He ohio s ate t university slide 23 ece 600 fir

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: NIVERSITY Slide # 21    ¤ £ ¤ ¤ £ ¤ Wgt   ¥ ¥ § ¦ § ¡ § ¡ § ¡ § ¡ § ¡  # ¥  ¡   ¡ ¡ ¨     ¢ ¤ £ ¤ ¤ £ ¤ ¤ £ ¤ ¥ § ¦ § ¦ § ¡ § ¡ § ¡ § ¡ ¥  ¡        " ¡ ¡ ¨ ¨   ¢  ¤ £ ¤ ¤ £ ¤ # ¢ ¢ ¥ § ¦ § ¡ § ¡ § ¡ Projection Theorem: Normal Equations ECE 600 FIR Design &  ¡ ¡  " " ¡ ¡ ¢ "  § ¡ § ¢ £¡ ¥ ¦¤ § ¢ £¡ ¥¤ § ¡  Wgt #$ ¨  ¡ ¥  ¡ ¡ ¨  ¢ ¤ £ ¤ ¥ § ¡ ¦  © ¨ § ¥ ¢  ¨  ¡ ¡  ¤ £ ¤ § ¡  £ ¡ ¢ ¦  © ¨ § ¥ © ¦   ¢ ¨ § ¨ ¢  ¡ ¢ ¥ ¦¤  £ ¢¡ ¥¤ § ¦  © ¨ § For example, consider # ! # #$ ¨ Wgt § Error  ¡ ¢¡ ¥¤ § % $ Equiripple Design even ( odd): OHIO S ATE T T.H.E UNIVERSITY Slide # 22 Equiripple Design: The Approximation Error ECE 600 FIR Design T.H.E OHIO S ATE T UNIVERSITY Slide # 23 ECE 600 FIR Design Wgt    ¢  ¤   ¦ ¢ §% ¤ ¤ ¨ ¨ ¥§ ¤ . . . $ # # # #  ¢ ¥§ ¥§ ¨ § ¨ "  © © © © ¥ ¨ "  % ¦ § ¨ ¨ ¥§   $ # # # # %  © © © © . . . ¨    ¥§   ¦  " " " ¡ ¨ ¦ § ¡ § # § ¡ § ¢ £¡ ¥ ¦¤   ¡ § ¥       ¢ ¢ ¢ "  ¨ ¨   £ £ ¤ § ¡ § ¡ ¡ ¡  & ¢ in ¥§ ¤ ¦ ¦    §   values   ¢ &  ¦ ¤ ¨ $ # # # # # #  . . . Wgt . . .   ¦  ¥§   ¨ Wgt ! ¤ ¨     ¥§ ¨ and ¦ . . .   ¥§   ¨     § ¡ Chebyshev 1882 There must exist at least  ¦  ¥§    © © © © © © . . .       Wgt ¡ ¡ ¨ ¥§  ¥ ¨ ¥§ . . . ¦ ¦ ' ( Equiripple Design such that , the th order Tchebyshev polynomial OHIO S ATE T T.H.E UNIVERSITY Slide # 24 ¨ ¡ # ¨  ¢  ¡   ¥  ¡ ¥ £ ¥ !    ¥   !    ¡  ¨ £ § ¢    ¡ ¡ § # & ¨ #  " & ¨  ¢ % 0 ¢  ¢ " pass band stop band  ¢ Equiripple Design Example: Lowpass Example: Equiripple FIR filter design 1.2 FIR design, L=15 tolerance 1 magnitude response, A 0.8 0.6 0.4 0.2 0 −0.2 −0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized frequency, theta/pi ECE 600 FIR Design T.H.E OHIO S ATE T UNIVERSITY Slide # 25 Multi-band Equiripple Design for each band; weight is $ 2. Specify tolerance #$ 1. Identify band edges and desired amplitudes 3. Compute approximate order (and specify type III or IV, if necessary) $ 4. Solve optimization using Remez exchange (Parks-McClellan algorithm) $ 5. If specifications are met, try reducing ; if specs are not met, increase . Return to 4. Equiripple design allows separate tolerance in different bands ECE 600 FIR Design T.H.E OHIO S ATE T UNIVERSITY Slide # 26 Example: Multi-band Equiripple Design ECE 600 FIR Design T.H.E OHIO S ATE T UNIVERSITY Slide # 27  © ¨ (GLP, type IV) £ £ ¥  ¢ £ §  ¥ © ¦ §¥ ¦ §¥ ¨ ¥  ¦ ¢ ¢  ¥ ¨ ¥¨ ¤¥ ¨ § £ ¥ ¡ ¤ ¡ ¨ ¤ ¡ § ¦¥ Example: Differentiator ¥    Example, >> h=remez(15,[0 1],[0 pi/T],'differentiator') N=15, fs=1000 N=15, fs=1000 3500 1500 3000 1000 impulse response magnitude response 2500 2000 1500 500 0 −500 1000 −1000 500 0 0 100 200 300 frequency (Hz) ECE 600 FIR Design 400 500 −1500 0 5 10 15 sample index T.H.E OHIO S ATE T UNIVERSITY Slide # 28...
View Full Document

This note was uploaded on 01/15/2014 for the course ECE 600 taught by Professor Clymer,b during the Winter '08 term at Ohio State.

Ask a homework question - tutors are online