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Table 1 Summary of main notations

From: Instantaneous cross-correlation function type of WD based LFM signals analysis via output SNR inequality modeling

Notation

Description

\(f,f_1,{\widehat{g}},{\widetilde{g}}\); n

Deterministic signals; random noise

\({\mathbf {A}}=(a,b;c,d),{\mathbf {A}}_1=(a_1,b_1;c_1,d_1), {\mathbf {A}}_2=(a_2,b_2;c_2,d_2)\)

Parameter matrices

\({\mathcal {K}}_{{\mathbf {A}}},{\mathcal {K}}_{{\mathbf {A}}_1}\)

Linear canonical domain kernel functions

\(F_{{\mathbf {A}}},F_{{\mathbf {A}}_1},F_{{\mathbf {A}}_2}\); \({\widehat{G}}_{{\mathbf {A}}_1}\); \({\widetilde{G}}_{{\mathbf {A}}_1}\)

LCTs of f; LCT of \({\widehat{g}}\); LCT of \({\widetilde{g}}\)

\(*\)

Complex conjugate

\(f\left( t+\frac{\tau }{2}\right) f^{*}\left( t-\frac{\tau }{2}\right)\)

Instantaneous autocorrelation function

\(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) f^{*}\left( t-\frac{\tau }{2}\right) ,{\widehat{G}}_{{\mathbf {A}}_1}\left( t+\frac{\pi }{2}\right) {\widetilde{g}}^{*}\left( t-\frac{\pi }{2}\right), {\widetilde{G}}_{{\mathbf {A}}_1}\left( t+\frac{\pi }{2}\right) {\widehat{g}}^{*}\left( t-\frac{\pi }{2}\right)\)

Linear canonical domain ICFs

\(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) F_{{\mathbf {A}}_2}^{*}\left( t-\frac{\tau }{2}\right)\)

Linear canonical domain CICF

\(\text {W}_f,\text {W}_n\)

WDs of f, n

\(\text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}},\text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}},\text {W}_{f+n}^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}}\)

CICFWDs of \(f,n,f+n\)

\(\text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}},\text {W}_{{\widehat{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}},\text {W}_{{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}},\text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}},\text {W}_{f+n}^{{\mathbf {A}}_1,{\mathbf {A}}}\)

ICFWDs of \(f,{\widehat{g}},{\widetilde{g}},n,f+n\)

\(\text {W}_{{\widehat{g}},{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}},\text {W}_{{\widetilde{g}},{\widehat{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}\)

Cross ICFWDs of \({\widehat{g}}\) and \({\widetilde{g}}\)

\(\text {ESNR}_{\text {WD}},\text {ESNR}_{\text {ICFWD}}^{{\mathbf {A}}_1,{\mathbf {A}}},\text {ESNR}_{\text {CICFWD}}^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}}\)

Expectation-based output SNRs of WD, ICFWD, CICFWD

\(\max\); \(\mathop {\arg \max }\)

Maximum; arguments of the maximum

Mean

Arithmetic mean or integral average

\(\text {E}[\cdot ]\); \(\text {Var}[\cdot ]\)

Expectation operator; variance operator

\(\delta (\cdot )\)

Dirac delta operator

\(\alpha ,{\widehat{\alpha }},{\widetilde{\alpha }}\); \(\beta ,{\widehat{\beta }},{\widetilde{\beta }}\)

Initial frequency; frequency rate

\(h_1,{\widehat{h}}_1,{\widetilde{h}}_1\)

\(\frac{1}{2\beta b_1+a_1},\frac{1}{2{\widehat{\beta }}b_1+a_1},\frac{1}{2{\widetilde{\beta }}b_1+a_1}\)

\(\widetilde{{\widehat{l}}},\widehat{{\widetilde{l}}}\)

\(\frac{a}{2b}+\frac{d_1-{\widehat{h}}_1}{8b_1}-\frac{{\widetilde{\beta }}}{4},\frac{a}{2b}+\frac{d_1-{\widetilde{h}}_1}{8b_1}-\frac{{\widehat{\beta }}}{4}\)

D

Power spectral density of the noise