DAC Reconstruction Filter Talk and Other Stuff

I don't understand. I'm a DSP bozo, but I thought a perfect digital reconstruction filter should have impulse response that looks like a sinc function. Yggdrasil impulse looks pretty much like a sinc function. Shouldn't that just result in a correctly band limited signal with flat FR, rather than ringing?

 

Hmm.
Quantization artifacts aren't the easiest fit to the measurement. Time domain blip with 24-bit input is only obvious at zero-crossing instead of all bits.

The filter does make for a good reconstruction as you piece together the continuous signal. It rings in that the impulse response shows a slow decaying oscillation (a sinc-like function), not so much that it adds ringing to whatever other signal you input.
Steep transitions in one Fourier term (e.g. frequency) are reflected by slower transitions in the opposite term (e.g. time), and vice versa.

 

Good, thanks, so "ringing" just refers to the correct behaviour of a digital reconstruction filter.

Ok, now I am confused again. The input to the reconstruction filter is, I guess, an impulse train, and by the Nyquist theorem, every such impulse train represents a unique band-limited signal. If you naively sample a synthetic square (or saw, triangle or step) wave then sure you will not get it back again after reconstruction because you failed to band-limit your input signal. But the error on reconstruction would be aliasing of the harmonics above Nyquist into the available bandwidth which would potentially look like a total inharmonic mess.

On the other hand if you did correctly band limit your input signal then wouldn't you have Gibbs phenomenon/ringing at the discontinuities just because that is what the band-limiting of a square wave looks like?

Tl;dr, isn't ringing always a sign that you have designed your system correctly?

 

Upsampling in the digital domain can cause aliasing too. Since the original signal is now represented by a train of impulses, it's turned into a signal that has energy above Nyquist: those are all the images of the baseband signal repeating to +- infinity. The interpolation filter in an up/oversampling filter has to lowpass filter the signal to less than Nyquist of the new sampling rate. For example, upsampling 44.1 kHz to 88.2 kHz, the upsampling filter has to bandwidth limit the 44.1 kHz sample stream to less than 44.1 kHz before doing the 88.2 kHz upsampling. A signal sampled at 44.1 kHz has aliases or copies of the baseband signal centered at every integer multiple of 44.1 kHz, so it has to be bandwidth limited before resampling otherwise those repeated images will alias into the audible bandwidth.

Quantization error, like chopping off the lowest bits of a sample word without dither, will also cause aliasing since that's a very non-linear discontinuous process that can produce high frequency energy (you are basically creating stairsteps in the signal). Those weird, low-level spikes below the noisefloor in the Stereophile measurements may be an artifact of this, but that's only a guess based on the limited information available.

Filter overload is another thing in a DAC that may cause aliasing. A while ago, the guys at TC Electronic wrote a paper that showed how highly compressed signals (in the loud radio ad sense, not in the MP3 sense) can cause peaks in the reconstructed signal well above 0 dBFS which can latch up or otherwise screw with digital filters unprepared to handle this. Here's a link to the paper if you're curious: http://www.tcelectronic.com/media/1018207/nielsen_lund_2000_0dbfs_le.pdf

In other words, even if your actual sample points are all below the maximum digital level of your filter, they can represent a signal which is higher than the max digital level in between the sample points. This is easy to imagine: think of a sine wave, and think of sampling the sine wave only at its lower amplitude points. When reconstructed, you're going to get a signal which has points that are higher than your original sample point.

When those in-between peaks are reconstructed, they present a higher than expected signal to whatever is downstream of the interpolater, and that can cause all sorts of issues especially on a signal that's so compressed that all of its sample points live near the peak digital level. I had a surround processor once whose surround sound matrixing algorithm (which worked in the digital domain) would get latched up by a certain song track. The TC Electronic paper has examples of more straightforward DACs whose filters were screwed up badly by highly compressed signals. Once your DSP algorithm goes non-linear, it can generate all sorts of out-of-band energy that's aliased, and so becomes more audible.

This is the phenomenon that Benchmark's DAC-2 and now DAC-3 claim to guard against when they talk about their high-headroom DSP.

BTW, this is why digital level meters that look at sample values only are worthless. You have to look at the interpolated values between the sample points to get a true sense of a signal's peak levels. It's also why simplistic methods of determining when a digital signal has overloaded (eg. 5 consecutive max samples values) are often wrong too.

 

just want to say thanks; this thread (or snippet...) is a perfect example of why SBAF is yummy. and moist.

 

Or indeed pretty much any kind of nonlinear signal processing in the digital domain!

This was quite fun to try and figure out for myself. I think I got it by an argument like this:

If I have a digital signal at 44.1kHz then I can view it as an analogue signal given by an impulse train at that frequency. The frequency domain transform of this signal is periodic with period 22.05kHz. To reconstruct the corresponding bandlimited signal I convolve with a sinc function which is the transform of a step function of width 22.05kHz.

If I upsample to 88.2kHz by inserting zeros, then the analogue impulse train to which it corresponds stays *the same*, because I inserted zeros. And now to reconstruct the corresponding bandlimited signal at the new sample rate, I convolve with a sinc function which is the transform of a step function of width 44.1kHz. This kills all the repeating copies in the frequency domain except for the first *two*. So I get one extra image of the frequency domain response in the region 22.05-44.1kHz.

Thinking about this also just brought home why one should care about the fact that the Schiit upsampling filter preserves the original sample values: because that is what's supposed to happen in the Nyquist theorem. It's sort of an amazing property for a convolution filter to have.

 

There are definitely words in this thread, and after careful examination I've determined that they're probably English.

I've given trying to understand this stuff a good go, and I do want to learn, but does anyone want to point me to some good starting reading material so I can understand things being said here a bit better? When I try to search the terms on Google it pretty much dumps me in the deep end, and I can't seem to find the wading pool.

 

I'm going to have to read this

 

Thanks guys! Gonna have me some reading to do.

 

Adding noise or dither is not non-linear: sorry, didn't mean to imply that if I did. However, quantization (chopping off the lowest bits) is non-linear, and any time you have a non-linear process in the digital domain, you can alias.

The definition of linearity I'm using is the mathematical one we learned back in high school algebra:

1. a*f(x) = f(a*x)
2. f(a) + f(b) = f(a+b)

How non-linearities can cause aliasing can be seen if you take some non-linear function like f(x) = x^2, and put x=sin(t) into it. Work through the trig identities, and you generate sin(2*t) terms, which are sine waves twice the frequency of the original signal. Similarly any power function will generate harmonics that are multiples of the highest powers: something with x^4 terms will generate 4th harmonics or sines with 4x frequency. You can see how with higher frequency sines, you can easily exceed the Nyquist limit of a sampled system.

Power series non-linearities show up in lots of signal processing too: level compression and limiting systems are inherently non-linear, and so often use oversampled algorithms to contain their non-linearities, eg. the compression process is run at 2x or 4x or higher sample rates.

In the case of undithered quantization, it's introducing a sharp discontinuity in the waveform by chopping the lowest bits off. Sharp discontinuities contain higher frequencies, and high frequencies will cause aliasing if they're above Nyquist. Undithered quantization also does not fulfill the linearity rules above.