Quantizers and Quantization Noise¶
A quantizer is a signal processing block, that maps a continuous amplitude to a discrete amplitude. The output of the quantizer is discrete, meaning that it can only output Q different values. Practically, the quantizer is an analog-to-digital converter, since it maps the continuous input amplitude to a digital representation of this value. Formally, the quantized output Q[x] of some input value x, is given by
Q[x] = \arg \min_{l\in\mathcal{S}} |l-x|.Here, the set \mathcal{S} contains all possible output values of the quantizer, which we name quantization levels. What does that mean? It means, that for a given input x, the quantizer returns the quantization level l, which is closest to the input value. Hence, a quantizer is completely defined by its set of quantization levels \mathcal{S}.
Let's assume the quantizer should be able to quantize values between the input amplitudes -U and +U, i.e. the peak-to-peak amplitude range equals 2U. Furthermore, the quantization levels should be encoded by b bits. This gives us a number of 2^q different quantization levels in \mathcal{S}. Then, a logical decision is to make the distance \Delta_s between the quantization levels constant, and equal to
\Delta_s = \frac{2U}{q}.Commonly, there are two different quantizers used, the ...