Have some queries about Fourier Transform?

Question

I am fond of Fourier Transform. I have some queries about Fourier Transform

1. In most of the cases,the Fourier transform of a signal is symmetric about positive and negative axis.I think the computational complexity increases because only half part of symmetric spectrum (i.e. spectrum except on negative axis) is of use. Also,while calculating in frequency domain we could get wrong value of energy /power due to the spectrum on negative axis .

2. In the Fourier transform formula the limits of integration are from -infinity to +infinity .But for a signal which is continuously or exponentially increasing with time,one can't compute it's Fourier transform.

3. After computation of Fourier transform of a signal, we get Phase and Frequency spectrum of the whole signal which is localised in frequency domain only . But from both these spectrums,we don't get any spatial component features like which frequency component is present at which time (and same with the phase value ).

4. If we think practically ,concept of negative frequency doesn't exists. But after computation of Fourier transform of signal,with dc and positive frequencies we also get unnecessary negative frequency components. I think concept of negative frequency doestn't exists practically.

So can anybody give explanation on any of the above doubts?

Answer 1

In most of the cases,the Fourier transform of a signal is symmetric about positive and negative axis. So i think the computational complexity increases. Also, energy on negative side unnecessarily gets calculated/wasted.

For real-valued signals, the Fourier transform is conjugate-symmetric about the y-axis.

However, it's entirely possible to use this information when calculating the transform (or estimating it numerically) and so there's no increase in computational complexity.

In signal processing, complex-valued signals are also considered, and when these are used then the transform is no longer conjugate-symmetric.

In the Fourier transform formula the limits of integration are from -infinity to +infinity .But for a signal which is continuously or exponentially increasing,one can't compute it's Fourier transform.

Yes. This is essentially why the Laplace transform exists.

My experience, however, is that the Laplace transform is rarely needed for practical engineering work (at least in my area of experties).

After computation of Fourier transform of a signal, we get Phase and Frequency spectrum of the whole signal which is localised in frequency domain only . But from both these spectrums,we don't get any spatial component features.

I'm not sure what you mean by this.

In image processing, they certainly do do Fourier transforms between the spatial domain and the spatial-frequency domain.

If we think practically, concept of negative frequency doesn't exists.

Negative frequency exists if you consider complex-valued functions and use the complex exponentials \$e^{j\omega{}t}\$ as your basis set. This allows you to keep track of in-phase and quadrature components without doing separate sine and cosine transforms.

As mentioned above, practical Fourier transform calculations take advantage of symmetry and don't do any extra work to determine the negative-frequency components.