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Beyond the LSB

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If we truncate a number, that is to throw away the LSBs (least significant bits) we loose resolution.

A 4 bit number truncated to 2 bits: numbers shown in binary (Base2).

0100
0101
0110
0111 

Would all become 01.

Introducing Dither

Effective dithering increase the accuracy beyond the LSB of the truncated values. Consider a small fractional value rounded to an integer. Starting with 0.5 with 1 bit random dither:

Input  Dither Sum & truncate
00.1   0.1    01
00.1   0.0    00
00.1   0.1    01
00.1   0.0    00
Average over 4 samples is 0.5

If the value is below 0.5 this dither scheme will not work, as it would never round to a full integer.

So we increase the dither to 2 bits. Inputting a constant 0.25, Dither cycling through all possible values

Input  Dither Sum & truncate
00.01  0.00   00
00.01  0.01   00
00.01  0.10   00
00.01  0.11   01
Average of 4 Samples 0.25

In the above example we have increased the resolution of the time averaged value by 2 bits.

For every doubling of the frequency we gain an effective LSB if dithered correctly. Adding more bits than log2(oversampling rate) of dithering will not gain accuracy. Although it may help breakup tonal behaviour of a system.

Summary

Apply dither to the bits to be truncated. Apply enough bits of dither so it is possible for the LSB of the required resolution to effect the truncated value.

The increase in resolution is limited to log2(oversampling rate).

Further Reading:

Quanatization and Dither: A Theoretical Survey. LIPSHITZ, WANNAMAKER and VANDERKOOY

Resolution Below the Least Significant Bit in Digital Systems with Dither. VANDERKOOY and LIPSHITZ

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