ag-gipp/GoUldI

{
  "definitions" : {
    "H" : [ "system function", "system response", "transfer function", { } ],
    "j" : [ {
      "Q193796" : "imaginary unit (square root of negative one, used to define complex numbers)"
    } ],
    "\\omega" : [ {
      "Q12916" : "real number (quantity along a continuous line)"
    } ],
    "\\mathcal{F}" : [ {
      "Q6520159" : "Fourier transform (mathematical transform that expresses a mathematical function of time as a function of frequency)"
    } ],
    "h" : [ "impulse response", { } ],
    "t" : [ {
      "Q11471" : "time (dimension in which events can be ordered from the past through the present into the future)"
    } ]
  },
  "constraints" : [ ],
  "math_inputtex" : "H(j \\omega) = \\mathcal{F}\\{h(t)\\}",
  "math_inputtex_semantic" : "\\wf{Q334269}{H}(\\w{Q193796}{j} \\w{Q12916}{\\omega}) = \\wf{Q6520159}{\\mathcal{F}} \\{\\wf{Q1139726}{h}(\\w{Q11471}{t})\\}",
  "correct_tex" : "H(j \\omega) = \\mathcal{F}\\{h(t)\\}",
  "correct_mml" : "<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"p1.1.m1.1\" class=\"ltx_Math\" alttext=\"H(j\\omega)=\\mathcal{F}\\{h(t)\\}\" display=\"inline\">\n  <semantics id=\"p1.1.m1.1a\">\n    <mrow id=\"p1.1.m1.1.14\" xref=\"p1.1.m1.1.14.cmml\">\n      <mrow id=\"p1.1.m1.1.14.1.2\" xref=\"p1.1.m1.1.14.1.1.cmml\">\n        <mi id=\"p1.1.m1.1.1\" xref=\"p1.1.m1.1.1.cmml\">H</mi>\n        <mo id=\"p1.1.m1.1.14.1.2a\" xref=\"p1.1.m1.1.14.1.1.cmml\">⁡</mo>\n        <mrow id=\"p1.1.m1.1.14.1.2.2\" xref=\"p1.1.m1.1.14.1.1.cmml\">\n          <mo stretchy=\"false\" id=\"p1.1.m1.1.2\" xref=\"p1.1.m1.1.14.1.1.cmml\">(</mo>\n          <mrow id=\"p1.1.m1.1.14.1.2.1\" xref=\"p1.1.m1.1.14.1.2.1.cmml\">\n            <mi id=\"p1.1.m1.1.3\" xref=\"p1.1.m1.1.3.cmml\">j</mi>\n            <mo id=\"p1.1.m1.1.14.1.2.1.1\" xref=\"p1.1.m1.1.14.1.2.1.1.cmml\">⁢</mo>\n            <mi id=\"p1.1.m1.1.4\" xref=\"p1.1.m1.1.4.cmml\">ω</mi>\n          </mrow>\n          <mo stretchy=\"false\" id=\"p1.1.m1.1.5\" xref=\"p1.1.m1.1.14.1.1.cmml\">)</mo>\n        </mrow>\n      </mrow>\n      <mo id=\"p1.1.m1.1.6\" xref=\"p1.1.m1.1.6.cmml\">=</mo>\n      <mrow id=\"p1.1.m1.1.14.2.2\" xref=\"p1.1.m1.1.14.2.1.cmml\">\n        <mi class=\"ltx_font_mathcaligraphic\" id=\"p1.1.m1.1.7\" xref=\"p1.1.m1.1.7.cmml\">ℱ</mi>\n        <mo id=\"p1.1.m1.1.14.2.2a\" xref=\"p1.1.m1.1.14.2.1.cmml\">⁡</mo>\n        <mrow id=\"p1.1.m1.1.14.2.2.2\" xref=\"p1.1.m1.1.14.2.1.cmml\">\n          <mo stretchy=\"false\" id=\"p1.1.m1.1.8\" xref=\"p1.1.m1.1.14.2.1.cmml\">{</mo>\n          <mrow id=\"p1.1.m1.1.14.2.2.1.2\" xref=\"p1.1.m1.1.14.2.2.1.1.cmml\">\n            <mi id=\"p1.1.m1.1.9\" xref=\"p1.1.m1.1.9.cmml\">h</mi>\n            <mo id=\"p1.1.m1.1.14.2.2.1.2a\" xref=\"p1.1.m1.1.14.2.2.1.1.cmml\">⁡</mo>\n            <mrow id=\"p1.1.m1.1.14.2.2.1.2.1\" xref=\"p1.1.m1.1.14.2.2.1.1.cmml\">\n              <mo stretchy=\"false\" id=\"p1.1.m1.1.10\" xref=\"p1.1.m1.1.14.2.2.1.1.cmml\">(</mo>\n              <mi id=\"p1.1.m1.1.11\" xref=\"p1.1.m1.1.11.cmml\">t</mi>\n              <mo stretchy=\"false\" id=\"p1.1.m1.1.12\" xref=\"p1.1.m1.1.14.2.2.1.1.cmml\">)</mo>\n            </mrow>\n          </mrow>\n          <mo stretchy=\"false\" id=\"p1.1.m1.1.13\" xref=\"p1.1.m1.1.14.2.1.cmml\">}</mo>\n        </mrow>\n      </mrow>\n    </mrow>\n    <annotation-xml encoding=\"MathML-Content\" id=\"p1.1.m1.1b\">\n      <apply id=\"p1.1.m1.1.14.cmml\" xref=\"p1.1.m1.1.14\">\n        <eq id=\"p1.1.m1.1.6.cmml\" xref=\"p1.1.m1.1.6\"/>\n        <apply id=\"p1.1.m1.1.14.1.1.cmml\" xref=\"p1.1.m1.1.14.1.2\">\n          <csymbol cd=\"latexml\" id=\"p1.1.m1.1.1.cmml\" xref=\"p1.1.m1.1.1\">Q334269</csymbol>\n          <apply id=\"p1.1.m1.1.14.1.2.1.cmml\" xref=\"p1.1.m1.1.14.1.2.1\">\n            <times id=\"p1.1.m1.1.14.1.2.1.1.cmml\" xref=\"p1.1.m1.1.14.1.2.1.1\"/>\n            <csymbol cd=\"latexml\" id=\"p1.1.m1.1.3.cmml\" xref=\"p1.1.m1.1.3\">Q193796</csymbol>\n            <csymbol cd=\"latexml\" id=\"p1.1.m1.1.4.cmml\" xref=\"p1.1.m1.1.4\">Q12916</csymbol>\n          </apply>\n        </apply>\n        <apply id=\"p1.1.m1.1.14.2.1.cmml\" xref=\"p1.1.m1.1.14.2.2\">\n          <csymbol cd=\"latexml\" id=\"p1.1.m1.1.7.cmml\" xref=\"p1.1.m1.1.7\">Q6520159</csymbol>\n          <apply id=\"p1.1.m1.1.14.2.2.1.1.cmml\" xref=\"p1.1.m1.1.14.2.2.1.2\">\n            <csymbol cd=\"latexml\" id=\"p1.1.m1.1.9.cmml\" xref=\"p1.1.m1.1.9\">Q1139726</csymbol>\n            <csymbol cd=\"latexml\" id=\"p1.1.m1.1.11.cmml\" xref=\"p1.1.m1.1.11\">Q11471</csymbol>\n          </apply>\n        </apply>\n      </apply>\n    </annotation-xml>\n    <annotation encoding=\"application/x-tex\" id=\"p1.1.m1.1c\">H(j\\omega)=\\mathcal{F}\\{h(t)\\}</annotation>\n  </semantics>\n</math>",
  "uri" : "https://en.formulasearchengine.com/w/index.php?oldid=8338#math8338.87",
  "title" : "LTI_system_theory",
  "type" : "relation"
}