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We propose a new distance metric for a radial basis functions (RBF) neural network. We consider a two-dimensional space of time and frequency. In the usual context of RBF, a two- dimensional space would imply a two-dimensional feature vector. In our paradigm, however, the input feature vector may be of any length, and is typically a time series (say, 512 samples). We also propose a rule for positioning the centers in time-frequency (TF) space, which is based on the well-known expectation maximization (EM) algorithm. Our algorithm, for which we have coined the term `log-on expectation maximization' (LEM) adapts a number of centers in TF space in such a way as to fit the input distribution. We propose two variants, LEM1, which works in one dimension at a time, and LEM2, which works in both dimensions simultaneously. We allow these `circles' (somewhat circular TF contours) to move around in the two-dimensional space, but we also allow them to dilate into ellipses of arbitrary aspect ratio. We then have a generalization which embodies both the Weyl-Hiesenberg (e.g., sliding window FFT) and affine (e.g., wavelet) spaces as special cases. Later we allow the `ellipses' to adaptively `tilt.' (In other words we allow the time series associated with each center to chirp, hence the name `chirplet transform.') It is possible to view the process in a different space, for which we have coined the term `bow-tie' space. In that space, the adaptivity appears as a number of bow-tie shaped centers which also move about to fit the input distribution in this new space. We use our chirplet transform for time-frequency analysis of Doppler radar signals. The chirplet essentially embodies a constant acceleration physical model. This model almost perfectly matches the physics of constant force, constant mass objects (such as cars with fixed throttle starting off at a stoplight). Our transform resolves general targets (those undergoing nonconstant acceleration) better than the classical Fourier Doppler periodogram. Since it embodies the constant velocity (Doppler periodogram) as a special case, its extra degrees of freedom better capture the physics of moving objects than does classical Fourier processing. By making the transform adaptive, we may better represent the signal with fewer transform coefficients.
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