Contents

1 Definition

Quantization is the process of mapping a continuous range of values into a finite range of discrete values. A quantizer, usually denoted by $Q$, is a system that:

1. In the analog case (see the Figure 1), a PAM signal $s(nT)$ (i.e. a sequence of analog samples where $s$ is an (analog) signal, $n\in \mathbb{Z}$ and $T$ is the sampling period) is transformed into a PCM signal $\mathbf{s}=s[n=0,1,\cdots ]$. Therefore, analog quantization is the process of mapping a continuous range of values (not necessarily countable) into a finite range of discrete values (necessarily countable) [4].

   Notice that, in general, analog signals are 1-dimensional, and that analog quantization is irreversible.

2. In the digital case, where $\mathbf{s}$ is already a PCM signal (a sequence of integers of finite precision), the output of a quantizer is a sequence of quantization indexes $\mathbf{k}$, and the inverse system, called a dequantizer or inverse quantizer (denoted by $Q^{-1}$), can only recovers an approximated version of $\mathbf{s}$ that it will be denoted by $\tilde{\mathbf{s}}$.

   If we define the cardinality operator $|\cdot |$ applied to a digital signal as the number of different values that such signal can take, i.e., the size of the signal alphabet, it ususally holds that $$\begin{array}{}\text{(1)}& |\mathbf{s}|\ge |\tilde{\mathbf{s}}|=|\mathbf{k}|.\end{array}$$ As a consequence of this property, the values that $\mathbf{k}$, individually, can take will require less bits to be represented than the values that the original signal $\mathbf{s}$ can.

   Finally, note that in the case of digital quantization, $Q^{-1}$ is only a formal notation and does not correspond to the reciprocal function of $Q$ since quantization is noninvertible [1] (Quantization is not linear transform).

2 Quantization error

Quantization is a lossy process that usually generates a distortion between $\mathbf{s}$ and $\tilde{\mathbf{s}}$. The quantization error in unpredectible (for this reason is also called quantization noise), and therefore, we cannot recover $\mathbf{s}$ after quantization.

We define the quantization error $$\begin{array}{}\text{(2)}& \mathbf{e}=\mathbf{s}-\tilde{\mathbf{s}},\end{array}$$ an the distortion, generally measured as the Mean Squared Error determined by $$\begin{array}{}\text{(3)}& \text{MSE}(\mathbf{s},\tilde{\mathbf{s}})=E((\mathbf{s}-\tilde{\mathbf{s}}{)}^2),\end{array}$$ where $E(\cdot )$ denotes the expectation operator.

3 Quantizer design

$Q$ should be designed to minimize the MSE, and this depends on how the mapping between the input signal $\mathbf{s}$ and the quantized signal $\tilde{\mathbf{s}}$ has been performed.

$Q$ is defined by a finite set of decision boundaries $\{{\mathbf{d}}_i;i\in \mathbb{Z}\}$ that describe a set of cells¹ in the signal domain, and a finite set of representation values $\{{\mathbf{r}}_i;i\in \mathbb{Z}\}$ (see Figure 1), both with the same cardinality (because there is one representation level per cell). The set $\mathbf{r}$ is called the codebook and to their elements, ${\mathbf{r}}_i$, codewords.

Given a finite number $K$ of cells, to minimize the MSE, $\mathbf{d}$ and $\mathbf{r}$ are selected depending on the characteristics of $\mathbf{s}$. In general, we need to consider the statistical distribution of the samples (or vectors of samples, depending on the way we process the input) in $\mathbf{s}$.

Notice that $K$ has an impact on the output bit-rate of the quantizer and therefore, we could be interested in minimizing the RD (Rate/Distortion) tradeoff instead of simply the MSE. However, such problem in general is addressed by using an entropy codec at the output of $Q$.

4 Scalar quantization

Scalar quantizers map each source sample independently from the other samples and therefore, a quantization index ${\mathbf{k}}_i$ is produced for each input sample ${\mathbf{s}}_i$ [2].

5 Vector quantization

In vector quantization, several source samples are quantized simultaneously and a single index is associated to a vector of source samples. Vector quantization allows to account for the correlation between source samples directly at the quantizer, which improves its RD efficiency [1, 3].

6 Uniform quantization

In uniform quantizers, the size of the cells is constant. For example, in a scalar quantizer, the quantization step size $\mathrm{\Delta }$ is constant and independent of the input signal.

Uniform quantizers are used in most A/D (analogic/digital) converters, where it is expected the generation of uniformely distributed sequences of samples. In the case of scalar uniform digital quantizers, it is common to have $r_i=i$ and the input intervals are of the form $]d_{i-1},d_i]=]i-1/2,i+1/2]$.

7 Non-uniform quantization

In order to minimize the MSE, the cells defined by the quantizer cound be of different size depending on the PDF of $\mathbf{s}$. When all the cells do not have the same size we have designed a non-uniform quantizer.

8 Resources