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Unformatted text preview: Assignment #5 m Solutions « 9.1
ECSEu24I 0 Signals & Systems — Fall 2006 Due Fri 09/15/06 1(30). Find the equation and sketch the resulting convolution, w(r) = a(t) * b{r), where a t) 5(5)
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BOSE2410 Signals & Systems  Fall 2006 Due Fri 09/15/06 1 Cont. Find the equation and sketch the resulting convolution, w(t) = (2(1) * b0), where :20) 5(1) i 0 l l 0 i MATLAB 13101:. t:{u3:.01 :3]; E1 zi>:(w2)&t<m( 1); g22t>(l )&t<} ; g3 392} &t<=2;
wm(0.5*(2+t)."2).*g1+(l~.5*t."2).*g2+(0.5*(2~t)."2).*g3;
plot(t,w,‘LineWidth‘,2); grid Assignment #5 — Solutions — 13.3
ECSE~2410 Signals & Systems  Fall 2006 Due Fri 09/15/06 2(35). The inputoutput relationship of an LTl system is govemed by the differential equation, 53:39 + 2w) m 2x(t) . x0) LTI W) dz Assume the system is causal and is in a condition of initial rest, so that y(0) = 0. (a) Solve this differential equation when the input is a unit step function. Recall that in classical solution
of differential equations, the step input is treated as a constant of unity for z 2 0 .
(b)Using the results from (a), find the system impulse response, [1(5), ((3) Find the step response of this system by convolution, y(t) = x(t) * 110) , using the Mt) found in (b).
(d)Is the following true or false? Given the system impulse reSponse, the step response found by solving the differential equation is the same as the one solved by convolution. in other words, convolution is
nothing more than the particular soiution of the differential equation! (a) w “3&1 M. +‘Zig “2.. 2..) “tea: at éﬁ:§g $§gqg ﬂaking) (9+Zlam2, "A? [cal “flak if?" Suii: Ae H W we
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ECSE—2410 Signals & Systems  Fall 2006 Due Fri 09/15/06 6131(3) 3(35). Solve the differential equation, r +2y(r) = sin(z), t 2 0, with y(0) = 0, using classical techniques. Express your answer in the form, y(t) m A?“ + B sin(t + 6) ‘ You need to ﬁnd the
numbers for the unknown constants, A,a,8,6 . Express the phase shift, 9 , in degrees. gﬁ léewgwm Sal‘s ‘jl‘ts—a Aa taro J $45“; :‘ KQSg'ﬂ{rdr k; Laﬁt— {gFaJ—alésﬁt lg @1935 Sal ﬁrst“? WM
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ECSE24l0 Signals & Systems  Fall 2006 Due Fri 09/15/06 3(35). Continued. Solve the differential equation, + 2 32(1) 3 sin(t), r 2 0, with y(0) m 0, using classical techniques. Express your answer in the form, y(t) m Aewo” + Bsin(r «t 49) . You need to ﬁnd the
numbers for the unknown constants, A, 0:, B, 6 , Express the phase shift, 6, in degrees. The solution is y(r) = 0.23""3‘ + 0.45 sin(r — 26.6”): 0262—Zl + 0.458in(z “— rad.) Need radians for
piotting. steady — state term
y(t) = 0.45 sin(I — 3i“— rad.) 57.3 transient term
y(t) x: 0.2% complete solution
y(t) m 0.29% + 0.45 sin(r w rad.) t={0:0.01:2*pi]; y1=0.2.*exp(—2_*t);
y2=0.45.*sin(t—(26.6/57.3));
y=0.2.*exp(2.*E)+0_45.*sin(t(26.6/57.3));
plot(t,y1,':‘,t,y2,'—’,t,y) grid ...
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This homework help was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.
 Spring '07
 WOZNY

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