SNAPSHOTS OF UNCERTAINTY
Source:Synergy 3
Posted:17 Oct 2000Snapshots of uncertainty - how wavelets provide a clearer focus on frequency and time.
Beneath all the noise, discontinuities and incomplete data that make many real-world signals so difficult to unravel, lies a dark secret that can never be cracked. Quantum physics doesn't have a monopoly on uncertainty principles, there's one at the very heart of signal processing. Just as you cannot simultaneously know the momentum and the position of a moving particle, you can never know the precise frequency components that exist in a given signal at a particular instant in time.
Yet the bulk of real world signals are non-stationary: their frequency components change over time. Moreover, for many applications the most interesting information is hidden in the time/frequency ratio of the signal. Whether you want to pick up the small scale changes in an ECG reading caused by myocardial ischemia, or to identify an individual whale from his characteristic song, it's the time/frequency signature that you'll be looking for.Traditionally, short-time Fourier transforms (STFT) have been used to slide a window across the different frequency and time bands within a signal and so glean information about what frequency bands exist at what time intervals. However, the problem with the STFT is that it gives a fixed resolution at all times. Once you have selected the size of the 'window' you wish to open on your data, you can't vary its time/frequency aspect ratio. It's a one-size-fits-all approach that frequently leaves engineers on the prongs of a dilemma: choose a wide window and gain good frequency resolution at the expense of poor time resolution, or choose a narrow window and gain good time resolution at the expense of poor frequency resolution.
For decades, researchers in fields as diverse as nuclear engineering and neurophysiology have been trying to box clever with uncertainty, to extract what trade-offs they can from a principle that dictates that more accurate time information can be obtained about higher frequencies, while more accurate frequency information can be obtained about lower frequencies. The result is multi-resolution analysis or wavelet theory.
In a sense wavelet analysis can be said to bow to the laws of physics, being specifically designed to offer good time resolution at high frequencies and good frequency resolution at low frequencies.
It can also be said to bow to the practicalities of signal processing in the real world, where high frequency components generally appear in intermittent bursts and low frequency components tend to be more persistent.This ability to analyse data at multiple resolutions, together with the fact that wavelet decomposition is invertible (a signal can be perfectly reconstructed from its wavelet coefficients) is what chiefly distinguishes wavelet analysis from other methods. In effect, it acts as an adjustable zoom lens, resolving a signal at each scale in terms of differences and averages, so that one can focus on different levels of detail.
Wavelet analysis breaks with the sinusoids of Fourier analysis. Basis functions are no longer prescribed, rather new ones, suitable for different datasets or operations, are continually being described. The key idea is that any signal can be expressed as a linear combination of functions, all of which are simply dilations or contractions of a single mother function - the prototype wavelet.
Having selected a mother wavelet appropriate to your particular data, this becomes the prototype for every window opened on the time/frequency plane during the transformation process. Basically each window is a scaled (dilated or contracted) and shifted version of the mother wavelet, with temporal analysis obtaining more accurate results with contracted short time windows and frequency analysis obtaining more accurate results with dilated long time windows.The discovery of the fast Fourier transform has had a profound impact on engineering. Similarly, the discovery of the fast wavelet transform marks the beginning of the wavelet era. This algorithm furnishes a simple and efficient way to make a wavelet decomposition. A fast wavelet decomposition is carried out with a cascade of two filters. The first filter acts as a low-pass filter (smoothing the signal) while the second high-pass filter furnishes the details of the signal (the wavelet coefficients). The details of the signal at a lower level of resolution are obtained by applying iteratively the same two filters to the smoothed signal. The overall picture that results could be called a stylisation of the original signal - a caricature, which highlights features of interest such as spikes, discontinuities and periodic components.
Not surprisingly, then, wavelet analysis is becoming an increasingly popular technique for denoising, data compression and pattern identification or matching. It's an approach that's rapidly finding a role in areas as diverse as earthquake prediction and biomedical signal processing. Wavelets have been used to watermark digital images and to increase the level of detail captured in satellite images. They are central to the compression capability of JPEG2000, and the FBI's central, searchable database of 30 million sets of fingerprints.
They make applications possible that were once far out of reach. Just how effective wavelets can be in homing in on a tiny signal amidst a sea of noise, has been demonstrated by engineers at the US Department of Energy's national laboratory, who used wavelet analysis within a new device capable of detecting the sound of a heartbeat even when a person is hidden inside a large vehicle - a technology that, had it been in use in Europe earlier this year, might have saved the tragic loss of 58 lives at Dover.
Wavelets on the Web
§ Amara Graps' Introduction to Wavelets
http://www.amara.com/IEEEwave/IEEEwavelet.html§ Robi Polikar's Engineer's Guide to Wavelet Analysis
http://www.public.iastate.edu/~rpolikar/WAVELETS/WTtutorial.html(Source:Synergy 3 Posted:17 Oct 2000)
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