## Communication TheoryThe properties of signals and transmission systems are discussed, and it is shown that the Fourier transform greatly simplifies the handling of commonly-occurring physical systems. Moreover, it explains the concepts of sampling and filtering. The description is then extended to unknown signals and noise. It is shown that known properties are well-described by probabilities. Further, probabilities define how to optimally decide under an observation, and lead to the concepts of communication source and channel. Building on these concepts, analogue signals are constructed such that a transmission of digital messages is possible. The Shannon measure of binary information is used to prove that a message source can be compressed up to its entropy, and that one may maximally transmit up to the channel capacity without error. The results are based on the identification of typical sets, i.e., precise sets of possibilities. Codes for the transmission of messages are described, and an efficient decoding method for good codes is presented. This decoder is based on residual sets, which are defined in a manner similar to typical sets. |

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### Contents

Introduction | 1 |

Signals and Systems | 8 |

Stochastic Signals and Maps | 33 |

Digital Transmission | 67 |

Information Theory | 87 |

Error Correction | 113 |

Advanced Topics | 143 |

TimeVariant Convolution | 156 |

TimeDiscrete Processes | 162 |

Digital Signalling | 180 |

Notation | 227 |

### Common terms and phrases

algorithm andthe arg max arg min assumed az(z bandwidth binary Bits bound ceE(c ceE(D CHEBYSHEV's inequality code E(C code word complexity computation considered convolution correlation decoding deﬁned definition delay depicted discrete FOURIER transform distribution doesnot elements entropy equivalent baseband error probability estimation Example Exercise exhibits ﬁlter ﬁrst follows forthe FOURIER series frequency GAUSSian density GAUSSian process HAMMING HAMMING distance Hence Hs(s hz(z implies impulse response inequality inverse Lemma length linear linear codes log2 matrix maximum mean value minimum ML decision Moreover multiplication mutual information noise Note observation obtains oo oo optimal parameters parity check polynomial prefix-free code Proof Ps(s px(x PZ(z random variable Reed-Solomon code Remark representation sampled seE(s shows signal s(t statement stationary process STC process stochastically independent symbol thecode thefollowing Theorem theprobability thesignal thestatement time-variant transmission trellis variance vector yields zero