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2. Prologue
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3. Introduction
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4. Characterization of Random Signals
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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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10. The Wiener filter
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The restoration case: noise
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Using the least mean-square error criterion
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Expressing the mean-square error
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Correlation returns
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The Wiener-Hopf equation
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As seen from the Fourier domain
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What is that least mean-square error?
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Classic example, classic result
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The more general restoration case: noise & distortion
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Example: Sound of (distorted) music
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Example: Cleaning up our act
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Determining the Wiener filter
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Problems
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Problem 10.1
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Problem 10.2
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Problem 10.3
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Problem 10.4
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Laboratory Exercise 10.1
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Laboratory Exercise 10.2
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Laboratory Exercise 10.4
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Laboratory Exercise 10.5
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<a href="#the-restoration-case-noise" class="md-nav__link">
The restoration case: noise
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<a href="#using-the-least-mean-square-error-criterion" class="md-nav__link">
Using the least mean-square error criterion
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Expressing the mean-square error
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<a href="#correlation-returns" class="md-nav__link">
Correlation returns
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The Wiener-Hopf equation
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<a href="#as-seen-from-the-fourier-domain" class="md-nav__link">
As seen from the Fourier domain
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What is that least mean-square error?
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Classic example, classic result
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<a href="#the-more-general-restoration-case-noise-distortion" class="md-nav__link">
The more general restoration case: noise & distortion
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Why we avoid the inverse filter
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Example: Sound of (distorted) music
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Example: Cleaning up our act
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Determining the Wiener filter
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Problems
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Problem 10.1
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Problem 10.2
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Problem 10.3
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Problem 10.4
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Laboratory Exercises
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Laboratory Exercise 10.1
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Laboratory Exercise 10.2
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Laboratory Exercise 10.3
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Laboratory Exercise 10.4
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Laboratory Exercise 10.5
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<h1 id="the-wiener-filter">The Wiener filter<a class="headerlink" href="#the-wiener-filter" title="Permanent link">¶</a></h1>
<p>In the previous chapter we showed that a desired effect, a maximized SNR, could be achieved by the suitable choice of a linear filter, a matched filter. We will now address a more difficult problem: the use of linear filters to estimate a <em>stochastic</em> signal in the presence of noise. We let <span class="arithmatex">\(x[n]\)</span> be a stochastic, ergodic signal from a process with known
statistics.</p>
<h2 id="the-restoration-case-noise">The restoration case: noise<a class="headerlink" href="#the-restoration-case-noise" title="Permanent link">¶</a></h2>
<p>In the presence of noise and using a linear filter, we wish to produce an <em>estimate</em> of <span class="arithmatex">\(x[n]\)</span> which we call <span class="arithmatex">\({x_e}[n].\)</span> This is frequently termed signal <em>restoration</em> because we are attempting to restore a signal <span class="arithmatex">\(x[n]\)</span> that has become damaged, corrupted, and/or noisy. We want the <em>best</em> estimate where the definition of “best” will be explained. We assume that <span class="arithmatex">\(x[n]\)</span> is real as is the noise process <span class="arithmatex">\(N[n].\)</span> The total signal <span class="arithmatex">\(r[n]\)</span> composed of <span class="arithmatex">\(x[n]\)</span> and <span class="arithmatex">\(N[n]\)</span> is to be processed to produce the estimate <span class="arithmatex">\({x_e}[n]\)</span> of the original <span class="arithmatex">\(x[n].\)</span> The model is shown in <a href="#fig:fig_Wiener1">Figure 10.1</a>.</p>
<figure class="figaltcap fullsize" id="fig:fig_Wiener1"><img src="images/Fig_10_1.gif" /><figcaption><strong>Figure 10.1:</strong> LTI filter <span class="arithmatex">\(h[n\rbrack\)</span> to estimate a stochastic signal <span class="arithmatex">\(x[n\rbrack\)</span> in the presence of additive noise.</figcaption>
</figure>
<p>We start with:</p>
<div class="" id="eq:Weq1">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.1)</td>
<td class="eqTableEq">
<div>$${x_e}[n] = \sum\limits_{m = - \infty }^{ + \infty } {r[n - m]} h[m]$$</div>
</td>
</tr>
</table>
</div>
<p>and we define “best” in terms of a measure of the difference (error) between the “true” stochastic signal <span class="arithmatex">\(x[n]\)</span> and the estimate <span class="arithmatex">\({x_e}[n].\)</span></p>
<h2 id="using-the-least-mean-square-error-criterion">Using the least mean-square error criterion<a class="headerlink" href="#using-the-least-mean-square-error-criterion" title="Permanent link">¶</a></h2>
<p>Specifically we look at the mean (expected) square error:</p>
<div class="" id="eq:Weq2">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.2)</td>
<td class="eqTableEq">
<div>$$e = mean\text{-}squared\;error = E\left\{ {{{\left| {{x_e}[n] - x[n]} \right|}^2}} \right\}$$</div>
</td>
</tr>
</table>
</div>
<p>and we propose to choose <span class="arithmatex">\(h[n]\)</span> in order to minimize this error. We will need a basic result from least mean-square estimation theory and this can be found in <a href="Chap_13.html#appendix-i-mean-square-error-minimization">Appendix I</a>. </p>
<p>To determine a minimum, we look at:</p>
<div class="" id="eq:Weq3">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.3)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{\frac{{\partial e}}{{\partial h}}}&{ = \frac{\partial }{{\partial h}}E\left\{ {{{\left| {{x_e}[n] - x[n]} \right|}^2}} \right\}}\\
{\,\,\,}&{ = \frac{\partial }{{\partial h}}E\left\{ {{{\left( {\sum\limits_{m = - \infty }^{ + \infty } {r[n - m]} h[m] - x[n]} \right)}^2}} \right\} = 0}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>That is, we vary the filter to choose the one that gives the minimum mean-square error. Note that the restriction to real signals and systems ensures that we can replace the non-differentiable absolute value operation <span class="arithmatex">\({\left| \bullet \right|^2}\)</span> with the differentiable operation <span class="arithmatex">\({\left( \bullet \right)^2}\)</span> in <a href="Chap_10.html#eq:Weq3">Equation 10.3</a>. </p>
<p>The equation above, of course, yields an extremum but it can be shown that this is a minimum; see <a href="#problem-101">Problem 10.1</a>. The development of this approach follows those in the references Castleman<sup id="fnref:castleman1996"><a class="footnote-ref" href="#fn:castleman1996">1</a></sup>, Lee<sup id="fnref:lee1960"><a class="footnote-ref" href="#fn:lee1960">2</a></sup>, and Papoulis<sup id="fnref:papoulis1977"><a class="footnote-ref" href="#fn:papoulis1977">3</a></sup>.</p>
<h2 id="expressing-the-mean-square-error">Expressing the mean-square error<a class="headerlink" href="#expressing-the-mean-square-error" title="Permanent link">¶</a></h2>
<p>Based upon <a href="Chap_13.html#appendix-i-mean-square-error-minimization">Appendix I</a>, and starting from <a href="Chap_10.html#eq:Weq1">Equation 10.1</a> and <a href="Chap_10.html#eq:Weq2">Equation 10.2</a>, we have:</p>
<div class="" id="eq:Weq4">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.4)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
e&{ = E\left\{ {{{\left( {{x_e} - x} \right)}^2}} \right\} = E\left\{ {{{\left( {{x_e}[n] - x[n]} \right)}^2}} \right\}}\\
{\,\,\,}&{ = E\left\{ {{{\left( {r[n] \otimes h[n] - x[n]} \right)}^2}} \right\}}\\
{\,\,\,}&{ = E\left\{ {{{\left( {\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]h[k]} - x[n]} \right)}^2}} \right\}}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>We might have expected that the left-hand side of <a href="Chap_10.html#eq:Weq4">Equation 10.4</a> would be <span class="arithmatex">\(e[n]\)</span> instead of <span class="arithmatex">\(e.\)</span> The expectation operation, <span class="arithmatex">\(E\left\{ \bullet \right\},\)</span> combined with the assumption of ergodicity assures that the result <span class="arithmatex">\(e\)</span> is independent of <span class="arithmatex">\(n.\)</span></p>
<p>We now apply the key result—orthogonality—from <a href="Chap_13.html#eq:Ap1eq3">Equation 13.3</a> and rewrite this as:</p>
<div class="" id="eq:Weq5">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.5)</td>
<td class="eqTableEq">
<div>$$0 = \frac{{de}}{{d{h_i}}} = \,2E\left\{ {\left( {\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]h[k]} - x[n]} \right)r[n - i]} \right\}$$</div>
</td>
</tr>
</table>
</div>
<h2 id="correlation-returns">Correlation returns<a class="headerlink" href="#correlation-returns" title="Permanent link">¶</a></h2>
<p>Rearranging this expression gives:</p>
<div class="" id="eq:Weq6">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.6)</td>
<td class="eqTableEq">
<div>$$E\left\{ {x[n]r[n - i]} \right\} = E\left\{ {r[n - i]\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]} h[k]} \right\}$$</div>
</td>
</tr>
</table>
</div>
<p>The term on the left side of the equation is the cross-correlation function between <span class="arithmatex">\(x[n]\)</span> and <span class="arithmatex">\(r[n].\)</span> Because both signals are real, we have from <a href="Chap_5.html#eq:autoeven">Equation 5.5</a> and <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a>:</p>
<div class="" id="eq:Weq7">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.7)</td>
<td class="eqTableEq">
<div>$$E\left\{ {x[n]r[n - i]} \right\} = {\varphi _{xr}}[ - i] = \varphi _{rx}^*[i] = {\varphi _{rx}}[i]$$</div>
</td>
</tr>
</table>
</div>
<p>Because the signal <span class="arithmatex">\(r[n - i]\)</span> is not a function of <span class="arithmatex">\(k\)</span> and the order of the operations of summation and expectation can—with all the usual caveats—be reversed, the term on the right side of the equation can be rewritten as:</p>
<div class="" id="eq:Weq8">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.8)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
E\left\{ {\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]r[n - i]} h[k]} \right\} = \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{k = - \infty }^{ + \infty } {E\left\{ {r[n - k]r[n - i]} \right\}} h[k]
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>Again we have made use of <a href="Chap_4.html#eq:additive">Equation 4.13</a> which says that expectation as an operator distributes over sums.</p>
<h2 id="the-wiener-hopf-equation">The Wiener-Hopf equation<a class="headerlink" href="#the-wiener-hopf-equation" title="Permanent link">¶</a></h2>
<p>The result is an implicit expression for the optimum filter <span class="arithmatex">\(h[n]\)</span>:</p>
<div class="" id="eq:Weq9">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.9)</td>
<td class="eqTableEq">
<div>$${\varphi _{rx}}[k] = \sum\limits_{m = - \infty }^{ + \infty } {{\varphi _{rr}}[k - m]} h[m] = {\varphi _{rr}}[k] \otimes h[k]$$</div>
</td>
</tr>
</table>
</div>
<p>Note the way the variable names “<span class="arithmatex">\(m\)</span>” and “<span class="arithmatex">\(k\)</span>” are used in order to be
consistent with earlier notation, for example, <a href="Chap_5.html#eq:autoeven">Equation 5.5</a> and <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a>.</p>
<p>We distinguish between two cases of this famous equation, the <a href="https://en.wikipedia.org/wiki/Wiener%E2%80%93Hopf_method">Wiener-Hopf
equation</a>. The variable <span class="arithmatex">\(k\)</span> represents the interval over which the process is observed. In the first case, <span class="arithmatex">\(k > 0\)</span> and this represents the case of <em>causal</em> observations. That is, in principle, it is only possible to estimate <span class="arithmatex">\(x[n]\)</span> from data that have already been gathered. The estimate <span class="arithmatex">\({x_e}[n]\)</span> can only have a causal dependence on <span class="arithmatex">\(r[n]\)</span> and the (causal) filter choice <span class="arithmatex">\(h[n].\)</span> With this condition the solution of the Wiener-Hopf equation is extremely
difficult, far beyond what we introduce here.</p>
<p>We will concentrate, instead, on the case <span class="arithmatex">\(- \infty \le k \le + \infty\)</span> which admits a direct solution through application of Fourier techniques. While this case is somewhat less realistic for temporal signals (but not for spatial signals), it will enable us to develop some insights into the character of filters developed as solutions to the Wiener-Hopf equation. Such filters, independent of the conditions on <span class="arithmatex">\(k,\)</span> are known
as <em>Wiener filters</em> after a mathematical giant of the 20<sup>th</sup> century <a href="https://en.wikipedia.org/wiki/Norbert_Wiener">Prof. Norbert Wiener</a> (1894-1964).</p>
<h2 id="as-seen-from-the-fourier-domain">As seen from the Fourier domain<a class="headerlink" href="#as-seen-from-the-fourier-domain" title="Permanent link">¶</a></h2>
<p>We start by taking the Fourier transform of both sides of <a href="Chap_10.html#eq:Weq9">Equation 10.9</a> to produce:</p>
<div class="" id="eq:Weq10">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.10)</td>
<td class="eqTableEq">
<div>$${S_{rx}}(\Omega ) = {S_{rr}}(\Omega )H(\Omega )$$</div>
</td>
</tr>
</table>
</div>
<p>The desired filter is, of course, <span class="arithmatex">\(H(\Omega )\)</span> and this is given by:</p>
<div class="" id="eq:Weq11">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.11)</td>
<td class="eqTableEq">
<div>$$H(\Omega ) = \frac{{{S_{rx}}(\Omega )}}{{{S_{rr}}(\Omega )}}$$</div>
</td>
</tr>
</table>
</div>
<h2 id="what-is-that-least-mean-square-error">What is that least mean-square error?<a class="headerlink" href="#what-is-that-least-mean-square-error" title="Permanent link">¶</a></h2>
<p>Through a series of manipulations (following section 10.3 in Papoulis<sup id="fnref2:papoulis1977"><a class="footnote-ref" href="#fn:papoulis1977">3</a></sup>),
we show that the total minimum error <span class="arithmatex">\(e\)</span> is:</p>
<div class="mainresult" id="eq:Weq12">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.12)</td>
<td class="eqTableEq">
<div>$$e = \frac{1}{{2\pi }}\int\limits_{ - \pi }^{ + \pi } {\left( {{S_{xx}}(\Omega ) - {S_{rx}}(\Omega )H(\Omega )} \right)} d\Omega$$</div>
</td>
</tr>
</table>
</div>
<p>We start with <a href="Chap_10.html#eq:Weq4">Equation 10.4</a> and using some algebra and the definition of the autocorrelation function we find:</p>
<div class="" id="eq:Weq13">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.13)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
e&{ = E\left\{ {{{\left( {{x_e}[n] - x[n]} \right)}^2}} \right\} }\\
{\,\,\,}&{ = E\left\{ {{x^2}[n]} \right\} + E\left\{ {x_e^2[n]} \right\} - 2E\left\{ {{x_e}[n]x[n]} \right\}}\\
{\,\,\,}&{ = {\varphi _{xx}}[0] + E\left\{ {x_e^2[n]} \right\} - 2E\left\{ {{x_e}[n]x[n]} \right\}}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>Continuing with the use of <a href="Chap_10.html#eq:Weq1">Equation 10.1</a> this becomes a lengthy—somewhat inelegant— expression:</p>
<div class="" id="eq:Weq14">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.14)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
e&{ = {\varphi _{xx}}[0] + E\left\{ {x_e^2[n]} \right\} - 2E\left\{ {{x_e}[n]x[n]} \right\}}\\
{}&{ = {\varphi _{xx}}[0]\,\, + }\\