# Approximate the desired response by combinations of

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Unformatted text preview: stems I; Win 09-10, Pauly 16 Improved Pole Lowpass Filter • Improve single pole lowpass ﬁlter by adding a complex pole pair – Flattens passband – Sharpens transition – Repeated pole would be 9 dB down at corner, this is 3 dB down. Bode Plot 20 Magnitude (dB) ! !c × −60 dB/decade 0 −3 dB !20 !40 !60 !1 10 0 1 10 10 ω/ωc × 200 × 100 Phase (deg) ! 0 !100 −!c !200 !1 10 0 1 10 Frequency (rad/s) 10 ω/ωc EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 17 • Another complex pole pair further improves passband, transition – Repeated pole would be 15 dB down at corner, this is 3 dB down. – Poles lie on arc from −ωp to ωp (Butterworth Filter) Bode Plot 20 × × Magnitude (dB) ! !c −3 dB !40 !60 !1 10 × −100 dB/decade 0 !20 0 10 ω/ωc 1 10 200 × 100 Phase (deg) ! × 0 !100 −!c !200 !1 10 0 10 Frequency (rad/s) 1 10 ω/ωc EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 18 Notch Filter Combine a complex pole pair and a complex zero pair at the same frequency ω0, but with diﬀerent decay rates. Far from ω0, the two cancel At ω0, we get the ratio of the real parts Example: p1,2 = −0.2 ± j , z1,2 = −0.02 ± j Bode Plot 0 Magnitude (dB) ! ×◦ !5 !10 !15 !20 !1 10 ! 0 10 1 10 150 Phase (deg) 100 ×◦ 50 0 !50 !100 !150 !1 10 0 10 Frequency (rad/s) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 1 10 19 Allpass Filter LHP poles with mirror RHP zeros (s − 1)(s − 3) s2 − 4s + 3 H (s) = = (s + 1)(s + 3) s2 + 4s + 3 |Hs(j ω )| = 1 (obvious from graphical interpretation) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 20 (called all-pass ﬁlter or phase ﬁlter since gain magnitude is one for all frequencies) Bode plot: ∠H ( jω01+.#)*,#2/3)4+2&amp;56%,#)*,7/ (|H ( jω)|) ) 20 log10 78,#)95+2&quot;+:. ) ; &lt;=&gt; &lt; !&lt;=&gt; !; ) /ωc &lt; !;&lt;&lt; !@&lt;&lt; !?&lt;&lt; &lt;=&lt;&lt;; &lt;=&lt;; &lt;=; ; ω/ωc !&quot;#\$%#&amp;'()*&quot;+,-.#'/ ;&lt; ;&lt;&lt; EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly ;&lt;&lt;&lt; 21 What Does This Filter Do? ! × ◦ ! × EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly ◦ 22 What Does This Filter Do? Bode Plot 20 × Magnitude (dB) ! ◦ 15 10 5 0 !1 10 0 10 1 10 200 ! × Phase (deg) 100 ◦ 0 !100 !200 !1 10 0 10 Frequency (rad/s) 1 10 • Narrowband bandpass ﬁlter • 360◦ phase shift across the passband EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 23 Circuit Examples for Second Order Filters A driven series RLC circuit can serve as a lowpass ﬁlter. i L x(t ) + − R C + y(t ) − EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 24 Circuit Examples for Second Order Filters A driven series RLC circuit can serve as a lowpass ﬁlter. i Time L R x(t ) + − vL(t ) = Li￿(t ) vR(t ) = Ri(t ) C Z + 1t y(t ) vC (t ) = i(!)! C0 − EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 25 Circuit Examples for Second Order Filters A...
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## This note was uploaded on 09/28/2013 for the course EE 108b taught by Professor Mukamel during the Fall '13 term at Singapore Stanford Partnership.

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