Should I combine the negative part of the spectrum with the positive one?

Question

When filtering sound I currently analyse only the positive part of the spectrum. From the mathematical point of view, will discarding the negative half of the spectrum impact significantly on my analysis?

Please consider only samples that I will actually encounter, not computer generate signals that are designed to thwart my analysis.

I know this question involves physics, biology and even music theory. But I guess the required understanding of mathematics is deeper than of those other fields of study.

Answer 1

Sound processing is achieved through Real signal samples. Therefore there is no difference in the phase and magnitude of the FFT, or DFT coefficients, from positive to negative part of the found spectrum.

So, to save us or the machine the burden of saving/analyzing twice the same information/data, one looks only to the positive side of the FFT/DFT. However, do take notice that when figuring out spectral energy, you must remember to multiply the density by two (accounting for the missing, yet equal, negative part).

Answer 2

The general Fourier transform is defined for complex functions (signals), and in that case all the frequencies are meaningful. For most common applications, we have a real signal, and then the reality condition implies that the values of the transform for negative frequencies is the complex conjugate of those corresponding for positive frequencies, hence they are redundant and one does not need to compute/store them. It's in this scenario and in this sense, that we can "ignore" the negative frecuencies, safely, with no error. But be aware that this does NOT mean that we consider them to be zero.