A novel detection approach of linear FM (LFM) signals, with single or multiple components, in the time-frequency plane of Teager-Huang (TH) transform is presented. The detection scheme that combines TH transform and Hough transform is referred to as Teager-Huang-Hough (THH) transform. The input signal is mapped into the time-frequency plane by using TH transform followed by the application of Hough transform to recognize time-frequency components. LFM components are detected and their parameters are estimated from peaks and their locations in the Hough space. Advantages of THH transform over Hough transform of Wigner-Ville distribution (WVD) are: 1) cross-terms free detection and estimation, and 2) good time and frequency resolutions. No assumptions are made about the number of components of the LFM signals and their models. THH transform is illustrated on multicomponent LFM signals in free and noisy environments and the results compared with WVD-Hough and pseudo-WVD-Hough transforms
Let z=(x,y)∈Rd×RN−d, with 1≤d<N. We prove a priori estimates of the following type :
\|\Delta_x^\frac \alpha 2 v \|_L^p(\R^N) \le
c_p
\Big \| L_x v + \sum_i,j=1^Na_ijz_i\partial_z_j v \Big \|_L^p(\R^N),
\;\; 1
for v∈C0∞(RN),
where Lx is a non-local operator comparable with the Rd-fractional Laplacian Δx2α in terms of symbols, α∈(0,2).
We require that when Lx is replaced by the classical Rd-Laplacian Δx, i.e., in the limit local case α=2, the operator
\Delta_x + \sum_i,j=1^Na_ijz_i\partial_z_j satisfy
a weak type H\"ormander condition with invariance by suitable dilations. Such estimates were only known
for α=2.
This is
one of the first results on Lp estimates for degenerate non-local operators under H\"ormander type conditions.
We complete our result on Lp-regularity for L_x + \sum_i,j=1^Na_ijz_i\partial_z_j by proving
estimates like
\beginequation* \labelnew
\|\Delta_y_i^\frac \alpha_i 2 v \|_L^p(\R^N) \le
c_p
\Big \| L_x v + \sum_i,j=1^Na_ijz_i\partial_z_j v \Big \|_L^p(\R^N),
\endequation*
involving fractional Laplacians in the degenerate directions yi (here αi∈(0,1∧α) depends on α and on the numbers of commutators needed to obtain the yi-direction). The last estimates are new even in the local limit case α=2 which is also considered
Sterling Emmal is author of the sci-fi fantasy The Executioner of Rawule and L. S. Goulet is author of the fantasy book Sword of Dragonblood. Tundra Kill is Stan Jones' latest Nathan Active mystery. His other books include White Sky, Black Ice; Shaman Pass, Frozen Sun; Village of the Ghost Bears, and the nonfiction classic, The Spill: Personal Stories from the Exxon Valdez Disaster, coauthored with Sharon Bushell