scholarly journals A Bijective Proof of Garsia's $q$-Lagrange Inversion Theorem

10.37236/1364 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Dan W. Singer
Keyword(s):  
Lagrange Inversion ◽  
Inversion Theorem ◽  
Bijective Proof

A $q$-Lagrange inversion theorem due to A. M. Garsia is proved by means of two sign-reversing, weight-preserving involutions on Catalan trees.

scholarly journals An Improved Generalized Chirp Scaling Algorithm Based on Lagrange Inversion Theorem for High-Resolution Low Frequency Synthetic Aperture Radar Imaging

2019 ◽  
Vol 11 (16) ◽  
pp. 1874 ◽  
Author(s):  
Xing Chen ◽  
Tianzhu Yi ◽  
Feng He ◽  
Zhihua He ◽  
Zhen Dong
Keyword(s):  
High Resolution ◽  
Stationary Phase ◽  
Low Frequency ◽  
Synthetic Aperture ◽  
Phase Coupling ◽  
Lagrange Inversion ◽  
Inversion Theorem ◽  
Coupling Phase ◽  
Large Bandwidth ◽  
Chirp Scaling Algorithm

The high-resolution low frequency synthetic aperture radar (SAR) has serious range-azimuth phase coupling due to the large bandwidth and long integration time. High-resolution SAR processing methods are necessary for focusing the raw data of such radar. The generalized chirp scaling algorithm (GCSA) is generally accepted as an attractive solution to focus SAR systems with low frequency, large bandwidth and wide beam bandwidth. However, as the bandwidth and/or beamwidth increase, the serious phase coupling limits the performance of the current GCSA and degrades the imaging quality. The degradation is mainly caused by two reasons: the residual high-order coupling phase and the non-negligible error introduced by the linear approximation of stationary phase point using the principle of stationary phase (POSP). According to the characteristics of a high-resolution low frequency SAR signal, this paper firstly presents a principle to determine the required order of range frequency. After compensating for the range-independent coupling phase above 3rd order, an improved GCSA based on Lagrange inversion theorem is analytically derived. The Lagrange inversion enables the high-order range-dependent coupling phase to be accurately compensated. Imaging results of P- and L-band SAR data demonstrate the excellent performance of the proposed algorithm compared to the existing GCSA. The image quality and focusing depth in range dimension are greatly improved. The improved method provides the possibility to efficiently process high-resolution low frequency SAR data with wide swath.

scholarly journals The Lagrange inversion theorem in the smooth case

2008 ◽  
Vol 340 (2) ◽  
pp. 1263-1270 ◽  
Author(s):  
Steven G. Krantz ◽  
Harold R. Parks
Keyword(s):  
Lagrange Inversion ◽  
Inversion Theorem ◽  
Smooth Case

scholarly journals Focus Improvement of Airborne High-Squint Bistatic SAR Data Using Modified Azimuth NLCS Algorithm Based on Lagrange Inversion Theorem

2021 ◽  
Vol 13 (10) ◽  
pp. 1916
Author(s):  
Chuang Li ◽  
Heng Zhang ◽  
Yunkai Deng
Keyword(s):  
Large Range ◽  
Cross Coupling ◽  
Synthetic Aperture ◽  
Phase Point ◽  
Lagrange Inversion ◽  
Inversion Theorem ◽  
Range Cell ◽  
Higher Order Terms ◽  
Bistatic Synthetic Aperture Radar ◽  
Range Cell Migration

In this paper, a modified azimuth nonlinear chirp scaling (NLCS) algorithm is derived for high-squint bistatic synthetic aperture radar (BiSAR) imaging to solve its inherent difficult issues, including the large range cell migration (RCM), azimuth-dependent Doppler parameters, and the sensibility of the higher order terms. First, using the Lagrange inversion theorem, an accurate spectrum suitable for processing airborne high-squint BiSAR data is introduced. Different from the spectrum that is based on the method of series reversion (MSR), it is allowed to derive the bistatic stationary phase point while retaining the double square root (DSR) of the slant range history. Based the spectrum, a linear RCM correction is used to remove the most of the linear RCM components and mitigate the range-azimuth coupling, and, then, bulk secondary range compression is implemented to compensate the residual RCM and cross-coupling terms. Following this, a modified azimuth NLCS operation is applied to eliminate the azimuth-dependence of Doppler parameters and equalize the azimuth frequency modulation for azimuth compression. The experimental results, with better focusing performance, prove the high accuracy and effectiveness of the proposed algorithm.

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