One of the deep truths of maths, physics and the universe in general, is that ANY mathematical system of sufficient complexity (loosely speaking, a C* algebra of bounded operators on a Hilbert space) has an isomorphism to another space via a representation based upon the characters of the algebra. ***THERE IS NOT JUST ONE FOURIER TRANSFORM*** Without going too deep into it, my suspicion as to where you went wrong is this: On one hand, I think it's great that you explained some technical points, and mentioned the Kronecker and Dirac deltas, but my concern is that these are VERY technical subjects, and your simplification, whilst making things easier to read, simplified out some CRUCIAL details. The next thing I feel I should discuss is your original post.
Ok, so that will make for good test functions. Likewise instructive is scaled gaussian functions - look what the rho in one domain does to the rho in the other.) The sharper the resonance, the longer it takes for the sinewave which is its Fourier representation to decay (verify this by taking the Fourier transform of decaying exponentials * sin(x). This being so, when we have these discussions about echoes, and the consequences of things on impulse responses, we should naturally look for easy examples, and sharp resonances make for great examples. Sharp in one domain is soft in the other. But your intuition should always lead you in this direction. I'll spare you the proof here, because it's long, technical and irrelevant.
This is a fundamental consequence of the correspondance between a domain and its Fourier representation. Sharp peak -> long exponentially decaying sinewaves. What is SHARP in one domain is LONG in another! As your email suggested, I CAN explain all these phenomena, but doing so in the way that's the most clear to all readers always poses a challenge!įirst things first, let's talk about the uncertainty principle! (Yes, this is the same as in Quantum Mechanics!) I've had a read through - there are a LOT of points which are absolutely worth addressing in here, and I'm not sure what's the best way to answer them all without responding to each post individually! I have to go out in a bit, so I'll try and throw in a few key points to add as much clarity in as little space as possible. Sometimes I hear really bad pre echo's and "glitches" that anticipate or are bad representations of transients in mp3s and wonder if linear phase eq used in mixing or mastering would make these artifacts more likely? it certainly can't help! I've observed it a million times tweaking bass sunds and viewing them through s(m)exoscope. A single cycle of a low frequency sawtooth or something put through a HPF at 40hz or something will clearly show up pre / post ringing and the delayed high frequencies you mention. I think it's easiest to focus on the low frequencies and an HPF of at least 24dB/oct. Not only does this technique work, but it works really well (used it myself several times in mastering). BManic's tip for setting equilibrium to a short window (lower quality) linear phase topology is a technique employed to tame overly clicky hihats by smearing the transient out. Linear phase is great for parallel processing though.Īs for linear phase eq blurring transients, it can and does. Something about linear phase is very boring and sterile when trying to inject life into the source and I prefer min phase 99 times out of 100. I really don't like the sound of it and always use min phase for such tasks. LP HPF on bass cuts can create a huge amount of pre ringing ime.