In this module we will talk about what is really the second half of the story in digital signal processing. The first half is the discretization of time, that is, sampling; we need sampling because computing devices can only process sequences of numbers and not continuous-time signals. But digital devices, which always store input values using a finite number of bits, can in fact only manipulate integer values.

Of course we have a lot of sophisticated ways to approximate real-valued quantities in a digital device using internal representations such as fixed point, floating point, double precision etc. But, ultimately, if variables are physically encoded by $$R$$ bits each, the number of possible distinct values for a sample cannot be larger than $$2^R$$.

In order to have a digital device perform signal processing on a real-world signal, after sampling the signal must be quantized , that is, each sample value must be mapped onto one of the possible values that the device can accept as input.

Quantization introduces an irreversible loss of information, which we want to keep as small as possible. Mathematically, it is convenient to model the quantization error as an additive source of noise and therefore the performance analysis of a quantizer will be analyzed using the stochastic signal processing results that we studied in DSP2 , module 2.3.

The easiest way to visualize what a quantizer does is to think about a simple, old fashioned thermometer: when we read the temperature, we are helped by a graduated scale in which each tick represents a degree or a fraction of a degree. While the height of the mercury column can be arbitrary, we look at the tick on the scale that is closest to the tip of the column and use that value as the temperature value, thereby discarding the micro-variations around the tick.

The open questions for a signal quantizers are

• how to best design the implicit "graduated scale"

• what is the error (that is, the loss of information) introduced by the quantization

We will only examine some simple cases but, in so doing, we will introduce the classic framework used in quantizer analysis: signals are considered stochastic and, under reasonable hypotheses, quantization noise can be modeled as an additive white noise source.