Hey,
isn't there already a way to just emulate the behaviour of an anti-aliased [phasor~] (..to just limit the problem to a ramp-shape (or other simple shapes)..) ??
-If not, do you think it's possible to achieve that??
The procedure shown in the "J.08 classicsynth"-example is a CPU-killer with all those filters and the "times 16"-samplerate... can't one somehow predict or approximate the shape of the, again downsampled, signal?

Bye
Flipp

Right...I guess I'm just not sure why you have a problem with that. I mean, we're talking about aliasing here. That's a frequency phenomenon. Why do you want to describe what happens in the frequency domain using the time domain?

...f-domain is so handy because the pc does all the calculations, and there are used approximations, interpolations or leaky integrators as well... And that's already the reason: Calcuations in the time-domain are faster, while harder to "design", most of the time. I already have some things that work pretty well in terms of alias-suppression. But the math behind it is just "1st order" and of course using higher math uses more cpu-power.. (I also got a saw-osc that doesn't produce these wobble-sweep-harmonics while it's f gradually changes. It works with smoothing the jump.. But i won't post it yet, 'cause the "math" are just values from experiments i did with the osc, and yet i don't know completely where these numbers come from... )

That's a frequency phenomenon.

...yeah OR a time-phenomenon... one doesn't exclude the other...

...hmm, the patch... ...I tried it again, and it didn't work as well. But that's not our fault!! (Everything points to a dsp-bug i mentioned in an other post.. or it's a bug with the [vline~] ..if i remember correctly that object is a little buggy anyways..)

Soooo I did a second one, that can hande the "crazy situation" and posted it here.
Please have a second try. ...Actually i just didn't change the calculations, but rearranged the dsp-order...
From time to time if i load the patch it doesn't work from the start! If that happens to you click the [-0.5( !! ...but it would be best not to happen at all...
--> It seems the ramp sometimes doesn't start at "0" (..0, 0.2, 0.4...) but at the next value "0.2". So we have to use [0.5( or [-0.5(...

Well, if everything works as it should, one can see that there is NO DC! But it slowly comes back due to numerical errors. That's why we have to sync, or rather "restart" the oscs from time to time.

Simply making a waveform perfectly symmetrical doesn't mean there is no DC offset

I don't think so! Mathematically it does just mean that.
E.g. if I invert a waveforms amplitude and play it time-reversed, and the waveform still looks the same, there can't be any DC-offset. If there was, it wouldn't look the same anymore...!!!

No, this is because of the "reverse saw" spectrum. The loudest frequencies that alias are just above Nyquist

...but that's f-domain again. Lets imagine we could hear far beyond the samplingrate (say 44100). Now as we sweep up a sine, it starts as a sine and at 22050Hz ends up as a square! And if ther was no "smear", that is a s&h-like behavior in time, the output of the sweeped osc at 22050Hz would rather be a tri-shape since we just have points instead of "lines" in time (and we assume "physics to be steady outside the pc", so we kind of interpolate between those points).
...this just as one reason for aliasing seen in time-domain!

An example where timedomain is really "better": Imagine we have to build an ANALOG "aliasing-device". It simulates aliasing of a given waveform/spectrum:
-As a freq.-domain approach we would take some sine-oscs and tune each according to the foldover-theorems etc. (and the waveform itself) ...as well as setting each amlitude of course.
So we create a "additive-aliasing-synth" that beside the harmonics has to calculate even the disharmonics... That's faaar too much...
-As a time-domain approach we would just use... well..a pc.. ...rather a sample'n'hold-unit ((..like in d'n'b, -hihi . )), since it's part of any "adc-dac".
Well if that wasn't easier, then idk...!!!

http://www.pdpatchrepo.info/hurleur/saw-alias-comp1b.pd

@Flipp said:

...f-domain is so handy because the pc does all the calculations, and there are used approximations, interpolations or leaky integrators as well... And that's already the reason: Calcuations in the time-domain are faster, while harder to "design", most of the time.

The frequency domain is also a useful tool for designing time-domain solutions. The easiest way to design an arbitrary FIR filter, for example, is to do plot the desired spectrum and perform an inverse Fourier transform on it.

It works with smoothing the jump.. But i won't post it yet, 'cause the "math" are just values from experiments i did with the osc, and yet i don't know completely where these numbers come from... )

Have you looked at BLEP synthesis? It's basically that. Might help you with the math.

I don't think so! Mathematically it does just mean that.
E.g. if I invert a waveforms amplitude and play it time-reversed, and the waveform still looks the same, there can't be any DC-offset. If there was, it wouldn't look the same anymore...!!!

It doesn't mean symmetrical. It means it's centered around the average value. Again, look at a pulse train.

Anyway, of course with a saw the average value is also right there in the middle. I was just assuming your patch was right and trying to come up with a possible explanation as to why there was still some DC. It does work now.

...but that's f-domain again.

One doesn't exclude the other.

...this just as one reason for aliasing seen in time-domain!

No it isn't. When you're dealing with discrete time, you don't make assumptions about what happens between samples. Those square steps only occur at the dac before the filter, which smoothes out those steps. The raw data itself doesn't have that, but the aliasing is still there. Pushing a sine wave toward Nyquist doesn't make it less of a sine wave. It doesn't alias.

The reason for aliasing is because there simply isn't enough resolution in discrete-time signals to represent a frequency above Nyquist. The fluctuations between samples get lost, and you end up with a lower frequency. That's it. That's the time-domain explanation.

So we create a "additive-aliasing-synth" that beside the harmonics has to calculate even the disharmonics... That's faaar too much...

That's also technically still in the time domain.

The easiest way to design an arbitrary FIR filter, for example, is to do plot the desired spectrum and perform an inverse Fourier transform on it.

I never digged deeper into digital filterdesign, but i made a pseudo-digital analog filter ...uhh it just dawned on me to implement a simulation of it in "slowed down time" (<-see the attached patch). .."realtime" won't work cause the analog filter needs a "driver-freq." higher than 44100.. ..I need a better soundcard...
...anyways that's another line of work, as if there wasn't enough already...

...but if it is like drawing a spectrum relative to the filters f_c and "multiplying" it with an input-spectrum, to get a filtered input, I did that some time ago. It's cool since you can record the spectrum of e.g. a saw or a vowel and use it as a tuneable filtershape for other audio.

BLEP-synthesis.. I heared of it... I'll have a closer look at it, thx! (...it's not much but what else should I say...)

Those square steps only occur at the dac before the filter, which smoothes out those steps.

Just to make shure are u speaking of an input- or an output-filter?

The reason for aliasing is because there simply isn't enough resolution in discrete-time signals to represent a frequency above Nyquist. The fluctuations between samples get lost, and you end up with a lower frequency. That's it. That's the time-domain explanation.

Yeah shure, that causes those sub-non-harmonics and that stuff. But I think there are different types of aliasing in t-domain: the subharmonics-stuff caused by descrete time, and a kind of waveshaper that brings out more overtones due to the louder or harder (cause higher) jump from step to step as the frequency of a given signal raises. (<- see patch)
And of course in a pc there is no need to store a value for EVERY single point in real-time if the following points would just be the same. So we just take "the beginning of a line"
But at the digital-to-analog-converter (dac) there is ALWAYS a voltage (not just at certain timepoints and between them there is like nothing..). And this voltage is steppy (if not filtered afterwards). ..Anyways, filters already reduce the aliasing, so they can be excluded from "aliasing-production"..

Pushing a sine wave toward Nyquist doesn't make it less of a sine wave. It doesn't alias.

If I get you right, this is correct in "slowed down time" as well: Have a look at the attached patch, (there's even a little more ...text... inside). I can see and hear a squarewave there...

That's also technically still in the time domain.

okok.. technically everything that IS is in timedomain..
..but the calculations could be in f-domain.. ...to me everything that deals with sinusoids and harmonics belongs to the f-domain.. ..(<- does it look too feminine without a nose?!)...

http://www.pdpatchrepo.info/hurleur/slow_sr_aliasing1.pd

@Flipp said:

...but if it is like drawing a spectrum relative to the filters f_c and "multiplying" it with an input-spectrum, to get a filtered input, I did that some time ago.

It's not quite like that. When you do the inverse Fourier transform of a desired filter's frequency response, you end up with its impulse response, which is in the time domain. The coefficients of an FIR filter are exactly the same as the impulse response. So you end up with time-domain solution. And that's the point I was trying to make. Just because you're looking for a time-domain solution doesn't mean you should ignore the frequency domain.

Just to make shure are u speaking of an input- or an output-filter?

Output.

But at the digital-to-analog-converter (dac) there is ALWAYS a voltage (not just at certain timepoints and between them there is like nothing..). And this voltage is steppy (if not filtered afterwards). ..Anyways, filters already reduce the aliasing, so they can be excluded from "aliasing-production"..

Exactly. The steppiness you're referring to happens AFTER the data is converted to voltage and is no longer a discrete time signal. You can make it as steppy as you want at that point, it's not going to alias. It will generate high frequencies above Nyquist, but the filter should get rid of those.

If I get you right, this is correct in "slowed down time" as well: Have a look at the attached patch, (there's even a little more ...text... inside). I can see and hear a squarewave there...

Yeah, and look at the spectrum of that. It's not aliasing around the artificial sampling rate set by [samphold~]. The distortion extends up to the global sampling rate's Nyquist frequency and aliases there. What you're basically showing is that harmonics are being generated past the artificial sampling rate, just like what would happen when the signal is converted to analog. Except once it's analog, there is no global sampling rate anymore, so those harmonics just extend without aliasing.

..(<- does it look too feminine without a nose?!)...

It kind of looks like a frog to me. That's cool, though. I like frogs.

We might have different definitions of aliasing, so we maybe mean the same.. I think of aliasing as being everything that doesn't belong to the original, intended spectrum. I hope I got you right, you mean everything that is related to folding over and distinguish it from waveshaping-effects?! Since you said:

Pushing a sine wave toward Nyquist doesn't make it less of a sine wave. It doesn't alias.

So just compare analog vs. digital and name the errors that occur:
quantized, steppy time: foldover at f_sr
steppy values (raw voltage at the output): kind of square-waveshaper..

It's not aliasing around the artificial sampling rate set by [samphold~]. The distortion extends up to the global sampling rate's Nyquist frequency and aliases there.

...NOnono.
I doupt that's the reason. This works at any samplerate, sothat the real one is negligible (if you don't want to make things more complicated). And the sin does what it would do in "realtime" as well.
E.g. Set the "new" sr to 5512.5 Hz (sr/8) and set the tablesize of your spectrum to 64.
Now watch the spectrum as the sine-freq goes beyond f_ny (=2756.25Hz): It comes back folded over!! Just like in realtime.. (beyond f_sr the whole procedure restarts...)

...the only thing that's different now is that "WE ARE DOGS" now (idk how frogs tend hear...), so we can hear past f_ny (..and see if you put the spectrums tablesize back to 512...)!
And what we see and hear is that the normal waveform get shaped more and more to a square when reaching f_ny! Or: the sqr-shape depends on f_ny -> the higher f_ny (rel. to f_osc), the less sqr-shape.
-So we have a sqare at f_ny. Itself has odd harmonics. So in the end we have sines at 1, 3, 5,... times f_ny!
Now as we sweep a sine up: from 0Hz to f_ny -> just sine;; from f_ny to f_sr (=2*f_ny! -> even!!) -> folded over (inverse) sine;; from f_sr to 3*f_ny (=odd ->"from sqr") -> "normal" sine again, and so on..
(Seen from time-domain: there is nothing past f_sr. Every oscillation past f_sr is just an osc "past 0Hz", thats due to the nature of a periodic signal..)

...Anyways, all those bandlimit- and "sinc"-approaches have one disadvantage, and that is the "ripple" of the last partial.
If I waveshape those these ripples can produce more undesired sounds than an aliasing-saw...
..So better no bandlimiting but the "upsample-filter"-method-> still aliasing but rather inaudible...

It seems in discrete time (and spectrum) in general one has to decide whether to keep harmonics OR the waveform as natural as possible.
Since I want both, I'll have to live with approximations...

@Flipp said:

We might have different definitions of aliasing, so we maybe mean the same.. I think of aliasing as being everything that doesn't belong to the original, intended spectrum. I hope I got you right, you mean everything that is related to folding over and distinguish it from waveshaping-effects?!

Perhaps we are talking about two different things. However, it useful to use the correct definitions, as you can avoid long, perhaps unnecessary, discussion.

Aliasing is when frequencies outside of the bandwidth of a digital signal (i.e. 0Hz to Nyquist) foldover. After reading this post, it looks like what you're referring to would fall under the general umbrella of "distortion" or non-linear processing. I'm not sure if waveshaping is the right term, though.

...NOnono.
I doupt that's the reason. This works at any samplerate, sothat the real one is negligible (if you don't want to make things more complicated). And the sin does what it would do in "realtime" as well.

Yesyesyes.
The new frequencies that spring up due to the sine wave becoming "squarer" extend up to the global samplerate in Pd and alias there, so it's not really negligible (it never is). I don't know what you mean be "realtime" in this instance. This is all realtime.

As far as your example, that illustrates exactly what I'm talking about (though I don't know why you use a spectrum size of 64). Yes, the sinewave ITSELF aliases at 2756.25 Hz, but those new frequencies caused by the distortion don't. They alias at the global sample rate. So in a dac when the signal is converted to analog (which is only when this would happen), the new frequencies generated are above the sample rate and are then filtered out. No aliasing.

Take a look at the attached patch. The notes should hopefully clarify what I mean about the distortion.

from f_ny to f_sr (=2*f_ny! -> even!!)

This is really just an illusion. Those even "harmonics" are odd harmonics aliasing above the global sample rate (so, it isn't so negligible).

(Seen from time-domain: there is nothing past f_sr.

This is not something you can see from the time domain.

Every oscillation past f_sr is just an osc "past 0Hz", thats due to the nature of a periodic signal..)

If you really want to get technical, in discrete time everything between 0 Hz and SR is the SAME THING as everything between SR and SR*2, and SR*2 and SR*3, and so on. A discrete-time signal can't determine the distance, so the spectrum becomes infinitely periodic. But since it is periodic, it is easier to think of it as being finite between 0 Hz and SR, as everything you change in that range will change the rest of it (in some applications, like upsampling, you can't ignore it). It is not the nature of a periodic signal time-domain signal, however. It's the nature of discrete time.

...Anyways, all those bandlimit- and "sinc"-approaches have one disadvantage, and that is the "ripple" of the last partial.

What do you mean by this? Are you talking about the last partially suddenly dropping out when it hits Nyquist? I solved this by simply cross-fading between two oscillators--one with one more partial than the other--as the frequency approached Nyquist. It seems to work fine.

It seems in discrete time (and spectrum) in general one has to decide whether to keep harmonics OR the waveform as natural as possible.
Since I want both, I'll have to live with approximations...

It just depends on the application. If you are trying to make a bandlimited oscillator, then who cares if the saw looks like a perfect saw in an oscilliscope? Hell, even if all the harmonics weren't phase-aligned so that it doesn't look like a saw at all, it will still sound the same. If you're trying to make an LFO, then you would probably want something that looks like a perfect saw, and you wouldn't care about bandlimiting.

http://www.pdpatchrepo.info/hurleur/slow_sr_aliasing.mmb.pd

...After reading this post, it looks like what you're referring to would fall under the general umbrella of "distortion" or non-linear processing. I'm not sure if waveshaping is the right term, though.

Waveshaping always implies non-linearities. Linear (..no offs!) shaping would just be an amplification..

Yesyesyes.
The new frequencies that spring up due to the sine wave becoming "squarer" extend up to the global samplerate in Pd and alias there, so it's not really negligible (it never is). I don't know what you mean be "realtime" in this instance. This is all realtime

Hmhmhm...
But that "becoming squarer" is closely related to the "new" samplerate (of that s&h~), or rather its f_ny (=f_ny_new) (..cause it "becomes" a squarewave with f_sqr=f_ny_new)!!!
And how do you explain that the "harmonics" always meet at the f_ny_new (or at f_sr_new), if there was no relation to it... (choosing 5512.5 (=sr/8) was a little bit of an unfortunate accident, since new foldover fits right into the original (sr=44100..) foldover!)
It's obvious: Just change the new samplerate and watch the loundest partials fold at the new sr (and its multiples)...

The more quiet a sound the more negligible! And by using a low-freq. wave, the overtones that fold over are more quiet. That easy..
And the aliasing I hear is far from being quiet!!! ...beside being related to the new f_ny.
By "realtime" I still mean the contrast to the "slowed down time" of that patch.

As far as your example, that illustrates exactly what I'm talking about (though I don't know why you use a spectrum size of 64)

You shouldn't take a spectrum size of 64, just change the tablesize to "zoom in" so that it looks the same as in "realtime"..
...and beside that negligible "crap" from the original sr-aliasing, it just behaves nicely 'relative' to the f_sr_new.

(Seen from time-domain: there is nothing past f_sr.

This is not something you can see from the time domain.

You don't like the timedomain, do you?
...since shure it is: Sampling each "0.2 period" is the same as sampling each "1.2 period" since an offset of 1 period doesn't change anything..

What do you mean by this? Are you talking about the last partially suddenly dropping out when it hits Nyquist?

More or less. I mean that "sine on top of the waveform" (...that makes all values not being the same as before...)

It just depends on the application. If you are trying to make a bandlimited oscillator, then who cares if the saw looks like a perfect saw in an oscilliscope? Hell, even if all the harmonics weren't phase-aligned so that it doesn't look like a saw at all, it will still sound the same. If you're trying to make an LFO, then you would probably want something that looks like a perfect saw, and you wouldn't care about bandlimiting.

...but maybe about "upsample-filter-stuff"...
...I wonder why the saw is called SAW... (and not like "shark-tooth" or sth.)
A synth with just one osc (and filters) is a little lame, don't you think?!
So when there is another osc to mix with, phase becomes important, too. And with waveshaping the shape of the wave (...what else... ) is important! So a saw is a saw and not just "overtones decaying with 1/n"...

@Flipp said:

Waveshaping always implies non-linearities.

Non-linearity, however, does not imply waveshaping. AM and FM, for example, are non-linear processes. But they aren't waveshapers.

Also, if you use a straight line as the transfer function, it's still a waveshaper, but it can be linear.

Hmhmhm...
But that "becoming squarer" is closely related to the "new" samplerate
<snip>
And how do you explain that the "harmonics" always meet at the f_ny_new (or at f_sr_new), if there was no relation to it...

The harmonics begin at the new SR, an extend beyond it. The only frequency that aliases at the new Nyquist is that of the sine wave (the fundamental of the distorted wave).

Take a look at my patch again. Starting at 0 Hz, as you increase frequency you will immediately see the first harmonic pop up at the "new SR". As you increase, some partials will increase in frequency while others decrease. Those decreasing once are folding over from the global sample rate, not the [samphold~] one. Once the [osc~] frequency goes above 2756.25, it aliases, and you see that frequency moving back toward zero. Everything above 2756.25 is distortion, and the only reason there are partials that fall below 5512.5 is because they were aliased at 22050. You can see this because the partials between 2756.25 and 5512.5 always go in the opposite direction of the lowest frequency, whether it has been aliased or not.

Take a look at the modified attached patch. I added a high-order Chebychev filter to act like an anti-aliasing filter would at the dac. It's not perfect because 1) no anti-aliasing filter is, and 2) there's still a global sampling rate, which would not be the case in a good dac. But it does illustrate some important points. First, the anti-aliasing filter fixes those steps so that the sinewave is reconstructed. Second, it shows that only the [osc~] frequency aliases at the new SR. If the other frequencies were aliasing, the level of that one remaining unfiltered partial would start to decrease as you increased the frequency. But it doesn't. It stays at the same level (plus or minus the filter ripple and slope). You can push it well beyond the SR and it will be a sine wave at the same level (except if you push WAY too high, but that's because of floating-point madness).

It's obvious: Just change the new samplerate and watch the loundest partials fold at the new sr (and its multiples)...

The loudest partial is the fundamental, i.e. the [osc~] frequency. It's the only one that folds over at that new Nyquist.

The more quiet a sound the more negligible!

That's because they're quiet. The amplitude range is limited by the bit depth. Also, they're just quiet.

And the aliasing I hear is far from being quiet!!!

That's not the aliasing you hear at the new SR you hear. It's the distortion. That artificial SR created by [samphold~] does not mean everything between that Nyquist and the global Nyquist don't exist.

...and beside that negligible "crap" from the original sr-aliasing, it just behaves nicely 'relative' to the f_sr_new.

In that it never aliases at that SR.

You don't like the timedomain, do you?

Sure I do. Most of what I do ends up being in the time domain. I just think it's important to know which domain is useful for what purpose.

...since shure it is: Sampling each "0.2 period" is the same as sampling each "1.2 period" since an offset of 1 period doesn't change anything..

But in a signal with more than one harmonic, you can't tell which of those harmonics were aliasing in the time domain. Also, as I mention before, .2 and 1.2 are technically the same thing in discrete time. You don't really even know which one it is.

...I wonder why the saw is called SAW... (and not like "shark-tooth" or sth.)

Hmm...maybe that's a hard-synced saw?

A synth with just one osc (and filters) is a little lame, don't you think?!
So when there is another osc to mix with, phase becomes important, too.

Yeah, in that case it might matter. But it doesn't have to be a perfect saw shape. Just a collection of harmonics with 1/n amplitude up until Nyquist that are phase-aligned.

By the way, if you see an oversampled-and-filtered sawtooth that just has a smooth transition where the discontinuity would be and no ripple, that means it used a filter that doesn't have a linear phase, and so the phase alignment gets fucked up anyway. A linear phase FIR filter will have some ripple, similar to a limited sum of sine waves, and it won't look like a perfect saw, either.

http://www.pdpatchrepo.info/hurleur/anti-alias_with_filter.pd

@Flipp said:

...and beside that negligible "crap" from the original sr-aliasing, it just behaves nicely 'relative' to the f_sr_new.

Ah, sorry, I just realized what (I think) you meant by this. The aliased frequencies (from the global SR!) sit on top of each other, so they don't become a huge mess. If you change the new SR to, say, 5500, they don't overlap, you have nasty distortion all over the place, and so the anti-alias filter doesn't work in this case. You need a much higher global sampling rate for it to be worth it.