Front Matter and Index (Mertins - Signal Analysis (Revised Edition)), страница 2

. . . . . .6.6.2 Paraunitary Case . . . . . . . . . . . . . . . . . . . . . .6.6.3OversampledCosine-ModulatedFilterBanks.....6.6.4 Pseudo-QMFBanks . . . . . . . . . . . . . . . . . . . ..................6.7LappedOrthogonalTransforms113114116124124127130133133134141143144. 144. 147148148. 149150. 151. 155. 158. 159160. 162. 164. 164. 166.

168168170. 174. 175179. 183184. 186sform...v111Contents6.8 SubbandCoding of Images . . . . . . . . . . . . . . . . . . . .1886.9 Processing of Finite-Length Signals . . . . . . . . . . . . . . . . 1896.10 Transmultiplexers . . . . . . . . . . . . . . . . . . .

. . . . . . .1957 Short-Time Fourier Analysis7.1 Continuous-Time Signals . . . . . . . . . . . . . . . . . . . . . .7.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . .7.1.2 Time-FrequencyResolution . . . . . . . . . . . . . .7.1.3 TheUncertainty Principle . . . . . . . . . . . . . . .7.1.4 TheSpectrogram . . . . .

. . . . . . . . . . . . . . . . .7.1.5 Reconstruction . . . . . . . . . . . . . . . . . . . . . . .7.1.6 Reconstruction via Series Expansion . . . . . . . . .7.2 Discrete-Time Signals . . . . . . . . . . . . . . . . . . . . . . .7.3 Spectral Subtraction based on the STFT . . . . . . . . . .

.....196196196. 198. 200201202. 204205. 2078 Wavelet8.1The Continuous-Time Wavelet Transform . . . . . . . . . . .8.2 Wavelets for Time-ScaleAnalysis . . . . . . . . . . . . . . . .8.3IntegralandSemi-DiscreteReconstruction...........8.3.1 IntegralReconstruction.................8.3.2 Semi-DiscreteDyadic Wavelets . . . . . . .

. . . . . .8.4 WaveletSeries...........................8.4.1 DyadicSampling . . . . . . . . . . . . . . . . . . . . . .8.4.2 Better FrequencyResolution - Decompositionof Octaves . . . . . . . . . . . . . . . . . . . . . . . . . .8.5 The Discrete Wavelet Transform (DWT) . . . . . . . . . . . .8.5.1 Multiresolution Analysis . . . . . . .

. . . . . . . . . .8.5.2WaveletAnalysis by MultirateFiltering . . . . . . . .8.5.3 Wavelet Synthesis by Multirate Filtering . . . . . . . .8.5.4 The RelationshipbetweenFiltersandWavelets........................8.6 WaveletsfromFilterBanks8.6.1 GeneralProcedure . . . . . . . .

. . . . . . . . . . . . .8.6.2 Requirements to be Metby the Coefficients . . . . . .8.6.3 Partition of Unity . . . . . . . . . . . . . . . . . . . . .8.6.4 The Norm of Constructed Scaling Functionsand Wavelets . . . . . . . . . . . . . . . . . . . . . .

. .8.6.5 Moments . . . . . . . . . . . . . . . . . . . . . . . . . .8.6.6 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . .8.6.7 WaveletswithFinite Support . . . . . . . . . . . . . .8.7 WaveletFamilies . . . . . . . . . . . . . . . . . .

. . . . . . . .8.7.1 Design of BiorthogonalLinear-PhaseWavelets....210. 210. 214. 217. 217. 219223223225. 227. 227. 232. 233. 234.237237241241242243244. 245247. 247Contentsix8.7.2 TheOrthonormal DaubechiesWavelets........8.7.3 Coiflets . . . . . . . .

. . . . . . . . . . . . . . . . . . .8.8 The Wavelet Transform of Discrete-Time Signals . . . . . . .8.8.1 The A TrousAlgorithm . . . . . . . . . . . . . . . . .8.8.2 The Relationship between the Mallat and A TrousAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . .8.8.3 The Discrete-Time MorletWavelet . . . . . .

. . . . .8.9DWT-BasedImageCompression . . . . . . . . . . . . . . . ......................8.10 Wavelet-BasedDenoising. 2529 Non-LinearTime-FrequencyDistributions9.1The Ambiguity Function . . . . . . . . . . . . . . . . . . . . . .9.2 The WignerDistribution . . . . . . . . . . . . . . . . . . . . . .9.2.1 Definition andProperties . . . . .

. . . . . . . . . . .9.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . .9.2.3 Cross-Terms and Cross WignerDistributions . . . . .9.2.4 LinearOperations.....................9.3General Time-Frequency Distributions . . . . . . . . . . . . ......9.3.1Shift-InvariantTime-FrequencyDistributions9.3.2 Examples of Shift-Invariant Time-Frequency Distributions . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .9.3.3 Affine-Invariant Time-FrequencyDistributions....9.3.4 Discrete-Time Calculation of Time-Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.4 The Wigner-Ville Spectrum . . . . .

. . . . . . . . . . . . . . .265265269. 269274. 275279. 280. 281BibliographyIndex253. 255. 256259. 260. 261263283. 289290292299311PrefaceA central goal in signal analysis is to extractinformation from signals that arerelated to real-world phenomena. Examplesare theanalysis of speech, images,and signals in medical or geophysical applications. One reason for analyzingsuch signals is to achieve better understanding of the underlying physicalphenomena.Another is to find compactrepresentations of signals whichallow compact storage or efficient transmission of signals through real-worldenvironments.

The methods of analyzing signals are wide spread and rangefrom classical Fourier analysis to various types of linear time-frequency transforms and model-basedand non-linear approaches. This book concentrates ontransforms, but also gives a brief introduction to linear estimation theory andrelated signal analysis methods. The text is self-contained for readers withsome background in system theory and digital signal processing, as typicallygained in undergraduate courses in electrical and computer engineering.The first five chapters of this book cover the classical concepts of signalrepresentation, including integral and discrete transforms.

Chapter 1 containsanintroduction to signals and signal spaces. It explains the basic toolsforclassifying signals and describing theirproperties.Chapter2 gives anintroduction to integral signal representation.ExamplesaretheFourier,Hartley and Hilbert transforms. Chapter 3 discusses the concepts and toolsfor discrete signal representation. Examples of discrete transforms are givenin Chapter 4.

Some of the latter are studiedcomprehensively,while others areonly briefly introduced, to a level required in the later chapters. Chapter 5 isdedicated to the processing of stochastic processes using discrete transformsand model-based approaches. It explains the Karhunen-Lobve transform andthe whitening transform, gives an introduction to linear estimation theoryand optimal filtering, and discusses methods of estimating autocorrelationsequences and power spectra.The final four chapters of this book are dedicated to transforms thatprovide time-frequency signal representations. In Chapter 6, multirate filterbanks are considered.

They form the discrete-time variant of time-frequencytransforms. The chapter gives an introduction to the field and provides anoverview of filter design methods. The classical method of time-frequencyanalysis is the short-time Fourier transform, which is discussed in Chapter 7.This transform was introduced by Gabor in 1946 and is used in many applications, especially in the form of spectrograms. The most prominent exampleof linear transforms with time-frequencylocalization is the wavelet transform.This transform attracts researchers from almost any field of science, becausexixiiContentsit has many useful features: a time-frequency resolution that is matched tomany real-world phenomena, a multiscale representation, and a very efficientimplementationbasedonmultiratefilter banks.Chapter 8 discusses thecontinuous wavelet transform, thediscrete wavelet transform, and thewavelettransform of discrete-time signals.

Finally, Chapter 9is dedicated to quadratictime-frequency analysis tools like the Wigner distribution, the distributionsof Cohen’s class, and the Wigner-Ville spectrum.Thehistory of this book is relatively long. Itstarted in 1992whenI produced the first lecturenotes for courses on signal theoryand lineartime-frequency analysis at the Hamburg University of Technology, Germany.Parts of the material were included in a thesis (“Habilitationsschrift”) that Isubmitted in 1994.

In 1996, the text was published as a textbook on SignalTheory in German. This book appearedin a series on InformationTechnology,edited by Prof. Norbert J. Fliege and published by B.G. Teubner, Stuttgart,Germany. It was Professor Fliege who encouraged me to write the book, and Iwould like to thank him for that and for his support throughout many years.The present book is mainly a translation of the original German. However, Ihave rearranged some parts, expanded some of the chapters, and shortenedothers in order to obtain a more homogeneous andself-contained text. Duringthe various stages, from thefirst lecture notes,over the German manuscript tothe present book, many people helpedme by proofreading and commenting onthe text. Marcus Benthin, Georg Dickmann, Frank Filbir, Sabine Hohmann,Martin Schonle, Frank Seide, Ursula Seifert, and Jens Wohlers read portionsof the German manuscript. Their feedbacksignificantly enhanced the qualityof the manuscript.

My sister,IngeMertins-Obbelode,translated the textfrom German into English and also proofread the new material that was notincluded in the German book. Tanja Karp and JorgKliewer went through thechapters on filter banks and wavelets, respectively, in the English manuscriptand made many helpful suggestions. Ian Burnett went through a completedraft of the present text and made many suggestions that helped to improvethe presentation.

I wouldlike to thank them all. Without their effort andenthusiasm this project would not have been realizable.Alfred MertinsWollongong, December 1998Signal Analysis: Waveless, Filter Banks, Time-Frequency Transforms andApplications, Alfred MertinsCopyright © 1999 John Wiley & Sons LtdPrint ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4IndexAutocorrelation method, 131Autocorrelation sequenceestimation, 133, 134for deterministic signals, 9for random processes, 16Autocovariance functionfor random processes, 13, 14for wide-sense stationary processes,14, 15Autocovariance matrix, 18Autocovariance sequencefor random processes, 16Autoregressive process, 109, 128relationshipbetweenKLT andDCT, 94relationshipbetweenKLT andDST, 97AC matrix (see Modulation matrix)Admissibility condition, 211, 215, 222Affine invariancetime-frequency distributions, 289wavelet transform, 214Alias component (AC) matrix(see Modulation matrix)Aliasing, 146All-polesourcemodel(see Autoregressive process)allpass filters, 173Ambiguity function, 265ff.cross, 269generalized, 281Amplitude distortion, 149Analysis window, 197, 216Analytic signal, 36ff., 214Analytic wavelet, 214AR( 1) process (see Autoregressive process)Arithmetic coding, 188A trous algorithm, 256ff.Autocorrelation functionfor analytic processes, 44for deterministic signals, 8for ergodic processes, 18for random processes, 13for wide-sensestationary processes,14temporal, 269-271time-frequency, 267Autocorrelation matrix, 17, 18Balian-Low theorem, 205Banach space, 4Bandlimited white noise process, 19Bandpassfiltering, 39signal, spectrum, 35, 293stationary process, 298stationary processes, 43Bandpassanalysisand wavelet transform, 210Bandwidth, 211Bartlett method, 137-139Bartlett window, 133-135, 138, 141Basis, 48Bessel inequality, 53311312Best linear unbiased estimator (BLUE),114, 123Bezout’s theorem, 248Bias (see Unbiasedness)Biorthogonal filter banks, 149cosine-modulated, 174Biorthogonal lapped transforms, 186Biorthogonal wavelets, 224, 247Biorthogonality condition, 58, 224, 227Bit reversed order, 88Blackman window, 140, 141Blackman-Tukey method, 138, 139Block transforms, 47BLUE, 114, 123Burg method, 142Butterfly graph, 87Cauchy integral theorem, 77Cauchy-Schwarz inequality (see Schwarzinequality)Center frequency, 211Characteristic function, 13Chebyshev polynomials, 54Chinese remainder theorem, 93Chirp signal, 274Choi-Williams distribution, 288-291Cholesky decomposition, 64, 112Circulant matrix, 84Circular convolution, 83, 93, 99, 190Cohen’s class, 281ff.Cohen-Daubechies-Feauveau wavelets,247Coiflets, 253Complete orthonormal sets, 52Completeness relation, 53Complex envelope, 36ff., 293, 298Conditional probability density, 12Consistency, 133, 134, 136, 137, 140Constant-Q analysis, 211Correlation, 8ff.Correlation matrix, 17, 18Cosine-and-sine function, 29Cosine-modulated filter banks, 165,174ff.biorthogonal, 174critically subsampled, 175Indexoversampled, 183paraunitary, 179Counter identity matrix, 187Covariance method, 132Critical subsampling, 143Cross ambiguity function, 269Cross correlation functionfor deterministic signals, 9for random processes, 16Cross correlation matrix, 17, 18Cross correlation sequencefor deterministic signals, 10for random processes, 17Cross covariance matrix, 18Cross covariance sequencefor random processes, 17Cross energy density spectrumcontinuous-time signals, 9discrete-time signals, 10Cross power spectral densityforcontinuous-timerandom processes, 16for discrete-time random processes,17Cross Wigner distribution, 275Cross-terms, 275Cumulative distribution function, 11Cyclo-stationary process, 14, 292, 293,297Daubechies wavelets, 252DCT (see Discrete cosine transform)Decimation, 144Decomposition relation, 232Denoising, 208, 263Density function, 11DFT (see Discrete Fourier transform)DFT filter banks, 165, 170ff.oversampled, 172polyphase, 172, 207DHT (see Discrete Hartley transform)Dirac impulse, 19, 23Direct sum of subspaces, 49Discrete cosine transform (DCT), 93-95Discrete Fourier transform(DFT), 8285, 97, 99, 137IndexDiscrete Hartley transform(DHT), 9799Discrete polynomial transformsChebyshev polynomials, 54Hermite polynomials, 55Laguerre polynomials, 54, 55Legendre polynomials, 53Discrete sine transform (DST), 96, 97Discrete transforms, 47ff.Discrete wavelet transform (DWT), 188,197, 227ff.Discrete-time Fourier transform, 80, 81Downsampling, 143, 144Dual wavelet, 223DWT (see Discrete wavelet transform)Dyadic sampling, 223, 224ELT (see Extendedlappedtransform)Embedded zerotree coding, 188, 262Energy density spectrumcomputation via Hartley transform,33for continuous-time signals, 8for discrete-time signals, 9, 10Energy signal, 1Ergodic processes, 18Estimation, 113best linear unbiased, 114-116least-squares, 113, 114minimummeansquareerror(MMSE), 116-123of autocorrelationsequences,133,134of power spectral densities, 134-142Euclidean algorithm, 162Euclidean metric, 4Euclidean norm, 2, 4Expected value, 12Extended lapped transform (ELT), 181Fast Fourier transformGood-Thomas FFT, 92FastFouriertransform(FFT), 85-88,90-93, 97, 99, 137313Cooley-Tukey FFT, 92in place computation, 88radix-2 decimation-in-frequency, 88radix-2 decimation-in-time, 85radix-4 decimation-in-frequency, 90split-radix FFT, 91Fast Hartley transform (FHT), 97, 99FFT (see Fast Fourier transform)FHT (see Fast Hartley transform)Filter banks, 143ff.M-channel, 164ff.basic multirate operations, 144cosine-modulated, 174ff.DFT, 170ff.FIR prototype, 156general FIR, 167lattice structures, 158, 159, 168modulated, 170ff.polyphase decomposition, 147ff.tree-structured, 162two-channel, 148ff.two-channel paraunitary, 155ff.two-dimensional, 188First-order Markov process, 109Fourier series expansion, 53Fourier transformfor continuous-time signals, 26ff.for discrete-time signals, 80, 81Frame bounds, 224-226Frequency resolution, 199, 213, 286Gabor expansion, 204Gabor transform, 197Gaussian function, 23, 201,215,216, 268,274Gaussian signal, 274Gaussian white noise process, 19Generalizedlappedorthogonaltransforms, 186GenLOT (see Generalized lapped orthogonal transforms)Givens rotations, 65, 73Gram-Schmidt procedure, 51Grammian matrix, 58, 64Group delay, 42Index314Holder Regularity, 245Haar wavelet, 229, 240Hadamard transform, 100Halfband filter, 156Hamming distance, 4Hamming window, 140, 141Hanning window, 140, 141Hard thresholding, 263Hartley transform, 29ff.Hartley transform, discrete (DHT), 9799Hermite functions, 56Hermite polynomials, 55Hermitian of a vector, 5Hilbert space, 7Hilbert transform, 34ff.Householder reflections, 70Householder reflections, 65, 69-72Huffman coding, 188Laguerre polynomials, 54Lapped orthogonal transforms, 186ff.Lattice structurefor linear-phase filter banks, 159for paraunitary filter banks, 158for M-channel filter banks, 168Least-squares estimation, 113Legendre polynomials, 53Levinson-Durbin recursion, 131Lifting, 160maximum-delay, 178zero-delay, 177Linear estimation (see Estimation)Linear optimal filters, 124-132Linear prediction, 126-130Linear subspace, 48Linear-phase wavelets, 247LOT (see Lapped orthogonal transforms)Low-delay filter banks, 174IIR filters, 173In-phase component, 38Inner product spaces, 4ff.Instantaneous frequency, 274, 280Integral transforms, 22ff.Interference term, 277Interpolation, 144Joint Probability Density, 12JPEG, 95Karhunen-LoBve transform (KLT), 84for continuous-time processes, 101103for discrete-time processes,103-108forreal-valuedAR( 1) processes,109Kernelof an integral transform, 22reciprocal, 22, 23self-reciprocal, 24KLT (see Karhunen-LoBve transform)Kronecker symbol, 49Lagrange halfband filter, 257Mallat algorithm, 259Markov process, first-order, 109Matched-filter condition, 156Maximum-delay lifting, 178MDFT filter bank, 172Metric, metric space, 3Minimum mean square error estimation(MMSE), 116-123MLT (see Modulated lapped transform)MMSE estimation (see Minimum meansquare error estimator)Modulated filter banks, 170ff.Modulated lapped transform, 181Modulation matrix, 152, 154Moments, 243, 244Moments of a random variable, 12Moore-Penrose pseudoinverse, 64, 66-68Morlet wavelet, 215discrete-time, 260Moyal’s formula forauto-Wigner distributions, 273cross-Wigner distributions, 276MPEG, 95, 174, 192MRA (see Multiresolution analysis)Index315Multiresolution analysis, 227Noise reduction, 208Non-stationary process, 13, 292Norm, normed space, 2, 3Normal equationsof linear prediction, 128of orthogonal projection, 63Nullspace, 68, 69Nyquist condition, 151, 294Octave-band analysis, 211, 225Optimal filters, 124-132Orthogonal projection, 50, 60Orthogonal series expansions, 49Orthogonal sum of subspaces, 49Orthogonality principle, 117Orthonormal basis, 49Orthonormal wavelets, 224Orthonormality, 49, 225Oversampledcosine-modulatedfilterbanks, 183Oversampled DFT filter banks, 172Paraconjugation, 79Parameterestimation(see Estimation)Paraunitary filter banksM-channel, 168two-channel, 155, 237Parseval’s relationfor discrete representations, 51, 53for the Fourier transform, 28general, 25Parseval’s theorem, 28, 81Partition of unity, 241pdf (see Probability density function)Perfectreconstruction,143,149,150,165, 167Periodogram, 135-140Phase delay, 42Pitch, 202Polyphase decomposition, 147ff.type-l, 147type-2, 147type-3, 148Polyphase matrixM-channel filter bank, 166two-channel filter bank, 154Power signal, 2Power spectral densityestimation of, 134-142forcontinuous-timerandomprocesses, 15for discrete-time random processes,16Power-complementary filters, 156P R (see Perfect reconstruction)P R conditiontwo-channel filter banks, 148ff.Pre-envelope, 38Prediction, 126-130Prediction error filter, 130Probability density function, 11Probability, axioms of, 11Product kernel, 289Pseudo-QMF banks, 184Pseudo-Wigner distribution, 279Pseudoinverse (see Moore-Penrose pseudoinverse)QCLS (see Quadratic-constrained leastsquares)QMF (see Quadrature mirror filters)QR decomposition, 64, 73Quadratic-constrained least-squares, 176Quadrature component, 38Quadrature mirror filters, 149Radar uncertainty principle, 268, 269Radix-2 FFT, 85-88, 90Radix-4 FFT, 90, 91Raised cosine filter, 294Random process, 1Off.cyclo-stationary, 14non-stationary, 13stationary, 13transmissionthroughlinearsystems, 20wide-sense stationary, 14Random variable, 10Reciprocal basis, 58, 59IndexReciprocal kernel, 22, 23Region of convergence (ROC), 76Regularity, 244, 245, 247, 252Representation, 48Rihaczek distribution, 288Roll-off factor.