Elements of Modern Signal Processing Lecture 8 — January 28 , 2016

Last Time: We discussed quadrature rules, which are means of estimating integrals of functions using a weighted sum of function values. Among these, one of the simplest is the trapezoidal rule, which derives its name from the fact that it attempts to approximate the area under the curve using a trapezoid. Although it can be proved the accuracy of this rule increases quadratically with the number of quadrature points, in some cases the accuracy increases exponentially. We showed how this phenomenon can be understood in terms of the Fourier transform, as the accuracy can be related to the rate of decay of the Fourier coefficients of the function. For a smooth and periodic function these decay faster than any polynomial, which in turn explains the spectral accuracy of the trapezoidal rule for this class of functions.