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2. Prologue
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3. Introduction
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5. Correlations and Spectra
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6. Filtering of Stochastic Signals
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8. Characterizing Signal-to-Noise Ratios
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9. The Matched Filter
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10. The Wiener filter
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11. Aspects of Estimation
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12. Spectral Estimation
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12. Spectral Estimation
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The periodogram – unbiased?
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Windowed observations
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The periodogram – what about convergence?
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Other windows
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A family of windows
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Problem 12.2
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Problem 12.3
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Laboratory Exercise 12.1
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Laboratory Exercise 12.2
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Laboratory Exercise 12.3
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Laboratory Exercise 12.4
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The periodogram – unbiased?
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Windowed observations
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The periodogram – what about convergence?
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Other windows
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Laboratory Exercise 12.1
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Laboratory Exercise 12.2
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Laboratory Exercise 12.3
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Laboratory Exercise 12.4
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<h1 id="spectral-estimation">Spectral Estimation<a class="headerlink" href="#spectral-estimation" title="Permanent link">¶</a></h1>
<p>We may not have an analytical model of a signal, its autocorrelation function, or its power spectral density. An example of this problem was presented in <a href="Chap_5.html#predicting-the-natural-climate-a-case-study">Chapter 5</a>. It should be clear that the phenomenon of weather is far too complex to admit an analytical solution for the autocorrelation function of the temperature <span class="arithmatex">\({\varphi _{TT}}[k]\)</span> or its power spectral density <span class="arithmatex">\({S_{TT}}(\Omega ).\)</span> We must work with estimates of these quantities. </p>
<p>In the previous chapter we examined the central questions of estimation: bias and convergence applied to means, variances, and correlations. In this chapter we will look at the problem of estimating the power spectral density. </p>
<p>Our estimate of the power spectrum of a random process is <span class="arithmatex">\({I_N}(\Omega ).\)</span> That is <span class="arithmatex">\({I_N}(\Omega )\)</span> is the estimate of <span class="arithmatex">\({S_{xx}}(\Omega ) = {\mathscr{F}}\left\{ {{\varphi _{xx}}[k]} \right\}.\)</span> We begin with a stochastic signal whose power spectral density is shown in <a href="Chap_8.html#fig:stand_model2">Figure 8.2</a> and given, for <span class="arithmatex">\({S_o} = 1\)</span> and <span class="arithmatex">\(\alpha = 4/7,\)</span> by <a href="Chap_8.html#eq:snreq10">Equation 8.10</a>:</p>
<div class="" id="eq:SEeq1">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.1)</td>
<td class="eqTableEq">
<div>$${S_{xx}}(\Omega ) = \frac{{{S_o}}}{{1 + {\alpha ^2} - 2\alpha \cos \Omega }} = \frac{{49}}{{65 - 56\cos \Omega }}$$</div>
</td>
</tr>
</table>
</div>
<p>We use the spectral estimator:</p>
<div class="" id="eq:SEeq2">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.2)</td>
<td class="eqTableEq">
<div>$${I_N}(\Omega ) = {\mathscr{F}}\left\{ {{{\hat c}_{xx}}[k]} \right\} = \sum\limits_{k = 0}^{N - 1} {{{\hat c}_{xx}}[k]} \,{e^{ - j\Omega k}}$$</div>
</td>
</tr>
</table>
</div>
<p>But <span class="arithmatex">\({{{\hat c}_{xx}}[k]}\)</span> is, itself, formed with the aid of <a href="Chap_11.html#eq:Eeq39">Equation 11.39</a> from <span class="arithmatex">\(x[n]\)</span> and using basic Fourier theory:</p>
<div class="" id="eq:SEeq3">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.3)</td>
<td class="eqTableEq">
<div>$${I_N}(\Omega ) = \frac{1}{N}{\left| {X(\Omega )} \right|^2}$$</div>
</td>
</tr>
</table>
</div>
<h2 id="the-periodogram-unbiased">The periodogram – unbiased?<a class="headerlink" href="#the-periodogram-unbiased" title="Permanent link">¶</a></h2>
<p>This basic estimate is referred to in the literature as a <em>periodogram</em>. The issue we will now examine is whether the periodogram <span class="arithmatex">\({I_N}(\Omega )\)</span> is a good estimate of <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span> By now we understand that “good” relates to the issues of bias and convergence.</p>
<p>This bias is defined as:</p>
<div class="" id="eq:SEeq4">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.4)</td>
<td class="eqTableEq">
<div>$$B = E\left\{ {{I_N}(\Omega )} \right\} - {S_{xx}}(\Omega ) = B(\Omega )$$</div>
</td>
</tr>
</table>
</div>
<p>The term that must be investigated is:</p>
<div class="" id="eq:SEeq5">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.5)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{E\left\{ {{I_N}(\Omega )} \right\}}&{ = E\left\{ {\sum\limits_{k = 0}^{N - 1} {{{\hat c}_{xx}}[k]} \,{e^{ - j\Omega k}}} \right\}}\\
{\,\,\,}&{ = \sum\limits_{k = 0}^{N - 1} {E\left\{ {{{\hat c}_{xx}}[k]} \right\}\,} {e^{ - j\Omega k}}}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>As the only term inside the first set of braces that is stochastic is <span class="arithmatex">\({{{\hat c}_{xx}}[k]},\)</span> we can exchange the order of expectation and summation. From <a href="Chap_11.html#eq:Eeq40">Equation 11.40</a> we can replace the expectation as follows:</p>
<div class="" id="eq:SEeq6">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.6)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{E\left\{ {{I_N}(\Omega )} \right\}}&{ = \sum\limits_{k = 0}^{N - 1} {\left( {\frac{{N - \left| k \right|}}{N}} \right)} {\varphi _{xx}}[k]{e^{ - j\Omega k}}}\\
{\,\,\,}&{ = \sum\limits_{k = 0}^{N - 1} {\left( {1 - \frac{{\left| k \right|}}{N}} \right)} {\varphi _{xx}}[k]{e^{ - j\Omega k}}}\\
{\,\,\,}&{ = \sum\limits_{k = 0}^{N - 1} {{\varphi _{xx}}[k]{e^{ - j\Omega k}}} - \sum\limits_{k = 0}^{N - 1} {\frac{{\left| k \right|}}{N}{\varphi _{xx}}[k]{e^{ - j\Omega k}}} }
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>The first sum does not have the limits that we should expect from the formal definition of <span class="arithmatex">\({S_{xx}}(\Omega )\)</span>; the variable <span class="arithmatex">\(k\)</span> does not go from <span class="arithmatex">\(- \infty\)</span> to <span class="arithmatex">\(+ \infty.\)</span> See <a href="Chap_12.html#eq:SEeq9">Equation 12.9</a> below. The second sum in <a href="Chap_12.html#eq:SEeq6">Equation 12.6</a> shows clearly that the estimate is biased. Specifically, the term <span class="arithmatex">\(\left| k \right|/N\)</span> would seem to indicate that <span class="arithmatex">\(E\left\{ {{I_N}(\Omega )} \right\}\)</span> is an underestimate. Thus the periodogram based upon <span class="arithmatex">\({{{\hat c}_{xx}}[k]}\)</span> is a biased estimate of <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span></p>
<p>We might choose instead to consider the spectral estimate given by:</p>
<div class="" id="eq:SEeq7">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.7)</td>
<td class="eqTableEq">
<div>$${P_N}(\Omega ) = {\mathscr{F}}\left\{ {{c_{xx}}[k]} \right\} = \sum\limits_{k = 0}^{N - 1} {{c_{xx}}[k]} {e^{ - j\Omega k}}$$</div>
</td>
</tr>
</table>
</div>
<p>Once again our computation of the bias requires:</p>
<div class="" id="eq:SEeq8">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.8)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{E\left\{ {{P_N}(\Omega )} \right\}}&{ = E\left\{ {\sum\limits_{k = 0}^{N - 1} {{c_{xx}}[k]} {e^{ - j\Omega k}}} \right\}}\\
{\,\,\,}&{ = \sum\limits_{k = 0}^{N - 1} {E\left\{ {{c_{xx}}[k]} \right\}} {e^{ - j\Omega k}} = \sum\limits_{k = 0}^{N - 1} {{\varphi _{xx}}[k]{e^{ - j\Omega k}}} }
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>This last line follows from <span class="arithmatex">\({{c_{xx}}[k]}\)</span> being an unbiased estimate of <span class="arithmatex">\({{\varphi _{xx}}[k]}.\)</span> It might seem at this point that we have found an unbiased estimate of <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span> Our result <a href="Chap_12.html#eq:SEeq8">Equation 12.8</a>, however, is <em>not</em> the same as:</p>
<div class="" id="eq:SEeq9">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.9)</td>
<td class="eqTableEq">
<div>$${S_{xx}}(\Omega ) = {\mathscr{F}}\left\{ {{\varphi _{xx}}[k]} \right\} = \sum\limits_{k = - \infty }^{ + \infty } {{\varphi _{xx}}[k]} {e^{ - j\Omega k}}$$</div>
</td>
</tr>
</table>
</div>
<p>Once again the difference lies in the limits of the summation. In <a href="Chap_12.html#eq:SEeq8">Equation 12.8</a> the limits on <span class="arithmatex">\(k\)</span> are <span class="arithmatex">\(0 \le k \le N - 1\)</span> while in <a href="Chap_12.html#eq:SEeq9">Equation 12.9</a> the limits on <span class="arithmatex">\(k\)</span> are <span class="arithmatex">\(- \infty \le k \le + \infty.\)</span> The limits in <a href="Chap_12.html#eq:SEeq8">Equation 12.8</a> represent the fact that we have only a finite amount of data to estimate <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span></p>
<p>The conclusion is clear: Both estimates, <span class="arithmatex">\({I_N}(\Omega )\)</span> and <span class="arithmatex">\({P_N}(\Omega ),\)</span> are biased
estimates of the power spectral density <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span></p>
<h2 id="windowed-observations">Windowed observations<a class="headerlink" href="#windowed-observations" title="Permanent link">¶</a></h2>
<p>Our stochastic process is being observed through a “window” of width <span class="arithmatex">\(N.\)</span> This finite-length data record introduces a bias into our estimate of the power density spectrum. This window bias can be observed in both <span class="arithmatex">\({I_N}(\Omega )\)</span> (<a href="Chap_12.html#eq:SEeq6">Equation 12.6</a>) and <span class="arithmatex">\({P_N}(\Omega )\)</span> (<a href="Chap_12.html#eq:SEeq8">Equation 12.8</a>).</p>
<p>It is possible to rewrite our expression for <span class="arithmatex">\(E\left\{ {{I_N}(\Omega )} \right\}\)</span> and <span class="arithmatex">\(E\left\{ {{P_N}(\Omega )} \right\}\)</span> using the concept of a deterministic <em>window function</em>, <span class="arithmatex">\(w[n].\)</span></p>
<div class="" id="eq:SEeq10">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.10)</td>
<td class="eqTableEq">
<div>$$E\left\{ {{I_N}(\Omega )} \right\} = \sum\limits_{k = - \infty }^{ + \infty } {{w_I}[k]{\varphi _{xx}}[k]{e^{ - j\Omega k}}}$$</div>
</td>
</tr>
</table>
</div>
<div class="" id="eq:SEeq11">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.11)</td>
<td class="eqTableEq">
<div>$$E\left\{ {{P_N}(\Omega )} \right\} = \sum\limits_{k = - \infty }^{ + \infty } {{w_P}[k]{\varphi _{xx}}[k]{e^{ - j\Omega k}}}$$</div>
</td>
</tr>
</table>
</div>
<p>The two summations extend over an infinite data record and the influence of a finite data record, as well as the difference between the two estimates, has been transferred to the window function. </p>
<p><a class="smaller" href=""><em>Triangular (Bartlett) window:</em></a></p>
<div class="" id="eq:SEeq12">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.12)</td>
<td class="eqTableEq">
<div>$${w_I}[k] = \left\{ {\begin{array}{*{20}{c}}
{\frac{{N - \left| k \right|}}{N}}&{\left| k \right| < N}\\
0&{\left| k \right| \ge N}
\end{array}} \right.$$</div>
</td>
</tr>
</table>
</div>
<p><a class="smaller" href=""><em>Rectangular (Block) window:</em></a></p>
<div class="" id="eq:SEeq13">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.13)</td>
<td class="eqTableEq">
<div>$${w_P}[k] = \left\{ {\begin{array}{*{20}{c}}
1&{\left| k \right| < N}\\
0&{\left| k \right| \ge N}
\end{array}} \right.$$</div>
</td>
</tr>
</table>
</div>
<p>From Fourier theory we know that the influence of the window on the spectrum <span class="arithmatex">\({S_{xx}}(\Omega )\)</span> will be given by:</p>
<div class="" id="eq:SEeq14">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.14)</td>
<td class="eqTableEq">
<div>$$E\left\{ {{I_N}(\Omega )} \right\} = \frac{1}{{2\pi }}{W_I}(\Omega ) \otimes {S_{xx}}(\Omega )$$</div>
</td>
</tr>
</table>
</div>
<div class="" id="eq:SEeq15">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.15)</td>
<td class="eqTableEq">
<div>$$E\left\{ {{P_N}(\Omega )} \right\} = \frac{1}{{2\pi }}{W_P}(\Omega ) \otimes {S_{xx}}(\Omega )$$</div>
</td>
</tr>
</table>
</div>
<p>That is, multiplication in the time domain <span class="arithmatex">\(n\)</span> yields convolution in the frequency domain <span class="arithmatex">\(\Omega\)</span> where:</p>
<div class="" id="eq:SEeq16">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.16)</td>
<td class="eqTableEq">
<div>$${W_I}(\Omega ) = {\mathscr{F}}\left\{ {{w_I}[k]} \right\}\,\,\,\,\,\,\,\,\,{W_P}(\Omega ) = {\mathscr{F}}\left\{ {{w_P}[k]} \right\}$$</div>
</td>
</tr>
</table>
</div>
<p>Plots of the two windows <span class="arithmatex">\({w_I}[k]\)</span> and <span class="arithmatex">\({w_P}[k]\)</span> as well as their spectra <span class="arithmatex">\({W_I}(\Omega )\)</span> and <span class="arithmatex">\({W_P}(\Omega )\)</span> are given in <a href="#fig:fig_SEst1">Figure 12.1</a>. A comparison of various properties is useful.</p>
<p>The ideal window would have the form in the frequency domain of <span class="arithmatex">\(W(\Omega ) = 2\pi \,\delta \left( \Omega \right).\)</span> This is because the convolutions in <a href="Chap_12.html#eq:SEeq14">Equation 12.14</a> and <a href="Chap_12.html#eq:SEeq15">Equation 12.15</a> would then yield the desired spectrum, <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span> Instead we have spectra <span class="arithmatex">\({W_I}(\Omega )\)</span> and <span class="arithmatex">\({W_P}(\Omega )\)</span> that have wider <em>center</em> lobes than the impulse <span class="arithmatex">\(\delta \left( \Omega \right)\)</span> and that have <em>side</em> lobes that further contaminate the spectral estimate. Comparing the two spectra, we see that <span class="arithmatex">\({W_I}(\Omega )\)</span> has a wide center lobe but small side lobes while <span class="arithmatex">\({W_P}(\Omega )\)</span> has a narrower center lobe but larger side lobes, another example of a trade-off.</p>
<p>Once we have accepted the idea that windowing of the data is inevitable<sup id="fnref:windowed"><a class="footnote-ref" href="#fn:windowed">1</a></sup>, we can consider searching for other window shapes that might have certain desirable properties for spectral estimation.</p>
<figure class="figaltcap fullsize" id="fig:fig_SEst1"><img src="images/Fig_12_1.png" /><figcaption><strong>Figure 12.1:</strong> (<em>top</em>) Spectrum <span class="arithmatex">\({W_I}\left( \Omega \right)\)</span> associated with a triangular (Bartlett) window <span class="arithmatex">\({w_I}[n\rbrack\)</span>; (<em>bottom</em>) Spectrum <span class="arithmatex">\({W_P}\left( \Omega \right)\)</span> associated with a rectangular (Block) window <span class="arithmatex">\({w_P}[n\rbrack.\)</span> Note the trade-off between the width of the main lobe and the height of the side lobes.</figcaption>
</figure>
<p>The spectrum associated with windowing is not only affected by a window’s shape but by the window’s length as well. In <a href="#movie121">Movie 12.1</a> we show how the spectrum <span class="arithmatex">\({W_P}(\Omega )\)</span> is affected by the length (width) of the window <span class="arithmatex">\({w_P}[k].\)</span> </p>
<table class="imgtxt" style="margin:1em auto;" id="movie121">
<tr>
<td style="width:auto; vertical-align:middle; text-align:center;">
<video id="theVideo" src="media/Time-Bandwidth.m4v"
poster="media/PosterMovie_12.1.png" width=100% controls
onended="rewind()" style="border: 2px solid rgb(174,24,16)"></video>
</td>
</tr>
</table>
<figcaption style="margin: 0px 30px;">
<b>Movie 12.1:</b> Tradeoff between (<i>top</i>) window width and (<i>bottom</i>) spectral bandwidth. The window extends from $- N$ to $+ N.$ Notice, in particular, the width of the main, central spectral lobe as $N$ increases.
</figcaption>
<p>The width of this window is <span class="arithmatex">\(2N + 1\)</span> and, as <span class="arithmatex">\(N \to \infty,\)</span> the window approaches the ideal, impulse-like behavior, <span class="arithmatex">\(\delta \left( \Omega \right).\)</span> The exact form of <span class="arithmatex">\({W_P}(\Omega )\)</span> is given in <a href="Chap_12.html#eq:SEeq17">Equation 12.17</a>. </p>
<div class="" id="eq:SEeq17">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.17)</td>
<td class="eqTableEq">
<div>$${W_P}(\Omega ) = \frac{{\sin \left( {\Omega \left( {2N + 1} \right)/2} \right)}}{{\sin \left( {\Omega /2} \right)}}$$</div>
</td>
</tr>
</table>
</div>
<h2 id="the-periodogram-what-about-convergence">The periodogram – what about convergence?<a class="headerlink" href="#the-periodogram-what-about-convergence" title="Permanent link">¶</a></h2>
<p>In the previous section we looked at the issue of bias in spectral estimation. The issue of convergence, as we have seen earlier, is also pivotal. In general, however, the issue is too complicated to analyze here. The results for certain simplifying assumptions show that, as estimates, <span class="arithmatex">\({I_N}(\Omega )\)</span> and <span class="arithmatex">\({P_N}(\Omega )\)</span> are seriously deficient.</p>
<p>In other words, neither one provides an estimate whose variance goes to zero as <span class="arithmatex">\(N \to \infty.\)</span> This effect is illustrated in <a href="#fig:fig_SEst2">Figure 12.2</a> where the lack of convergence in the spectral estimate as <span class="arithmatex">\(N\)</span> increases is shown. As <span class="arithmatex">\(N\)</span> grows, the estimate of the spectrum at any given frequency <span class="arithmatex">\(\Omega\)</span> does not converge but, instead, continues to fluctuate and behave as an underestimate.</p>
<p>The theory to explain this has been worked out for Gaussian white noise with zero-mean and standard deviation <span class="arithmatex">\(\sigma.\)</span> It can be shown (Jenkins<sup id="fnref:jenkins1998"><a class="footnote-ref" href="#fn:jenkins1998">2</a></sup>) that:</p>
<div class="" id="eq:SEeq18">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.18)</td>
<td class="eqTableEq">
<div>$$Var\left\{ {{I_N}(\Omega )} \right\} = {\sigma ^4}\left( {1 + {{\left( {\frac{{\sin (N\Omega )}}{{N\sin \Omega }}} \right)}^2}} \right)$$</div>
</td>
</tr>
</table>
</div>
<p>Thus as <span class="arithmatex">\(N \to \infty\)</span> we have <span class="arithmatex">\(Var\left\{ {{I_N}\left( \Omega \right)} \right\} = {\sigma ^4}\)</span> and we conclude that the variance of the estimate does not go to zero. Neither the periodogram <span class="arithmatex">\({I_N}(\Omega )\)</span> nor <span class="arithmatex">\({P_N}(\Omega )\)</span> is a consistent estimate of <span class="arithmatex">\({S_{xx}}(\Omega ).\)</span> Further, as explained in Section 11.3.2 of Jenkins<sup id="fnref2:jenkins1998"><a class="footnote-ref" href="#fn:jenkins1998">2</a></sup> and illustrated in <a href="#fig:fig_SEst2">Figure 12.2</a>, the rapidity of the fluctuations in the spectral estimate increases as <span class="arithmatex">\(N\)</span> increases. </p>
<figure class="figaltcap fullsize" id="fig:fig_SEst2"><img src="images/Fig_12_2.png" /><figcaption><strong>Figure 12.2:</strong> The “true” power spectral density of a stochastic signal <span class="arithmatex">\({S_{xx}}(\Omega )\)</span> (in <strong><font color="darkred">dark red</font></strong>) and an <span class="arithmatex">\({I_N}(\Omega )\)</span> estimate based upon (<em>top</em>) <span class="arithmatex">\(N = 16\)</span> and <span class="arithmatex">\(N = 64\)</span> samples; (<em>middle</em>) <span class="arithmatex">\({I_N}(\Omega )\)</span> estimate based upon <span class="arithmatex">\(N = 256\)</span> and <span class="arithmatex">\(N = 1024\)</span> samples; (<em>bottom</em>) <span class="arithmatex">\({I_N}(\Omega )\)</span> estimate based upon <span class="arithmatex">\(N = 4096\)</span> and <span class="arithmatex">\(N = 16384\)</span> samples.</figcaption>
</figure>
<p>The most well-known technique for dealing with this problem is Bartlett’s procedure. We begin with <span class="arithmatex">\(M\)</span> samples of data, that is, <span class="arithmatex">\(\left\{ {x[0],x[1],x[2],...,x[M - 1]} \right\}.\)</span> We now split the <span class="arithmatex">\(M\)</span> samples up into <span class="arithmatex">\(K\)</span> non-overlapping, contiguous records, each of length <span class="arithmatex">\(N.\)</span> That is <span class="arithmatex">\(M = K \bullet N.\)</span> As an example we might have <span class="arithmatex">\(M = 2048\)</span> samples and choose <span class="arithmatex">\(K = 16\)</span> records with <span class="arithmatex">\(N = 128\)</span> samples/record.</p>
<p>For each record <span class="arithmatex">\(k = 1,2,...,16\)</span> we compute the periodogram <span class="arithmatex">\(I_N^{(k)}\left( \Omega \right),\)</span> that is, we compute <span class="arithmatex">\(K\)</span> different periodograms, one per record. We now form the Bartlett estimate of the spectrum:</p>
<div class="mainresult" id="eq:SEeq19">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.19)</td>
<td class="eqTableEq">
<div>$${B_I}(\Omega ) = \frac{1}{K}\sum\limits_{k = 1}^K {I_N^{(k)}(\Omega )}$$</div>
</td>
</tr>
</table>
</div>
<p>The Bartlett estimate is the arithmetic mean (average) of the <span class="arithmatex">\(K\)</span> periodograms. Of course, it is also possible to form the Bartlett estimate using <span class="arithmatex">\(P_N^{(k)}\left( \Omega \right)\)</span> giving:</p>
<div class="mainresult" id="eq:SEeq20">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.20)</td>
<td class="eqTableEq">
<div>$${B_P}(\Omega ) = \frac{1}{K}\sum\limits_{k = 1}^K {P_N^{(k)}(\Omega )}$$</div>
</td>
</tr>
</table>
</div>
<p>Because each of the <span class="arithmatex">\(K\)</span> records is independent of the other <span class="arithmatex">\(K - 1\)</span>
records—do you understand why?—we have:</p>
<div class="" id="eq:SEeq21">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.21)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
E\left\{ {{B_I}(\Omega )} \right\} = \frac{1}{K}\sum\limits_{k = 1}^K {E\left\{ {I_N^{(k)}(\Omega )} \right\}} = E\left\{ {I_N^{(k)}(\Omega )} \right\}\\
Var\left\{ {{B_I}(\Omega )} \right\} = \frac{1}{K}Var\left\{ {I_N^{(k)}(\Omega )} \right\}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<div class="" id="eq:SEeq22">
<table class="eqTable">
<tr>
<td class="eqTableTag">(12.22)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
E\left\{ {{B_P}(\Omega )} \right\} = \frac{1}{K}\sum\limits_{k = 1}^K {E\left\{ {P_N^{(k)}(\Omega )} \right\}} = E\left\{ {P_N^{(k)}(\Omega )} \right\}\\
Var\left\{ {{B_P}(\Omega )} \right\} = \frac{1}{K}Var\left\{ {P_N^{(k)}(\Omega )} \right\}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>Whether we choose <span class="arithmatex">\({{B_I}(\Omega )}\)</span> or <span class="arithmatex">\({{B_P}(\Omega )}\)</span> we can achieve a convergent
estimate of the spectrum by choosing a large value for <span class="arithmatex">\(K.\)</span> Thus the splitting of the <span class="arithmatex">\(M\)</span> data samples into <span class="arithmatex">\(K\)</span> records generates <span class="arithmatex">\(1/K\)</span> convergence as <span class="arithmatex">\(K \to M.\)</span> This convergence does not, however, come for free. For fixed <span class="arithmatex">\(M,\)</span> as we choose larger and larger values of <span class="arithmatex">\(K,\)</span> the value of <span class="arithmatex">\(N\)</span> (the number of samples per record) must decrease. Since the <em>bias</em> is only dependent on <span class="arithmatex">\(N\)</span> (and not <span class="arithmatex">\(K\)</span>), <a href="Chap_12.html#eq:SEeq21">Equation 12.21</a> and <a href="Chap_12.html#eq:SEeq22">Equation 12.22</a>, the bias will get worse as <span class="arithmatex">\(K\)</span> increases. Spectral estimation, like life, is filled with compromises.</p>
<p>Summarizing we see that for fixed <span class="arithmatex">\(M\)</span>:</p>
<p><span class="arithmatex">\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,N\)</span> large <span class="arithmatex">\(\Rightarrow\)</span> <span class="arithmatex">\(K\)</span> small <span class="arithmatex">\(\Rightarrow\)</span> good bias, poor convergence</p>
<p><span class="arithmatex">\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,N\)</span> small <span class="arithmatex">\(\Rightarrow\)</span> <span class="arithmatex">\(K\)</span> large <span class="arithmatex">\(\Rightarrow\)</span> poor bias, good convergence</p>
<p>An example of the application of this Bartlett technique to the data shown in <a href="#fig:fig_SEst2">Figure 12.2</a> is shown in <a href="#fig:fig_SEst3">Figure 12.3</a>.</p>
<figure class="figaltcap fullsize" id="fig:fig_SEst3"><img src="images/Fig_12_3.gif" /><figcaption><strong>Figure 12.3:</strong> Bartlett estimates of the spectrum for various values of <span class="arithmatex">\(M,\)</span> <span class="arithmatex">\(K,\)</span> and <span class="arithmatex">\(N\)</span> where <span class="arithmatex">\(M = K \bullet N.\)</span> The “true” spectrum is shown in <strong><font color="darkred">dark red</font></strong>.</figcaption>
</figure>
<p>The evolution of this Bartlett estimation of a “true” spectrum is shown in <a href="#movie122">Movie 12.2</a>. </p>
<table class="imgtxt" style="margin:1em auto;" id="movie122">
<tr>
<td style="width:auto; vertical-align:middle; text-align:center;">
<video id="theVideo" src="media/Bartlett_Estimate.m4v"
poster="media/PosterMovie_12.2.png" width=100% controls
onended="rewind()" style="border: 2px solid rgb(174,24,16)"></video>
</td>
</tr>
</table>
<figcaption style="margin: 0px 30px;">
<b>Movie 12.2:</b> Bartlett estimation with $M$ samples, $K$ records, and $N$ samples/record.
</figcaption>
<p>Once again we are confronted with a trade-off. The proper choice of <span class="arithmatex">\(M,\)</span> <span class="arithmatex">\(N,\)</span> and <span class="arithmatex">\(K\)</span> then becomes dependent on the specifics of the problem and what the goals and constraints of the estimation problem are.</p>
<h2 id="other-windows">Other windows<a class="headerlink" href="#other-windows" title="Permanent link">¶</a></h2>
<p>In the previous section we have looked at two windows: the Block (rectangular) window and the triangular window. The latter is also known as the Bartlett window. Once we have accepted the idea that windowing is inevitable, we might search for “optimum” windows, windows that give a “best” estimate of an underlying spectrum. In addition to the two
previously mentioned windows, we will describe the Gauss, Tukey, Hamming, Hann, Parzen, and Verbeek windows. The first five of these are well-described in the
<a href="https://en.wikipedia.org/wiki/Window_function#/">literature</a>; the last
one is described in <a href="#problem-122">Problem 12.2</a>.</p>
<p>These weighted windows have been proposed and analyzed and their various properties are summarized in <a href="#tbl:t01">Table 12.1</a> and <a href="#tbl:t02">Table 12.2</a>. It is important, however, to decide what criteria will be used to assess effectiveness of any specific window.</p>
<p>We would like to use as little data as possible to estimate a spectrum so this implies that the duration of the window should be as small as possible. In the frequency domain, however, the window spectrum is convolved with the true spectrum. This means that an ideal window has a spectral bandwidth that is as narrow, as impulse-like, as possible. Putting these two requirements together means using a window that has a small <em>time-bandwidth product</em>. A continuous-time Gaussian of infinite extent is known to fulfill this condition but we will be dealing with discrete-time windows of finite extent.</p>
<h2 id="a-family-of-windows">A family of windows<a class="headerlink" href="#a-family-of-windows" title="Permanent link">¶</a></h2>
<p>For purposes of comparison, we assume that all windows are real, even, and normalized such that <span class="arithmatex">\(w[n = 0] = 1.\)</span> As the windows are of finite extent and even, this means that they are identically zero for <span class="arithmatex">\(\left| n \right| > N\)</span> with an extent (duration) of <span class="arithmatex">\(2N + 1.\)</span> The windows are defined in <a href="#tbl:t01">Table 12.1</a> and their properties are compared in <a href="#tbl:t02">Table 12.2</a>.</p>
<div id="tbl:t01"></div>
<table class="T12_1">
<!--Header row -->
<thead>
<tr>
<th><i><b> Window</b></i></th>
<th><i><b>Definition w[n]</b></i></th>
<th><i><b>Graph w[n]</b></i></th>
</tr>
</thead>
<!--Footer row -->