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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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        <phrase>joint time-frequency analysis</phrase>
        <phrase>time-frequency analysis, joint</phrase>
        <phrase>time-frequency representations</phrase>
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        <phrase>time-frequency representations</phrase>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Example - The Choi-Williams Distribution</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 advantage of formulating a BTFR in terms of Cohen's general approach is that the kernels may be specified at will. A number of time-frequency representations have been invented by designing kernels that would result in representations with certain desired properties. The Choi-Williams representation is one such BTFR. It generates smoothed cross terms and satisfies the marginals. However, it does not have finite time and frequency support.</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 time-lag kernel for the Choi-Williams distribution is:</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">Look at the plots below. They represent the bat chirp signal for different values of damping. In general, as the dampening parameter goes to infinity, the Choi-Williams distribution approaches the Wigner distribution.</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">A third "streak" appears between the original two because the higher the damping factor, the more the Choi-Williams representation approximates the Wigner representation.</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 Born-Jordan and cone representations are related. The Born-Jordan distribution satisfies the time and frequency marginals, as well as the finite support property. The cone kernel representation satisfies only the time marginal property, but results in visually precise plots. The two representations are related by a scaling factor. The Born-Jordan is equal to the cone scaled by <region region-id="5108" left="204" top="928.5" width="29.25" height="34.5" align-x="216.75" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:plus /><ml:real>1</ml:real><ml:id xml:space="preserve">τ</ml:id></ml:apply></ml:apply></math><rendering item-idref="14" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr></f></p>
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        <phrase>Born-Jordan distribution</phrase>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - Custom Representations</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">When constructing a kernel matrix, consider how its size is associated to the physical dimensions of the kernel:</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
          <f size="14">
            <b>
              <c val="#008080">• </c>
            </b>
          </f>rows of kernel matrix: time variable</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
          <f size="14">
            <b>
              <c val="#008080">• </c>
            </b>
          </f>columns of kernel matrix: lag variable.</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The number of columns determines the number of frequencies calculated, approximately twice the number of columns. The kernel matrix is set up in this way because the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">timefreq</inlineAttr></b></f> function first calculates the local autocorrelation by internally calling the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">timecorr</inlineAttr></b></f> function, and then computes the FFT of the result.</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Finally, note that you need only provide the step size and the kernel at a minimum when calculating your custom distribution, The number of frequencies is determined from the number of columns of the kernel, and the string "custom" is optional.</p>
      </text>
      <indexes>
        <phrase>kernel matrix, constructing</phrase>
        <phrase>timefreq function</phrase>
        <phrase>function, timefreq</phrase>
        <phrase>timecorr function</phrase>
        <phrase>function, timecorr</phrase>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - timecorr Function</p>
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      <indexes>
        <phrase>timecorr function</phrase>
        <phrase>function, timecorr</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<f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"> timecorr</inlineAttr></b></f> function calculates the time-dependent autocorrelation function of a signal for a certain kernel. The time-dependent autocorrelation of a signal s(t) is defined as:</p>
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          <tab />
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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">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">where </inlineAttr>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> is the </inlineAttr>
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          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr font-style="normal" underline="false" line-through="false">kernel</inlineAttr>
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          </f>
          <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
            <f family="Times New Roman" charset="0" size="12">.</f>
          </inlineAttr> The above function is also called the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">deterministic generalized local autocorrelation function, </inlineAttr></i></f><inlineAttr font-style="normal" line-through="false"><f family="Times New Roman" charset="0">because it calculates the autocorrelation of s(t) locally using a deterministic formulation. The result is a matrix, whose rows indicate different values of the lag variable </f></inlineAttr><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">t</inlineAttr></f><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><f family="Times New Roman" charset="0" size="12"> and whose columns indicate the time t.</f></inlineAttr></p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">The kernel determines the type of autocorrelation.</inlineAttr>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> The meaning of the input parameters is identical to that for the</inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr font-style="normal" underline="false" line-through="false"> timefreq </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">function, except that the parameter f is the number of </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <i>
              <inlineAttr font-weight="normal" underline="false" line-through="false">lags, </inlineAttr>
            </i>
          </f>
          <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
            <f family="Times New Roman" charset="0" size="12">not the number of frequencies to compute</f>
          </inlineAttr>
          <i>
            <f family="Times New Roman" charset="0" size="12">
              <inlineAttr font-weight="normal" underline="false" line-through="false">. </inlineAttr>
            </f>
          </i>
          <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
            <f family="Times New Roman" charset="0" size="12">The results of the two functions are related: the </f>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-style="normal" underline="false" line-through="false">
              <b>timefreq</b>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> function calls </inlineAttr>
          </f>
          <inlineAttr font-style="normal" underline="false" line-through="false">
            <b>
              <f family="Times New Roman" charset="0" size="12">timecorr </f>
            </b>
          </inlineAttr>
          <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
            <f family="Times New Roman" charset="0" size="12">function to compute the results in the time-lag domain, and then takes the Fourier transform of the result matrix, along the lag dimension (along each columns individually).</f>
          </inlineAttr>
        </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
            <f family="Times New Roman" charset="0" size="12">Sometimes it is useful to display the time-dependent autocorrelation, because we may be interested in how a process changes over time. For example, time-varying ARMA processes, equalizer design, and Kalman filtering are instances in which the time-dependent autocorrelation function of a certain type may be useful.</f>
          </inlineAttr>
        </p>
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      <indexes>
        <phrase>time-dependent autocorrelation function</phrase>
        <phrase>autocorrelation function, time-dependent</phrase>
        <phrase>kernel</phrase>
        <phrase>deterministic generalized local autocorrelation function</phrase>
        <phrase>function, deterministic generalized local autocorrelation</phrase>
        <phrase>ARMA, time-varying processes</phrase>
        <phrase>Kalman filtering</phrase>
        <phrase>filtering, Kalman</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">The example below shows how to compute the autocorrelation using the kernel for the Wigner distribution.</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 resultant matrix has 64 columns and approximately <region region-id="4907" left="258" top="2068.5" width="22.5" height="33.75" align-x="264.75" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:div /><ml:id xml:space="preserve">N</ml:id><ml:id xml:space="preserve">step</ml:id></ml:apply></math><rendering item-idref="26" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> rows. We can observe the</inlineAttr></f> result on a 3-D perspective plot.</p>
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        <p style="Heading 1" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Acknowledgments</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We wish to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this tutorial.</p>
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