Well I believe the system is LTI just because the DFT is LTI by definition. The impulse response of a rectangular window I believe is that of a sinc function, which has ripple artifacts. Actually, the overlap-add method (sorry I don't have time to dig into the differences between overlap-add and overlap-save right now) minimizes artifacts depending on the windowing function. A sine window actually sums to 1, the proof of which can be found in audio coding theory. I suggest you check out the book by Bosi.
Thanks, -ez On Sun, Mar 8, 2020, 3:01 PM robert bristow-johnson < r...@audioimagination.com> wrote: > > > > On March 8, 2020 2:00 PM Zhiguang Eric Zhang <zez...@nyu.edu> wrote: > > > > it is not causal because the zero-phase system does not depend on past > samples > > > > > > On Sun, Mar 8, 2020 at 1:58 PM Zhiguang Eric Zhang <zez...@nyu.edu> > wrote: > > > the frequency response is a function of the windowing function > > > > > > > > what frequency response, Eric? if the only sample the "zero-phase system" > depends on is the present sample, not past nor future samples, then the > system is a wire or simply a scaler (if it's linear) or some other > memoryless function if it's nonlinear. we normally mean linear systems > when we discuss "frequency response". > > people here on this list are pretty competent about what linear system > theory is, what makes systems linear or not, what makes them time-invariant > or not, what makes for causal and acausal systems, what determines the > frequency response of a linear system (it's the impulse response), what > window functions are and their effect on the spectra of signals, what > linear-phase systems are and what hypothetical zero-phase systems are. > this is all in textbooks. > > one thing that needs to happen for anyone to truly assist anyone else > here, is that communication is unambiguous and accurate. that means we > really have to agree on the definitions of terms. > > an FIR filter is a linear, time-invariant system. it can be a > linear-phase system, but it doesn't have to be. that depends on the IR, > which is F. if and only if the IR is symmetrical, it's linear-phase. a > linear-phase FIR can be modeled as a hypothetical zero-phase FIR (so only > magnitude response is in its description) followed or preceded by a delay > of 1/2 of the length of the FIR. the phase response of a delay is a linear > function of frequency, hence "linear phase". a hypothetical zero-phase > filter is one with symmetry about the t=0 axis of the impulse response. > that means that samples in the future of the IR are a mirror image of > samples in the past. add some delay and you can move all of the non-zero > samples to the past (and one for the present input sample). > > using the FFT (and iFFT and frequency-domain multiplication of the > frequency response) is an operation we often call "fast convolution". fast > convolution is used for very long (but still finite) FIRs. for short FIRs > we just do it the simple way, a dot-product of the IR against the most > recent input samples. fast convolution is performed using one of two > techniques: "overlap-add" (OLA) and "overlap-scrap" (OLS). the older lit > will say "overlap-save" for OLS, but we mean the same thing. > > normally, the only windowing we do for fast convolution, is the > rectangular window. because of the overlapping and the linear-system > nature of the operation, there are no windowing effects. it *is* possible > to do something like fast convolution FIR with overlap-add using a > different window as long as it is complementary in the overlap (such as a > Hann window), but it makes the operation less "fast". if there is 50% > overlap, the number of net output samples computed in each frame is 1/2 > that for using rectangular given both windows having the same non-zero > length, so it would be half as efficient. > > lastly, this "overlap-add" method of fast convolution should not be > confused with the overlap-add method of the STFT processing music DSP > people often use for more general and sophisticated processing of audio > (such as a phase vocoder). generally more sophisticated than an FIR. in > that application, a rectangular window would be considered pretty bad. but > all this is not what "fast convolution" is, which we use to implement long > FIR filters (like reverberators). > > i'm just trying to be helpful, but none of us can really be helpful if we > cannot at least agree on the terminology. > > -- robert > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > > On Sun, Mar 8, 2020 at 10:34 AM robert bristow-johnson < > r...@audioimagination.com> wrote: > > > > > > > > > > > > > On March 8, 2020 10:05 AM Ethan Duni <ethan.d...@gmail.com> > wrote: > > > > > > > > > > > > > > > It is physically impossible to build a causal, zero-phase system > with non-trivial frequency response. > > > > > > > > a system that operates in real time. when processing sound files > you can pretend that you're looking at some "future" samples. i guess that > would be acausal, so you're right, Ethan. > > > > > > > > -- > > > > > > > > r b-j r...@audioimagination.com > > > > > > > > "Imagination is more important than knowledge." > > > > > > > > > > > > > > > > > > > On Mar 7, 2020, at 7:42 PM, Zhiguang Eric Zhang <zez...@nyu.edu> > wrote: > > > > > > > > > > > > Not to threadjack from Alan Wolfe, but the FFT EQ was > responsive written in C and running on a previous gen MacBook Pro from > 2011. It wouldn't have been usable in a DAW even without any UI. It was > running FFTW. > > > > > > > > > > > > As far as linear / zero-phase, I didn't think about the impulse > response but what you might get are ripple artifacts from the FFT windowing > function. Otherwise the algorithm is inherently zero-phase. > > > > > > > > > > > > > > > > > > On Sat, Mar 7, 2020, 7:11 PM robert bristow-johnson < > r...@audioimagination.com> wrote: > > > > > > > > > > > > > > > > > > > > > > On March 7, 2020 6:43 PM zhiguang zhang < > zhiguangezh...@gmail.com> wrote: > > > > > > > > > > > > > > > > > > > > > > > > Yes, essentially you do have the inherent delay involving a > window of samples in addition to the compute time. > > > > > > > > > > > > > > yes. but the compute time is really something to consider as > a binary decision of whether or not the process can be real time. > > > > > > > > > > > > > > assuming your computer can process these samples real time, > the delay of double-buffering is *twice* the delay of a single buffer or > "window" (i would not use that term, i might use the term "sample block" or > "sample frame") of samples. suppose your buffer or sample block is, say, > 4096 samples. while you are computing your FFT (which is likely bigger than > 4K), multiplication in the frequency domain, inverse FFT and overlap-adding > or overlap-scrapping, you are inputting the 4096 samples to be processed > for your *next* sample frame and you are outputting the 4096 samples of > your *previous* sample frame. so the entire delay, due to block processing, > is two frame lengths, in this case, 8192 samples. > > > > > > > > > > > > > > now if the processing time required to do the FFT, > frequency-domain multiplication, iFFT, and overlap-add/scrap exceeds the > time of one frame, then the process cannot be real time. but if the time > required to do all of that (including the overhead of interrupt I/O-ing the > samples in/out of the blocks) is less than that of a frame, the process is > or can be made into a real-time process and the throughput delay is two > frames. > > > > > > > > > > > > > > > > On Sat, Mar 7, 2020, at 6:00 AM, Zhiguang Eric Zhang > wrote: > > > > > > > > > ... FFT filtering is essentially zero-phase ... > > > > > > > > > > > > > > FFT filtering **can** be linear-phase (which is zero-phase > plus a constant delay) if the FIR filter impulse response is designed to be > linear-phase (or symmetrical). it doesn't have to be linear phase. > > > > > > > > > > > > > > -- > > > > > > > > > > > > > > r b-j r...@audioimagination.com > > > > > > > > > > > > > > "Imagination is more important than knowledge." > > > > > > > > > > > > > > > On Sat, Mar 7, 2020, 5:40 PM Spencer Russell < > s...@media.mit.edu> wrote: > > > > > > > > > On Sat, Mar 7, 2020, at 6:00 AM, Zhiguang Eric Zhang > wrote: > > > > > > > > > > Traditional FIR/IIR filtering is ubiquitous but > actually does suffer from drawbacks such as phase distortion and the > inherent delay involved. FFT filtering is essentially zero-phase, but > instead of delays due to samples, you get delays due to FFT computational > complexity instead. > > > > > > > > > > > > > > > > > > I wouldn’t say the delay when using FFT processing is due > to computational complexity fundamentally. Compute affects your max > throughput more than your latency. In other words, if you had an > infinitely-fast computer you would still have to deal with latency. The > issue is just that you need at least 1 block of input before you can do > anything. It’s the same thing as with FIR filters, they need to be causal > so they can’t be zero-phase. In fact you could interchange the FFT > processing with a bank of FIR band pass filters that you sample from > whenever you want to get your DFT frame. (that’s basically just a > restatement of what I said before about the STFT.) > _______________________________________________ > 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
