One more thing about bug hunting: If you are doing anything
involving Fourier transforms (including FFTs) you should make
it a habit, every time you do a transform, to check that it
upholds Parseval's identity:
∫ |V|^2 dt ≡ ∫ |Vhat|^2 df
If it fails that test, it means there's something messed up
with the normalization or the units or something.
This is cheap and easy to do.
Speaking of normalization and units: For example, if the
time-domain ordinates are in volts, the frequency-domain
ordinates should have units of volt-seconds. The typical
library functions know nothing about this, so you have to
do it ourself, every time you do a transform.
This is also cheap and easy to do. If you make a habit of
it, life is a lot simpler. It makes it easier to think
about what's going on. There are fewer bugs. The details
are spelled out at https://www.av8n.com/physics/fourier-refined.htm
Students were not born knowing this stuff. And if they
(gasp) read the typical textbooks they wouldn't find it
there, either.
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Turning now to the FWHM of the resonance lineshape:
On 11/22/2016 04:22 PM, Donald Polvani wrote in part:
I get a quartic equation
Right.
Therefore:
FWHM = f_U - f_L = f_0/Q
We all agree that is correct in the high-Q limit.
However, for smallish Q, it's messier. The quartic has at most
two positive roots, and it's easy enough to solve in terms of
the square roots of square roots, but it's ugly. The cleanest
form I've been able to obtain is
Δϕ = √(ϕ^2_peak + 1/|G|_peak)
- √(ϕ^2_peak - 1/|G|_peak) for Q ≥ Qmin
where the normalized FWHM is Δϕ = FWHM / f0. I checked this
graphically and numerically for rather small Q.
For some purposes, a Taylor series approximation suffices:
Δϕ = 1/Q + (1/4) (1/Q^3) + terms of order 1/Q^5
One can understand why introductory-level textbooks confine
the discussion to the high-Q limit.
The whole concept of a "peak" with a "FWHM" goes out the window
when the height of the peak is less than 2. One of the positive
roots of the quartic goes to zero. The critical value of Q is
Qmin = √[1 + √(1/2)]
≈ 1.30656
For lower Q, you can still define a "width" but the meaning
is different.