{"id":287,"date":"2012-05-01T10:50:18","date_gmt":"2012-05-01T10:50:18","guid":{"rendered":"http:\/\/ahay.org\/blog\/?p=287"},"modified":"2015-09-03T13:17:32","modified_gmt":"2015-09-03T13:17:32","slug":"program-of-the-month-sfderiv","status":"publish","type":"post","link":"https:\/\/ahay.org\/blog\/2012\/05\/01\/program-of-the-month-sfderiv\/","title":{"rendered":"Program of the month: sfderiv"},"content":{"rendered":"

sfderiv<\/a> applies the first derivative filter. <\/p>\n

The algorithm implemented in this program is described in the paper<\/a> <\/p>\n

Pei, S.-C., and P.-H. Wang, 2001, Closed-form design of maximally flat FIR Hilbert transformers, differentiators, and fractional delayers by power series expansion<\/em><\/strong>: IEEE Trans. on Circuits and Systems, v. 48, No. 4, 389-398. <\/p>\n

It is based on the Taylor expansion of the inverse sine function
\n$$\\arcsin{x} = \\displaystyle \\sum_{n=0}^{\\infty} \\frac{(2n)!}{4^n\\,(n!)^2\\,(2n+1)}\\,x^{2n+1}$$
\nwhich turns into an expansion of the ideal derivative filter into a chain of digital filters. The order=<\/strong> parameter controls the order of the expansion and the accuracy-efficiency trade-off. The following example from rsf\/rsf\/sfderiv<\/a> shows the frequency responses and the impulse responses for differentiators of different orders <\/p>\n

\"\" \"\"<\/p>\n

An alternative is sfigrad<\/a>, which implements a simple first-order derivative. igrad is more efficient and adequate when computing derivatives of smooth functions. <\/p>\n

\"\" \"\"<\/p>\n

Previous programs of the month<\/h3>\n