Digital Signa
197
Part A | 9.1
9. Digital Signal Processing
Nikolaos I. Xiros
In this chapter the concept of discrete-time signals
and sampled-data systems implemented on digital
hardware versus those of continuous-time signals
driving analog systems and processes is introduced
early on. The processes of signal sampling and analog signal reconstruction are then investigated,
and the Nyquist sampling rate to avoid aliasing
is explained by means of Fourier series analysis.
Then the Z-transform is introduced as the tool of
preference for analysis and synthesis of discretetime linear, time-invariant systems defined by
difference equations in the discrete time domain.
A detailed account of the most important and practical continuous-time system mapping techniques
to discrete-time ones is then presented. A brief account of digital filter structures and types is also
given along with a presentation of the fast Fourier
transform algorithm for the calculation of the discrete Fourier transform of discrete-time signals
with finite duration admitting periodic expansion.
The notions of waveform statistics as encountered
in random signals and stochastic processes are
then given in order to conclude with the effect
on a random signal’s statistics due to its propagation through a linear, time-invariant system.
Finally, concepts of optimal signal estimation and
the Wiener filter are presented, leading to matched
filter and zero-forcing equalizer parametric designs
in the discrete-time domain.
9.1 Discrete-Time Systems ........................... 197
9.1.1 Discrete-Time Signals
and Digital Systems ...................... 197
9.1.2 Signal Sampling ........................... 198
9.1.3 Analog Signal Reconstruction
Using a Discrete-Time Signal ......... 201
9.1.4 The Z-Transform .......................... 201
9.1.5 Discrete-Time LTI Systems.............. 202
9.1.6 Continuous-Time
System Mapping .......................... 203
9.2 Digital Filters ........................................ 206
9.2.1 Important FIR Filter Structures....... 206
9.2.2 Important IIR Filter Structures ....... 208
9.3 The Fast Fourier Transform (FFT)............. 211
9.3.1 Review of Integral Transforms ....... 211
9.3.2 The Discrete
Fourier Transform (DFT) ................. 212
9.4 Waveform Analysis ................................ 216
9.4.1 Definitions for Waveforms
and Random Signals .................... 216
9.4.2 Signal Power
and Power Spectral Density........... 218
9.4.3 Waveform Propagation
Through a Linear,
Time-Invariant System ................. 219
9.5 Optimal Signal Estimation ..................... 220
9.5.1 System Identification ................... 220
9.5.2 Discrete-Time Wiener–Hopf
Equation over a Finite-Duration
Window ...................................... 221
9.5.3 Signal Estimation
and the Wiener Filter ................... 222
9.6 Concluding Remarks.............................. 225
References................................................... 225
9.1 Discrete-Time Systems
The progress in integrated circuit manufacturing (Very
Large Scale Integration (VLSI)) combined with the
need for high complexity and accuracy signal analysis
and processing algorithms without development cost or
time increase are the basic factors that contributed to
the continuously increasing use of digital systems in industry and technology [9.1, 2].
9.1.1 Discrete-Time Signals
and Digital Systems
A digital processing system is a system where the
basic mathematical operations and functions, including signal transformations and data series analysis, are
implemented as an embedded program onboard real-
197
Part A | 9.1
9. Digital Signal Processing
Nikolaos I. Xiros
In this chapter the concept of discrete-time signals
and sampled-data systems implemented on digital
hardware versus those of continuous-time signals
driving analog systems and processes is introduced
early on. The processes of signal sampling and analog signal reconstruction are then investigated,
and the Nyquist sampling rate to avoid aliasing
is explained by means of Fourier series analysis.
Then the Z-transform is introduced as the tool of
preference for analysis and synthesis of discretetime linear, time-invariant systems defined by
difference equations in the discrete time domain.
A detailed account of the most important and practical continuous-time system mapping techniques
to discrete-time ones is then presented. A brief account of digital filter structures and types is also
given along with a presentation of the fast Fourier
transform algorithm for the calculation of the discrete Fourier transform of discrete-time signals
with finite duration admitting periodic expansion.
The notions of waveform statistics as encountered
in random signals and stochastic processes are
then given in order to conclude with the effect
on a random signal’s statistics due to its propagation through a linear, time-invariant system.
Finally, concepts of optimal signal estimation and
the Wiener filter are presented, leading to matched
filter and zero-forcing equalizer parametric designs
in the discrete-time domain.
9.1 Discrete-Time Systems ........................... 197
9.1.1 Discrete-Time Signals
and Digital Systems ...................... 197
9.1.2 Signal Sampling ........................... 198
9.1.3 Analog Signal Reconstruction
Using a Discrete-Time Signal ......... 201
9.1.4 The Z-Transform .......................... 201
9.1.5 Discrete-Time LTI Systems.............. 202
9.1.6 Continuous-Time
System Mapping .......................... 203
9.2 Digital Filters ........................................ 206
9.2.1 Important FIR Filter Structures....... 206
9.2.2 Important IIR Filter Structures ....... 208
9.3 The Fast Fourier Transform (FFT)............. 211
9.3.1 Review of Integral Transforms ....... 211
9.3.2 The Discrete
Fourier Transform (DFT) ................. 212
9.4 Waveform Analysis ................................ 216
9.4.1 Definitions for Waveforms
and Random Signals .................... 216
9.4.2 Signal Power
and Power Spectral Density........... 218
9.4.3 Waveform Propagation
Through a Linear,
Time-Invariant System ................. 219
9.5 Optimal Signal Estimation ..................... 220
9.5.1 System Identification ................... 220
9.5.2 Discrete-Time Wiener–Hopf
Equation over a Finite-Duration
Window ...................................... 221
9.5.3 Signal Estimation
and the Wiener Filter ................... 222
9.6 Concluding Remarks.............................. 225
References................................................... 225
9.1 Discrete-Time Systems
The progress in integrated circuit manufacturing (Very
Large Scale Integration (VLSI)) combined with the
need for high complexity and accuracy signal analysis
and processing algorithms without development cost or
time increase are the basic factors that contributed to
the continuously increasing use of digital systems in industry and technology [9.1, 2].
9.1.1 Discrete-Time Signals
and Digital Systems
A digital processing system is a system where the
basic mathematical operations and functions, including signal transformations and data series analysis, are
implemented as an embedded program onboard real-