable (many functions, particularly continuous ones, and quite a number of non-functions are), the transform preserves all information.

If the original function contains lots of transients with piecewise different frequencies, you can use shift and windowing theorems to calculate the transform and invert it. In particular, you can use the convolution theorem for convolving the whole function with some impulse response.

What the transients affect, however, is the interpretability of the frequency domain. The more transients are involved and the shorter the steady parts of the original function are, the more opaque the transform data becomes. That is the main reason that signal processing applications work with windowed overlapping short-time Fourier transforms: not because they would contain more data, but because the data becomes a lot more interpretable again.

In particular, for a spectrum analyzer you don't want to transform too long segments exactly because most real signals change characteristics over time. When they don't, like when analyzing static noise, you get improved results with arbitrary long transform times.

Also, if you work with piecewise changing and/or adaptive convolution kernels, the transitions can be better managed with short-time transforms.

Answer 3

Can an entire signal with transients be reconstructed from an inverse Fourier transform?

In theory, if you have a "well behaved" signal (see below) and you know the Fourier transform exactly, you will be able to reconstruct the original signal. Unfortunately, even slight deviations from the exact Fourier transform will result in significant deviations in the reconstructed signal, if the signal is complex, like a piece of music. So in practice, reconstructed music is not likely to be very faithful.

In practice, real signals are well behaved, but if we leave this fact aside, and consider all mathematical functions, we find that there are an infinite number of (discontinuous) functions that all have the same Fourier transform. I will give an example.

$$f(0) =1$$ $$f(x\ne0)=0$$

and

$$g(x)=0$$

Both of these functions have the same Fourier transform, i.e.

$$F(\omega)=0 +0j$$

It should be obvious that are an infinite number of such functions with the same Fourier transform. The same is true for any function.

Answer 4

Sadly, no. Fourrier transform takes the assumption that a signal is periodic over time. Music wpuld most likely result in a white noise source if repeated long enough.