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**Wavelets can be used to identify the time, the frequency, and the shape of a transient signal**** **

**F**ast Fourier Transforms (FFT) are in wide use and work very well if your signal stays at a constant frequency (“stationary”). But if the signal could vary, have pulses, “blips” or any other kind of interesting behavior then you need Wavelets. Wavelets are remarkable tools that can stretch and move like an amoeba to find the hidden “events” and then simultaneously give you their location, frequency, and shape. Wavelet Transforms allow this and many other capabilities not possible with conventional methods like the FFT

In this class you will learn how Wavelets can be used to identify the time, the frequency, and the shape of a transient signal (radar pulse, heartbeat, stock market tic, image edge detection in baggage, etc,) and do all 3 simultaneously! For example, in the figure above (a 3-D wavelet transform of a chirp signal), the horizontal axis indicates time, the vertical axis indicates frequency, and the colors indicate magnitude (at that particular time and frequency)

This course is vastly different from traditional math-oriented Wavelet courses or books in that we use examples, figures, and computer demonstrations to show how to understand and work with Wavelets. This is a comprehensive, in-depth, up-to-date treatment of the subject, but from an intuitive, conceptual point of view.

We do look at a few key equations from the traditional literature but only AFTER the concepts are demonstrated and understood. If desired, further study from scholarly texts and papers is then made much easier and more palatable when you already understand the fundamental equations and how they relate to the real world. Each student will receive extensive course slides, a CD with MATLAB demonstrations, and a copy of the instructor’s new book, Conceptual Wavelets.

**What you will learn:**** **

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**• How to use Wavelets as a “microscope” to analyze data that changes over time or has hidden “events” that would not show up on an FFT.

• How to understand and efficiently use the three types of Wavelet Transforms to better analyze and process your data using state-of-the-art methods and applications.

• How to compress and de-noise data using advanced Wavelet techniques. How to avoid potential pitfalls by understanding the concepts. A “safe” method if in doubt.

• How to increase productivity and reduce cost by choosing (or building) a Wavelet that best matches your particular application.

**Course Outline, Samplers, and Notes**** **

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**1. What is a Wavelet? Examples and Uses. “Waves” that can start, stop, move and stretch. Real-world applications in many fields: Signal and Image Processing, Internet Traffic, Airport Security, Medicine, JPEG, Finance, Pulse and Target Recognition, Radar, Sonar, etc.

2. Comparison with traditional methods. The concept of the FFT, the STFT, and Wavelets as all being various types of comparisons with the data. Strengths, weaknesses, optimal choices.

3. The Continuous Wavelet Transform (CWT). Stretching and shifting the Wavelet for optimal correlation. Predefined vs. Constructed Wavelets.

4. The Discrete Wavelet Transform (DWT). Shrinking the signal by factors of 2 through downsampling. Understanding the DWT in terms of correlations with the data. Relating the DWT to the CWT. Demonstrations and uses.

5. The Redundant Discrete Wavelet Transform (RDWT). Stretching the Wavelet by factors of 2 without downsampling. Tradeoffs between the alias-free processing and the extra storage and computational burdens. A hybrid process using both the DWT and the RDWT. Demonstrations and uses.

6. “Perfect Reconstruction Filters”. How to cancel the effects of aliasing. How to recognize and avoid any traps. A breakthrough method to see the filters as basic Wavelets. The “magic” of alias cancellation demonstrated in both the time and frequency domains.

7. Highly useful properties of popular Wavelets. How to choose the best Wavelet for your application. When to create your own and when to stay with proven favorites.

8. Compression and De-Noising using Wavelets. How to remove unwanted or non-critical data without throwing away the alias cancellation capability. A new, powerful method to extract signals from large amounts of noise. Demonstrations.

9. Additional Methods and Applications. Image Processing. Detecting Discontinuities, Self-Similarities and Transitory Events. Speech Processing. Human Vision. Audio and Video. BPSK/QPSK Signals. Wavelet Packet Analysis. Matched Filtering. How to read and use the various Wavelet Displays. Demonstrations.

10. Further Resources. The very best of Wavelet references.

Determine for yourself the value of this course before you sign up.

See Slide Samples of our Wavelets: A Conceptual Practical Approach

After attending the course you will receive a full set of detailed notes from the class for future reference, as well as a certificate of completion. Please visit our website for more valuable information.

**Comments from participants:**

“Your Wavelets course was very helpful in our Radar studies. We often use wavelets now instead of the Fourier Transform for precision de-noising.”

–Long To, NAWC WD, Point Wugu, CA

“I was looking forward to this course and it was very rewarding–Your clear explanations starting with the big picture immediately contextualized the material allowing us to drill a little deeper with a fuller understanding”

–Steve Van Albert, Walter Reed Army Institute of Research

“Good overview of key wavelet concepts and literature. The course provided a good physical understanding of wavelet transforms and applications.”

–Stanley Radzevicius, ENSCO, Inc.

**The Instructor**** **

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**Dr. Lee Fugal is the Founder and President of an independent consulting firm. He has over 30 years of industry experience in Digital Signal Processing (including Wavelets) and Satellite Communications.

He has been a full-time consultant on numerous assignments since 1991. Recent projects include Excision of Chirp Jammer Signals using Wavelets, design of Space-Based Geolocation Systems (GPS & Non-GPS), and Advanced Pulse Detection using Wavelet Technology.

He has taught upper-division University courses in DSP and in Satellites as well as Wavelet short courses and seminars for Practicing Engineers and Management. He holds a Masters in Applied Physics (DSP) from the University of Utah, is a Senior Member of IEEE, and a recipient of the IEEE Third Millennium Medal.

**About ATI**** **

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**Since 1984, the Applied Technology Institute (ATI) has provided leading-edge public courses and onsite technical training to DoD and NASA personnel, as well as contractors.

Our mission here at ATI is to provide expert training and the highest quality professional development in space, communications, defense, sonar, radar, and signal processing. We are not a one-size-fits-all educational facility.

Our short classes include both introductory and advanced courses. ATI’s instructors are world-class experts who are the best in the business. They are carefully selected for their ability to clearly explain advanced technology.

**Times, Dates, and Locations**** **

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**Wavelets: A Conceptual Practical Approach

Feb 22-24, 2011 San Diego, CA

Jun 7-9, 2011 Beltsville, MD

Sincerely,

The ATI Courses Team

**P.S** Call today for registration at 410-956-8805 or 888-501-2100 or access our website at www.ATIcourses.com. For general questions please email us at ATI@ATIcourses.com.