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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">SECTION 7 - JOINT TIME-FREQUENCY ANALYSIS</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Calculates the short-time Fourier transform (STFT) of a signal whose spectral characteristics change over time.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Returns a matrix whose columns contain the Fourier transform of the input signal at times separated by s samples. For a signal x(t), the STFT is defined in continuous time as:</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">where w(t) is a sliding real-valued window taper used to localize the influence of the transform around a specific time.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The following signal is composed of two sinusoids of different frequencies occurring at different times.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The STFT will contain information on the frequency content of the signal and the time at which the frequency is present. Calculate the STFT with 100 frequency samples, a time step of 30 and a Hanning window (window 4):</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The <f family="Times New Roman" charset="0"><b><inlineAttr font-style="normal" line-through="false">spectrogram</inlineAttr></b></f><f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><sp /></inlineAttr></i></f><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false">obtained can be</inlineAttr></f> plotted as the contour of the squared magnitude, with the normalized frequency on the x axis and time on the y axis:</p>
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        <phrase>spectrogram</phrase>
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    <region region-id="709" left="6" top="1326.75" width="298.5" height="114" align-x="6" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This plot shows not only the two frequencies, but also the time at which they occur. However, the spectrogram suffers from what is called the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">localization trade-off. </inlineAttr></i></f><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false">T</inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false">o achieve time localization (that is, the STFT pinpointing the times at which the sinusoids are active), requires a short window in the time domain. Conversely, good frequency localization requires a long time window. This trade-off is further dependent on the type of window used, the signal under consideration, time, and frequency. </inlineAttr></f></p>
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        <phrase>localization trade-off</phrase>
        <phrase>trade-off, localization</phrase>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To illustrate, calculate the STFT and the spectrogram of the same signal as above, but using a frequency window length of 300 instead of 100.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The STFT calculated with a long window has better frequency resolution, but its time resolution is not as good. This time-frequency trade-off can be avoided by using <b><u><link href="./timeFreq1.xmcd" title="Links to section" popup="false">other techniques in time-frequency analysis.</link></u></b></p>
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      <indexes>
        <phrase>time-frequency analysis</phrase>
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                <c val="#800000">Example - Custom Windowing</c>
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        <phrase>custom windowing</phrase>
        <phrase>windowing, custom</phrase>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The frequency window in this example is zero-padded up to the length of the Fourier transform. (Note: The length of the window cannot be longer than the length of the Fourier transform.)</p>
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        <phrase>Fourier transform</phrase>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Rescale the window coefficients so that the window has unit energy:</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Calculate the STFT and the spectrogram:</p>
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            <ml:id xml:space="preserve">stft</ml:id>
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              <ml:real>30</ml:real>
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