# Dither noise probability density explained

From:     Christopher Hicks

I don’t know if this is interesting or helpful, but here it is anyway…

Take a normal 6-sided die. Roll it. You get a number between 1 and 6, each with equal probability. Roll it again, you get another number, again between 1 and 6, and totally unrelated to the first one. Roll it again, and you get a third number, unrelated to the previous two. This is white rectangular dither. “Rectangular” because you get each possible value with equal probability (if you plot a graph of probability against score you get a rectangle), and “white” because successive numbers are completely unrelated.

Now take two such dice. Roll them together and add the spots. This time you get a number from 1 to 12, and you are more likely to get the “middle” numbers (ie 5,6,7,8) than the extremes (2 or 12). Pick them up, and roll them again. You get another number, unrelated to the first, again with the middle scores more likely than the extremes. This is white triangular dither: white because successive numbers are unrelated, and triangular because the numbers in the middle are more likely to come
up than the extreme numbers (now if you plot a graph of probability against score you get a triangle).

Now, take two dice of different colours, red and blue say. Roll them both, and write down the total. Now pick up the red one (leave the blue as it is), roll it and write down the total. Now roll the blue one (leaving the red as it is) and write down the total. Continue in this way, rolling the red and blue alternately, and you’ll end up with a sequence of numbers from 2 to 12. This sequence has a triangular probability graph, but each number is now related to the one immediately before it; if the blue is showing a five or six as we roll the red, thenthe total we get on this roll is likely to be high, and vice versa. If you plot a graph of the successive scores it will look a little smoother than a similar graph drawn from the “white” sequence of the previous paragraph. This is low-pass triangular dither, not much used in audio.

To get high-pass triangular dither is a bit more complicated still. Roll both dice, and write down the blue score (B) minus the red score (R) to get B-R (which may be negative). Now roll the red, and write down R-B. Now roll the blue, and write down B-R. Repeat this sequence, rolling the dice in turn, and subtracting the one you left on the table from the one you just rolled. The probability distribution is triangular again, with numbers around 0 being the most likely, and +/-5 being relatively unlikely. Successive numbers tend to be at opposite ends of the possible range because of the alternating subtraction, and a graph of the sequence of numbers comes out relatively jagged. This is triangular high-pass dither.

This really is exactly how these dither signals are generated in practice, except that the electronic dice in use generally have more sides, and we take a little more care over the scaling and dc bias. Basically, rectangular dither comes up with its different numbers with equal probability, whereas triangular dither covers a larger total range
of values, but the extreme values are less likely to come up than the middle values. This property is called the “distribution” or “probability density function (pdf)” of the sequence. The dither “colour”, “spectrum”, or “autocorellation” tells you about how
successive dither samples are related to each other, regardless of their distribution.

P.S. –  By the way, it is a common misconception that “white” and “Gaussian” are synonymous when talking about noise. This is totally untrue – “white” is a noise colour, and “Gaussian” is a distribution, and any given noise signal can be either white, or Gaussian, or both, or neither. You can get reasonably close to a white, Gaussian sequence by taking a large number of dice (20, say) rolling them all together and writing down their sum. Repeat ad infinitum.

-Chris Hicks

By the way, Chris, can I take you along with me on my next trip to Vegas? Enhancing the explanations of the complex stuff in easy-to-understand ways is what  my book is all about! And your explanation beats the pants off of everyone else. I would include a graph picture of gaussian versus triangular in order to increase the explanation, as in “based on this shape, it is more likely that the middle number ranges of the dice will come up. In other words, it’s just as unlikely to roll snake eyes (two 1s) as it is to roll boxcars (two 6s).”

– Regards,
Bob