Editing Quantization (signal processing) (section)

===Mid-riser and mid-tread uniform quantizers===
Most uniform quantizers for signed input data can be classified as being of one of two types: ''mid-riser'' and ''mid-tread''. The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a [[stairway]]. Mid-tread quantizers have a zero-valued reconstruction level (corresponding to a ''tread'' of a stairway), while mid-riser quantizers have a zero-valued classification threshold (corresponding to a ''[[Stair riser|riser]]'' of a stairway).<ref name=Gersho77>{{cite journal | last=Gersho | first=A. |author-link=Allen Gersho| title=Quantization | journal=IEEE Communications Society Magazine | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=15 | issue=5 | year=1977 | issn=0148-9615 | doi=10.1109/mcom.1977.1089500 | pages=16–28| s2cid=260498692 }}</ref>

Mid-tread quantization involves rounding. The formulas for mid-tread uniform quantization are provided in the previous section.

:<math>Q(x) = \Delta \cdot  \left\lfloor \frac{x}{\Delta} + \frac{1}{2} \right\rfloor</math>,

Mid-riser quantization involves truncation. The input-output formula for a mid-riser uniform quantizer is given by:
:<math>Q(x) = \Delta\cdot\left(\left\lfloor \frac{x}{\Delta}\right\rfloor + \frac1{2}\right)</math>,
where the classification rule is given by
:<math>k = \left\lfloor \frac{x}{\Delta} \right\rfloor</math>
and the reconstruction rule is
:<math>y_k = \Delta\cdot\left(k+\tfrac1{2}\right)</math>.

Note that mid-riser uniform quantizers do not have a zero output value – their minimum output magnitude is half the step size. In contrast, mid-tread quantizers do have a zero output level. For some applications, having a zero output signal representation may be a necessity.

In general, a mid-riser or mid-tread quantizer may not actually be a ''uniform'' quantizer – i.e., the size of the quantizer's classification [[interval (mathematics)|intervals]] may not all be the same, or the spacing between its possible output values may not all be the same. The distinguishing characteristic of a mid-riser quantizer is that it has a classification threshold value that is exactly zero, and the distinguishing characteristic of a mid-tread quantizer is that is it has a reconstruction value that is exactly zero.<ref name=Gersho77/>