https://www.xiph.org/video/vid2.shtml
Great video with simple explanations of digital sampling, bit depth, dither, band limitation and timing.
https://www.xiph.org/video/vid2.shtml
Great video with simple explanations of digital sampling, bit depth, dither, band limitation and timing.
This is interesting too...
192kHz considered harmful
192kHz digital music files offer no benefits. They're not quite neutral either; practical fidelity is slightly worse. The ultrasonics are a liability during playback.
Neither audio transducers nor power amplifiers are free of distortion, and distortion tends to increase rapidly at the lowest and highest frequencies. If the same transducer reproduces ultrasonics along with audible content, any nonlinearity will shift some of the ultrasonic content down into the audible range as an uncontrolled spray of intermodulation distortion products covering the entire audible spectrum. Nonlinearity in a power amplifier will produce the same effect. The effect is very slight, but listening tests have confirmed that both effects can be audible.
Source: http://people.xiph.org/~xiphmont/demo/neil-young.html
The output of the DAC goes through a filter that "connects the dots" between the stair steps and reconstructs the waveform.
Yes, the reconstruction filter is mathematically part of the process. Check any DSP textbook or resource.
But note that in practice, most audio DACs are oversampling delta-sigma affairs anyway, so there's not actually 2^N possible output values internally (with N being the bit depth).
Take some time and watch the mentioned video, especially the 4 minutes from about 3:37 to 7:30.
The Wikipedia piece isn't strictly correct. Not all DACs produce a zero-order hold output. Some produce a narrow pulse of the required current or voltage value, rather than holding it steady for the whole sample period. Even for zero-order hold, it's not all bad. It results in a slight low-pass filter effect which simplifies the reconstruction filter.
The following is over-simplified but will hopefully make sense:
What do the "stair steps" look a bit like? Square wave steps.
What is a square wave? It's made up of a (sine wave) fundamental frequency and harmonics.
Now filter off all the harmonics. You're left with the fundamental sine wave.
It's the same thing with the unfiltered output of a DAC. Filter off all harmonics higher in frequency higher than half the sampling rate, and you're left with a smooth curve.
The following is over-simplified but will hopefully make sense:
What do the "stair steps" look a bit like? Square wave steps.
What is a square wave? It's made up of a (sine wave) fundamental frequency and harmonics.
Now filter off all the harmonics. You're left with the fundamental sine wave.
It's the same thing with the unfiltered output of a DAC. Filter off all harmonics higher in frequency higher than half the sampling rate, and you're left with a smooth curve.
Yup.
For another take, think about what it takes to get a sharp, abrupt change in the level like on a staircase. You need very high frequencies. To get very fast changes you need very high frequencies.
Think of the filter as a block of electronics that are preventing those very high frequencies from passing through, filtering them out. It's continually processing the input (what's from the DAC) and generating a processed version for the output based on the characteristics of that block. If it sees a very sudden change in the input, that change (which contains very high frequencies) is getting filtered and slowed down, smoothing out the response of the output compared to the input.
What with all the handwaving, hopefully that didn't create extra confusion.
Yup.
For another take, think about what it takes to get a sharp, abrupt change in the level like on a staircase. You need very high frequencies. To get very fast changes you need very high frequencies.
Think of the filter as a block of electronics that are preventing those very high frequencies from passing through, filtering them out. It's continually processing the input (what's from the DAC) and generating a processed version for the output based on the characteristics of that block. If it sees a very sudden change in the input, that change (which contains very high frequencies) is getting filtered and slowed down, smoothing out the response of the output compared to the input.
What with all the handwaving, hopefully that didn't create extra confusion.
A third take is to say that the stair-steps was never there, only the data points exist.
And in the same way as there is only one possible straight line that can connect two given points, as long as there are more than two data points per cycle, there is only one possible sine wave that can fit them all.
The actual data is discrete time, but you can say the staircase exists like that in some systems—just as an intermediary step in a process.
The solutions aren't always just pure sign waves though. If there are multiple frequencies, you need to specify the phase for uniqueness. e.g. an impulse (containing many frequencies) will result in a different looking continuous-time representation for a minimum phase filter than a linear phase one. That is, unless I need some coffee / hitting the books really bad and am forgetting something.
Also, if there are fewer than two data points per cycle, that still gets represented with the same kind of uniqueness, just using the wrong frequencies...
It was more an attempt of a conceptual description of how one can make smooth curves from discrete points, rather than describing how it's actually done in a DAC.
But I believe, given the right conditions, or rules, for drawing the curves, it's theoretically reasonably sound.
Should be pretty easy to do yourself. All you'd need is Audacity (or something similar) to whisk up a couple of samples and some ABX software to test yourself.
As to what kind of samples, I'd probably go with some pure 1kHz sine tones, and start with maybe a .5dB difference and gradually reduce it until I can no longer reliably tell the difference.