Revision #1660 → #1813 · back to history
modifiedWavelet (informal)0dfd30822186
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| note | No notion of wavelet exists in Mathlib. | No notion of wavelet exists in Mathlib (loogle for "wavelet" returns 0 hits). |
modifiedUncertainty lower bound5fca52d0ba6a
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| note | No Heisenberg/Fourier uncertainty principle is present in Mathlib (grep for uncertainty/Heisenberg found nothing). | No Heisenberg/Fourier uncertainty principle is present in Mathlib (loogle for "Heisenberg" returns 0 hits). |
modifiedMother wavelet5345642d0da1
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| anchors | [{"section":"Continuous wavelet transforms (continuous shift and scale parameters)","snippet":"This subspace in turn is in most situations generated by the shifts of one generating function"},{"type":"math_alttext","value":"{\\displaystyle \\psi (t)=2\\,\\operatorname {sinc} (2t)-\\,\\operatorname {sinc} (t)={\\frac {\\sin(2\\pi t)-\\sin(\\pi t)}{\\pi t}}}"}] | — |
modifiedExamples of mother wavelets13626f77f29c
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| anchors | [{"section":"Continuous wavelet transforms (continuous shift and scale parameters)","snippet":"That, Meyer's, and two other examples of mother wavelets are:"},{"type":"math_alttext","value":"{\\displaystyle \\psi (t)=2\\,\\operatorname {sinc} (2t)-\\,\\operatorname {sinc} (t)={\\frac {\\sin(2\\pi t)-\\sin(\\pi t)}{\\pi t}}}"}] | — |
modifiedChild wavelets6d407eb882fa
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| anchors | [{"section":"Continuous wavelet transforms (continuous shift and scale parameters)","snippet":"is generated by the functions (sometimes called child wavelets )"},{"type":"math_alttext","value":"{\\displaystyle \\psi _{a,b}(t)={\\frac {1}{\\sqrt {a}}}\\psi \\left({\\frac {t-b}{a}}\\right),}"}] | — |
modifiedProjection and wavelet coefficients92018c88ce12
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| anchors | [{"section":"Continuous wavelet transforms (continuous shift and scale parameters)","snippet":"The projection of a function x onto the subspace of scale a then has the form"},{"type":"math_alttext","value":"{\\displaystyle x_{a}(t)=\\int _{\\mathbb {R} }WT_{\\psi }\\{x\\}(a,b)\\cdot \\psi _{a,b}(t)\\,db}"},{"type":"math_alttext","value":"{\\displaystyle WT_{\\psi }\\{x\\}(a,b)=\\langle x,\\psi _{a,b}\\rangle =\\int _{\\mathbb {R} }x(t){\\psi _{a,b}(t)}\\,dt.}"}] | — |
| mathlib.decl | orthogonalProjection | Submodule.orthogonalProjection |
modifiedAffine system701d43902d1e
| Field | From #1660 | To #1813 |
|---|
| anchors | [{"section":"Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)","snippet":"One such system is the affine system for some real parameters"},{"type":"math_alttext","value":"{\\displaystyle \\psi _{m,n}(t)={\\frac {1}{\\sqrt {a^{m}}}}\\psi \\left({\\frac {t-nba^{m}}{a^{m}}}\\right).}"}] | — |
modifiedSufficient condition for reconstruction2078d1725080
| Field | From #1660 | To #1813 |
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| anchors | [{"section":"Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)","snippet":"A sufficient condition for the reconstruction of any signal x of finite energy by the formula"},{"type":"math_alttext","value":"{\\displaystyle x(t)=\\sum _{m\\in \\mathbb {Z} }\\sum _{n\\in \\mathbb {Z} }\\langle x,\\,\\psi _{m,n}\\rangle \\cdot \\psi _{m,n}(t)}"}] | — |
modifiedDaubechies 4-tap wavelet6e828c1d6263
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| note | Daubechies wavelets are not in Mathlib. | Daubechies wavelets are not in Mathlib (loogle for "Daubechies" returns 0 hits). |
modifiedSubspaces from mother and father wavelets3042906a1b64
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| anchors | [{"section":"Multiresolution based discrete wavelet transforms (continuous in time)","snippet":"From the mother and father wavelets one constructs the subspaces"},{"type":"math_alttext","value":"{\\displaystyle V_{m}=\\operatorname {span} (\\phi _{m,n}:n\\in \\mathbb {Z} ),{\\text{ where }}\\phi _{m,n}(t)=2^{-m/2}\\phi (2^{-m}t-n)}"},{"type":"math_alttext","value":"{\\displaystyle W_{m}=\\operatorname {span} (\\psi _{m,n}:n\\in \\mathbb {Z} ),{\\text{ where }}\\psi _{m,n}(t)=2^{-m/2}\\psi (2^{-m}t-n).}"}] | — |
modifiedOrthogonal complement W_m451c64173c55
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| anchors | [{"section":"Multiresolution based discrete wavelet transforms (continuous in time)","snippet":"W m is the orthogonal complement of V m inside the subspace"},{"type":"math_alttext","value":"{\\displaystyle \\{0\\}\\subset \\dots \\subset V_{1}\\subset V_{0}\\subset V_{-1}\\subset V_{-2}\\subset \\dots \\subset L^{2}(\\mathbb {R} )}"},{"type":"math_alttext","value":"{\\displaystyle V_{m}\\oplus W_{m}=V_{m-1}.}"}] | — |
modifiedRefinement equationa3ae5dd3d2ee
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| anchors | [{"section":"Multiresolution based discrete wavelet transforms (continuous in time)","snippet":"is a refinement equation for the father wavelet"},{"type":"math_alttext","value":"{\\displaystyle g_{n}=\\langle \\phi _{0,0},\\,\\phi _{-1,n}\\rangle }"},{"type":"math_alttext","value":"{\\displaystyle h_{n}=\\langle \\psi _{0,0},\\,\\phi _{-1,n}\\rangle }"}] | — |
modifiedOrthogonal decomposition of L232d392867512
| Field | From #1660 | To #1813 |
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| anchors | [{"section":"Multiresolution based discrete wavelet transforms (continuous in time)","snippet":"From the multiresolution analysis derives the orthogonal decomposition of the space"},{"type":"math_alttext","value":"{\\displaystyle L^{2}=V_{j_{0}}\\oplus W_{j_{0}}\\oplus W_{j_{0}-1}\\oplus W_{j_{0}-2}\\oplus W_{j_{0}-3}\\oplus \\cdots }"},{"type":"math_alttext","value":"{\\displaystyle S=\\sum _{k}c_{j_{0},k}\\phi _{j_{0},k}+\\sum _{j\\leq j_{0}}\\sum _{k}d_{j,k}\\psi _{j,k}}"},{"type":"math_alttext","value":"{\\displaystyle c_{j_{0},k}=\\langle S,\\phi _{j_{0},k}\\rangle }"},{"type":"math_alttext","value":"{\\displaystyle d_{j,k}=\\langle S,\\psi _{j,k}\\rangle .}"}] | — |
modifiedSpace L2(R)f7fd20b0ecfa
| Field | From #1660 | To #1813 |
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| anchors | [{"section":"Mother wavelet","snippet":"This is the space of Lebesgue measurable functions that are both absolutely integrable and square integrable"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }|\\psi (t)|\\,dt<\\infty }"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }|\\psi (t)|^{2}\\,dt<\\infty .}"}] | — |
modifiedZero mean and unit norm conditions933e47187528
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| anchors | [{"section":"Mother wavelet","snippet":"one can formulate the conditions of zero mean and square norm one"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }\\psi (t)\\,dt=0}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }|\\psi (t)|^{2}\\,dt=1}"}] | — |
modifiedVanishing moments5cec4ef11b7c
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| anchors | [{"section":"Mother wavelet","snippet":"to restrict ψ to be a continuous function with a higher number M of vanishing moments"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }t^{m}\\,\\psi (t)\\,dt=0.}"}] | — |
modifiedSTFT kernelf0e0fd8f417a
| Field | From #1660 | To #1813 |
|---|
| anchors | [{"section":"Comparisons with Fourier transform (continuous-time)","snippet":"one may think of the STFT as a transform with a slightly different kernel"},{"type":"math_alttext","value":"{\\displaystyle \\psi (t)=g(t-u)e^{-2\\pi it}}"},{"type":"math_alttext","value":"{\\displaystyle E=\\int _{-\\infty }^{\\infty }|\\psi (t)|^{2}\\,dt={\\frac {1}{2\\pi }}\\int _{-\\infty }^{\\infty }|{\\hat {\\psi }}(\\omega )|^{2}\\,d\\omega }"},{"type":"math_alttext","value":"{\\displaystyle \\sigma _{u}^{2}={\\frac {1}{E}}\\int |t-u|^{2}|\\psi (t)|^{2}\\,dt}"}] | — |
modifiedWavelet energy via Parseval37563d975542
| Field | From #1660 | To #1813 |
|---|
| anchors | [{"section":"Comparisons with Fourier transform (continuous-time)","snippet":"Using Parseval's theorem , one may define the wavelet's energy as"},{"type":"math_alttext","value":"{\\displaystyle \\psi (t)=g(t-u)e^{-2\\pi it}}"},{"type":"math_alttext","value":"{\\displaystyle E=\\int _{-\\infty }^{\\infty }|\\psi (t)|^{2}\\,dt={\\frac {1}{2\\pi }}\\int _{-\\infty }^{\\infty }|{\\hat {\\psi }}(\\omega )|^{2}\\,d\\omega }"},{"type":"math_alttext","value":"{\\displaystyle \\sigma _{u}^{2}={\\frac {1}{E}}\\int |t-u|^{2}|\\psi (t)|^{2}\\,dt}"}] | — |
modifiedTemporal support of windowc7989830af66
| Field | From #1660 | To #1813 |
|---|
| anchors | [{"section":"Comparisons with Fourier transform (continuous-time)","snippet":"the square of the temporal support of the window offset by time u is given by"},{"type":"math_alttext","value":"{\\displaystyle \\psi (t)=g(t-u)e^{-2\\pi it}}"},{"type":"math_alttext","value":"{\\displaystyle E=\\int _{-\\infty }^{\\infty }|\\psi (t)|^{2}\\,dt={\\frac {1}{2\\pi }}\\int _{-\\infty }^{\\infty }|{\\hat {\\psi }}(\\omega )|^{2}\\,d\\omega }"},{"type":"math_alttext","value":"{\\displaystyle \\sigma _{u}^{2}={\\frac {1}{E}}\\int |t-u|^{2}|\\psi (t)|^{2}\\,dt}"}] | — |
modifiedSpectral support of window1f4edc98857b
| Field | From #1660 | To #1813 |
|---|
| anchors | [{"section":"Comparisons with Fourier transform (continuous-time)","snippet":"the square of the spectral support of the window acting on a frequency"},{"type":"math_alttext","value":"{\\displaystyle {\\hat {\\sigma }}_{\\xi }^{2}={\\frac {1}{2\\pi E}}\\int |\\omega -\\xi |^{2}|{\\hat {\\psi }}(\\omega )|^{2}\\,d\\omega }"}] | — |
addedQuadrature mirror filterb2cd4a8f059b
addedHaar wavelet9db52108eecb