Digital signal processing. From theory to applications

ISBN 978-83-206-1640-8
Author: Tomasz P. Zieliński
Publisher: WKŁ

About the book

The book has been made accessible in an accessible way
from the mathematical basis of analogue signal theory
for modern applications of signal analysis and processing
digital. The necessary mathematical considerations are illustrated
numerous computational examples, drawings and computer programs, written in Matlab language.
In addition to classic themes, such as analogue filtration
and digital and continuous and discrete Fourier transforms are described
also more advanced issues: adaptive filtration,
recursive estimation and modern methods of analysis
frequency and time-frequency signals,
including wavelet transformation and filter assemblies. Also given
basics: coding and recognition of speech signal, compression
MP3 audio signal, image analysis and processing as well as digital multi-tone modulation applied, among others in fast phone ADSL modems and in local wireless Wi-Fi computer networks.
The book is an academic textbook. In the author's intention, each of the chapters is a closed whole, suitable for separate reading, which is why some of the presented material will be repeated to a small extent, but usually in a slightly different form.

The book is addressed to senior researchers
universities, PhD students, students exploring the secrets of digital signal processing and practicing engineers interested in their own development.


Preface
The purpose of this book is to provide a comprehensive overview of the basics of digital signal processing. Their knowledge is extremely important in times when there is a strong preference for solutions based on digital and not analogue technology. This trend has been clearly observed for many years and is the result of the increasing availability, also price, of highly efficient digital circuits (microprocessors and microcontrollers, memory and peripheral devices such as analog-digital and digital-analog converters) and the advantages of digital over analog processing ( the unchanged time of data processing, caused by the lack of dependence on aging and changing the properties of electronic components). Therefore, more and more often electronic design is observed, where the analogue signals are converted to digital as early as possible and the processing algorithm is completely implemented in the form of arithmetic operations on numbers representing the instantaneous values of the sampled analog signals. This scenario is common everywhere. An example can be various control and supervision systems: industrial, military, medical. Additionally, in the multimedia era, digital signals of speech, music (audio), images and their sequences (video, television) are widely processed and analyzed.
Wherever there is a processor that processes digital measurement data we deal with digital signal processing. It turns out, however, that regardless of the source of these signals, the basic methods of their processing and analysis are identical or very similar.
Why is this happening? Because in each case we look at the signal as a function variable in time or space, and we use known, generally available tools for mathematical analysis of these functions, i.e. we use, for example, the eighteenth-century Fourier transform or just a dozen years of wavelet transformation for frequency analysis of signals.
According to the author, one can not talk about the processing of digital signals without referring to the theory (analysis and processing) of analog signals, or the theory of continuous functions. The analysis and processing of digital data are inextricably linked to the analysis and processing of analog signals. Most often digital data is a sampled version ("copy") of analog data and their analysis is to give us information not about "copy" but about "original." The tools used in both cases and their properties are mutually permeating each other. from analog filters, the discrete Fourier transform implemented on computers (used, for example, in the currently very popular mp3 music compression standard) arose from the "integral" Fourier series, and the transformation Z has a similar role in the world of linear, discrete time-unchanging discrete circuits like Laplace's transformation in the world of analog circuits.
For this reason, this book will be a journey across many continents. It will include: elements of the theory of analog signals and electrical circuits (chapters 1-6), basic (chapters 7-13) and more advanced (chapters 14-18) of digital signal analysis and processing methods and their selected, interesting applications (chapters 19 - 23). The author will always be guided by the primary goal to show the connections and interpenetration of the "analog" and "digital" world. At the same time, the emphasis will be placed not on the existing "mnemonic designs", giving the engineer a ready-made prescription, how to live "today, but closing the way for further development tomorrow, only for a careful mathematical explanation of the issues under consideration, which will allow further, independent, conscious movement of the reader in the fields new to him.
The first explanation will always be as simple as possible. The main intention of the author is to "disenchant" seemingly difficult subjects and "throwing bridges" between seemingly distant banks.
There is nothing new in this book. Everything was already there. In large part, it consists of simple derivations and convincing explanations that have been extracted from hundreds of books and articles - sea of words - and carefully remembered. So why did she write it? The author regretfully states that he has been discovering some simple truths for many years. And just like during climbing in the mountains, after each "approach" a new view was revealed to him. This trek has been going on for over twenty years. And for sure the summit is still very far away. But maybe it is worth showing the way to "shortcuts", try to match the elements of the "puzzle" and the synthesis of your own thoughts.
The book is an academic textbook. In the author's intention, each of the chapters is a closed whole, suitable for separate reading, which is why some of the presented material will be repeated to a small extent, but usually in a slightly different form.
...
The author hopes that although he managed to achieve his ambitious goals in a small part. Therefore, he presents the modest result of his work with embarrassment and humbleness.
Cracow, September 2005 Tomasz P. Zieliński

Table of Contents:

Preface
List of markings
List of shortcuts
1. Signals and their parameters 1
1.1. Basic concepts 1
1.2. Classification of signals 2
1.3. Constitutional signals 4
1.3.1. Parameters 4
1.3.2. Examples 7
1.3.3. Complex signals 13
1.3.4. Distribution of signals into components 14
1.3.5. The function of self and mutual correlation 14
1.3.6. Weave of signals 17
1.3.7. Fourier transformation 22
1.4. Random signals 24
1.4.1. Random variables 24
1.4.2. Random processes, stationary, ergodicity 26
1.4.3. Correlation and covariance functions, spectral density
28th power
1.4.4. Estimators of parameters and functions 30
1.4.5. Filtration of random signals 34
1.5. An example of a computer exercise 35
2. Mathematical basis of signal analysis
deterministic 39
2.1. Spaces of deterministic signals 39
2.2. Discrete representations of continuous signals
deterministic 41
2.3. Continuous representations of continuous deterministic signals
- integral transformations 47
2.4. Representations of discrete signals - spaces
vector 50
2.5. An example of a computer exercise 60
3. Fourier series 63
3.1. Orthogonal base functions 63
3.2. Harmonics complex basic functions 65
3.3. Harmonical real base functions 66
3.4. Calculation example 67
3.5. An example of a computer exercise 68
3.6. Fourier series of discrete signals - discrete
Fourier conversion 71
4. Integral Fourier transform 74
4.1. Definition 74
4.2. Basic properties 75
4.3. Fourier transforms of selected signals 79
4.4. Spectrum of the product and convolution of two signals 87
4.5. The sampling theorem
4.6. Spectrum of the sampled signal 97
4.7. An example of a computer exercise 101
5. Analogue systems 103
5.1. Analog LTI 103 systems
5.2. Transmission of analog circuit, zero and poles 107
5.3. Laplace transform, Laplace transfer function 112
5.4. Choke 116 charts
5.5. Complex LTI analog systems 118
5.6. Mathematical analysis of selected systems
electric 120
5.7. Design examples 124
5.8. An example of a computer exercise 129
6. Butterworth and Czebyszew analog filters 131
6.1. General rules for designing analog filters 132
6.2. Frequency transformation 139
6.3. Butterworth filters 146
6.4. Chebyshev filters type I 157
6.5. Chebyshev type II filters 161
6.6. Hardware implementation of analog filters 165
7. Discretization of analog signals 173
7.1. Basics 173
7.2. Analog-to-digital converters 179
7.3. Digital to analogue converters 184
7.4. The analogue-digital processing path
and digital-analog 185
8. Frequency analysis of discrete signals 192
8.1. Fourier spectrum of discrete signals 192
8.1.1. Fourier transform for continuous signals 193
8.1.2. Fourier series for continuous signals 193
8.1.3. Fourier transform for discrete signals 194
8.1.4. Fourier series for discrete signals, i.e. discrete
Fourier transform 198
8.2. Examples of discrete Fourier transforms
signals 202
8.3. Interpretation of the discrete Fourier transform 206
8.4. Signal processing during analysis
frequency 210
8.5. Discrete time windows 212
8.5.1. Nonparametric windows 212
8.5.2. Parametric windows 217
8.6. Examples of frequency analysis using
window functions 220
8.7. Fast determination of autocorrelation function and density function
spectral power 226
9. Algorithms for determining discrete transformation
Fourier 231
9.1. Direct method 231
9.2. Goertzel's algorithm 234
9.3. Recursive determination of discrete sequences
Fourier transform 236
9.4. Snake transformation - a magnifier in the field
239 frequencies
9.5. Fast Fourier transform - radix-2 241 algorithms
9.5.1. Division in the field of time - DIT ( Decimation
in Time ) 241
9.5.2. Division in the frequency domain - DIF ( Decimation
in Frequency ) 252
9.6. Fast Fourier transform for signals
real 255
9.7. Two-dimensional discrete Fourier transform 257
9.8. Determination of DCT by rapid transformation
Fourier 258
10. Silent systems 260
10.1. Discrete LTI 260 systems
10.2. The algorithm of filtration of signals by means of discrete
LTI 265 systems
10.3. Transformation Z 267
10.4. Reverse transformation Z 270
10.5. Transformation properties Z 274
10.6. Transmission of discrete systems 275
10.7. Examples of discrete circuit design
using the "zeros and poles" method 280
10.8. Example of computer exercise 284
11. Designing recursive digital filters 288
11.1. Requirements for digital filters 289
11.2. Yule-Walker method 291
11.3. Method of invariability of impulse response 291
11.4. The method of matching transformation Z 293
11.5. The Biline Transformation Method 293
11.6. Examples of filter design in Matlab 297
11.7. Example of computer exercise 304
12. Designing non-recursive digital filters 307
12.1. Introduction 308
12.2. Sampling method in the frequency domain 313
12.3. Root-square optimization method 317
12.4. Chebyshev approximation method (algorithm
Remez) 321
12.5. Window method 325
12.6. Special filters 339
12.6.1. Hilbert filter 339
12.6.2. Differential filter 345
12.6.3. Interpolator and digital decimeter filter 347
12.6.4. Example of computer exercise 351
12.7. Synchronization of input and output samples
353 filter
13. Digital filtration algorithms 356
13.1. Classic structure of digital filters 356
13.2. Structure of state variables 361
13.3. Other digital filter structures 363
13.4. Linear and circular weave 364
13.5. Algorithms of the quick detachment of discrete signals 371
13.6. The algorithms of the sectioned fast splice of signals
discreet 373
13.7. An example of computer exercise 376
14. Adaptive filters 379
14.1. Introduction 379
14.2. Basics of adaptive filtration 380
14.3. Optimal filtration - Wiener filter 382
14.4. Gradient adaptive filters 384
14.5. LSM Adaptive Filters - without 386 memory
14.6. LS adaptive filters (RLS) - filters with 388 memory
14.7. Application examples 391
14.8. Example of a computer exercise - adaptive filter
(N) LMS 394
15. Linear recursive estimation 399
15.1. Least squares method. RLS filters
and WRLS 399
15.2. The minimum-average square method. Kalman filter 408
16. Advanced frequency analysis methods
signals 420
16.1. Introduction 420
16.2. Parametric modeling AR, MA and ARMA 423
16.2.1. Basics 423
16.2.2. Model AR 426
16.2.3. Model MA 427
16.2.4. Model ARMA 429
16.2.5. Summary 430
16.3. Subspace methods 430
16.3.1. Basics 430
16.3.2. The Pisaren 432 method
16.3.3. Derived methods: MUSIC, EV and MV 435
16.3.4. The ESPRIT 437 method
16.3.5. Subspace subspace methods (components
major) 439
16.4. Example of computer exercise 440
17. Time-frequency analysis methods
signals 443
17.1. The problem of time-frequency analysis 444
17.2. Transformation of Gabor 450
17.3. Short-term Fourier transform STFT 455
17.4. Wavelet transformation 459
17.5. Wigner-Ville transformation 472
17.6. Time-frequency representations from the class
Cohen 477
17.7. Examples of applications 486
17.8. An example of computer exercise 493
18. Filter assemblies 496
18.1. Introduction 496
18.2. Basic concepts 500
18.2.1. Decimator and interpolator 500
18.2.2. Polyphase decomposition of 503 signals
18.2.3. Decimator and interpolator in polyphase record 506
18.3. Mathematical description of the 507 filter assembly
18.3.1. Analysis of one branch 507
18.3.2. Analysis of all branches 511
18.3.3. Polyphase record of filter assembly 512
18.3.4. The condition of perfect reconstruction 514
18.4. Filter assemblies with complex modulation 515
18.4.1. DFT as a modulated filter set 516
18.4.2. Short-term Fourier STFT transformation
as a modulated filter set 518
18.4.3. Generalized modulated filter assembly
based on DFT 519
18.5. Filter assemblies with cosine modulation 527
18.5.1. Equations, construction 527
18.5.2. Designing of 533 prototype filters
18.6. Software implementation of the filter assembly
MPEG audio 539 standard
19. LPC-10 project: the basics of compression and recognition
speech signal 545
19.1. Introduction 545
19.2. Speech signal generation model 549
19.3. The decision-making system "speech voiced / unvoiced" 551
19.4. Designation of the voice string filter 557
19.5. Encoder and speech decoder algorithm of LPC-10 563 standard
19.6. An example of a computer program 566
19.7. From coding to speech recognition 569
20. LPC-10 project: speech signal compression - methods
advanced 577
20.1. The Durbin-Levinson method 577
20.2. Lattice filters 581
20.3. An exemplary computer program 590
21. MPEG AUDIO project: psychoacoustic compression
sound 592
21.1. Introducing the 593 MPEG audio standard
21.2. Basics of psychoacoustic modeling 594
21.3. Psychoacoustic models of the MPEG audio 603 standard
21.3.1. Psychoacoustic model I 603
21.3.2. Psychoacoustic model II 604
21.3.3. Computer program 612
21.4. Filter sets in the MPEG audio 618 standard
21.5. Audio coding at MP1 and MP2 631 levels
21.5.1. Compression and decompression algorithm 631
21.5.2. Computer program 638
22. PICTURE project: basic analysis and processing
two-dimensional signals 647
22.1. Introduction to the world of 2D and 3D 649
22.2. 2D orthogonal transformations of images 658
22.2.1. Discrete Fourier transform 658
22.2.2. Discrete cosine transformation 663
22.2.3. Any orthogonal transformation - interpretation
coefficients 665
22.3.4. Computer program 668
22.3. 2D image filtering 670
22.3.1. Weave 2D 670
22.3.2. Designing 2D 674 filters
22.3.3. Examples of 2D 683 filters
22.3.4. Computer program 686
22.4. Wave decomposition of 2D images 690
22.4.1. One-dimensional predictive transformation
wavelet 691
22.4.2. Relationships between classical and predictive t
wave wavelet 697
22.4.3. A computer program for wavelet decomposition
700 images
22.5. Application examples 707
22.5.1. JPEG and MPEG 707 compression
22.5.2. Watermarks in 715 images
22.5.3. Matching digital images to each other 718
22.5.4. Line detection in materials engineering - transformation
Hough 730
22.2.5. Algorithmic image stabilization in applications
Medical 733
22.5.6. Navigation systems supporting treatments
Medical 737
23. MODEM ADSL project: fast Internet access
on the 740 telephone line
23.1. Fundamentals of modulation 741
23.2. Digital multi-tone modulations 745
23.3. Standard ADSL 748
23.4. Modulator-demodulator DMT 751
23.5 Distortion and interference sources 754
23.6 Selected implementation issues 759
23.6.1. Identification of pulse response of channel 759
23.6.2. Time correction of the channel - shortening the duration
impulse response 764
23.6.3. Block Synchronization 767
23.6.4. Frequency correction of channel 769
23.6.5. Bit rate estimation 770
23.6.6. The right choice of the time corrector 773
23.7. An example of computer exercise 773
24. PHASE project: estimation of the temporary shift
phase 778
24.1. Simple estimators 778
24.2. Complex estimators 781
24.3. Examples of algorithms 782
24.4. An exemplary computer program 786
25. EPILOGUE: implementation of DSP algorithms
on 787 signal processors
25.1 Introduction to construction and programming
DSP 788 processors
25.2. Weave signals on the DSP 791 processor
25.3. Selected implementation issues 796
25.3.1. The specifics of the construction and applications of processors
signal 796
25.3.2. Basics of writing and running 800 programs
25.3.3. Advanced tools 803
25.3.4. Example of IIR 805 filter design
25.4. An example application of DSP 807 processor
25.5. DSP processors and FPGA 808 programmable devices
25.6. Future - are we trendy? 810
Literature 813
Additions 823
D.1. List of programs 823
D.2. The electronic version of programs 824
Index 825