Bounds of the difference of a bounded band-limited function - Signal Processing Stack Exchange most recent 30 from dsp.stackexchange.com 2019-12-14T02:38:08Z https://dsp.stackexchange.com/feeds/question/52953 https://creativecommons.org/licenses/by-sa/4.0/rdf https://dsp.stackexchange.com/q/52953 5 Bounds of the difference of a bounded band-limited function alwaystudent https://dsp.stackexchange.com/users/38528 2018-10-29T17:24:36Z 2018-10-31T17:50:23Z <p>For a continuous signal (function), we have Bernstein inequality : <span class="math-container">$$|{df(t)}/dt| \le 2AB\pi$$</span> where <span class="math-container">$A=\sup|f(t)|$</span> and <span class="math-container">$B$</span> is the bandwidth of <span class="math-container">$f(t)$</span>. The question is: is there a relationship for a discrete function <span class="math-container">$x[n]$</span> like this? <span class="math-container">$$|x[n] -x[n-1] | \le\ \mu\ W$$</span> where <span class="math-container">$$X[k] = \sum\limits_{n = 0}^{N - 1} {x[n]{e^{ - j\frac{{2\pi }}{N}nk}}}$$</span> is the DFT for <span class="math-container">$x[n]$</span>, <span class="math-container">$X[k]=0$</span> for <span class="math-container">$k&gt; W$</span>.</p> https://dsp.stackexchange.com/questions/52953/-/52977#52977 3 Answer by Matt L. for Bounds of the difference of a bounded band-limited function Matt L. https://dsp.stackexchange.com/users/4298 2018-10-31T10:10:08Z 2018-10-31T17:50:23Z <p>As shown in the answers to <a href="https://dsp.stackexchange.com/q/51617/4298">this question</a>, for continuous-time signals, the bound predicted by Bernstein's inequality is achieved with equality by a sinusoidal signal with a frequency equal to the upper frequency limit. </p> <p>In analogy to this, in this answer I'll show a bound on <span class="math-container">$\big|x[n]-x[n-1]\big|$</span> for a discrete-time sinusoidal signal <span class="math-container">$x[n]$</span> at angular frequency <span class="math-container">$W$</span>:</p> <p><span class="math-container">$$x[n]=A\sin(nW+\phi)\tag{1}$$</span></p> <p>For the signal <span class="math-container">$x[n]$</span> given by <span class="math-container">$(1)$</span>, the largest value of <span class="math-container">$\big|x[n]-x[n-1]\big|$</span> for all <span class="math-container">$n\in\mathbb{Z}$</span> occurs for two values <span class="math-container">$x[k]$</span> and <span class="math-container">$x[k-1]$</span> symmetrical to the point of the largest derivative of a (continuous) sinusoid, i.e., for <span class="math-container">$x[k]=A\sin(W/2)$</span> and <span class="math-container">$x[k-1]=A\sin(-W/2)$</span>. Consequently, the bound for a sinusoid with frequency <span class="math-container">$W$</span> is given by</p> <p><span class="math-container">$$\big|x[n]-x[n-1]\big|\le 2A\sin\left(\frac{W}{2}\right)\tag{2}$$</span></p> <p>This bound is achieved with equality for the signals</p> <p><span class="math-container">$$x[n]=A\sin\left(nW+\frac{(2l+1)W}{2}\right),\qquad l\in\mathbb{Z}\tag{3}$$</span></p> <p>Note that the constant <span class="math-container">$A$</span> does not generally equal the maximum amplitude <span class="math-container">$B=\max |x[n]|$</span> of <span class="math-container">$x[n]$</span>. Depending on the sampling phase, the maximum amplitude <span class="math-container">$B$</span> lies in the following interval:</p> <p><span class="math-container">$$A\cos\left(\frac{W}{2}\right)\le B\le A\tag{4}$$</span></p> <p>Consequently, we have</p> <p><span class="math-container">$$A\le\frac{B}{\cos\left(\frac{W}{2}\right)},\qquad 0&lt;W&lt;\pi\tag{5}$$</span></p> <p>Combining <span class="math-container">$(2)$</span> and <span class="math-container">$(5)$</span> we get</p> <p><span class="math-container">$$\big|x[n]-x[n-1]\big|\le 2B\tan\left(\frac{W}{2}\right),\qquad B=\max_n\big|x[n]\big|\tag{6}$$</span></p> <p>Note, however, that <span class="math-container">$(6)$</span> is only useful for <span class="math-container">$\tan(W/2)&lt;1$</span>, i.e., for <span class="math-container">$W&lt;\pi/2$</span> because for any <span class="math-container">$x[n]$</span> with <span class="math-container">$\max|x[n]|=B$</span> we must have</p> <p><span class="math-container">$$\big|x[n]-x[n-1]\big|\le 2B\tag{7}$$</span></p> <p>I believe that the bound <span class="math-container">$(6)$</span> holds for all discrete-time band-limited signals with a maximum frequency <span class="math-container">$W$</span> (i.e., with <span class="math-container">$X(e^{j\omega})=0$</span> for <span class="math-container">$|\omega|\in (W,\pi]$</span>), but I don't know how to show it.</p>