DigitalComm_ Baroz Farang - spring 2010

DigitalComm_ Baroz Farang - spring 2010 - Digital...

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Unformatted text preview: Digital Communications Behrouz Farhang-Boroujeny Department of Electrical and Computer Engineering University of Utah 2010, Behrouz Farhang-Boroujeny, ECE Department, University of Utah, MEB Room 3280, Salt Lake City, UT 84112, USA Contents Contents 1 Introduction 1 1.1 The Big Picture . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A Brief History of Modems . . . . . . . . . . . . . . . . . . 4 1.3 Signal Processing in Modems . . . . . . . . . . . . . . . . . 5 2 Continuous-Time Signals and Systems 9 2.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . 18 2.3.1 Convolution integral . . . . . . . . . . . . . . . . . . 18 2.3.2 Transfer function . . . . . . . . . . . . . . . . . . . . 19 2.4 Autocorrelation Function and Power Spectral Density . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Energy-type signals . . . . . . . . . . . . . . . . . . 21 2.4.2 Power-type signals . . . . . . . . . . . . . . . . . . . 21 2.4.3 Passing a signal through an LTI system . . . . . . . 22 2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Sampling Theorem, DFT, and Discrete-Time Signals and Systems 25 3.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Reconstruction of x ( t ) from the samples x ( nT s ) . . . 26 3.1.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.3 Antialiasing filter . . . . . . . . . . . . . . . . . . . . 28 3.1.4 Sampling in the Frequency Domain . . . . . . . . . . 28 3.2 Numerical Computation of the Fourier Transform: Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . 29 3.2.1 Derivation of DFT . . . . . . . . . . . . . . . . . . . 30 3.2.2 Properties of DFT . . . . . . . . . . . . . . . . . . . 32 3.2.3 Fast Fourier transform (FFT) . . . . . . . . . . . . . 33 3.2.4 Time and frequency scales . . . . . . . . . . . . . . . 34 i Contents ii 3.2.5 Improving the frequency resolution of the spectrum via zero padding . . . . . . . . . . . . . . . 36 3.3 Discrete-Time Signals and Systems . . . . . . . . . . . . . . 36 3.3.1 The z-transform and Fourier transform of discrete-time signals . . . . . . . . . . . . . . . . . . 36 3.3.2 Precautionary notes . . . . . . . . . . . . . . . . . . 38 3.4 Autocorrelation Function and Power Spectral Density . . . . . . . . . . . . . . . . . . . . 40 4 Random Variables and Random Processes 43 4.1 Some Useful Random Variables . . . . . . . . . . . . . . . . 43 4.1.1 The Bernouli random variable . . . . . . . . . . . . . 43 4.1.2 The binomial random variable . . . . . . . . . . . . 44 4.1.3 The uniform random variable . . . . . . . . . . . . . 44 4.1.4 The Gaussian (normal) random variable . . . . . . . 44 4.1.5 The Rayleigh random variable . . . . . . . . . . . . 46 4.1.6 The Ricean random variable . . . . . . . . . . . . . . 46 4.2 The Central Limit Theorem . . . . . . . . . . . . . . . . . .4....
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This note was uploaded on 02/24/2011 for the course ECE 5520 taught by Professor Shamir,g during the Spring '08 term at University of Utah.

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DigitalComm_ Baroz Farang - spring 2010 - Digital...

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