Representation Of Signals
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CHAPTER 1
Representation of Signals 1.1
Introduction
1.1
1.2
Periodic Signals and Fourier Series
1.2
1.2.1 Periodic signals
1.2
1.2.2 Fourier series
1.4
1.2.3 Convergence of Fourier series and Gibbs phenomenon
1.19
Aperiodic Signals and Fourier Transform
1.22
1.3.1 Fourier transform
1.24
1.3.2 Dirichlet conditions
1.29
1.4
Properties of the Fourier Transform
1.31
1.5
Unified Approach to Fourier Transform
1.49
1.5.1 Unit impulse (Dirac delta function)
1.49
1.5.2 Impulse response and convolution
1.60
1.5.3 Signum function and unit step function
1.69
Correlation Functions
1.78
1.6.1 Cross-correlation functions
1.78
1.6.2 Autocorrelation function
1.85
Hilbert Transform
1.95
1.7.1 Properties of the Hilbert transform
1.100
Bandpass Signals
1.104
1.8.1 Pre-envelope
1.107
1.8.2 Complex envelope
1.111
Bandpass (BP) Systems
1.119
Appendix A1.1: Tabulation of sinc ( λ )
1.122
Appendix A1.2: Fourier transform of Rx p y p ( τ )
1.124
Appendix A1.3: Complex envelope of the output of a BP System
1.127
References
1.130
1.3
1.6
1.7
1.8
1.9
CHAPTER 2
PROBABILITY AND RANDOM VARIABLES 2.1
Introduction
2.1
2.2
Basics of Probability
2.3
2.2.1 Terminology
2.3
2.2.2 Probability of an event
2.5
Random Variables
2.13
2.3.1 Distribution function
2.13
2.3.2 Probability density function
2.17
2.3.3 Joint distribution and density function
2.19
2.3.4 Conditional density
2.22
2.3.5 Statistical independence
2.23
Transformation of Variables
2.27
2.4.1 Functions of one random variable
2.28
2.4.2 Functions of two random variables
2.37
Statistical Averages
2.48
2.5.1 Variance
2.50
2.5.2 Covariance
2.53
Some Useful Probability Models
2.59
2.6.1 Discrete random variables
2.59
2.6.2 Continuous random variables
2.63
2.3
2.4
2.5
2.6
Appendix A2.1: Proof of Eq. 2.36
2.75
Appendix A2.2: Q (
2.78
)
Function Table
(
Appendix A2.3: Proof that N m X , σ2X
References
) is a valid PDF
2.80 2.81
CHAPTER 3
Random Signals and Noise 3.1
Introduction
3.1
3.2
Definition of a Random Process
3.3
3.3
Stationarity
3.8
3.4
Ensemble Averages
3.13
3.4.1 Properties of ACF
3.16
3.4.2 Cross-correlation
3.26
3.5
Systems with Random Signal Excitation
3.28
3.6
Power Spectral Density
3.30
3.6.1 Properties of Power Spectral Density
3.31
3.7
Cross Spectral Density
3.37
3.8
Gaussian Process
3.41
3.9
Electrical Noise
3.46
3.9.1 White noise
3.50
3.10 Narrowband Noise
3.61
3.10.1 Representation of narrowband noise
3.64
3.10.2 Properties of narrowband noise
3.66
3.10.3 PDF of the envelope of narrowband noise
3.72
3.10.4 Sine wave plus narrowband noise
3.74
Appendix A3.1 Properties of Narrowband Noise: Some proofs
3.78
References
3.82
CHAPTER 4
LINEAR MODULATION 4.1
Introduction
4.1
4.2
DSB-SC Modulation
4.5
4.2.1 Modulation
4.5
4.2.2 Coherent Demodulation
4.14
4.2.3 Carrier recovery for coherent demodulation
4.25
DSB-LC modulation (or AM)
4.29
4.3.1 Tone modulation
4.32
4.3.2 Spectra of AM Signals
4.36
Generation of AM and DSB-SC signals
4.41
4.4.1 Generation of AM
4.42
4.4.2 Generation of DSB-SC
4.47
4.5
Envelope Detector
4.52
4.6
Theory of Single Sideband
4.57
4.7
Generation of SSB Signals
4.64
4.7.1 Frequency discrimination method
4.64
4.7.2 Phase discrimination method
4.68
4.8
Demodulation of SSB
4.71
4.9
Vestigial SideBand (VSB) Modulation
4.80
4.9.1 Frequency domain description of VSB
4.80
4.9.2 Time domain description of VSB
4.86
4.10
Envelope Detection of VSB+C
4.92
4.11
Superheterodyne Receiver
4.97
4.3
4.4
Appendix A4.1: Analysis of ED with Tone Modulation
4.106
References
4.111
CHAPTER 5
Angle Modulation 5.1
Introduction
5.1
5.2
Bandwidth of FM
5.13
5.2.1 NarrowBand FM (NBFM)
5.14
5.2.2 WideBand FM (WBFM)
5.15
Tone Modulation
5.19
5.3.1 NBFM
5.19
5.3.2 WBFM
5.21
5.4
Phase Modulation
5.30
5.5
Generation of FM
5.36
5.5.1 Narrowband FM
5.37
5.5.2 WBFM: Indirect and direct methods
5.37
Demodulation of FM
5.49
5.6.1 FM-to-AM conversion
5.49
5.6.2 Phase shift discriminator
5.57
5.6.3 Zero-crossing detection
5.61
5.6.4 FM demodulation using PLL
5.63
5.7
Bandpass Limiter (BPL)
5.68
5.8
Broadcast FM
5.71
5.8.1 Monophonic FM reception
5.71
5.8.2 Two-channel (stereo) FM
5.73
Appendix A5.1: Table of Bessel Functions
5.76
Appendix A5.2: Phase Shift Discriminator
5.77
5.3
5.6
A5.2.1: Foster-Seely discriminator
5.77
A5.2.2: The ratio detector
5.82
Appendix A5.3: Multitone FM
5.85
Appendix A5.4: RMS Bandwidth of WBFM
5.86
Appendix A5.5: Modulation techniques in TV
5.91
References
5.97
CHAPTER 6
DIGITAL TRANSMISSION OF ANALOG SIGNALS: PCM, DPCM AND DM 6.1
Introduction
6.1
6.2
The PCM system
6.2
6.3
Sampling
6.4
6.3.1 Ideal impulse sampling
6.4
6.3.2 Sampling with a rectangular pulse train
6.9
6.3.3 Flat topped sampling
6.11
6.3.4 Undersampling and the problem of aliasing
6.13
Quantization
6.26
6.4.1 Uniform quantization
6.26
6.4.2 Quantization noise
6.31
6.4.3 Non-uniform quantization and companding
6.41
6.5
Encoding
6.56
6.6
Electrical waveform representation of binary sequences
6.59
6.7
Bandwidth requirements of PCM
6.61
6.7.1 Unipolar format
6.62
6.7.2 Polar format
6.63
6.7.3 Bipolar format
6.64
6.7.4 Manchester format
6.65
6.8
Differential Pulse Code Modulation (DPCM)
6.70
6.9
Delta Modulation
6.80
6.9.1 Linear Delta Modulation (LDM)
6.81
6.9.2 Adaptive delta modulation
6.92
6.4
Appendix A6.1: PSD of a waveform process with discrete amplitudes 6.97 References
6.99
CHAPTER 7
Noise Performance of Various Modulation Schemes 7.1
Introduction
7.1
7.2
Receiver Model and Figure of Merit: Linear Modulation
7.2
7.2.1 Receiver model
7.2
7.2.2 Figure-of-merit
7.3
Coherent Demodulation
7.8
7.3.1 DSB-SC
7.8
7.3.2 SSB
7.11
Envelope Detection
7.12
7.4.1 Large predetection SNR
7.13
7.4.2 Weak predetection SNR
7.14
7.5
Receiver Model: Angle Modulation
7.21
7.6
Calculation of FOM
7.22
7.6.1 Large predetection SNR
7.22
7.6.2 Weak predetection SNR
7.33
7.7
Pre-Emphasis and de-Emphasis in FM
7.40
7.8
Noise Performance of a PCM System
7.49
7.3
7.4
Appendix A7.1: PSD of noise for angle modulated signals
7.59
References
7.63
Representation of Signals 1.1
Introduction
1.1
1.2
Periodic Signals and Fourier Series
1.2
1.2.1 Periodic signals
1.2
1.2.2 Fourier series
1.4
1.2.3 Convergence of Fourier series and Gibbs phenomenon
1.19
Aperiodic Signals and Fourier Transform
1.22
1.3.1 Fourier transform
1.24
1.3.2 Dirichlet conditions
1.29
1.4
Properties of the Fourier Transform
1.31
1.5
Unified Approach to Fourier Transform
1.49
1.5.1 Unit impulse (Dirac delta function)
1.49
1.5.2 Impulse response and convolution
1.60
1.5.3 Signum function and unit step function
1.69
Correlation Functions
1.78
1.6.1 Cross-correlation functions
1.78
1.6.2 Autocorrelation function
1.85
Hilbert Transform
1.95
1.7.1 Properties of the Hilbert transform
1.100
Bandpass Signals
1.104
1.8.1 Pre-envelope
1.107
1.8.2 Complex envelope
1.111
Bandpass (BP) Systems
1.119
Appendix A1.1: Tabulation of sinc ( λ )
1.122
Appendix A1.2: Fourier transform of Rx p y p ( τ )
1.124
Appendix A1.3: Complex envelope of the output of a BP System
1.127
References
1.130
1.3
1.6
1.7
1.8
1.9
CHAPTER 2
PROBABILITY AND RANDOM VARIABLES 2.1
Introduction
2.1
2.2
Basics of Probability
2.3
2.2.1 Terminology
2.3
2.2.2 Probability of an event
2.5
Random Variables
2.13
2.3.1 Distribution function
2.13
2.3.2 Probability density function
2.17
2.3.3 Joint distribution and density function
2.19
2.3.4 Conditional density
2.22
2.3.5 Statistical independence
2.23
Transformation of Variables
2.27
2.4.1 Functions of one random variable
2.28
2.4.2 Functions of two random variables
2.37
Statistical Averages
2.48
2.5.1 Variance
2.50
2.5.2 Covariance
2.53
Some Useful Probability Models
2.59
2.6.1 Discrete random variables
2.59
2.6.2 Continuous random variables
2.63
2.3
2.4
2.5
2.6
Appendix A2.1: Proof of Eq. 2.36
2.75
Appendix A2.2: Q (
2.78
)
Function Table
(
Appendix A2.3: Proof that N m X , σ2X
References
) is a valid PDF
2.80 2.81
CHAPTER 3
Random Signals and Noise 3.1
Introduction
3.1
3.2
Definition of a Random Process
3.3
3.3
Stationarity
3.8
3.4
Ensemble Averages
3.13
3.4.1 Properties of ACF
3.16
3.4.2 Cross-correlation
3.26
3.5
Systems with Random Signal Excitation
3.28
3.6
Power Spectral Density
3.30
3.6.1 Properties of Power Spectral Density
3.31
3.7
Cross Spectral Density
3.37
3.8
Gaussian Process
3.41
3.9
Electrical Noise
3.46
3.9.1 White noise
3.50
3.10 Narrowband Noise
3.61
3.10.1 Representation of narrowband noise
3.64
3.10.2 Properties of narrowband noise
3.66
3.10.3 PDF of the envelope of narrowband noise
3.72
3.10.4 Sine wave plus narrowband noise
3.74
Appendix A3.1 Properties of Narrowband Noise: Some proofs
3.78
References
3.82
CHAPTER 4
LINEAR MODULATION 4.1
Introduction
4.1
4.2
DSB-SC Modulation
4.5
4.2.1 Modulation
4.5
4.2.2 Coherent Demodulation
4.14
4.2.3 Carrier recovery for coherent demodulation
4.25
DSB-LC modulation (or AM)
4.29
4.3.1 Tone modulation
4.32
4.3.2 Spectra of AM Signals
4.36
Generation of AM and DSB-SC signals
4.41
4.4.1 Generation of AM
4.42
4.4.2 Generation of DSB-SC
4.47
4.5
Envelope Detector
4.52
4.6
Theory of Single Sideband
4.57
4.7
Generation of SSB Signals
4.64
4.7.1 Frequency discrimination method
4.64
4.7.2 Phase discrimination method
4.68
4.8
Demodulation of SSB
4.71
4.9
Vestigial SideBand (VSB) Modulation
4.80
4.9.1 Frequency domain description of VSB
4.80
4.9.2 Time domain description of VSB
4.86
4.10
Envelope Detection of VSB+C
4.92
4.11
Superheterodyne Receiver
4.97
4.3
4.4
Appendix A4.1: Analysis of ED with Tone Modulation
4.106
References
4.111
CHAPTER 5
Angle Modulation 5.1
Introduction
5.1
5.2
Bandwidth of FM
5.13
5.2.1 NarrowBand FM (NBFM)
5.14
5.2.2 WideBand FM (WBFM)
5.15
Tone Modulation
5.19
5.3.1 NBFM
5.19
5.3.2 WBFM
5.21
5.4
Phase Modulation
5.30
5.5
Generation of FM
5.36
5.5.1 Narrowband FM
5.37
5.5.2 WBFM: Indirect and direct methods
5.37
Demodulation of FM
5.49
5.6.1 FM-to-AM conversion
5.49
5.6.2 Phase shift discriminator
5.57
5.6.3 Zero-crossing detection
5.61
5.6.4 FM demodulation using PLL
5.63
5.7
Bandpass Limiter (BPL)
5.68
5.8
Broadcast FM
5.71
5.8.1 Monophonic FM reception
5.71
5.8.2 Two-channel (stereo) FM
5.73
Appendix A5.1: Table of Bessel Functions
5.76
Appendix A5.2: Phase Shift Discriminator
5.77
5.3
5.6
A5.2.1: Foster-Seely discriminator
5.77
A5.2.2: The ratio detector
5.82
Appendix A5.3: Multitone FM
5.85
Appendix A5.4: RMS Bandwidth of WBFM
5.86
Appendix A5.5: Modulation techniques in TV
5.91
References
5.97
CHAPTER 6
DIGITAL TRANSMISSION OF ANALOG SIGNALS: PCM, DPCM AND DM 6.1
Introduction
6.1
6.2
The PCM system
6.2
6.3
Sampling
6.4
6.3.1 Ideal impulse sampling
6.4
6.3.2 Sampling with a rectangular pulse train
6.9
6.3.3 Flat topped sampling
6.11
6.3.4 Undersampling and the problem of aliasing
6.13
Quantization
6.26
6.4.1 Uniform quantization
6.26
6.4.2 Quantization noise
6.31
6.4.3 Non-uniform quantization and companding
6.41
6.5
Encoding
6.56
6.6
Electrical waveform representation of binary sequences
6.59
6.7
Bandwidth requirements of PCM
6.61
6.7.1 Unipolar format
6.62
6.7.2 Polar format
6.63
6.7.3 Bipolar format
6.64
6.7.4 Manchester format
6.65
6.8
Differential Pulse Code Modulation (DPCM)
6.70
6.9
Delta Modulation
6.80
6.9.1 Linear Delta Modulation (LDM)
6.81
6.9.2 Adaptive delta modulation
6.92
6.4
Appendix A6.1: PSD of a waveform process with discrete amplitudes 6.97 References
6.99
CHAPTER 7
Noise Performance of Various Modulation Schemes 7.1
Introduction
7.1
7.2
Receiver Model and Figure of Merit: Linear Modulation
7.2
7.2.1 Receiver model
7.2
7.2.2 Figure-of-merit
7.3
Coherent Demodulation
7.8
7.3.1 DSB-SC
7.8
7.3.2 SSB
7.11
Envelope Detection
7.12
7.4.1 Large predetection SNR
7.13
7.4.2 Weak predetection SNR
7.14
7.5
Receiver Model: Angle Modulation
7.21
7.6
Calculation of FOM
7.22
7.6.1 Large predetection SNR
7.22
7.6.2 Weak predetection SNR
7.33
7.7
Pre-Emphasis and de-Emphasis in FM
7.40
7.8
Noise Performance of a PCM System
7.49
7.3
7.4
Appendix A7.1: PSD of noise for angle modulated signals
7.59
References
7.63
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