A#12_Solutions - Assignment#12 w Solutions pd ECSE—2410...

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Unformatted text preview: Assignment #12 w Solutions - pd ECSE—2410 Signals & Systems — Fall 2006 Fri 10/24/06 1(5). Consider the signal x(t)$3 cos(:rt)+—2cos(3ar)+ces(8m). Find the Nyquist sampling frequency, ms, which guarantees that x(t) can be recovered from its samples. ii—gfieri 75% we > 258???) file??? i Marge melee. afar?” £12549: 8??“ I j 2(10). Find the largest sampling frequency, 605, so that the signal x(t)mcos(10z), when sampled and appropriately filtered, i.e.,with cutoff frequency, (06 2 32-603. has an alias frequency of ca=5 rad/sec. Assignment #12 .... Solutions - p.2 ECSE-2410 Signals & Systems - Fall 2006 Fri 10/24/06 4(10). Find the equation of the frequency response function H(co) whose straightniine Bode magnitude plot is shown below. EvaIuate all constants. Assume the system is minimum phase, i.e., ail terms in H( co) have positive signs" [H(a;)| in dB +20dB/decade ~40dB/decade A w Assignment #12 ~— Solutions -» 13.3 ECSE~2410 Signals & Systems - Fall 2006 Fri 10/24/06 5(10). Use the Bode magnitude approximation to find the approximate magnitude of 100[1+j—w] EM $6663} Y" ........ _ H(jm):“mm""fl‘fla:“ atmmle/I-(l. m Gm)”. gm mil}. E“ (1+jw)£1+’;—0] wig it? E M witfifi? 6(10). Find an expression for H(a)) so that the straight-line Bode magnitude diagram shown crosses the 9 dB —axis at a} = 100 . Calculate all terms in your expression. Assume H ((0) is minimum phase. H(m)[ in dB 20 a’Bfi'decacle 40 dBr'decade ‘ 20 dflidecade 10K 40 ddeecade Assignment #12 - Soiutions « p.4 ECSE-2410 Signals & Systems — Fail 2006 Fri i0/24/06 7(30). Indicate on all your sketches: DC gain (1H (0)| in dB), break (corner) frequencies, and Slopes. 10 (a) Sketch the straight~line Bode Magnitude plot and sketch the Phase plot for H(w) m 5 _ . +103 2. Metzwg; ,2 zewwédg é ir’ifasii ég i (b) Find the gain crossover frequency, can, , of the straight line magnitude plot in (a), i.e., when |H((oc)|=1. '2. (Wm ‘3 e 2‘ mas-3 * {We g. ”at. gwaeisi e Assignment #12 — Solutions n 13.5 ECSE-24IO Signals & Systems - Fall 2006 Fri 10/24/06 7(30). Indicate on all your sketches: DC gain (1H(O)J in dB), break (corner) frequencies, and slopes. (0) Sketch the straight-line Bode Magnitude plot and sketch the Phase plot for jaJSO H :—————-—————. (w) (1+jw5)(10+jw) (d) Find the phase crossover frequency, 606, of the phase piot in (c), 1.8., when LH(a)C) = 0. "L WM; wigs Z- igaW' 5‘? ifiéfifigéffi ' § .. figmemgzwss an, %m% $me 9 Assignment #12 ~— Solutions - 9.6 BCSE~2410 Signals & Systems — Fail 2006 Fri 10/24/06 7(30). indicate on ali your sketches: DC gain (!H(O)! in (18), break (comer) frequencies, and slopes. 10(1+;'a)) . ja) a) 3+— [ 19] (6) Sketch the straight~iine Bode Magnitude and sketch the Phase plot for HUG) m (f) Find the gain crossover frequency, we , of the straight—line magnitude plot in (e), i.e., when Vitae) =E . Assignment #12 —— Solutions ~ [3.7 ECSE-2410 Signals & Systems - Fall 2006 Fri 10/24/06 8(10). For the configuration design a compensator G(a)) -— Le. find the frequency response, 6(a)) -— so that the overall cascade system H(w)mG( (0)8“;(03) satisfies both the following specifications: (:1) 1516:3121 for all frequencies M1000. (b) The straight line Bode magnitude plot ofH(a)) has a slope of —40 dB/decade for all frequencies (@1000. Sketch the straight line Bode magnitude plots G( (0), H(w), and H1(co) all on the same plot. Label all critical values. ...
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