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 |