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Unformatted text preview: Chapter 3 Quantization 3.1 Introduction to quantization The previous chapter discussed coding and decoding for discrete sources. Discrete sources are a subject of interest in their own right (for text, computer files, etc.) and also serve as the inner layer for encoding analog source sequences and waveform sources (see Figure 3.1). This chapter treats coding and decoding for a sequence of analog values. Source coding for analog values is usually called quantization . Note that this is also the middle layer for waveform encoding/decoding. waveform input sampler quantizer discrete encoder analog sequence symbol sequence reliable binary channel waveform output analog filter table lookup discrete decoder Figure 3.1: Encoding and decoding of discrete sources, analog sequence sources, and waveform sources. Quantization, the topic of this chapter, is the middle layer and should be understood before trying to understand the outer layer, which deals with waveform sources. The input to the quantizer will be modeled as a sequence U 1 , U 2 , , of analog random variables ··· (rv’s). The motivation for this is much the same as that for modeling the input to a discrete source encoder as a sequence of random symbols. That is, the design of a quantizer should be responsive to the set of possible inputs rather than being designed for only a single sequence of 63 Cite as: Robert Gallager, course materials for 6.450 Principles of Digital Communications I, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 64 CHAPTER 3. QUANTIZATION numerical inputs. Also, it is desirable to treat very rare inputs differently from very common inputs, and a probability density is an ideal approach for this. Initially, U 1 , U 2 , . . . will be taken as independent identically distributed (iid) analog rv’s with some given probability density function (pdf) f U ( u ). A quantizer, by definition, maps the incoming sequence U 1 , U 2 , , into a sequence of discrete ··· rv’s V 1 , V 2 , , where the objective is that V , for each m in the sequence, should represent U ··· m m with as little distortion as possible. Assuming that the discrete encoder/decoder at the inner layer of Figure 3.1 is uniquely decodable, the sequence V 1 , V 2 , will appear at the output of ··· the discrete encoder and will be passed through the middle layer (denoted ‘table lookup’) to represent the input U 1 , U 2 , . The output side of the quantizer layer is called a ‘table lookup’ ··· because the alphabet for each discrete random variables V m is a finite set of real numbers, and these are usually mapped into another set of symbols such as the integers 1 to M for an M symbol alphabet. Thus on the output side a look-up function is required to convert back to the numerical value V m ....
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