A DAC is a circuit that converts digital measures of audio amplitude in discrete steps into a continuous analog electrical equivalent of the sound to be reproduced. The amplitude is a digital number (like a 16 bit word) and the steps occur based on the sampling rate (like 48,000 times per second). This process is very much like an endless conveyer belt with empty one gallon jugs on it, moving by a filling station. The size of the jug is fixed, the rate they pass by is determined by the sampling rate. The goal of the DAC is to fill each jug to exactly the right level specified by the music. There are three techniques used to accomplish this; Delta Sigma, Ladder, and the MSB Sign Magnitude Ladder.

Delta Sigma DACs (Single Bit)

Each sample or jug is filled to the right level with many measuring cups, poured to reach the target level.

A “one bit” measuring cup is either full or empty. With 64 times oversampling, the cup is only 1/64th the volume of the jug. This rather crude cup does not come close to being accurate enough. The cup would need to be 1/16,777,216th the volume of the jug to be accurate.

Line speed is like Sampling Rate (48,000 per second)

By keeping a running tally of the error and by going a little over and a little under in many samples, a very accurate average is reached, but only with an aggressive filter applied to the output.

Ladder DAC

A ladder DAC is different in that instead of a single measuring cup (or bit), a whole array of cups are available, from very small to very large.

Any combination of cups can be used to fill each jug to exactly the right level.

No filtering is required but the accuracy is defined by the cumulative error of all the cups used.

MSB Sign Magnitude Ladder DAC

The MSB Sign Magnitude Ladder DAC is like a Ladder DAC but refined in two ways.

Because the jars are typically 1/2 full when finished, we start with very accurately 1/2 filled containers instead of empty ones. From there, we again use the wide range of measuring cups to either add or take away from each jar.

Again each jar is filled exactly so no filtering is needed, but because we only had to add or subtract a little bit our accuracy is higher. Because our hearing is most sensitive to low level sound (sound near the 1/2 full zero crossing) our DAC is most accurate near 1/2 full where we use the smallest and most accurate measuring cups.

What about Upsampling?

Synchronous Upsampling like MSB does just means adding more jars between the existing jars and moving them faster down the line. By looking at many jars before and after the new empty one, we calculate how full to make it. The result is smaller steps as shown in the waveform This is very significant when used with a Delta Sigma DAC but less so with a Ladder DAC.