FFT filterbanks are time variant due to framing effects and the circular convolution property. They exhibit “perfect reconstruction” if you design the windows correctly, but this only applies if the FFT coefficients are not altered between analysis and synthesis. If you alter the FFT coefficients (i.e., “filtering”), it causes time domain aliasing.
So, the operation of such a system can’t be reduced down to an equivalent LTI frequency response. We have the more basic issue of limiting the aliasing to acceptable levels. This depends partially on the frame size, overlap, and window shape, as these determine how any aliasing is distributed in a time/frequency sense. But more fundamentally, you have to put constraints on the response curves to limit the aliasing. I.e. the steeper the transitions in the frequency response, the longer the implied impulse response, and so the time domain aliasing gets worse. It is certainly possible to run any filter bank offline and compensate for its latency, in order to get a “zero phase” processing. But fundamentally they have framing delay given by the frame size, and algorithmic latency given by the overlap. These are the delays that you’d compensate when running offline. Ethan > On Mar 8, 2020, at 2:04 PM, zhiguang zhang <zhiguangezh...@gmail.com> wrote: > > > The system is memoryless just because it is based on the DFT and nothing > else, which is also why it's time-invariant. unless you alter certain > parameters in real-time like the window size or hop size or windowing > function, etc, any input gives you the same output at any given time, which > is the definition of time-invariance. > > well, you're RBJ and I see that you used to work at Kurzweil until 2008. > that's cool and what have you been up to since then? incidentally i was in > California until 2008. > > As you might be able to tell, i don't care too much about the fact that time > domain filtering theory is brought up often because I did my master's thesis > with this frequency domain FFT filter, which I believe was rather novel at > the time of completion. I do know that the field is steeped in tradition, > which might be why I'm writing to the mailing list and to you in particular. > but bringing up traditional FIR/IIR filtering terminology to describe FFT > filtering doesn't make sense in my mind. I'm not in the audio field. but > yes, I do believe that the system is time invariant, but I don't have time to > prove myself to you on this forum at this time, nor do I have any interest in > meeting Dr Bosi at AES. > > -ez > > > >> On Sun, Mar 8, 2020 at 4:42 PM robert bristow-johnson >> <r...@audioimagination.com> wrote: >> >> >> > On March 8, 2020 3:07 PM zhiguang zhang <zhiguangezh...@gmail.com> wrote: >> > >> > >> > Well I believe the system is LTI just because the DFT is LTI by definition. >> >> TI is nowhere in the definition of the DFT. L is a consequence of the >> definition of the DFT, but the DFT is not an LTI system. it is an operation >> done to a finite segment of samples of a discrete-time signal. >> >> > The impulse response of a rectangular window I believe is that of a sinc >> > function, >> >> window functions do not have impulse responses. >> >> both window functions and impulse responses can be Fourier transformed. the >> Fourier transform of the latter is what we call the "frequency response" of >> the system. i am not sure what they call the fourier transform of a window >> function. what is done with the frequency response (multiplication) is >> *not* what is done with the fourier transform of a window function >> (convolution). >> >> > which has ripple artifacts. >> >> there are no ripple artifacts in fast convolution using a rectangular >> window. you need to learn what that is. >> >> > Actually, the overlap-add method (sorry I don't have time to dig into the >> > differences between overlap-add and overlap-save right now) >> >> what you need is time to learn the basics and learn the proper terminology >> of things so that confusion in communication is minimum. >> >> > minimizes artifacts depending on the windowing function. >> >> again, there are no ripple artifacts in fast convolution using a rectangular >> window. none whatsoever. >> >> > A sine window ... >> >> i think you might mean the "Hann window" (sometimes misnamed "Hanning", but >> that is an old misnomer). i have never heard of a "sine window" and i have >> been doing this for 45 years. perhaps the classic Fred Harris paper on >> windows has a "sine window". >> >> > ... actually sums to 1, >> >> that's what we mean by "complementary". >> >> > the proof of which can be found in audio coding theory. I suggest you >> > check out the book by Bosi. >> >> i didn't even know Marina did a book, but i am not surprized. i've known >> (or been acquainted with) Marina since she was with Digidesign back in the >> early 90s. before the Dolby Lab days. before her injury at the New York >> Hilton in 1993. would you like me to introduce you to her at the next AES? >> >> Eric, you gotta get the basics down right and you gotta learn the correct >> terminology if you're going to communicate with other people about this >> topic matter. Neologisms are frowned on but people do them anyway. However >> you just cannot change the meanings of terms that have existed since the >> 1960s (and some as far back as the 1930s). >> >> -- >> >> r b-j r...@audioimagination.com >> >> "Imagination is more important than knowledge." > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp
_______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
