Understanding the Complex Analytic Signal: Benefits and Uses | Forum Discussion

[fisico30]

hello forum,

I might need some help understanding the usefulness of the complex analytic signal.

The world is made of real valued signals like x(t). Its Fourier transform can be one-sided, or, if we used complex sinusoids, two-sided and symmetric. So a real signal is only made of positive and equal amount of negative (complex) sinusoids. The negative sinusoids don't really a physical meaning, I guess.

The complex sinusoids seem to be useful. But then we come up with the complex analytic signal, which transforms a real signal into a complex signal with only the positive part of the frequency.
Ok, but what do we gain? We have been using the double sided spectrum. If we don't like the negative frequencies we could just filter them out.

thanks for any clarification
fisico30

 

[marcusl]

Wait a minute, a real signal always has a two-sided spectrum, and it's Hermitian (that means that the negative half of the spectrum is the odd complex conjugate of the positive half). The negative half does have a physical meaning, as seen when you modulate a carrier with the real sinusoid. You get a two-sided spectrum around the carrier.

The real part of the analytic signal representation is the real signal, but the analytic signal has only a one-sided (positive) spectrum as you note. There are two advantages. First, many operations and analyses are easier to perform on complex signals than on real ones. Consider modulation of a carrier by an exponential as a very simple example. In complex notation it is trivial to see that the output is at the sum frequency

y=e^{i\omega_m t}e^{i\omega_c t} = e^{i (\omega_m+\omega_c) t}

whereas this requires a little work when expressed as real sinusoids. You can imagine the simplifications when dealing with truly complicated signals. Second, certain communications modulations such as single-sideband are naturally described by a (one-sided) analytic spectrum.