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When an analog signal is converted to digital, or a digital signal is generated directly in a digital synthesizer, the result is a sequence of numbers. A CD-quality 16-bit signal has 65,536 possible different numbers to describe the signal.
An analog signal changes smoothly, but a digital signal jumps from one allowable number to the next. There are no 'in between' numbers. The only way to get to these intermediate positions is to increase the bit count to 20-bit or 24-bit. But even then, the digital signal must make that jump from one number to the next, they are just closer.
The result is that there is always a slight inaccuracy in the digital signal. If it were truly accurate, it would follow the analog signal precisely. Instead it follows it closely, but in a 'stair-step' fashion.
The inaccuracy in a digital signal is heard as noise. In a complex signal, such as a music signal, that noise is mostly innocuous, and it is anyway very low in level.
But there are situations that can occur, even in a music signal, where the noise takes on a harsh, grainy quality. And this can happen often. An example is in a reverb tail. As the tail decays, eventually it will get to the point where it crosses the boundary between the smallest possible digital number and absolute zero. Analog signals being as they are, the reverb tail will probably cross and recross this boundary many times.
The result is a chain of digital pulses at random intervals. Although this is very low in level, it is very harsh and therefore easily heard.
To get round this problem, digital systems add 'dither' noise to the signal. This can be simple white noise, or it can be cleverly designed to optimize its effect. Dither is added in analog to digital converters, systems that generate digital signals directly, and after any processing has been carried out, which negates the effect of any previously added dither.
Dither has the effect of randomizing the transitions between digital numbers and thus concealing the step from one to another. It works very well.
Dither has the additional advantage that it allows signals that are lower in level than the lowest possible level that can be described digitally to be heard.
It seems odd, but it is the random nature of noise that allows this. When the signal is extremely low in level, it can 'ride on the back' of the white noise and pretend to be bigger than it is. Even though it is lower in level than the dither signal, it can still be heard.
In digital audio workstations, dither is not added after every stage of processing, otherwise it would build up in level. Instead it is added right at the end of the chain, at the output.
In some systems, you have to enable dither manually. If you don't, you will get the digital harshness buzzing along at the lowest levels of your audio. This is something that is definitely to be avoided. Fortunately, if your accidentally dither-free recording gets as far as a mastering studio, the mastering engineer will spot this and add dither at this late stage.