I'm a Biology Ph.D. student with some math literacy (multivariable calc, discrete math, linear algebra, quantum chemistry) under my belt, so apologies if my knowledge is substandard for the work I'm trying to do. I am currently looking at some time series of environmental data sampled every 20 minutes (temperature, etc). With help from a Geology professor, I am teaching myself some time-series analysis techniques and had some quick questions about the Fourier Transform.

My understanding: F.T. involves plotting your time series in the complex plane in polar coordinates, and averaging your points into a vector. Changing a multiplier (radians per timepoint) changes the vector, because… no correlation would cause the points to average out to zero, but any 'resonance' would cause the data to pull the vector off of 0.

The point where I'm having trouble, theoretically, is understanding what the imaginary component of the frequencies represents. I understand that in negative frequencies, imaginary components are odd and reals are even. I also understand that when you're looking at the One-sided power spectrum, you end up using the magnitude of the vector to determine power. This also makes sense to me.

Is having an imaginary component to the data just an incidental happenstance that results from utilizing the complex plane? Using this makes sense to me in that you're taking advantage of Euler's equations (which I'm sure helps with a lot of the math), but is there any sort of real-world meaning to imaginary components of frequencies?

Also, my conceptual understanding of Fourier comes from using my data, which does not have an imaginary component to it. How does plotting the data in the complex plane work when you start thinking about an input with both real and imaginary components?

  • $\begingroup$ If you are talking about the complex plane as $\mathbb{C}^1$, then I can tell you this. $\mathbb{C}^1\cong\mathbb{R}^2$. This allows you to simplify two sets of variables, $(x,y)\in\mathbb{R}^2$ to just one in $\mathbb{C}^1$. $\endgroup$ – user122283 Apr 3 '14 at 22:50
  • $\begingroup$ Also, the frequency $\xi$ is usually real, giving us the Fourier transform of a function $f:\mathbb{R}^1\to\mathbb{C}^1\cong\mathbb{R}^2$ as $$\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx$$ $\endgroup$ – user122283 Apr 3 '14 at 22:55

The easier way to think of the Fourier transform of a function is in terms of magnitude and complex argument (corresponding to the "polar" representation $r e^{i \theta}$ rather than the "rectangular" representation $a + bi$).

To try to put things in the terms you've used: let $x(t)$ be the time series, and let $X(\omega)$ be the Fourier transform. The magnitude $|X(\omega)|$ at a particular frequency $\omega$ measures the power of the "resonance" of the time series with the reference oscillation $e^{i\omega t}$. Along those lines, $\angle X(\omega)$ (the argument) tells you the phase of the oscillation with which $x(t)$ oscillates the strongest.

  • $\begingroup$ Thanks for answering my question, and for following along with the coarse nonmathematical language I am using. I guess thinking of it in terms of the real and imaginary components is somewhat flawed in that they are the cartesian coordinate representation of a polar coordinate vector that we use to find the power and phase. Does this sound correct? $\endgroup$ – Umi Apr 4 '14 at 15:06
  • $\begingroup$ I wouldn't say that thinking of it in terms of real and imaginary components is "flawed". It is, however, generally more difficult less useful to do so. I think there may be some physical interpretation of the real and imaginary components, probably having to do with the extent of constructive/destructive interference with a reference real-oscillation $\cos(\omega t)$. $\endgroup$ – Omnomnomnom Apr 4 '14 at 16:08

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