Statistical Signal Processing · Lecture # 1 Signal Hierarchy Examples of Statistical Experiments...
Transcript of Statistical Signal Processing · Lecture # 1 Signal Hierarchy Examples of Statistical Experiments...
ECE 600-03Statistical Signal Processing
Aly A. FaragUniversity of Louisville
Spring 2010www.cvip.uofl.edu
Lecture # 1
Signal Hierarchy Examples of Statistical Experiments Outcome
– Seismic recordings– EEG signal– EMG signal– Stock market readings– Radar return signal– Temperature readings in an area over a period
Processing vs. Analysis– ECE 600 will do mix of processing and analysis
Reading Material
Signal Hierarchy
• Deterministic– Can be represented by a specific equations
Example: y = 5 cos 2π t
• Random (stochastic)– Cannot be described by a specific equation; yet
can be modeled by a certain parametric form
Example: y[n] = A + w[n]; where w[.] is an innovation process and A is a constant value.
Examples of Signals( as Outcomes of Statistical Experiments)
Seismic Signals
may register volcanic eruption, an earthquake and other internal earth activities due natural or artificial stimulus (e.g., oil exploration)
From NOVA
Goal: to predict volcanic eruption, for example
Biomedical SignalsEEG Recordings
Goal: to infer thefunctionality of the brainIn response to a stimulus, for example
EMG Recordings
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1455479/figure/F1/
http://www.emglab.net/emglab/Figures/figures.html
Goal: to infer thefunctionality of a muscle in response to a stimulus, for example
The electrocardiogram (ECG) is a technique of recording bioelectric currents generated by the heart. Clinicians can evaluate the conditions of a patient's heart from the ECG and perform further diagnosis. ECG records are obtained by sampling the bioelectric currents sensed by several electrodes, known as leads. A typical one-cycle ECG tracing is shown in Figure 1.
Figure 1: A typical one-cycle ECG tracingGenerally, the recorded ECG signal is often contaminated by noise and artifacts that can be within the frequency band of interest and manifest with similar characteristics as the ECG signal itself. In order to extract useful information from the noisy ECG signals, you need to process the raw ECG signals
ECG Recordings
http://zone.ni.com/devzone/cda/tut/p/id/6349
Goal: to infer thefunctionality of the heartfrom surface measurements.
DOW JONES STOCK MARKET READINGS
http://www.quote.com/beta/chart.action?s=%24INDU
Goal: to infer the “state of the economy”from the trading activities.
The Meltdown!
New President!
Demographic charts
http://www.swivel.com/charts/2396-Number-of-children-born-to-women-aged-15-to-49-in-various-OECD-countries
Goal: to infer trends in certain behavioral issues such as birth rate within a population.
Speech Signal
http://cnx.org/content/m0049/latest/
Goal: to understand someAspects of the speech signals;e.g., type, automatic reading,Identification, etc.
Radar Detection
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From the delay of the received signal, the location of the target can be estimated
Goal: to infer the location of a targetOr set of targets real-time. Radar signals have various uses in tracking and navigation.
Time Series
A Time Series -- A continuous plot of an experiment(stock market reading in our case)
Processing vs. Analysis
• Processing – conditions the outcome to a desired form. For example remove the DC value or enhance the oscillations (remove noise)
• Analysis – interprets the information content in the signal. For example, deciding that the given signal belongs to a particular individual
• The domains of signal processing and analysis is huge! We have our focus in ECE 600 to do a little bit of both which will give us some tools to better understand outcomes of certain engineering problems.
Elements of Our Approach
• Study statistical models– Overview of probability– Overview of Signal Representations– Overview of the Linear Statistical Model
• Examples of Processing– Weiner and Kalman Filters
• Examples of Modeling– Speech– Biomedical Signals– Stock market readings
• 2D Examples– Random Fields– Image Analysis
Random Variate Generation
• The building block is uniform random variable
• Use the inverse formula to generate other distributions
Example: 1D Gaussian from 1D Uniform using the Box-Muller formula:
y1 = sqrt( - 2 ln(x1) ) cos( 2 pi x2 )
y2 = sqrt( - 2 ln(x1) ) sin( 2 pi x2 )
The polar form of the Box-Muller transformation is both faster and more robust numerically. The algorithmic description of it is:
float x1, x2, w, y1, y2;
do { x1 = 2.0 * ranf() - 1.0; x2 = 2.0 * ranf() - 1.0;
w = x1 * x1 + x2 * x2; }
while ( w >= 1.0 );
w = sqrt( (-2.0 * ln( w ) ) / w );
y1 = x1 * w;
y2 = x2 * w;
http://www.taygeta.com/random/gaussian.html
Random Number Generation
1 & 2 Dimensional Uniform Data Streams using the MATLAB command: rand
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Random Number Generation
1 & 2 Dimensional Uniform Histograms
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Random Variable (x)
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Random Variable (X)
Histogram of 2D Uniform
Random Variable (Y)
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Random Number Generation
1 & 2 Dimensional Normal Data Streams using the MATLAB command: rand and the Box-Mueller Method
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1 & 2 Dimensional Normal Histograms
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Random Variable (x)
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Histogram Plot 2D of normal distribution
Random Variables andRandom Process
A Random Process is an indexed set of random variables. There are four possibilities based on whether the random variables are continuous/discrete and the index is continuous/discrete. Random processes may be stationary, e.g., white noise and non-stationary, e.g., speech signals.
A Random Variable is a mapping from the sample space of an experiment into the real-line. Evens (subsets of the sample space) are mapped into Borel sets on the real line.
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Average for all January months: 1970-2009
Random Variables: X1,X2,X3,XnX1 X2 X3 Xn
Week 1 Week 2 Week 3 Week 4
Illustration of a Random Process from an ensemble of readings over the month of January for 39 years (January 1970 – January 2009)
Readings from a certain day form realizations of a random variable
The collections {X1, X2, X3,…, Xn} form a nth
degree random process
Reading materials
1. G. E. Box and G. M. Jenkins, Time Series Analysis, 2nd
Edition, Holden-Day, CA, 1976.2. D. Brillinger, Time-Series, Expanded Edition,
McGraw-Hill, 1975.3. Thomas Kailath, Lectures on Weiner and Kalman
Filtering, Springer, 1981.4. S. Kay, Statictical Signal Processing, Vol 1., Prentice-
Hall, 1993.5. D. Manolakis, V. Ingle and S. Kogon, Statistical and
Adaptive Signal Processing, McGraw-Hill, 2000.6. J. Lim, Two-dimensional Signals and Image
Processing, Prentice-Hall, 1981.