Digital Signal Analysis Techniques: Time, Frequency, and Spatial Algorithms

TECHNOLOGY FOCUS
Conventional tools for analyzing and extracting the feature content of signals are filters (time content), Fourier transforms (frequency content) and beamforming (spatial content). Alternative tools to these conventional techniques are able to produce signal analysis results with finer temporal detail and higher spectral and spatial resolution. This course goes beyond available textbooks and extends these signal analysis tools to higher dimensions and multiple sensor channels.

COURSE CONTENT 
This course introduces practical implementation of the latest research approaches to time series and spectral analysis. The course uses a systematic approach based on a signal modelling theme, while focusing on fast computational procedures that make the alternative higher performance analysis techniques feasible to implement in practical applications. Of special focus are the extensions of fast computational algorithms for high performance signal analysis techniques to multichannel, multi-dimensional, and non-stationary instantaneous time-frequency analysis applications, where there is little published implementation literature.

The first two days focus on one-dimensional data sources. The following two days cover data from multi-channel and two-dimensional sources, and single channel data with highly non-stationary features. Application examples using telecom, sonar, radar, seismic, and biomedical data have been incorporated into the course. Emerging signal processing techniques in modern time series and spectral analysis, including bandwidth extrapolation methods, non-stationary signal time-frequency analysis, and multi-resolution (wavelet-based analysis) analysis, will be covered. 
An extensive software toolbox (MATLAB and C) is provided for all the algorithms and computational techniques introduced in the course.

WHO SHOULD ATTEND
The expected background for students is some fundamental knowledge of Fourier transforms, basic digital signal processing (filters, convolution, Fast Fourier Transform [FFT]), basic matrix mathematical operations (matrix inverse, eigenvectors), and introductory random signals (correlation, spectral density). There is a review during the first day on these topics to introduce notation and to remind students of basic concepts. Some experience applying time and frequency domain techniques to signals would be helpful. This course provides alternative time and frequency domain analysis techniques. A number of data cases are used to demonstrate the practical use of the theoretical concepts.

Monday
Signal Analysis Tools and Classical Spectral Analysis 
The first day reviews all the relevant concepts of traditional temporal and frequency domain analysis of signals. We will establish some important concepts for the advanced alternative temporal and frequency (spectral) techniques covered in sequel course days.

• Complex Signal Representations
• Analytic Signals
• Issues in Spectral Estimation
• Tutorial Review: Fourier Transform Theory
• Tutorial Review: Random Signal Theory
• Tutorial Review: Matrix Algebra Theory
• Resolution and Time-Bandwidth Uncertainty Principle
• Autocorrelation and Cross Correlation
• Power Spectral Density
• Window Selection
• Periodogram Method
• Blackman-Tukey Method

Tuesday
Parametric and Autoregressive Methods
The most successful high performance analysis approach is based on autoregressive and linear prediction modelling and estimation. We explore the basis for the high performance via a systematic modelling viewpoint, and illustrate the performance with actual data.

• Parametric Time Series Models: Autoregressive (AR), Moving Average (MA), and Autoregressive Moving Average (ARMA)
• Parametric Relationships among AR, MA, and ARMA Models
• Autocorrelation Relationships among Parametric Models
• AR, Linear Prediction, and Lattice Filters
• Levinson-Durbin Algorithm
• Maximum Entropy Analysis
• ARMA Spectral Estimation
• MA Spectral Estimation
• AR Spectral Estimation: Yule-Walker algorithm, Burg algorithm, least squares linear prediction algorithms
• AR Model Order Selection

Exponential Frequency Estimation and Minimum Variance Spectra 
Even higher performance for finding signal feature extraction is achieved using minimum variance and eigenanalysis approaches, at a computational cost increase. We explore the tradeoffs.

• Prony's Method
• Damped Exponential Parameter Estimation
• Relationship to AR Methods
• Least Squares Prony Algorithms
• Noise Excision by Eigenanalysis/Principal Components Analysis
• Minimum Variance Estimation: Derivation and relationship to AR spectral estimation, signal and noise subspace concepts
• Pisarenko's Technique
• MUSIC and ESPRIT Algorithms

Wednesday
Multi-Channel and Two-Dimensional Spectral Analysis
Some of the techniques discussed the first two days are extended to multi-channel (multiple sensors) and higher dimensional data situations. We cover many of the important extensions and show that higher estimation performance is achieved, sometimes with unexpected artefacts for which a user needs to be aware.

• Multi-Channel (MC) Spectral Analysis: MC transform theory, MC random process theory
• MC Classical Spectral Estimators
• MC AR/MA/ ARMA Processes
• MC Block-Levinson Algorithm
• MC AR Spectral Estimators
• MC Minimum Variance Spectral Analysis
• Two-Dimensional (2D) Spectral Analysis: 2D transform theory, 2D random process theory
• 2D Minimum Variance Spectral Algorithm
• 2D Autoregressive and Linear Prediction
• Relationship of Temporal and Spatial Spectral Analysis Techniques
• High Resolution Direction Finding and Beamformation

Thursday
Non-Stationary Time-Frequency Analysis (TFA) 
Some additional performance tradeoffs and issues concerning non-stationary (time-varying) signal analysis will be summarized, again using actual data to illustrate the tradeoffs.

• Time-Recursive AR Estimation
• Short Time Fourier Transform
• Linear and Quadratic Time-Frequency Representations
• Short Time Fourier Transform (STFT)
• Wigner-Ville Distribution
• 2D Methods of TFA
• Ambiguity Functions in TFAs

Multi-Resolution Analysis

• Alternative Multi-Resolution Basis Functions to Fourier Transform Frequency Representation
• Scaling (time compression/time expansion) Properties of Analyzing Wavelets for Time-Frequency Analysis
• Scalograms vs Time-Frequency Grams

Said about the course from previous participants:
"Very well presented course topics by the leading expert in the field."
"A very practically oriented course with a lot of examples."
"Overview of many techniques, m-functions and demonstrations."
"MatLab example programs on CD-ROM and a  variety of algorithms and "real life" test examples."
"Good balance between mathematical and physical."