>So in the digital sense, or in combination with the analog domain, >is it reasonable to think about "correctable" operations, which as >it were can be inverted, so that applying a digital signal transformation >*and* it's converse, we end up with the same signal or something similar.
I think you mean to write "inverse" rather than "converse" there, but the basic answer is yes. >Of course that first of all leads to the necessary condition that or signal >transformations are a bijection, which is hard considering most of these >operations will be filters, and in the digital domain, except for a few pathetic >cases, these will have bit depth issues. The bijection condition for LTI filters basically means that none of the roots are on the unit circle. Which you'd want anyway, since otherwise the inverse filter is going to be unstable. That said, you *can* actually invert filters with roots on the unit circle (or anywhere else) by using z-transform techniques (i.e., evaluate all of your integrals on a contour other than the unit circle which doesn't contain any of the roots), although there are various practical limitations associated with that as well. Moreover, indeed, bit-depth issues are the main thing here. No digital system can ever be truly linear, due to finite word-length. Typically we assume that the bit depth is big enough that we can ignore that, but you can certainly construct cases where it shows up. For example, suppose we hit a signal with a very aggressive (but invertible) low pass filter, which drives the high frequency content below the noise floor implied by the bit depth. Then if we run the output through the inverse of the filter, we won't get back the original high frequency content - instead we'll get end up amplifying the noise floor back up so it becomes audible. Running the filters in the opposite order might well work, though (supposing the first filter doesn't run out of headroom to amplify the band in question). This effect can come up even with relatively mild filters, depending on the content of the signals being processed - if the high frequency content is only a few bits above the noise floor to begin with, then even a gentle low-pass response is going to submerge that content below the noise floor and render it unrecoverable. >So if we have N digital poles, can we create N digital zeros at the same >frequencies, convolve those two filters and arrive at a digital wire ? Convolving an all-pole filter's response with its inverse is always going to result in the identity system (theoretically). And not a delayed identity system, but the canonical one with zero delay. But that doesn't necessarily imply that we can run a signal through an all-pole filter, and then run the output through the inverse filter, and end up with a digital wire (even allowing for a few LSBs error to account for round-off effects). For this to work, we need a further assumption that finite word length effects are not having a significant impact in either of the two filters. If they are, then the system is no longer approximately linear, and so its output is not explained by the convolution integral, and so the convolution of the two filter responses then fails to describe the operation of the chain of the two filters. And that assumption depends on both the response of the filter in question, and the content of the signals being processed. In particular, signals and/or filters with a large spectral dynamic range require special care. In practice, this all generally works fine if the bit-depth used for intermediate processing is significantly larger than the one used for final rendering, and we avoid filters with aggressive frequency responses. I.e., a gentle EQ that compensates for deviations in a microphone frequency response is unlikely to cause any problems. But attempting to invert a system with (for example) a strong low-pass response is asking for trouble. E On Thu, May 8, 2014 at 7:23 AM, Theo Verelst <theo...@theover.org> wrote: > Hi all, > > In the analog domain, where most interesting DSP originates since, well, > the time of radio and early telephone, it is a commonality to search for > "signal neutrality" in certain reasonableness. So a phone line would be > specified to transfer certain frequencies with a certain amplitude and > phase reliability, and a specified absence of noise and linear distortion, > echo damping, etc. > > In the digital age, of course in principle a connection or file with > sufficient number of bits and well qualified sampling frequency is > considered pretty neutral as is. That's not all a correct hypothesis, for > instance think about the sampling issues like the intended reconstruction > filtering, but enough about that. Also, there are people seemingly more > concerned with adding and dealing with dithers than actually passing a > signal from A to B, but that aside too (even though that is an interesting > subject for other professional reasons). > > So in the digital sense, or in combination with the analog domain, is it > reasonable to think about "correctable" operations, which as it were can be > inverted, so that applying a digital signal transformation *and* it's > converse, we end up with the same signal or something similar. Of course > that first of all leads to the necessary condition that or signal > transformations are a bijection, which is hard considering most of these > operations will be filters, and in the digital domain, except for a few > pathetic cases, these will have bit depth issues. > > So give or take a few LSB errors, are digital filters like filters in the > analog domain? So if we have N digital poles, can we create N digital zeros > at the same frequencies, convolve those two filters and arrive at a digital > wire ? Of course there may be some delay here... > > Practical ? Well, this week I was playing with my Lexicon AD convertors > and a good microphone setup, driving my large monitoring system with my > latest high quality ground-seperated 384 kHz DA convertor in a real-time > situation, and wanted to compensate the small (few dBs here and there) the > frequency sensitivity unevenness of the microphone I used, and applied some > jack/jack-rack/ladspa Linux filters for that. Worked great. > > T.V. > > -- > dupswapdrop -- the music-dsp mailing list and website: > subscription info, FAQ, source code archive, list archive, book reviews, > dsp links > http://music.columbia.edu/cmc/music-dsp > http://music.columbia.edu/mailman/listinfo/music-dsp > -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
