Meaning of negative frequencies in Baseband non sinusoidal, non periodic signal

I can understand the meaning of negative frequencies in a sine or cosine signal, since by using Euler's identities, you have two complex phasors moving to different directions, which when added, give a real (cosine or sine) signal.

However if I have a random baseband (real, not periodic) signal, what does it mean for it to have negative frequencies?

• Since the signal is still the sum (or integral) of a number of complex sinusoids, the same idea applies individually to each signal component (all operations involved are linear). – MBaz Jul 2 '20 at 19:00
• What does it mean for a non-periodic signal to have a frequency whatsoever? A well enough behaved real valued signal can be represented as a sum of real sinusoidals. Each real sinusoidal is the sum of a positive and negative frequency complex sinusoidal. – Cedron Dawg Jul 2 '20 at 19:30

It means the same thing.

The negative frequencies are there to cancel out the complex portion of the signal in the time domain, so that the time domain signal is constrained to the real axis.

The Fourier Transform decomposes a signal, $$x(t)$$ onto a set of complex basis functions: $$e^{-j\omega t}$$ for $$\omega \in (-\infty, \infty)$$.

If $$x(t)$$ is real, then the decomposition, i.e. the Fourier Transform $$X(\omega)$$, must be such that the complex portions cancel out, when the inverse transform is performed to recompose the signal from it's constituent parts.

This is a conceptual explanation of why the Fourier Transform of a real signal is symmetric about $$0$$ Hz.

If you have a random baseband real values time domain signal, then the frequency domain representation is symmetric around 0hz. The reason is simple. Fourier transform basically computes inner product of the signal with Fourier basis function $$e^{j2\pi ft}$$, this is done to get the projection of time domain signal onto a complex sinusoidal of frequency f. Why do we project? Because projection gives is a measure of similarity. So, in other words by taking Fourier transform we are actually computing how similar is our input to a particular frequency complex exponential $$e^{j2\pi ft}$$, meaning what is the composition of this time domain signal in terms of different frequencies.

All signals are made up of sinusoidals. Sines and Cosines. The exact terms are given by appropriate Fourier representations. A periodic continuous time signal has Fourier series, a continuous time signal has Fourier transform, a discrete infinite sequence has DTFT, a discrete periodic sequence has DFS and a discrete finite length sequence has DFT. All of these only differ in Fourier Basis of the signal space.

Now, if you took only those complex exponentials which are having positive f, and then you try to construct a real values signal, you cannot because, complex valued exponential will actually expand to $$e^{j2\pi ft} = cos2\pi ft +jsin2\pi ft$$, which is having an imaginary part at different frequencies f. To cancel these imaginary components out, you need negative frequency complex exponentials at same frequencies f of exactly the same magnitude. That is why a real valued time domain signal (continuous or discrete) always has a symmetric Frequency domain representation across origin.