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<blockquote data-quote="Frank Koenig" data-source="post: 147984" data-attributes="member: 416"><p>Re: FIR filters</p><p></p><p></p><p></p><p>That's exactly right. We might call this a "short frequency Fourier transform". The rationale being that windowing on the frequency side allows the windows to be of logarithmic width and spacing to better match perception. The Hilbert transform is implicit in the (inverse) Fourier transform back to the time domain -- we just set the negative frequencies to zero.</p><p></p><p>Dave Gunness presented a paper* describing essentially this representation. His twist was that, for computational expediency, he computed the time-domain convolution of the impulse response with an an approximate Gaussian by using multiple passes of a first order IIR filter in both directions. This converges very closely to a Gaussian after just a few passes. My implementation uses brute force in the frequency domain. It takes a few seconds to compute in R on a 10 year old desk top PC, so who cares?</p><p></p><p>Mark, I appreciate your perspective on all this, it gets me, and others I assume, thinking. I have "Signal Analysis" by Papoulis sitting on my shelf and he covers FM, the method of stationary phase, instantaneous frequency, ambiguity, etc., but some of it is tough sledding -- maybe too tough for this old dog. Thanks to computers and tools like R I can take a more empirical approach. All for now. Thanks everyone.</p><p></p><p>--Frank</p><p></p><p> </p><p>*"A Spectrogram Display for Loudspeaker Transient Response"</p><p>David W. Gunness, William R. Hoy</p><p>Convention Paper 6568, AES 119th Convention, October 7-10, 2005</p></blockquote><p></p>
[QUOTE="Frank Koenig, post: 147984, member: 416"] Re: FIR filters That's exactly right. We might call this a "short frequency Fourier transform". The rationale being that windowing on the frequency side allows the windows to be of logarithmic width and spacing to better match perception. The Hilbert transform is implicit in the (inverse) Fourier transform back to the time domain -- we just set the negative frequencies to zero. Dave Gunness presented a paper* describing essentially this representation. His twist was that, for computational expediency, he computed the time-domain convolution of the impulse response with an an approximate Gaussian by using multiple passes of a first order IIR filter in both directions. This converges very closely to a Gaussian after just a few passes. My implementation uses brute force in the frequency domain. It takes a few seconds to compute in R on a 10 year old desk top PC, so who cares? Mark, I appreciate your perspective on all this, it gets me, and others I assume, thinking. I have "Signal Analysis" by Papoulis sitting on my shelf and he covers FM, the method of stationary phase, instantaneous frequency, ambiguity, etc., but some of it is tough sledding -- maybe too tough for this old dog. Thanks to computers and tools like R I can take a more empirical approach. All for now. Thanks everyone. --Frank *"A Spectrogram Display for Loudspeaker Transient Response" David W. Gunness, William R. Hoy Convention Paper 6568, AES 119th Convention, October 7-10, 2005 [/QUOTE]
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