Digital Drive: Re: This Wasn't as Bad as Expected... by csown

> I think what you mean by a "damping envelope" as the filter function itself. But the function does the *opposite* of damp. It **induces** "ringing." As stated before, to reduce modulation, which flattens HF response. And the "ringing" helps reject artifacts above half the sample rate frequency.

It's a matter of definition here. You are restricting "damping" to time domain artifacts only. But it can be applied to any domain. For 1x sampling, this occurs in the frequency domain. In transmission electron microscopy, for example, there is a thing called the contrast transfer function. The chromatic aberration of the objective lens yields an attenuation of the contrast transfer out to high frequencies. This is a damping envelope also.

> If you took a 20-bit 8x oversampling DAC (very common in the late '90's), and removed the actual D/A portion of the DAC following the oversampling section, and just used the oversampled signal as the output, you would have created an 20/352 upsampler. And aside from synchronous conversion in this case, the filter process is basically the same. Or in other words, an upsampler is an oversampling DAC minus the D/A section.

This is not true. Upsampling is asynchronous oversampling, i.e., it is oversampling using non-integer multiples of the sampling frequency. This is what gives rise to all the effects Doug specifies in his paper; it doesn't have anything to do with the analog post-filter at all.

> I've heard from the late Julian Dunn, the upsample conversion can be "synchronous" by being 320x oversampled followed by 147x downconversion. I think in reality, there are *two* conversions, which is somewhat "lossy" compared to synchronous conversion.

How is this synchronous? Explain this in more detail. Upsampling is by definition asynchronous to f _s .

> You cannot have a steeper rolloff than the "sinc"/"brickwall" filter function. The "sinc" function and the "brickwall" filter are one and the same. If you widened the FIR window, covering more of the "sinc" function, the "more-perfect" the brickwall filter's steep cutoff becomes.

The brickwall term describes the analog post-filter, not the behavior of the digital filter. It's called brickwall because it has to be steep enough to wipe out the first image starting at about 24KHz. It does not have to reflect the same slope of the image stop, in fact it can't. Remember, you supposedly can't hear above 20KHz, but the 44KHz-sampled audio can represent 22KHz signals. The response of CD thus goes up to 22KHz, then mirrors itself immediately above this band. The target stop band starts at 20KHz, but there is still information above. The brickwall filter thus tries to attenuate starting at 20KHz to near-nothing by 24KHz. That's why it's called brickwall, not because it looks like the stop band effect of the FIR filter.

If you use the FIR without the gradual post-filter, there is flat response to 22KHz, then nothing, then stuff starting at x* f _s , where x=the oversampling coefficient. This is not brickwall behavior, this is just oversampling.

> You are getting carried away with "damping." In digital audio, there is *no* damping going on. It may *appear* damped relative to a "sinc" filter, but it is *not* damped in the classic sense.
> The ringing is not "reduced," but *induced* by the digital filter. The FIR function is what *injects* a ringing pattern into the signal. A "time-resolute" filter (Wadia), which has a more-gradual rolloff, injects less "ringing" than a classic "sinc" filter. As I said earlier, this induced "ringing" attains a flatter response by reducing HF modulation and to attain rejection of artifacts above half the sample frequency.

You're still mixing up time domain effects with frequency domain effects - in other words, limiting your analysis only to how fourier coefficients comprise impulse response. Sinc rolloff is *not* the FIR impulse curve. The FIR impulse curve is described by a sinc function which arises due to the finite number of fourier components. Again, this is not the sinc rolloff.

"which means you are getting ringing at frequencies quite closer to the audible band than you would have gotten if you simply used standard FIR"
> The ringing frequency is half the base sample rate frequency. No way to get around that. Otherwise one would not get the proper "brickwall" response.

Not true with an upsampler. Due to asynchronicity, the ringing frequencies are not half the sample rate. The fourier coefficients in the upsampler's reconstructed impulse are different from those in the FIR's reconstructed impulse.

> The non oversampling DAC just takes the *raw* CD data, and converts it directly to an analog signal. The image about half the sample frequency is attenuated solely by the analog post filter, as opposed to mostly by the digital oversampling filter. The analog post filter should have a steeper cutoff slope to reject most of that image.

See Werner's post below. "convert directly to analog signal" means impulses. This is NOT what happens in non-OS; the non-OS conversion uses ZOH.

"Incidentally, I think the crux of this issue is that your interpretation of time smear is that it is due to poor impulse response (i.e., pre-ringing). I believe it is also due to absolute phase error, but these two issues are inextricably linked with each other."
> The phase error is more of a function of the *symmetry* of the FIR coefficients than the amount of ringing that takes place. One FIR DAC could be a classic sinc with ringing, another FIR DAC could be a "time-resolute" function with almost no ringing, yet both have zero average phase error.

This is obvious. My point was that you are associating time smear only with impulse response. Impulse response contains information about the phase error, but it doesn't explicitly tell you anything about real world musical transient response. You need to see the behavior of the phase response across the audible band to understand the effect on time smear (as per my definition of time smear above).

"To me, time smear is a much broader term which looks to be different from your perspective."
> "Time smear" is simply the ringing components of the DAC's time response.

Defined by whom? Ringing components are supposedly inaudible. Like I said in the previous message, transients in real music are not anything like the impulse function (or the FIR-filtered approximation), they are spectrally complex. Group delay of specific, individual components (not the sum of all fourier components) contributes to the "time smear", not just supersonic ringing!

"Brickwall filter is an ambiguous term."
> No it is not. It is a sharp cutoff filter made expressly to reject the images above half the sample rate frequency.

My point above (The brickwall term describes the analog post-filter..., etc.).

"The point of the post-filter ("brickwall" or not) is to roll off steeply enough to exclude high frequency images."
> The post-filter is almost *never* a "brickwall" filter. (Unless if it's also the main filter, like on the first Sony players.) Because the "brickwall" filter has already taken place in the oversampling DAC or upsampler.

Which is the brickwall to you, the post-filter or the oversampler? You're inconsistent about this, and it completely brings into question your previous points about "brickwall" being in every DAC. Here, you say that brickwall filter takes place in the digital filter itself. Prior to this, you said that all DACs are using brickwall filters nowadays (except non-OS). Any way you think about it, there is a contradiction. And saying that an upsampler is a brickwall filter is patently wrong.

"A gradual rolloff filter with oversampling can be construed as a brickwall filter as well, because it rolls of quickly enough to exclude the first image."
> If it rolls off quickly enough to reject the first image, it is *not* a gradual filter. A "brickwall" filter is *not* a gradual filter. The near-perfect vertical slope is where the term "brickwall" comes from. Conversely, a gradual filter does *not* completely reject the first image.

Yes it does! The gradual rolloff filter does reject the first image of the FIR filter.

"I think you're still missing what linearizing means. I believe you're thinking about something like a trapezoidal filter. It's not the same as interpolation, i.e., connecting the samples."
> All digital filters have interpolation. Once can create a "trapezoidal" response by altering the filter coefficients in the DAC's "kernel." But the DAC would *still* interpolate in the "trapezoidal" case.

About interpolation, that is obvious and has nothing to do with how to obtain differential linearity with "frequency-dependent high-frequency image" dither, which you still aren't understanding. I know that you understand classic dither, which I termed noise floor dither in my previous post, same thing as "hiding the LSB in noise". Perhaps this is a better interpretation of differential linearization:

Noise floor is a DC phenomenon, and you put a random LSB signal on top of it to "linearize" the LSB error. Now extrapolate this to a high frequency tone, say a constant zero order hold at X volts. Now during each ZOH timeslice, the upsampler modulates this with a series of dither levels, made up of frequencies that are non-integer multiples of the sample output frequency. This is what I termed "plateau dither" previously.

But dither has no use whatsoever in the D/A process itself, unless one wants to merely raise the noise floor.

Yes it does have use in the D/A process, because it linearizes the noise floor. With an upsampler and also with 1x sampler, it also linearizes the D/A conversion.

It is impossible for the max phase error of a symmetric FIR filter to *exceed* 90 degrees (which is pi/2). With an analog filter, full pass band to full stop band shifts the phase angle -180 degrees. (This is what I meant by "subtractive.")

Additive and subtractive are the same thing. With a symmetric FIR filter, the phase at the stop band is -Pi/2. A super-steep high-order analog post-filter will add a phase shift of up to -Pi. You have already gone non-linear (unstable) with this system way before the stop band. I'm sure lots of mid-80's CDP's sounded terrible because of this. Not to mention the phase shift in the ultrasonics, which can intermodulate into the audible band *out of phase*.

Best,

-Chris