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<blockquote data-quote="Frank Koenig" data-source="post: 147972" data-attributes="member: 416"><p>Re: FIR filters</p><p></p><p></p><p></p><p></p><p>Indeed. Good post. We're using the tools and language of signal analysis for system analysis by treating the impulse response as a signal, but it has few of the statistical properties, such a stationarity, that make signals amenable to analysis. I've got a fat book on time-frequency analysis (Carmon, Wen-Liang, and Torresani) that is entirely concerned with signals.</p><p></p><p>The current TFE distribution has the ridge appear as a (single valued) function of frequency. One can imagine a distribution with different concentration where the ridge would be a function of time. There's likely some sort of duality there.</p><p></p><p>As for the properties of the ridge of the current TFE distribution, I can make an intuitive argument that it resembles the group delay. The group delay of a particular frequency can be thought of as the delay of the envelope of an amplitude modulated signal (with modulation frequency << carrier frequency so that the sidebands are close in) with the carrier being the frequency in question. For each frequency band the distribution passes the impulse response through a band pass filter. Converting back to the time domain we can think of the resulting signal as amplitude modulated with a carrier frequency at the filter center frequency. The carrier frequency is familiar as the ringing we hear when passing audio through a narrow band-pass, as results, for example, from a sound system on the verge of feedback. The ridge of the distribution shows us the time at which the bulk of that frequency's energy arrives.</p><p></p><p>Imagine an impulse response that has a short burst of 1 kHz at 5 ms. We can agree that this impulse response represents a group delay of 5 ms at 1 kHz. The burst will be selected by the 1 kHz filter but its place in time will be preserved (the filter bank has uniform time delay due to its symmetrical impulse response). By displaying the magnitude of the analytic signal we're essentially displaying the envelope of that 500 Hz burst which will make a bump in the TFE representation at that time and frequency. How's that for hand waving? I added a smoothed GD calculation to my TFE code and plotted the results side-by side for the same data, so the experimentalist in me thinks I'm on the right track -- for the moment.</p><p></p><p>Best,</p><p></p><p>--Frank[ATTACH]154510[/ATTACH]</p></blockquote><p></p>
[QUOTE="Frank Koenig, post: 147972, member: 416"] Re: FIR filters Indeed. Good post. We're using the tools and language of signal analysis for system analysis by treating the impulse response as a signal, but it has few of the statistical properties, such a stationarity, that make signals amenable to analysis. I've got a fat book on time-frequency analysis (Carmon, Wen-Liang, and Torresani) that is entirely concerned with signals. The current TFE distribution has the ridge appear as a (single valued) function of frequency. One can imagine a distribution with different concentration where the ridge would be a function of time. There's likely some sort of duality there. As for the properties of the ridge of the current TFE distribution, I can make an intuitive argument that it resembles the group delay. The group delay of a particular frequency can be thought of as the delay of the envelope of an amplitude modulated signal (with modulation frequency << carrier frequency so that the sidebands are close in) with the carrier being the frequency in question. For each frequency band the distribution passes the impulse response through a band pass filter. Converting back to the time domain we can think of the resulting signal as amplitude modulated with a carrier frequency at the filter center frequency. The carrier frequency is familiar as the ringing we hear when passing audio through a narrow band-pass, as results, for example, from a sound system on the verge of feedback. The ridge of the distribution shows us the time at which the bulk of that frequency's energy arrives. Imagine an impulse response that has a short burst of 1 kHz at 5 ms. We can agree that this impulse response represents a group delay of 5 ms at 1 kHz. The burst will be selected by the 1 kHz filter but its place in time will be preserved (the filter bank has uniform time delay due to its symmetrical impulse response). By displaying the magnitude of the analytic signal we're essentially displaying the envelope of that 500 Hz burst which will make a bump in the TFE representation at that time and frequency. How's that for hand waving? I added a smoothed GD calculation to my TFE code and plotted the results side-by side for the same data, so the experimentalist in me thinks I'm on the right track -- for the moment. Best, --Frank[ATTACH=CONFIG]154510.vB5-legacyid=14063[/ATTACH] [/QUOTE]
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