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DSP Filters and slopes?
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<blockquote data-quote="Phil Graham" data-source="post: 27281" data-attributes="member: 430"><p>Re: DSP Filters and slopes?</p><p></p><p></p><p></p><p>Douglas,</p><p></p><p>The perfect rectangular/brickwall low pass filter does exist, and its impulse response is the the sinc function. See the wikipedia article on the <a href="http://en.wikipedia.org/wiki/Sinc_filter" target="_blank">sinc filter</a>.</p><p></p><p>The catch with the sinc filter is that its impulse response is symmetric about time T=0 and mathematically extends to infinity in both the "positive" and "negative" time directions. Finite frequency behavior requiring infinite time behavior is a mathematical consequence of Fourier's work.</p><p></p><p>Since we can't have negative time in the real world, where stuff ends after it begins, we have to delay the final output to account for the fraction of the impulse response in the negative time direction necessary for the desired filter behavior.</p><p></p><p>The sinc function is therefore windowed and/or truncated to zero at some finite time length in positive and negative time. The width of this window in the negative time direction sets the necessary amount of total output delay to "fill" the filter in the time domain and make it function. This is where the tradeoff between brick wall behavior, lowest frequency for FIR, and the filter side artifacts (from windowing) comes into play. Its not simply a matter of programming limitations, there is physics in play.</p><p></p><p>Dan Lavry takes this the other way in his <a href="http://www.lavryengineering.com/documents/Sampling_Theory.pdf" target="_blank">sampling paper</a>, by showing how the reconstructed waveform may be viewed as the summation of sinc functions about each sample point. Here the sinc functions arise as a consequence of the reconstruction filter's theoretically perfect low pass behavior at the Nyquist frequency.</p></blockquote><p></p>
[QUOTE="Phil Graham, post: 27281, member: 430"] Re: DSP Filters and slopes? Douglas, The perfect rectangular/brickwall low pass filter does exist, and its impulse response is the the sinc function. See the wikipedia article on the [URL="http://en.wikipedia.org/wiki/Sinc_filter"]sinc filter[/URL]. The catch with the sinc filter is that its impulse response is symmetric about time T=0 and mathematically extends to infinity in both the "positive" and "negative" time directions. Finite frequency behavior requiring infinite time behavior is a mathematical consequence of Fourier's work. Since we can't have negative time in the real world, where stuff ends after it begins, we have to delay the final output to account for the fraction of the impulse response in the negative time direction necessary for the desired filter behavior. The sinc function is therefore windowed and/or truncated to zero at some finite time length in positive and negative time. The width of this window in the negative time direction sets the necessary amount of total output delay to "fill" the filter in the time domain and make it function. This is where the tradeoff between brick wall behavior, lowest frequency for FIR, and the filter side artifacts (from windowing) comes into play. Its not simply a matter of programming limitations, there is physics in play. Dan Lavry takes this the other way in his [URL="www.lavryengineering.com/documents/Sampling_Theory.pdf"]sampling paper[/URL], by showing how the reconstructed waveform may be viewed as the summation of sinc functions about each sample point. Here the sinc functions arise as a consequence of the reconstruction filter's theoretically perfect low pass behavior at the Nyquist frequency. [/QUOTE]
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