Re: Of stereo miking, Fourier analysis, and Ambisonics ***excerpt from previous post: ...Instead of the required condition for the latter - the signal being limited in bandwidth - the condition for the discrete spectrum being a complete (i.e. invertible) description of the signal is that the signal is limited in time. This is just the dual of the Nyquist sampling theorem. The factor of 2 that appears only in one case is because we are using real-valued signals but complex-valued spectra.
What this shows is that you don’t need any infinities to exactly describe a signal that is limited both in bandwidth and time. Ciao, [*] More correctly, K/2-1 complex ones and two real-valued ones at 0Hz and Fs/2. -- FA *** Hello Fons, Thank you for your very clear explanation. I suppose the classic interpretation of assuming the signal is cyclical is what I’ve analyzed or attempted to understand. As I see, a train of impulses will reveal a fundamental frequency whose frequency is simply the reciprocal of the time/period between pulses. Constituent frequencies or overtones will depend on duty cycle, or pulse width, and actual shape of *repetitive* waveform. In the digital domain, the highest spectral frequency that can be analyzed (or at least recorded) without aliasing will be half the sampling rate (Nyquist theory). But for low frequencies, number of samples (equivalent to time) appear to dictate analysis or low-frequency resolution (?). If the ear were to be viewed as a Fourier analyzer (loosely taught in Speech and Hearing because it suffices as a model), then we would expect the place-respective receptor cells (the IHCs) to respond to the frequencies that Fourier decomposition yields. When there is no repetitive wave, how do we go about such analysis without making it cyclical for sake of math? [I'm pretty sure this is what you answered.] Of course, with brief transients, the response of middle ear will impart a huge effect because of compliance, mass, inertia, etc. What happens in the inner ear is, well, admittedly complicated... I guess my initial question, then, was whether a single impulse (limited in time) can be reconstructed from sine waves when there is no luxury of time expansion in the physical world; in other words, there isn’t enough *time* (or samples in digital representation) in the width of an impulse to construct as much as a fraction of lower-frequency, constituent sine waves. Increasing the sampling rate and Nyquist’s theorem addresses the high frequencies, but increasing the sampling rate doesn’t give us any more *time* to reconstruct a low-frequency wave or wavefront. So, in a sense, my reservation was based on the summation of partial or incomplete sine waves (whose frequencies are determined via Fourier analysis) to construct an impulsive wave. Only partial waves can be constructed in the time allotted to creation of the impulse. This *allotted time* is all that existed in the physical world: The transient happened, and that was it. No extra theoretical time given, plus or minus. Making an impulse from complete, multi-repetitive waves is sort-of like making time that didn’t exist in the physical stimulus, but is added to make the analysis do-able. But of course, classical Fourier mathematics does state that waveform has to be repetitive. Given the scenario of an analyzer that is built on parallel, or contiguous, high-Q filters, would we then (or should we) expect same results in frequency and amplitude as an FFT analyzer would yield? (I’m guessing answer is yes, but transients still elude me.) Really high-Q filters, at least if analog, would probably *ring* in response to brief transients, and this could persist for a time longer than the transient itself. But ignoring this, would we expect the relative amplitudes at various frequencies to be the same for a repetitive impulse, a single impulse (same width), and mathematical decomposition of frequencies? If answer is yes, then Fourier analyzer would be a reasonable inner-ear model. Plus I'll feel better about speaker calibration using broadband impulses. It’s quite easy to make the assumption that the many available spectral analysis plug-ins (VSTs) and apps are accurate, even when we ignore windowing, smoothing type, number of samples, etc. that are easy to manipulate in MATLAB or similar software tools. With filters, even of the same order, not all filters will yield same results (look at output of an elliptic filter--it's bouncy). Similarly, not all spectrum analyzers yield the same results. Again, thanks for your time and explanations. I keep learning in order to make the best of available tools. Best, Eric C. -------------- next part -------------- An HTML attachment was scrubbed... URL: <https://mail.music.vt.edu/mailman/private/sursound/attachments/20130628/b2e9b8e9/attachment.html> _______________________________________________ Sursound mailing list [email protected] https://mail.music.vt.edu/mailman/listinfo/sursound
