Obtaining Phase and Amplitude from FFT

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Discussion Overview

The discussion revolves around the feasibility of calculating the phase and amplitude of a signal from data obtained through the Fast Fourier Transform (FFT). Participants explore theoretical and practical considerations, including the implications of signal characteristics and computational constraints.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the example signal is time-varying or constant, suggesting that if it is constant, the problem may be easier to solve.
  • Another participant notes that while it is theoretically possible to calculate phase and amplitude from FFT data, practical limitations exist, such as the amplitude being less than a least significant bit (LSB) or the sampling rate being insufficient for the frequency of interest.
  • A participant asks if having a good estimate of the angular frequency (ω) would aid in the calculations, reiterating the form of the signal as B + A*cos(ωt + φ).
  • It is mentioned that the output of the FFT is a complex number for each frequency bin, which can be converted from rectangular to polar coordinates to obtain magnitude and phase angle.
  • There is a question about the reference point for time (t) in relation to the FFT bins, specifically whether it is relative to the first bin (often referred to as the "zeroeth" bin).

Areas of Agreement / Disagreement

Participants express varying views on the feasibility of extracting phase and amplitude from FFT data, with some acknowledging theoretical possibilities while others highlight practical challenges. No consensus is reached on the best approach or the implications of the signal characteristics.

Contextual Notes

Participants note limitations related to signal characteristics, such as amplitude and sampling rate, which may affect the ability to accurately calculate phase and amplitude. The discussion also raises questions about the reference time for FFT bins, which remains unresolved.

henryd
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Is it possible to calculate the phase and signal amplitude from data gained from FFT?

For instance, if I have a samples from a signal B+A*cos(ψ), is it possible to obtain A and ψ?

Extra challenge: is it possible to do so without division? (I am looking to put this on a DSP and division is expensive)

Thanks!
 
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Which is the time varying signal? I think the example you gave might be a constant. If so your problem is much easier to solve. :)

Basically: In theory, sure. In practice, maybe. For example, if A is less than a LSB or the sampling rate is slower than the frequency you want to observe, then no. And then when measuring phase there is the whole problem of, what exactly is t0?

You might want to look at this primer. I think it looks pretty good.
http://www.google.com/url?sa=t&rct=...3oipn3Fy-ZK4CYszA&sig2=4Mk4C5H4T1Ug3mHVbrYeMQ
 
Does it help if I have a pretty good estimate of ω?

The signal would be:

B + A\cos(\omega t + \phi)

where B and A are constant
 
The output of your FFT is a complex number for each frequency bin.
Complex number is rectangular coordinates. You will use trig to convert these to polar (magnitude/phase angle).
 
henryd said:
Does it help if I have a pretty good estimate of ω?

The signal would be:

B + A\cos(\omega t + \phi)

where B and A are constant

t relative to what? what is the origin of your time axis? is it relative to the very first bin (sometimes called the "zeroeth" bin) x[0]. i.e. is bin 0:

x[0] = B + A\cos(\omega 0 + \phi) \ \ ?
 

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