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He ohio s ate t university slide 23 ece 600 fir

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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...
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