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A#08_Solutions

# A#08_Solutions - Assignment#08-— Solutions — p.1...

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Unformatted text preview: Assignment #08 -— Solutions — p.1 ECSE—2410 Signals & Systems - Spring 2009 Due Fri 02/13/09 1(30). Perform the convolution, C(t) = a(t) * [9(1), where a(t) 61(7) (a) Use the graphical approach. Evaluate the integrals for each range and denote the range of t where the limits on the integrals are valid. Write your final answer in bracket notation. For whim \$190 :7 09¢“ “t (‘3 [9(1) Hr ~1+~b70 @‘E42 :7 l4't42— 'l: (LU-)3 S deT ~l+ i: F0, ,;+t42.<; t>z=7 241243 2_ 't' —.‘+t '2.— Assignment #08 — Solutions - p.2 ECSE-2410 Signals & Systems — Spring 2009 Due Fri 02/13/09 1.C0ntinued. (b) Plot C(t) by hand, and then plot again using Matlab to get a Clean plot. Tao» t Data gym: it,” ° + 4b 7’ Z (is-ﬁlm 1 L xctéz. SiTciT: :‘F‘E' =. if?» at .. 24h“: —H~t vett- ‘f. 2. i: 2 z _ L ‘ —-L = ftija-j} {,2— -~—(3E-3}+! 2at<3 SiM’r—tgldﬁ-+1‘ +{: 4 at + ’2. mist. 2, 1‘. BLt‘Lt Sidt=baGI+Q=L ~L+Jc 4:.th 81¢¢=4~(~1++); "' «L-t-t Assignment #08 — Solutions — p.3 ECSE~2410 Signals & Systems — Spring 2009 Due Fri 02/13/09 1.C0ntinued. (b) Continued. Plot C(t) by hand, and then plot again using Matlab to get a clean plot. 0.75 0.5” t==[0: .0525]; uO=t>=0; u1=t>=1 ; u2=t>=2; u3=t>=3; u4=t>=4; u5=t>=5; g01=u0~u1 ; g12=u1—u2; g23=u2~u3; g34=u3—u4; g45=u4~u5; X01=O.25*t.’\2; x12=0.5*(t~0.5); x23=-O.25*(t—3).’\2+1; x34=1; X45=—t+5; X=X01.*g01+x12.*g12+X23.*g23+x34.*g34+x45.*g45; p10t(t,x); grid axis([—.5 5.5 0 1.1]) Assignment #08 — Solutions — p.4 ECSE—2410 Signals & Systems — Spring 2009 Due Fri 02/13/09 2(40). For the periodic waveform shown, 75 .1. (a)(5). Find the DC value, a0. T24, (00 = 3-, and a0 = ‘ ' = 0 (b)(20). Find a general expression for the other exponential Fourier coefficients, a k , k i O. Assignment #08 —~ Solutions - p.5 ECSE—2410 Signals & Systems — Spring 2009 Due Fri 02/ 13/09 2. Continued. (c)(15).Use Matlab to plot the magnitude spectrum, lak! vs. k. 1 1%! n clear all k=[~10:10]; 0.35 a1=sin(k*pi/2)./(k*pi/2); a2=exp(-i*k*pi/2); ak=(a1-aZ)./(j*k*pi); 0.3 a=abs(ak) stem(k,a); hold on 0.25 stem(0,0) ; grid 02 hold off 0 - k -10 —8 —6 —4 -2 O 2 4 6 8 10 We should check our answer against Matlab to make sure we are right. For k =1, Matlab says k=1 a1=sin(k*pi/2)./(k*pi/2); aZ=exp(—i*k*pi/2); ak=(a1—a2)./(j >"l<"‘pi); a=abs(ak) a = 0.3773 . 71' +75 SlnC[5] —— e 2 1 2 1 2 2 2 Our equation says, for k =1, a = —-—————————————— = —— '—- z [w] +[w] = 0.3773! OK! The results check. Assignment #08 — Solutions —— p.6 ECSE-24IO Signals & Systems - Spring 2009 Due Fri 02/13/09 3(30). (a) Find the trigonometric (Sines and/or cosines) Fourier series for the periodic waveform, Assignment #08 — Solutions -— p.7 ECSE—2410 Signals & Systems ~ Spring 2009 Due Fri 02/13/09 3(30). Continued. (b) Use Matlab to plot the Fourier series approximation consisting of the first three non— zero harmonics, plus DC. t=[—1 :.O1 :9]; a0=1/3; A=sqrt(3)/pi; w=pi/3; a1=A*cos(w.*(t-1 )); aZ=O.5*A*cos(2*w.*(t-1)); a4=~0.25*A*cos(4*w.*(t—1 )); at=aO+a1+a2+a4; x=t>=0&t<=2|t>=6&t<=8; plot(t,aO,t,a1,t,a2,t,a4,t,at,t,x); grid ...
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