---------------------------- Original Message ---------------------------- Subject: Re: [music-dsp] Anyone using Chebyshev polynomials to approximate trigonometric functions in FPGA DSP From: "Ethan Duni" <ethan.d...@gmail.com> Date: Thu, January 21, 2016 2:34 am To: "A discussion list for music-related DSP" <music-dsp@music.columbia.edu> -------------------------------------------------------------------------- >>given the same order N for the polynomials, whether your basis set are >> the Tchebyshevs, T_n(x), or the basis is just set of x^n, if you come up >>with a min/max optimal fit to your data, how can the two polynomials be >>different? > > Right, if you do that you'll end up with equivalent answers (to within > numerical precision). > > The idea is that you avoid the cost of doing the iterative algorithm to get > the optimal polynomial, and instead you simply truncate the Chebyshev > expansion to the desired order to get an approximation. For well-behaved > target functions it should be quite close. The justification is that the > Chebyshev polynomials each look like solutions to the minimax problem (they > oscillate between +-1 and the Nth polynomial has N+1 extrema), and the > error from truncating a series is approximately proportional to the last > retained term, so truncating a Chebyshev expansion should resemble the > optimal polynomial. got it. �thanks for the explanation, Ethan. -- � r b-j � � � � � � � � � r...@audioimagination.com � "Imagination is more important than knowledge."
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