hamming

DIGITAL FILTERS—I 103Figure 14.IIFigure 14.IIIIf we now shift these two least squares derived smoothing formulas to their proper places about the pointn we would haveWe now ask what will come out if we put in a pure eigenfunction. We know because the equations arelinear they should give the eigenfunction back, but multiplied by the eigenvalue corresponding to theeigenfunction’s frequency, the transfer function value at that frequency. Taking the top of the twosmoothing formulas we haveHence the eigenvalue at the frequency ω (the transfer function) is, by elementary trigonometry,In the parabolic smoothing case we will getThese are easily sketched along with the 2k+1 smoothing by straight line curves, Figure 14.IV.