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BIT REDUCTION
Figure 14. Sinusoidal Input Bit Reduction for Various Values of �������ℎ
BIT REDUCTION
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Bit reduction,or ‘bit crushing’as it’s more commonly referredto in the audio effectliterature, processes an input signal such that the resulting output can only retain certain possible amplitude values (Tarr, 2019). Reduction of the bit resolution results in the discarding of information previously available to represent the signal, in this case amplitude information. Conversely, increasing the bit resolution better represents the sound’s amplitude profile. Decreasing the bit depth gives a stepped character to the signal (fig 14), which is reminiscent of a signal after the quantisation stage in an analogue to digital converter. For this reason, bit reduction can be thought of as re-quantizing a signal, almost always at a lower bit depth, to create destructive distortion.
Figure 15. Bit Reduction Characteristic Curves for Various Values of �������ℎ
Bit reduction is achieved by rounding the samples of a signal to the nearest available amplitude value, the nearest available amplitude value is dictated by � in the formula:
To achieve an explicit relationship between bit depth and amplitude values the rounding function used here is ceil().Ceiling functionsalways roundup instead of rounding up or down based on the sample value. A limitation of MATLAB’s rounding functions is that they round to the nearest integer,therefore, a sinusoid of ±1 is only going to have the ability to round off to −1,0 or +1. To get around this, the amplitude of the signal can be increased by a scaling factor �, rounded and then divided by the same scaling factor, returning the signal to be within the range ±1.
The user has control over the bit depth of the bit reduction. Lowering the bit depth results in fewer amplitude values at which to represent the signal, therefore, a lower resolution and a more extreme, destructive distortion. Bit reduction creates harmonic distortion at even and odd multiples of the input frequency, these are more prominent at odd multiples of the input due to the relative oddness of the characteristic curve. The magnitudes of the harmonics are dictated by the bit depth, with lower bit depths producing harmonics with greater magnitudes (fig 16).
Figure 16. Harmonic Distortion of Bit Reduction as Bit Depth Decreases
The noise introduced by the bit crusher is called quantisation noise. Smaller amounts of quantisation noise for moderately reduced bit depths i.e., �������ℎ = 8 can give the sound a warmer tone, whilst drastically reducing the bit depth (fig 16) gives a noisier, harsher quality to the sound (Tarr, 2019).
In a distortion effect the quantization noise is a welcome characteristic and adds to the overall harmonic distortion profile. However, in analogue to digital converters this quantisation noise is the unwelcome product of quantization error (the difference between the continuous signal and the nearest available digital amplitude at which to represent it). In these systems quantization error can be corrected with dither, this involves adding noise to the signal pre-quantization resulting in quantization noise which is not harmonically related to the signal and therefore not as perceptible (Pohlmann, 2011). The relationship between bit depth and amplitude values is as follows.
Thus, a bit depth of 4 allows for 16 possible amplitude values (fig 17).
