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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalized spectrogram</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In order to view a signal (taken to be a function of time) represented over both time and frequency axis, <a href="Time%E2%80%93frequency_representation" title="Time–frequency representation">time–frequency representation</a> is used. <a href="Spectrogram" title="Spectrogram">Spectrogram</a> is one of the most popular time-frequency representation, and <b>generalized spectrogram</b>, also called "two-window spectrogram", is the generalized application of spectrogram.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>The definition of the spectrogram relies on the Gabor transform (also called short-time Fourier transform, for short STFT), whose idea is to localize a signal <span class="texhtml"><i>f</i></span> in time by multiplying it with translations of a window function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>w</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle w(t)}</annotation>
</semantics>
</math></span><img src="./327b8c0fa7ba996b29c530b495c276a7ebdea2ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.313ex; height:2.843ex;" alt="{\displaystyle w(t)}" loading="lazy"></span>.
</p><p>The definition of spectrogram is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}}</annotation>
</semantics>
</math></span><img src="./47fb0e930813033f871d4b2782046d5648ea4a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.769ex; height:3.509ex;" alt="{\displaystyle S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}}" loading="lazy"></span>,</dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {G_{x,{w_{1}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>G</mi>
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<annotation encoding="application/x-tex">{\displaystyle {G_{x,{w_{1}}}}}</annotation>
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</math></span><img src="./def868f28a2f4b5e199a4f2531c894dc365e126e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.465ex; height:2.843ex;" alt="{\displaystyle {G_{x,{w_{1}}}}}" loading="lazy"></span> denotes the <a href="Gabor_transform" title="Gabor transform">Gabor Transform</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>
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</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span>.
</p><p>Based on the spectrogram, the <b>generalized spectrogram</b> is defined as:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S{P_{x,{w_{1}},{w_{2}}}}(t,f)={G_{x,{w_{1}}}}(t,f)G_{_{x,{w_{2}}}}^{*}(t,f)}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle S{P_{x,{w_{1}},{w_{2}}}}(t,f)={G_{x,{w_{1}}}}(t,f)G_{_{x,{w_{2}}}}^{*}(t,f)}</annotation>
</semantics>
</math></span><img src="./fd45c50e795d01ba685a2d341c9e26c6903c5a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.657ex; height:3.009ex;" alt="{\displaystyle S{P_{x,{w_{1}},{w_{2}}}}(t,f)={G_{x,{w_{1}}}}(t,f)G_{_{x,{w_{2}}}}^{*}(t,f)}" loading="lazy"></span>,</dd></dl>
<p>where:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {G_{x,{w_{1}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{1}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {G_{x,{w_{1}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{1}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}</annotation>
</semantics>
</math></span><img src="./d9b5e1c01c458af751ccfee910b0f882c4f7cb11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.413ex; height:6.009ex;" alt="{\displaystyle {G_{x,{w_{1}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{1}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {G_{x,{w_{2}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{2}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {G_{x,{w_{2}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{2}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}</annotation>
</semantics>
</math></span><img src="./dbfcd9d13552cb78c2b6378a30e15ea115da6aa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.413ex; height:6.009ex;" alt="{\displaystyle {G_{x,{w_{2}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{2}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}" loading="lazy"></span></dd></dl>
<p>For <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}(t)=w_{2}(t)=w(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
<mi>w</mi>
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<annotation encoding="application/x-tex">{\displaystyle w_{1}(t)=w_{2}(t)=w(t)}</annotation>
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</math></span><img src="./2978706f8b8bcdc826e3bd3e914063faec2a5527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.245ex; height:2.843ex;" alt="{\displaystyle w_{1}(t)=w_{2}(t)=w(t)}" loading="lazy"></span>, it reduces to the classical spectrogram:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>,</mo>
<mi>w</mi>
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</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}}</annotation>
</semantics>
</math></span><img src="./47fb0e930813033f871d4b2782046d5648ea4a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.769ex; height:3.509ex;" alt="{\displaystyle S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}}" loading="lazy"></span></dd></dl>
<p>The feature of Generalized spectrogram is that the window sizes of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{1}(t)}</annotation>
</semantics>
</math></span><img src="./b3c4089e3ee708e9e727b5fb059ea4d338bbb6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.367ex; height:2.843ex;" alt="{\displaystyle w_{1}(t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{2}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{2}(t)}</annotation>
</semantics>
</math></span><img src="./10d1cacecdfdb1feb8521dc16efd97e221e7f062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.367ex; height:2.843ex;" alt="{\displaystyle w_{2}(t)}" loading="lazy"></span> are different. Since the time-frequency resolution will be affected by the window size, if one choose a wide <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{1}(t)}</annotation>
</semantics>
</math></span><img src="./b3c4089e3ee708e9e727b5fb059ea4d338bbb6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.367ex; height:2.843ex;" alt="{\displaystyle w_{1}(t)}" loading="lazy"></span> and a narrow <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{1}(t)}</annotation>
</semantics>
</math></span><img src="./b3c4089e3ee708e9e727b5fb059ea4d338bbb6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.367ex; height:2.843ex;" alt="{\displaystyle w_{1}(t)}" loading="lazy"></span> (or the opposite), the resolutions of them will be high in different part of spectrogram. After the multiplication of these two Gabor transform, the resolutions of both time and frequency axis will be enhanced.
</p>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<dl><dt>Relation with <a href="Wigner_distribution_function" title="Wigner distribution function">Wigner Distribution</a></dt>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)=Wig(w_{1}',w_{2}')*Wig(t,f)(x,w),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>W</mi>
<mi>i</mi>
<mi>g</mi>
<mo stretchy="false">(</mo>
<msubsup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mo>′</mo>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">)</mo>
<mo>∗<!-- ∗ --></mo>
<mi>W</mi>
<mi>i</mi>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)=Wig(w_{1}',w_{2}')*Wig(t,f)(x,w),}</annotation>
</semantics>
</math></span><img src="./6e298e059a406ab0e49de8dfe05e475d414cadd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.341ex; height:3.009ex;" alt="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)=Wig(w_{1}',w_{2}')*Wig(t,f)(x,w),}" loading="lazy"></span></dd>
<dd>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}'(s):=w_{1}(-s),w_{2}'(s):=w_{2}(-s)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<msubsup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mo>′</mo>
</msubsup>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{1}'(s):=w_{1}(-s),w_{2}'(s):=w_{2}(-s)}</annotation>
</semantics>
</math></span><img src="./60383e60b4fe28a33146b9b7a13a15f5620448fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.613ex; height:3.009ex;" alt="{\displaystyle w_{1}'(s):=w_{1}(-s),w_{2}'(s):=w_{2}(-s)}" loading="lazy"></span></dd>
<dt>Time marginal condition</dt>
<dd>The generalized spectrogram <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}</annotation>
</semantics>
</math></span><img src="./7b79bccf777856e8b44d28645dc92a043142f2c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.615ex; height:3.009ex;" alt="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}" loading="lazy"></span> satisfies the time marginal condition if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}w_{2}'=\delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msubsup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mo>′</mo>
</msubsup>
<mo>=</mo>
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{1}w_{2}'=\delta }</annotation>
</semantics>
</math></span><img src="./8dfa92d5218beb9fb05caf0a719f8c49b37518e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.584ex; height:3.009ex;" alt="{\displaystyle w_{1}w_{2}'=\delta }" loading="lazy"></span>,</dd>
<dd>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>
</semantics>
</math></span><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> denotes the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd>
<dt>Frequency marginal condition</dt>
<dd>The generalized spectrogram <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}</annotation>
</semantics>
</math></span><img src="./7b79bccf777856e8b44d28645dc92a043142f2c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.615ex; height:3.009ex;" alt="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}" loading="lazy"></span> satisfies the frequency marginal condition if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}w_{2}'=\delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msubsup>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mo>′</mo>
</msubsup>
<mo>=</mo>
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{1}w_{2}'=\delta }</annotation>
</semantics>
</math></span><img src="./8dfa92d5218beb9fb05caf0a719f8c49b37518e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.584ex; height:3.009ex;" alt="{\displaystyle w_{1}w_{2}'=\delta }" loading="lazy"></span>,</dd>
<dd>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>δ<!-- δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>
</semantics>
</math></span><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> denotes the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd>
<dt>Conservation of energy</dt>
<dd>The generalized spectrogram <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">S</mi>
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}</annotation>
</semantics>
</math></span><img src="./7b79bccf777856e8b44d28645dc92a043142f2c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.615ex; height:3.009ex;" alt="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}" loading="lazy"></span> satisfies the conservation of energy if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (w_{1},w_{2})=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (w_{1},w_{2})=1}</annotation>
</semantics>
</math></span><img src="./b63eaabbb8c612f23932cc3602da0bc78caa4bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.541ex; height:2.843ex;" alt="{\displaystyle (w_{1},w_{2})=1}" loading="lazy"></span>.</dd>
<dt>Reality analysis</dt>
<dd>The generalized spectrogram <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {SP}}_{w_{1},w_{2}}(t,f)(x,w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<ul><li><a rel="nofollow" class="external text" href="http://djj.ee.ntu.edu.tw/TFW.htm">Class notes of Time frequency analysis and wavelet transform -- from Prof. Jian-Jiun Ding's course website </a></li>
<li>P. Boggiatto, G. De Donno, and A. Oliaro, “<a rel="nofollow" class="external text" href="https://scholar.google.com.tw/scholar_url?url=http://nozdr.ru/data/media/biblioteka/kolxo3/M_Mathematics/MC_Calculus/MCf_Functional%2520analysis/Schulze%2520B.W.,%2520Wong%2520M.W.%2520(eds.)%2520Pseudo-differential%2520operators..%2520Complex%2520analysis%2520and%2520partial%2520differential%2520equations%2520(Birkhauser,%25202010)(ISBN%25203034601972)(O)(294s)_MCf_.pdf%23page%3D254&hl=zh-TW&sa=X&scisig=AAGBfm3k6US20Q9vL3As15cbbJLAFBdsDA&nossl=1&oi=scholarr&ei=gKW8VLroC5Ht8gWvr4DICw&ved=0CBoQgAMoADAA">Two window spectrogram and their integrals</a>," Advances and Applications, vol. 205, pp. 251–268, 2009.</li></ul>
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