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2. Prologue
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5. Correlations and Spectra
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7. The Langevin Equation – A Case Study
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How big is big? How small is small?
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The Langevin velocity equation
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Power spectral density
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The Langevin position equation
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Expected value
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Autocorrelation function
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Power spectral density
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Tethered particle motion
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How big is big? How small is small? – redux
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Choosing the sampling period
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Expected value
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Autocorrelation function
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Power spectral density
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Descriptive statistic – variance
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One step further
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Problem 7.5
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<h1 id="the-langevin-equation-a-case-study">The Langevin Equation – A Case Study<a class="headerlink" href="#the-langevin-equation-a-case-study" title="Permanent link">¶</a></h1>
<p>In 1908 <a href="https://en.wikipedia.org/wiki/Paul_Langevin">Prof. Paul Langevin</a> described how a
physical system modeled by a differential equation responds to a stochastic input. His work resulted in the concept of the <a href="https://en.wikipedia.org/wiki/Langevin_equation">Langevin equation</a>. A particle of mass <span class="arithmatex">\(m\)</span> is subject to random collision forces, <span class="arithmatex">\(f(t),\)</span> from many small, molecules. The term <span class="arithmatex">\(f(t)\)</span> is frequently described as the driving force. Further, the assumption is usually made that <span class="arithmatex">\(f(t)\)</span> results from a stationary random process with a Gaussian probability density function and an autocorrelation function of the form <span class="arithmatex">\({\varphi _{ff}}(\tau) = {F_o}\delta (\tau ).\)</span> Non-coincident time samples are uncorrelated. The mathematical description of <span class="arithmatex">\(f(t)\)</span> is a Gaussian, white-noise, ergodic process.</p>
<p>The motion of the particle is held back (retarded) by a viscous, frictional force that is proportional to the velocity of the particle. The equation of motion can be written for the velocity as a function of time or the position as a function of time:</p>
<div class="" id="eq:langEqs1">
<table class="eqTable">
<tr>
<td class="eqTableTag">(7.1)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
m\frac{{dv(t)}}{{dt}} = f(t) - \lambda v\\
m\frac{{{d^2}x(t)}}{{d{t^2}}} = f(t) - \lambda \frac{{dx(t)}}{{dt}}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>where <span class="arithmatex">\(\lambda\)</span> is the viscous, damping coefficient and both <span class="arithmatex">\(m\)</span> and <span class="arithmatex">\(\lambda\)</span> are
greater than zero. This <em>stochastic differential equation</em> describes <a href="https://en.wikipedia.org/wiki/Brownian_motion">Brownian motion</a>. We shall begin with the velocity, <span class="arithmatex">\(v(t),\)</span> and consider only one spatial dimension.</p>
<p>In this case-study our intention is to illustrate how the tools we have developed can be used to analyze this random process. (See also <a href="#tethered-particle-motion">TPM</a>.) The tools, however, have been developed in the context of discrete-time so the first step is to transform this first-order differential equation into a difference equation. As so many problems are studied, simulated, measured, and controlled with digital systems, this is not an unreasonable first step.</p>
<h2 id="from-t-to-n">From <i><font face="Georgia">t</font></i> to <i><font face="Palatino">n</font></i><a class="headerlink" href="#from-t-to-n" title="Permanent link">¶</a></h2>
<p>There are several procedures to transform a continuous-time LTI system—one that is based upon a linear differential equation with constant coefficients with input <span class="arithmatex">\(f(t)\)</span> and output <span class="arithmatex">\(v(t)\)</span> or <span class="arithmatex">\(x(t)\)</span>—into a discrete-time difference equation with input <span class="arithmatex">\(f[n]\)</span> and output <span class="arithmatex">\(v[n]\)</span> or <span class="arithmatex">\(x[n].\)</span> These include 1) impulse-invariant sampling, 2) backward-difference equation, and 3) bilinear transformation. For a thorough discussion of this topic see Section 10.8 of Oppenheim<sup id="fnref:oppenheim1993"><a class="footnote-ref" href="#fn:oppenheim1993">1</a></sup>.</p>
<p>Which one of the three techniques should we use? All three have advantages and disadvantages. For the purpose of this textbook, we will use the impulse-invariant sampling as it is easy to understand and leads to expressions that are easy to analyze in a case-study such as this.</p>
<h3 id="impulse-invariant-sampling">Impulse-invariant sampling<a class="headerlink" href="#impulse-invariant-sampling" title="Permanent link">¶</a></h3>
<p>If the impulse response is bandlimited, we sample the impulse response at an appropriate rate (above the Nyquist frequency) and then use <em>z</em>-transform techniques to produce the difference equation. Thus, if <span class="arithmatex">\(h(t)\)</span> is the bandlimited impulse response of a Langevin equation whose samples are <span class="arithmatex">\(h[n] = h\left( {n{T_s}} \right),\)</span> the transfer function will be given by:</p>
<div class="" id="eq:zfilt_langEqs1">
<table class="eqTable">
<tr>
<td class="eqTableTag">(7.2)</td>
<td class="eqTableEq">
<div>$$H(z) = \sum\limits_{n = - \infty }^{ + \infty } {h[n]{z^{ - n}} = } \sum\limits_{n = - \infty }^{ + \infty } {h(n{T_s}){z^{ - n}}}$$</div>
</td>
</tr>
</table>
</div>
<p>where <span class="arithmatex">\({T_s}\)</span> is the suitably-chosen sampling period. The associated sampling frequency is, thereby, <span class="arithmatex">\({\omega _s} = 2\pi /{T_s}.\)</span></p>
<p>The impulse response of the differential equation for velocity in <a href="Chap_7.html#eq:langEqs1">Equation 7.1</a> is causal, stable, straightforward to determine, and given by:</p>
<div class="" id="eq:impr_langEqs1">
<table class="eqTable">
<tr>
<td class="eqTableTag">(7.3)</td>
<td class="eqTableEq">
<div>$${h_v}(t) = \left( {\frac{1}{m}} \right){e^{ - \left( {\lambda t/m} \right)}}u(t) = \left( {\frac{1}{m}} \right){e^{ - \left( {t/\theta } \right)}}u(t)$$</div>