What is the exact meaning of the output of the Discrete Fourier Transform

Question

I'm fairly new to the subject, but so far my understanding that this would be a transform you could use to go from a discrete set of data, say [1, 0, 1, 2] to a continuous sinusoidal function in the frequency domain, if I have fundamentally got this wrong maybe this is where my confusion comes from but once you have applied the DFT to this signal, and then find its magnitude. What is the meaning of the outputs, are they coefficients of a sinusoidal function? Or is the just the representation of that data as an amplitude in accordance to its frequency?

Answer 1

The Discrete Fourier Transform (DFT) can be understood in different contexts. What the meaning of the values are depend on the context. Ideally you should understand them all.

The DFT is usually taught as a Dirac delta sampled case of the continuous FT with a rectangular window. This can be confusing if you don't know Calculus, or haven't had a little bit of Real Analysis. In this case, the values of the DFT can be considered coefficients of complex exponential functions that will reconstitute your original signal.

Another approach is to understand the DFT as a matrix multiplication in a Linear Algebra sense. In this case the DFT values can be considered coordinates in a N dimensional vector space.

My favorite, in terms of a concrete understanding, particularly for real valued signals, is the DFT bin value is a weighted average of a set of points (Roots of Unity) on the unit circle on the complex plane. You can also think of it as a center of mass. You can find a write up of this interpretation in my blog artice "DFT Graphical Interpretation: Centroids of Weighted Roots of Unity".

If you are not familiar with complex numbers, the Roots of Unity, or Euler's Equation, I recommend that you read my first blog as well. Understanding these concepts is key to understanding the DFT in all contexts.

Hope this helps,

Ced