Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Some Essential Notation
- 2 Signals, Integrals, and Sets of Measure Zero
- 3 The Inner Product
- 4 The Space L2 of Energy-Limited Signals
- 5 Convolutions and Filters
- 6 The Frequency Response of Filters and Bandlimited Signals
- 7 Passband Signals and Their Representation
- 8 Complete Orthonormal Systems and the Sampling Theorem
- 9 Sampling Real Passband Signals
- 10 Mapping Bits to Waveforms
- 11 Nyquist's Criterion
- 12 Stochastic Processes: Definition
- 13 Stationary Discrete-Time Stochastic Processes
- 14 Energy and Power in PAM
- 15 Operational Power Spectral Density
- 16 Quadrature Amplitude Modulation
- 17 Complex Random Variables and Processes
- 18 Energy, Power, and PSD in QAM
- 19 The Univariate Gaussian Distribution
- 20 Binary Hypothesis Testing
- 21 Multi-Hypothesis Testing
- 22 Sufficient Statistics
- 23 The Multivariate Gaussian Distribution
- 24 Complex Gaussians and Circular Symmetry
- 25 Continuous-Time Stochastic Processes
- 26 Detection in White Gaussian Noise
- 27 Noncoherent Detection and Nuisance Parameters
- 28 Detecting PAM and QAM Signals in White Gaussian Noise
- 29 Linear Binary Block Codes with Antipodal Signaling
- A On the Fourier Series
- Bibliography
- Theorems Referenced by Name
- Abbreviations
- List of Symbols
- Index
7 - Passband Signals and Their Representation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Some Essential Notation
- 2 Signals, Integrals, and Sets of Measure Zero
- 3 The Inner Product
- 4 The Space L2 of Energy-Limited Signals
- 5 Convolutions and Filters
- 6 The Frequency Response of Filters and Bandlimited Signals
- 7 Passband Signals and Their Representation
- 8 Complete Orthonormal Systems and the Sampling Theorem
- 9 Sampling Real Passband Signals
- 10 Mapping Bits to Waveforms
- 11 Nyquist's Criterion
- 12 Stochastic Processes: Definition
- 13 Stationary Discrete-Time Stochastic Processes
- 14 Energy and Power in PAM
- 15 Operational Power Spectral Density
- 16 Quadrature Amplitude Modulation
- 17 Complex Random Variables and Processes
- 18 Energy, Power, and PSD in QAM
- 19 The Univariate Gaussian Distribution
- 20 Binary Hypothesis Testing
- 21 Multi-Hypothesis Testing
- 22 Sufficient Statistics
- 23 The Multivariate Gaussian Distribution
- 24 Complex Gaussians and Circular Symmetry
- 25 Continuous-Time Stochastic Processes
- 26 Detection in White Gaussian Noise
- 27 Noncoherent Detection and Nuisance Parameters
- 28 Detecting PAM and QAM Signals in White Gaussian Noise
- 29 Linear Binary Block Codes with Antipodal Signaling
- A On the Fourier Series
- Bibliography
- Theorems Referenced by Name
- Abbreviations
- List of Symbols
- Index
Summary
Introduction
The signals encountered in wireless communications are typically real passband signals. In this chapter we shall define such signals and define their bandwidth around a carrier frequency. We shall then explain how such signals can be represented using their complex baseband representation. We shall emphasize two relationships: that between the energy in the passband signal and in its baseband representation, and that between the bandwidth of the passband signal around the carrier frequency and the bandwidth of its baseband representation. We ask the reader to pay special attention to the fact that only real passband signals have a baseband representation.
Most of the chapter deals with the family of integrable passband signals. As we shall see in Corollary 7.2.4, an integrable passband signal must have finite energy, and this family is thus a subset of the family of energy-limited passband signals. Restricting ourselves to integrable signals—while reducing the generality of some of the results—simplifies the exposition because we can discuss the Fourier Transform without having to resort to the L2-Fourier Transform, which requires all statements to be phrased in terms of equivalence classes. But most of the derived results will also hold for the more general family of energy-limited passband signals with only slight modifications. The required modifications are discussed in Section 7.7.
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- A Foundation in Digital Communication , pp. 101 - 142Publisher: Cambridge University PressPrint publication year: 2009