signal processing technques for software radio

Author:

Behrouz Farhang-Boroujeny

Department of Electrical and Computer Engineering

University of Utah

c 2007, Behrouz Farhang-Boroujeny, ECE Department, University of Utah,

USA

Contents

Contents

1 Introduction 1

2 Fourier Analysis and

Linear Time-Invariant Systems 3

2.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Convolution integral . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Energy and Power Spectral Density . . . . . . . . . . . . . . . . . . . 15

2.4.1 Energy-type signals . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Power-type signals . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.3 Random signals . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.4 Passing a signal through an LTI system . . . . . . . . . . . . 19

3 Digital Transmission Systems 21

3.1 Pulse Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Pulse-Shape Designs for Band-Limited

Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Raised-cosine filter . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Matched filtering and square-root raised-cosine filter . . . . . 27

3.2.3 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Modulation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Carrier-amplitude modulation . . . . . . . . . . . . . . . . . . 32

3.3.2 Quadrature amplitude modulation . . . . . . . . . . . . . . . 35

3.3.3 Carrier-phase modulation . . . . . . . . . . . . . . . . . . . . 39

3.4 Binary to Symbol Mapping . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Differential Encoding and Decoding . . . . . . . . . . . . . . . . . . 43

3.6 Baseband Equivalent of a Passband Channel . . . . . . . . . . . . . 43

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ii Contents

4 Sampling and Discrete Time Systems 53

4.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Reconstruction of x(t) from the samples x(nTs) . . . . . . . . 54

4.1.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.3 Antialiasing filter . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.4 Nyquist criterion for intersymbol interference free

communication . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.5 Sampling in the Frequency Domain . . . . . . . . . . . . . . . 56

4.2 Numerical Computation of the Fourier Transform: Discrete Fourier

Transform (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Derivation of DFT . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.2 Properties of DFT . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.3 Fast Fourier transform (FFT) . . . . . . . . . . . . . . . . . . 62

4.2.4 Time and frequency scales . . . . . . . . . . . . . . . . . . . . 62

4.2.5 Improving the frequency resolution of the spectrum via zero

padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Discrete-Time Signals and Systems . . . . . . . . . . . . . . . . . . . 63

4.3.1 The z-transform and Fourier transform of discrete-time

signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Energy and power spectral density . . . . . . . . . . . . . . . 67

4.3.3 Passing a signal through an LTI system . . . . . . . . . . . . 70

4.3.4 Precautionary notes . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.1 Filter specifications . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.2 Filter design using windowing method . . . . . . . . . . . . . 73

4.4.3 Equiripple filters . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.4 Nyquist (M) and square-root Nyquist (M) filters . . . . . . . 83

5 Multirate Signal Processing 97

5.1 M-fold Decimator and L-fold Expander . . . . . . . . . . . . . . . . 97

5.1.1 M-fold decimator . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.1.2 L-fold expander . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Rate Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.1 L-fold interpolation . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.2 M-fold decimation . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.3 L/M-fold rate change . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Commutative Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4 The Polyphase Representations . . . . . . . . . . . . . . . . . . . . . 111

5.5 Efficient Structures for Decimation and

Interpolation Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5.1 Polyphase structure for decimator filterss . . . . . . . . . . . 113

5.5.2 Polyphase structure for interpolator filters . . . . . . . . . . . 116

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5.5.3 Commutator Models . . . . . . . . . . . . . . . . . . . . . . . 117

5.5.4 L/M-fold resampling . . . . . . . . . . . . . . . . . . . . . . . 118

5.5.5 The polyphase identity . . . . . . . . . . . . . . . . . . . . . . 121

5.6 Multistage Implementation . . . . . . . . . . . . . . . . . . . . . . . 122

5.6.1 Interpolated FIR (IFIR) technique . . . . . . . . . . . . . . . 122

5.6.2 Multistage realization of decimation and interpolation filters 127

5.7 Cascaded Integrator-Comb Filters . . . . . . . . . . . . . . . . . . . 128

5.7.1 M-fold CIC interpolator . . . . . . . . . . . . . . . . . . . . . 128

5.7.2 M-fold CIC decimator . . . . . . . . . . . . . . . . . . . . . . 132

5.8 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.8.1 Timing recovery . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.8.2 All digital modulator . . . . . . . . . . . . . . . . . . . . . . . 137

5.8.3 All digital demodulator . . . . . . . . . . . . . . . . . . . . . 141

5.8.4 Parallel polyphase filtering for very fast sampling rates . . . . 143

6 An Overview of Transceiver Systems 151

6.1 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.2 MATLAB Simulation of Digital Transmission

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.3 Baseband PAM transceiver . . . . . . . . . . . . . . . . . . . . . . . 154

6.4 Eye Patterns in PAM Systems . . . . . . . . . . . . . . . . . . . . . 156

6.5 QAM Transceiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.6 Eye Patterns in QAM Systems . . . . . . . . . . . . . . . . . . . . . 159

6.7 The Impact of Frequency Offset on the Baseband Equivalent of Passband

Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7 Adaptive Systems 171

7.1 Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.1.1 The real-valued case . . . . . . . . . . . . . . . . . . . . . . . 173

7.1.2 Principle of orthogonality . . . . . . . . . . . . . . . . . . . . 178

7.1.3 Extension to the complex-valued case . . . . . . . . . . . . . 180

7.2 The LMS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.2.1 Range of μ, stability and misadjustment . . . . . . . . . . . . 185

7.2.2 The normalized LMS algorithm . . . . . . . . . . . . . . . . . 188

7.3 The Method of Least-Squares . . . . . . . . . . . . . . . . . . . . . . 188

7.3.1 Formulation of the Least-Squares Estimation . . . . . . . . . 189

7.3.2 The standard recursive least-squares algorithm . . . . . . . . 191

7.4 Sampling with Automatic Gain Control . . . . . . . . . . . . . . . . 198

8 Phase-Locked Loop 201

8.1 Continuous-Time PLL . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8.1.1 Linear model of PLL and its analysis . . . . . . . . . . . . . . 203

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8.2 Discrete-Time PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.2.1 Linear model of PLL and its analysis . . . . . . . . . . . . . . 213

8.2.2 Designing discrete-time PLL from continuous-time PLL . . . 217

8.3 Maximum Likelihood Phase Detection . . . . . . . . . . . . . . . . . 219

8.3.1 Cost function and the optimum phase . . . . . . . . . . . . . 219

8.3.2 The LMS algorithm for phase detection: a derivation of PLL 220

8.3.3 Alternative stochastic gradient . . . . . . . . . . . . . . . . . 222

8.3.4 A note on the step-size parameter μ . . . . . . . . . . . . . . 222

8.3.5 Higher order PLLs . . . . . . . . . . . . . . . . . . . . . . . . 224

8.4 A PLL with Extended Lock Range . . . . . . . . . . . . . . . . . . . 225

9 Carrier Acquisition and Tracking 231

9.1 Non-Data Aided Carrier Recovery Methods . . . . . . . . . . . . . . 232

9.1.1 Binary PSK with a rectangular pulse-shape . . . . . . . . . . 232

9.1.2 Binary PSK with a band-limited pulse-shape . . . . . . . . . 233

9.1.3 Quadrature Amplitude Modulation . . . . . . . . . . . . . . . 234

9.2 Non-Data Aided Carrier Acquisition and Tracking Algorithms . . . 240

9.2.1 Coarse carrier acquisition . . . . . . . . . . . . . . . . . . . . 240

9.2.2 Fine carrier acquisition and tracking . . . . . . . . . . . . . . 241

9.2.3 Costas Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

9.3 Pilot Aided Carrier Acquisition Method . . . . . . . . . . . . . . . . 255

9.4 Data Aided Carrier Tracking Method . . . . . . . . . . . . . . . . . . 257

10 Timing Recovery 271

10.1 Non-Data Aided Timing Recovery Methods . . . . . . . . . . . . . . 272

10.1.1 Fundamental results . . . . . . . . . . . . . . . . . . . . . . . 272

10.1.2 The timing recovery cost function . . . . . . . . . . . . . . . 274

10.1.3 The optimum timing phase . . . . . . . . . . . . . . . . . . . 274

10.1.4 Improving the cost function . . . . . . . . . . . . . . . . . . . 278

10.2 Non-Data Aided Timing Recovery Algorithms . . . . . . . . . . . . . 279

10.2.1 Early-late gate timing recovery . . . . . . . . . . . . . . . . . 280

10.2.2 Gradient-based algorithm . . . . . . . . . . . . . . . . . . . . 284

10.2.3 Tone extraction algorithm . . . . . . . . . . . . . . . . . . . . 285

10.3 Data Aided Timing Recovery Methods . . . . . . . . . . . . . . . . . 288

10.3.1 Muller and Muller’s method . . . . . . . . . . . . . . . . . . . 290

10.3.2 Decision directed method . . . . . . . . . . . . . . . . . . . . 292

11 Channel Equalization 301

11.1 Continuous-Time Channel Model . . . . . . . . . . . . . . . . . . . . 301

11.2 Discrete-Time Channel Model . . . . . . . . . . . . . . . . . . . . . . 303

11.2.1 Symbol-spaced equalizer . . . . . . . . . . . . . . . . . . . . . 303

11.2.2 Fractionally-spaced equalizer . . . . . . . . . . . . . . . . . . 303

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11.2.3 Symbol-spaced vesus fractionally-spaced equalizer . . . . . . 304

11.3 Performance Study of Equalizers . . . . . . . . . . . . . . . . . . . . 305

11.3.1 Wiener-Hopf equations . . . . . . . . . . . . . . . . . . . . . . 305

11.3.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 311

11.4 Adaptation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 311

11.5 Cyclic Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311