Dear Ben,
I will try and answer your questions first. You wrote:
Are you suggesting that it is better to use a cosine than a fourier transform because of causality issues?
-- MPEG coding already does so, not because the experts understood and considered the causality issue but simply because MDCT is most efficient. In this case, STFT is an detour.
I recall having discussions about causality issues and also about negative frequencies during my studies, and that neg. and pos. frequency should be considered together.
-- That is correct if we decided using FT. Unfortunately virtually nobody seems to as: Why do we use FT at all?
The fact that the transformation is augmented with a complex part was just for mathematical reason.
-- Hm. Oliver Heaviside introduced a trick how to integrate from minus infinity to plus infinity even though future data are definitely not available in advance. Accordingly I see the reason not in the mathematics but in the religious anchored notion of time and god from eternity to eternity. Did you read M290?
After multiplication with a transfer function, the complex part at the output could be removed.
-- You certainly meant the imaginary part. Yes. Performing inverse transform back to the real domain anyway removes it. Do not confuse the overly beneficial use of complex calculus including transfer function with the spectral analysis of a single function of elapsed time alone.
I think you are suggesting to use a cosine transform (CT). Can you keep the Fourier transfer function or does it need to be modified?
-- I feel a bit disappointed like a poor teacher being unable to convey my message: CT is tailor-made for analysis of reality. Use of FT is alternatively possible. It just requires correct interpretation. Let me tell you an even more simple example: Three people entered an empty room. Then five people left it. How many have then to enter in order to make the room empty again? My grandson gave to me the mathematically correct answer. Since his mother educated him as a believer, he has no problem to imagine negative people. Hopefully he will not become one of the theorists.
You say: "no explanation was available to the strange phenomenon of an audible missing fundamental". Could you explain this a bit more?
-- I refer to a dispute between Ohm and Seebeck in 1844. Seebeck claimed having identified by ear a tone that was not at all existent within the spectrum of the sound of a siren. Ohm did not believe him.
What is represented in Fig 1? Is that a similation of a sound traveling in a cochlea?
-- It is indeed similar to simulations as well as measurements. However, it is simply a superposition of cosine transforms for any sample that approximates the sound. The analyzed signal is given below.
On page 3, you refer to Heisenberg's uncertainty relation which, in my view, is a mathematical artifact and therefore it is not necessarily physical. In fact, based on QFM, there is no uncertainty in discrete time dt or discrete displacement dx of a particle. This then results in Edt >=h , pdx >=0 (because dx can be arbitrary small), and p x lamda = h (for p>0, lambda is de Broglie's wavelength).
-- Admittedly I did not yet deal with QFM because I doubt that the particles can be adequately described in terms of moving points as did Born. Nonetheless, the view I tried to to express in my essay seems to be close to your view. I wonder why the paper "Heisenberg's Uncertainty Principle" by P. Busch, T. Heinonen, and P. J. Lathi omits the fundamental mathematical flaw which is best addressed as Buridan's ass.
On page 9, I did not understand the statement: Circular frequency omega = 2 pi f parallels momentum p.
-- Doesn't omega multiplied with h equal to energy?
About the local/global issue that you raised earlier: this is in space (one domain). In the frequency-time analysis, two different domains are compared: small pulse gives wide frequency spectrum and v.v.
-- Spatial frequencies are called wave numbers.
Thank you for your questions. Do not hesitate requesting further references.
Regards,
Eckard