and the other times it is −0.8. This still results in a repeating, quantifiable error. Another plausible solution would be to take 4.8 and round it so that the first four times out of five it is rounded up to 5, and the fifth time it is rounded to 4. This would average out to exactly 4.8 over the long term. Unfortunately, however, it still results in repeatable and determinable errors, and those errors still manifest themselves as distortion to the ear. This leads to the dither solution. Rather than predictably rounding up or down in a repeating pattern, it is possible to round up or down in a random pattern. If a series of random numbers between 0.0 and 0.9 (ex: 0.6, 0.1, 0.3, 0.6, 0.9, etc.) are generated and added to the 4.8, two times out of ten the result will truncate back to 4 (if 0.0 or 0.1 are added to 4.8) and eight times out of ten it will truncate to 5. Each given situation has a random 20% chance of rounding to 4 or 80% chance of rounding to 5. Over the long haul, these results will average to 4.8 and their quantization error will be random noise. This noise is less offensive to the ear than the determinable distortion that other solutions would produce. Usage Dither is added before any quantization or re-quantization process, in order to de-correlate the quantization noise from the input signal and to prevent non-linear behavior (distortion). Quantization with lesser bit depth requires higher amounts of dither. The result of the process still yields distortion, but the distortion is of a random nature so the resulting noise is, effectively, de-correlated from the intended signal. In a seminal paper published in the AES Journal, Lipshitz and Vanderkooy pointed out that different noise types, with different probability density functions (PDFs) behave differently when used as dither signals, and suggested optimal levels of dither signal for audio. Gaussian noise requires a higher level of added noise for full elimination of distortion than noise with rectangular or triangular distribution. Triangular distributed noise also minimizes noise modulationaudible changes in the volume level of residual noise behind quiet music that draw attention to the noise. Dither can be useful to break up periodic limit cycles, which are a common problem in digital filters. Random noise is typically less objectionable than the harmonic tones produced by limit