sander99
Headphoneus Supremus
Is waveshape the first thing that is analog instead of digital? Or is it how air moves, at the very end of the signal chain? I'm wondering. And since I am definitely not talented in DSP and electronics, I'm curious if someone could "stress" a DAC by using an all pass filter (?), like, adding more time information for whatever reason. I'm afraid the idea attracts me because I know nothing about DACs.
Yes, the waveshape is analogue and it contains information about frequency, amplitude and phase. Even if we start with frequencies limited to 23 kHz, the combination of all those informations into the structure we call the waveshape will result in frequencies much higher than 23 kHz. If we sample it at 44.1 kHz, for example, a significant amount of information encoded in the wave shape will be missing when digital is converted back to analogue. This produces distorsion.
A waveshape is simply the shape or form of a periodical function depicted as a graph.
An analog audio signal is (in most cases) a voltage changing over time.
(And if a microphone is subjected to sound, it is actually subjected to air pressure changing over time, and a graph depicting the air pressure as a function of time would be "analogues" to the graph depicting the output voltage of the microphone as a function of time, so it would have the same or similar shape.)
If you depict it as a 2 dimensional graph with time on one axis and voltage on the other axis that graph has a shape: the waveshape.
Examples of periodical waveshapes a sine wave, a square wave, a triangle wave.
In a way you could also call a complete recording one big complicated waveshape.
Important to know is that every waveshape can be decomposed in (or written as the summation of) a finite or infinite number of sine and/or cosine waves.
(Discovered and proved - for functions in general, maybe he didn't think about sound at all - by a mathematician called Fourier 2 centuries ago!)
If we say a recording contains certain frequencies it means that the decomposition into sine and/or cosine waves contains sine and/or cosine waves at those frequencies.
For example a perfect square wave (that does not exist in the real world) is the sum of an infinite number of sine waves:
sin(x) + 1/3sin(3x) + 1/5sin(5x) + ...
If you listen to a square wave with a fundamental frequency of 10 kHz, the next harmonic will be at 30 kHz, the next at
Inside the inner ear are hairs "tuned" to small frequency bands. For some hairs that small frequency band will include 10 kHz. They will resonate with the 10 kHz fundamental and as a result neurons will fire to signal to the brain that something happens at 10 kHz.
There are no hairs tuned to frequency bands that include 30 kHz, or
So you won't be able to hear the difference between a sine wave at 10 kHz and a square wave at 10 kHz.
The sine/cosine decomposition of a band limited signal (with 20 kHz upper limit) will only contain sine/cosine waves below 20 kHz.
The sampling theorem is a proven mathematical statement. What is proven is that if you have samples of the signal at least every half cycle time of the highest frequency in the signal then you can completely and precisely reconstruct that signal. All sine/cosine waves of the decomposotion can be reconstructed including the exact phase and timing! That is if the samples are exact. If the samples only have a limited precision, for example 16 bit, then the reconstruction will contain added noise. (However, 16 bits is enough to keep the noise far below audible levels, assuming you don't listen at such a high volume that the loudest possible parts of the recording will destroy your hearing.)
If you listen to a square wave with a fundamental frequency of 3 kHz there will be some hairs reacting to 3 kHz, some to 9 kHz, if you are lucky and not to old some to 15 kHz, and that's it.
If this signal is recorded and played back digitally with 44.1 kHz sampling rate then the 3, 9 and 15 kHz will be perfectly reconstructed, with the correct timing/phase.
If you listen to this recording you will not be able to distinguish it from the original square wave because all the audible components of the square wave are there (and the same hairs in your inner ear will be reacting).
(That is: unless your amp and speaker will cause audible distortion when fed with the original square wave, which is very likely because no amp can change it's output voltage infinitely fast, and no speaker membrane can move infinitely fast. In this case using the digital recording - or low pass filtering the original square wave - would lead to a audibly better reproduction of the square wave.)
[Edit: corrected a few wrong numbers...]
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