hw4soln - 3’“ WILL; l" 2.57. (a) Realizing that...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3’“ WILL; l" 2.57. (a) Realizing that m2[n] = y1[n], we may eliminate these from the two given difference equations. This would give us min] = -ay2[n - 1] + box1[n1+ 12mm ~— 1]. This is the same as the overall difference equation. bu 'Afnl (L) at 6 6 (d) 6 ‘ ~+—‘ w out“) ‘1 [W] “W b o '2! {I’ll if») (1°) Figure 52.57 (b) The figures corresponding to the remaining parts of this problem are shown in the Figure 82.57. 59. (a) Integrating the given differential equation once and simplifying, we get y(t) = —-9—9 t y(r)dr + 92ft m(r)d¢ + 915cm. 0‘1 —oo 0‘1 ——oo 0'1 Therefore, A = —ao/a.1, B = bl/a1,C' = bo/al. (b) Realizing that m2(t) = y1(t), we may eliminate these from the two given integral equa- tions. This would give us t yg(t) =: 14/: y2(7)dT + Bflmmlhmr + Cmflt). (c) The figures corresponding to the remaining parts of this problem are shown in Figure 82.59. a 3 em the) 1971i gym I C ) (C 3 Lb 3(4- 1 £84 3 i I, I a A ~Ld) l 2 l Cf) E Figure 82.59 9-}— 2.61. (a) (i) From Kirchoff’s voltage law, we know that the input voltage must equal the sum of the voltages across the inductor and capacitor. Therefore, d2y(t) at? m(t) = LC + y(t). % E: T. i, 3, f Using the values of L and C we get d2y(t) dt2 (ii) Using the results of Problem 2.53, we know that the homogeneous solution of the differential equation + y(t) = d2y(t) dy(t) dt2 dt will have terms of the form Klesot+Kzesll where so and .91 are roots of the equation + agy(t) = bx(t). +0.1 32+als+a2=0.. (It is assumed here that so # 31.) In this problem, (11 = O and a2 = 1. Therefore, the root of the equation are so = j and 31 = —~ j .. The homogeneous solution is yh(t) = Klejt +K2€wjt. And, 021 = 1 = wz. (iii) If the voltage and current are restricted to be real, then K1 = K2 = K. Therefore, yh(t) = 2K cos(t) = 2K sin(t + 7r/2). t+2, ~2<Zt<A-1 a:(t)= 1, —1<t<1 2—m 1<t<2 T: 6, (10 = 1/2, and __ 0, k even “"‘ fifi$m%qmfi%x komi (iii) T = 3, a0 :21, and ak = 3] [aim/3 Sin(k27r/3) + 2ejk"/3 sin(k7r/3)], 273192 fiflT=z%=—U1%=%—bflhk¢u (v) T = 6, we = 7r/3, and _ cos(2k7r/3) —— cos(k7r/3) “k" jkw/3 ' Note that (10 = 0 and (1;, mm = 0. (vi) T = 4, mg = 7r/2, a0 = 3/4 and min/2 mam/2) + «rm/4 sinUmr/4) ale 2 W’ km (b) T = 2, ak : mfigfijk —- (3—1] for all k. % problen1 3 clear; x : [zeros(l,lOO) ones(l,10l) zeros(l,99)]; x1 = [x —X 0]; dt = 0.01; t = [—3:0.0l:3]; for k : 0:5 a(k+l) = dt/6*sum(xl.*eXp(—i*k*2*pi*t/6)); ? end k¢u Vk. CM: 0 h a, :JOQSQ?” (a) Wehave 1 1 1 2 ao=—/ tdt+—/ (2—t)dt:1/2. 2 0 21 (b) The signal _q(t) : dac(t)/dt is as shown in Figure 83.24. Figure 83.24 i The F8 coefficients bk of g(t) may be found as follows: 1]l 12 b:— —— dt: 0 2V/odt 2/1. 0 bk = l/-le_j7”°‘(1t— 3[2:3‘3'7'Mdt 2 0 2 1 and = — 67”]. (c) Note that g(t) = (1:?) (-55—) bk = jkwak. Therefore, ah = j—1;bk = —;21—k—2—{1 - e'j"k}. 7' 3.25. (a) The nonzero FS coefficients of 33(t) are a1 2 (L1 = 1/2. (b) The nonzero FS coefficients of x(t) are bl = bi] = 1/2j. (c) Using the multiplication property, we know that 00 2a) = mm) +55 at = 2 am l=~oo Therefore, 1 1 c =a *b =——.6k—2 —-——,6k+2. k k k 4Jl l 4Jl l This implies that the nonzero Fourier series coefficients of z(t) are (:2 = 6:2 2 (1 /4j ). (d) We have 1 '2- Sll’l(8t). Therefore, the nonzero Fourier series coefficients of z(t) are c2 = 6-2 = (1/43"). z(t) = Sin(4t) cos(4t) : ...
View Full Document

This note was uploaded on 04/30/2008 for the course BIOMEDE 311 taught by Professor Steel during the Winter '06 term at University of Michigan.

Page1 / 4

hw4soln - 3’“ WILL; l" 2.57. (a) Realizing that...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online