Discussion Overview
The discussion revolves around the behavior of Fourier transforms (FT) when dealing with sine waves that have the same frequencies but different amplitudes. Participants explore the implications of this scenario in both continuous and discrete Fourier transforms, focusing on how these amplitudes affect the final result.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant asks whether sine waves with similar frequencies but different amplitudes will be summed in the final Fourier transform result.
- Another participant seeks clarification on whether the question pertains to the difference between functions with Fourier transforms that differ by a constant or the effect of different amplitudes within the same transform.
- A participant explains that sine waves of different frequencies will produce peaks in the Fourier transform corresponding to their amplitudes, but the similarity of frequencies does not affect this outcome.
- It is noted that two Fourier transforms with identical frequencies but different amplitudes correspond to different time-domain signals, and that the Fourier transform is a linear operator allowing for the addition of transforms frequency by frequency.
- A later reply confirms that for a discrete Fourier transform, "similar" means close enough to be in the same bin, while for continuous Fourier transforms, "similar" means exact.
Areas of Agreement / Disagreement
Participants express differing views on the clarity of the original question and the implications of amplitude differences in Fourier transforms. There is no consensus on the interpretation of the effects of similar frequencies and differing amplitudes.
Contextual Notes
The discussion includes assumptions about the definitions of "similar" in the context of discrete versus continuous Fourier transforms, which remain unresolved.
Who May Find This Useful
Individuals interested in signal processing, Fourier analysis, or the mathematical properties of transforms may find this discussion relevant.