book_6 - Chapter 6 Channels modulation and demodulation 6.1...

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Unformatted text preview: Chapter 6 Channels, modulation, and demodulation 6.1 Introduction Digital modulation (or channel encoding) is the process of converting an input sequence of bits into a waveform suitable for transmission over a communication channel. Demodulation (channel decoding) is the corresponding process at the receiver of converting the received waveform into a (perhaps noisy) replica of the input bit sequence. Chapter 1 discussed the reasons for using a bit sequence as the interface between an arbitrary source and an arbitrary channel, and Chapters 2 and 3 discussed how to encode the source output into a bit sequence. Chapters 4 and 5 developed the signal-space view of waveforms. As explained there, the source and channel waveforms of interest can be represented as real or complex 1 L 2 vectors. Any such vector can be viewed as a conventional function of time, x ( t ). Given an orthonormal basis { φ 1 ( t ) , φ 2 ( t ) , . . . , } of L 2 , any such x ( t ) can be represented as x ( t ) = x j φ j ( t ) . (6.1) j Each x j in (6.1) can be uniquely calculated from x ( t ), and the above series converges in L 2 to x ( t ). Moreover, starting from any sequence satisfying j | x j | 2 < ∞ there is an L 2 function x ( t ) satisfying (6.1) with L 2 convergence. This provides a simple and generic way of going back and forth between functions of time and sequences of numbers. The basic parts of a modulator will then turn out to be a procedure for mapping a sequence of binary digits into a sequence of real or complex numbers, followed by the above approach for mapping a sequence of numbers into a waveform. In most cases of modulation, the set of waveforms φ 1 ( t ) , φ 2 ( t ) , . . . , in (6.1) will be chosen not as a basis for L 2 but as a basis for some subspace 2 of L 2 such as the set of functions that are baseband limited to some frequency W or passband limited to some range of frequencies. In some cases, it will also be desirable to use a sequence of waveforms that are not orthonormal. 1 As explained later, the actual transmitted waveforms are real. However, they are usually bandpass real waveforms that are conveniently represented as complex baseband waveforms. 2 Equivalently, φ 1 ( t ) , φ 2 ( t ) , . . . , can be chosen as a basis of L 2 but the set of indices for which x j is allowed to be nonzero can be restricted. 167 Cite as: Robert Gallager, course materials for 6.450 Principles of Digital Communications I, Fall 2006. MIT OpenCourseWare (, Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 168 CHAPTER 6. CHANNELS, MODULATION, AND DEMODULATION We can view the mapping from bits to numerical signals and the conversion of signals to a waveform as separate layers. The demodulator then maps the received waveform to a sequence of received signals, which is then mapped to a bit sequence, hopefully equal to the input bit sequence. A major objective in designing the modulator and demodulator is to maximize the rate at which bits enter the encoder, subject to the need to retrieve the original bit stream with...
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This note was uploaded on 02/21/2012 for the course ECON 830 taught by Professor Staff during the Spring '08 term at Michigan State University.

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book_6 - Chapter 6 Channels modulation and demodulation 6.1...

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