ࡱ> OQ N~[ Fbjbj !jj6YA lhhh|(((8(tb)|Z)+^ , , -E3N33@ZBZBZBZBZBZBZ$K\ k^fZh32^E333fZE , -{ZEEE3 ,h -@ZE3@ZEEKX|26h Z -) 0OLmdI|:%(o<Y Z4Z0Z"Y^C^ ZE||SIGNALS AND SYSTEMS LABORATORY 4: Polynomials, Laplace Transforms and Analog Filters in MATLAB  EMBED "Equation" "Word Object1" \* mergeformat  INTRODUCTION Laplace transform pairs are very useful tools for solving ordinary differential equations. Most applications involve signals that are exponential in the time domain and rational in the frequency domain. MATLAB provides tools for dealing with this class of signals. Our goal in this lab is to get acquainted with these tools and develop some familiarity with the classical Butterworth and Chebychev lowpass and bandpass filters. POLYNOMIALS The rational functions we will study in the frequency domain will always be a ratio of polynomials, so it is important to be able understand how MATLAB deals with polynomials. Some of these functions will be reviewed in this Lab, but it will be up to you to learn how to use them. Using the MATLAB help facility, or a MATLAB manual, study the functions roots, polyval, and conv. Then try these experiments: Finding the roots of a polynomial a=[1 10 35 50 24]; r=roots(a) r = -4.0000 -3.0000 -2.0000 -1.0000 The row vector a contains the coefficients of the polynomial (1)  EMBED "Equation" \* mergeformat . The roots function returns the zeros of this polynomial. This function is not bulletproof, however. It, along with any root finder, will have some difficulty when roots are repeated several times. (We will demonstrate this later.) The roots function will return complex values when appropriate: roots([1 0 1]) ans = 0 + 1.0000i 0 - 1.0000i In other words  EMBED "Equation" \* mergeformat , where j is the square root of -1. The roots function may have trouble when there is a root of very large multiplicity. To see this, try the following. a=poly(eye(3)); % constructs the polynomial coefficient of (s-1)^3 roots(a) ans = 1.0000 1.0000 + 0.0000i 1.0000 - 0.0000i a=poly(eye(10)); % constructs the polynomial coefficients of (s-1)^10 roots(a) ans = 1.0474 1.0376 + 0.0284i 1.0376 - 0.0284i 1.0130 + 0.0445i 1.0130 - 0.0445i 0.9846 + 0.0428i 0.9846 - 0.0428i 0.9633 + 0.0256i 0.9633 - 0.0256i 0.9555 The result looks good for a third order root, but is quite bad for a 10th order root. Use help and type to find out how poly(eye(3)) constructs the polynomial  EMBED Equation.3 . Multiplying Polynomials The conv function in MATLAB is designed to convolve time sequences, the basic operation of a discrete time filter. But it can also be used to multiply polynomials, since the coefficients of C(s)=A(s)B(s) are the convolution of the coefficients of A and the coefficients of B. For example: a=[1 2 1];b=[1 4 3]; c=conv(a,b) c = 1 6 12 10 3 In other words, (2)  EMBED "Equation" \* mergeformat . In MATLAB, try roots(a), roots(b), and roots(c). What happens? Adding Polynomials If a and b are row vectors representing polynomials, and C(s)=A(s)+B(s) is the sum of polynomials, then it is tempting to think that in MATLAB we need only write c=a+b. But this will work only when a and b have the same length. MATLAB will generate an error message when they have different lengths. There needs to be a left justification before the addition. The following homebrew function polyadd.m, found on the web page under Functions for Lab 4, could be used to add polynomials: function z=polyadd(x,y) % z=polyadd(x,y) % for finding the coefficient vector for the sum % of polynomials: z(s)=x(s)+y(s) m=length(x);n=length(y); if m>=n z=x+[zeros(1,m-n),y]; else z=y+[zeros(1,n-m),x]; end Using this function and the function conv one can now do polynomial algebra numerically. To construct D(s)=A(s)B(s)+3C(s), for example, we could write d=polyadd(conv(a,b),3*c); Evaluating Polynomials When you need to compute the value of a polynomial at some point, you can use the built-in MATLAB function polyval. The evaluation can be done for single numbers or for whole arrays. For example, to evaluate  EMBED Equation.3  at s = 1,2, and 3, type a=[1 2 1]; polyval(a,[1:3]) ans = 4 9 16 to produce the vector of values A (1) = 4, A(2) = 9, and A(3) = 16. The variables need not be real. For example, they can be complex: a=[1 0 1]; z=sqrt(-1); polyval(a,[z,z+1]) ans = 0 1.0000 + 2.0000i IMPULSE RESPONSE The impulse response, h(t), of an LTI system is its response to an impulse function, EMBED Equation.3 . This is illustrated as follows:  EMBED Word.Picture.8  The response of the system H to a signal  EMBED Equation.3 is  EMBED Equation.3 , where the complex impedance, or transfer function, H(s) is the Laplace transform of h(t). The response to  EMBED Equation.3 is the complex frequency response  EMBED Equation.3 , which is the Fourier transform of h(t) and also the complex impedance evaluated at s = j(. LAPLACE TRANSFORMS The Laplace transform provides an s-domain or frequency domain version of a time signal. Consider the common Laplace transform pair (3)  EMBED "Equation" \* mergeformat  where u(t) is the unit step function. The time function h(t) and the frequency function H(s) are alternate ways of describing the same signal. In the time domain, h(t) is exponential. In the frequency domain, H(s) is rational. The choice of the letter h for the above signal is commonly used for filter functions. In the time domain, h(t) is the impulse response function of the filter. In the frequency domain, H(s) is the transfer function of the filter. If we set s = j(, then we get the complex frequency response function H(j(). A rational function is, by definition, the ratio of two polynomials. (4)  EMBED "Equation" \* mergeformat , where (5)  EMBED "Equation" \* mergeformat  and  EMBED "Equation" \* mergeformat . This method of numbering the coefficients is not standard in mathematics, but it is standard for MATLAB. If one constructs a row vector  EMBED "Equation" \* mergeformat  in MATLAB, and uses it for polynomial coefficients, then the polynomial B(s) is what MATLAB thinks you are talking about. The first element is the coefficient of the highest power of s, and this power is the size of the vector minus one. Thus the vectors [0 1 2] and [1 2] both represent the polynomial s+2. Leading zeros have no effect, but trailing zeros change things. The vector [1 2 0] represents the polynomial s(s+2), but the vector [1 2] represents s+2. It is a good idea to trim leading zeros since they can cause trouble when used with some of the MATLAB tools. MATLAB provides a variety of tools for resolving Laplace transform pairs when H(s) is rational. We begin with the polynomial tools. Rational Functions The rational function H(s)=B(s)/A(s) requires polynomials to describe the numerator and denominator. Therefore we need two polynomial coefficient row vectors for a parameterization. The denominator should be normalized in the sense that the leading coefficient should be one. After all, if both B(s) and A(s) are multiplied by the same constant, H(s) will not change. Thus we can force the coefficient of the highest power in the denominator polynomial to be one. For example, the rational function (6)  EMBED "Equation" \* mergeformat  has normalized representation b=[.25, 0, .25], and a=[1, 1, .5, .25]. Using the MATLAB help facility, study the MATLAB functions residue, freqs, bode, nyquist, and rlocus. We will use only the first two of these, but they all deal with rational H(s) in some way. Evaluating rational functions, and the frequency response To evaluate H(s), we evaluate the numerator and denominator and then divide. For vectors, the MATLAB element-by-element division operator ./ can be used. Suppose we want to evaluate H(s)=B(s)/A(s) on the frequency range zero to fmax. The following will do it: fmax=1000; wmax=2*pi*fmax; w=[0:wmax/1000:wmax]; H=polyval(b,j*w)./polyval(a,j*w); There are some important things to recognize here. First, we must evaluate H at s=j(. By this, we are converting frequency values to angular frequency values by using ( =2( f. In the above sequence of commands, a, b, and j, have not been defined. The vectors a and b represent H(s) and have been previously constructed, but unless you have defined them to be something else, the symbols i and j will be assumed to be the square root of -1. The vector H will contain all the evaluations of H(j() (1001 of them), and will be complex. The functions abs and angle can be used to put things into polar form. The functions real and imag extract the real and imaginary parts. Actually, there is an easier way to do the evaluation, once the vector w is constructed. The MATLAB statement H=freqs(b,a,w); does the same thing as the statement above using polyval, except it uses the built-in MATLAB function freqs. This brings up a final comment. In MATLAB, the functions that need numerator and denominator polynomials follow the convention that the numerator comes first. If we typed freqs(a,b,w) we would get values of 1/H(s). Keep this in mind when things dont work. The poles and zeros of H(s) and pole-zero plots The poles of H(s) are the roots of the denominator polynomial A(s). At a pole, H becomes infinite. The zeros of H(s) are the roots of the numerator polynomial B(s). At a zero, H is zero. A pole-zero plot of H(s) simply places the poles (using the symbol x) and the zeros (using the symbol o) on the complex plane. This plot reveals a lot about the Laplace transform pair  EMBED Equation.3 , after you know what to look for. Download the m-file pzd.m from the web page under Functions for Lab 4. Use help and type to see what the file does. Note that we are using the MATLAB convention that the numerator comes first. For example, try using pzd.m to graph the poles and zeros of the transfer function in equation (6). Type a=[1 1 .5 .25];b=[.25 0 .25]; pzd(b,a) The result is shown below. You may think of this as the complex s-plane. The horizontal axis is the real axis (( ), and the vertical axis is the imaginary axis (j().  Partial Fraction Expansion If B(s) has degree less than that of A(s), and if A(s) does not have repeated roots, then the Laplace transform pair for H(s) = B(s) /A(s) is (7)  EMBED "Equation" \* mergeformat . The right hand side of the above is the partial fraction expansion of H(s). The problem is to compute the poles (the alphas) and the residues (the gammas) knowing only the coefficients of the polynomials A(s) and B(s). The poles are the roots of the polynomial A(s). The residues can be computed with pencil and paper via the formula (8)  EMBED Equation.3 . But the MATLAB function residue can compute the parameters for you. For example, Let (9)  EMBED "Equation" \* mergeformat . To do a partial fraction expansion, type b=[1 0 1]; % B(s) a=[1 6 11 6]; % A(s) [gamma,alpha,k]=residue(b,a) gamma = 5.0000 -5.0000 1.0000 alpha = -3.0000 -2.0000 -1.0000 k = [] This means that H(s) has the partial fraction expansion  EMBED "Equation" \* mergeformat . The parameter k is empty when the degree of B(s) is less than the degree of A(s). Partial fraction expansion in MATLAB will vary, and not be reliable, when there are repeated poles. A tool for plotting a Laplace Transform Pair We can put together what we have developed to construct a tool for exhibiting the elements of a Laplace transform pair. The function plotLTP.m, found under Functions for Lab 4 on the web page, will make a 4-panel plot: a pole-zero diagram, a graph of h(t) from zero to tmax, and graphs of the magnitude and phase of the complex frequency response function H(j( ) from zero to fmax. Type help plotLTP plotLTP(b,a,tmax,fmax) Plot Laplace transforms First, plot a pole zero diagram of H(s)=B(s)/A(s). Then plot the inverse Laplace transform h(t) from 0 to tmax, and the phase and magnitude of H from 0 to fmax. Note: any impulsive parts in h(t) will not be plotted. Try plotLTP.m for a pair of polynomials representing b and a. LOWPASS AND BANDPASS FILTERS There are ways of choosing the poles and zeros of a filter H(s) so that it acts as a lowpass or bandpass filter. A lowpass filter with cutoff frequency 1 kHz should pass all sinusoids whose frequency is less than 1 kHz, and stop those with frequencies above 1 kHz. Stop means that the output will be zero. There are a few classical filter designs that are used extensively for bandpass filters. These include the Butterworth and Chebychev filters, named for their respective inventors. To get an impression of what these filters do, use the tool plotLTP.m (plot Laplace Transform Pair). Use the MATLAB help command to investigate the functions butter, cheby1, and cheby2. Then build some filters and display the results using plotLTP.m. To construct a Butterworth lowpass filter with 10 poles, whose cutoff frequency is 1 Hz, type the following commands. [b,a]=butter(10,2*pi*1,'s'); plotLTP(b,a,10,4) Note that the function butter asks for radian frequency while plotLTP.m uses actual frequency. This explains the appearance of 2( in the first line. The string s will request an analog (rather than digital) filter. You may have to play with the parameter tmax to get a useful display of the impulse response. To construct a bandpass filter whose pass band is 10 Hz to 20 Hz, use the following: [b,a]=butter(4,2*pi*[10,20],'s'); plotLTP(b,a,1,40) This example shows the utility of transfer function zeros on the imaginary axis, in this case at  EMBED "Equation" "Word Object3" \* mergeformat . Whenever this happens, the frequency response function goes to zero. Thus the zeros of bandpass filters occur in the stop bands. Since our second example is a bandpass filter, the zeros at frequency zero force the zero-frequency or DC response to be zero.    PAGE 1 Page  PAGE 10 of  NUMPAGES 10 Page  PAGE 1 of  NUMPAGES 10 Assignment: How does poly(eye(3)) construct the polynomial  EMBED Equation.3 ? For each of the following functions, find a partial fraction expansion, and the inverse Laplace transform. Then use the function plotLTP.m to get graphical information. (Make a hardcopy only after you determine what tmax and fmax ought to be so that h(t) and H(j() use most of the graph, but are not truncated.) (1)  EMBED "Equation" \* mergeformat  (2)  EMBED "Equation" \* mergeformat  Verify that h(t) is the time derivative of g(t), from the graphs. Now consider the transfer function  EMBED Equation.3 . Choose an interesting value of  EMBED Equation.3 , like  EMBED Equation.3 , and re-work problem 2. What do you see and why do you see it? Can you see that normalized transfer functions can be frequency scaled to suit any application? Estimating the bandwidth of a lowpass filter from the impulse response An ideal lowpass filter with cutoff frequency  EMBED "Equation" "Word Object6" \* mergeformat  has a non-causal impulse response of the form  EMBED "Equation" "Word Object6" \* mergeformat . This signal has a central peak whose height is 2 f c. It will then cross zero 1/( 2 f c ) seconds to the right of the position of the peak. How well do these properties hold for realizable lowpass filters? Using Butterworth lowpass filters, designed with butter and displayed with plotLTP.m, make these measurements for filter orders of 2, 4, 6 and cutoff frequencies of 1, 10, and 100 Hz. Compare the actual values with the expected ones. (Using these rules, one can turn the tables and estimate the bandwidth from the impulse response.) Assignment: The figure below is the output of plotLTP.m for some filter. Find the command that designed the filter. It is either butter, cheby1, or cheby2. The command will look like [b,a]= (,'s') where . is either butter, cheby1, or cheby2. Determine which, and find the input parameters. (Use the help facility for these three functions.) Remember, s will request the analog filter.  Assignment Consider the transfer function  EMBED Equation.3  Use plotLTP.m to compute and plot the impulse response, pole-zero diagram, and Bode plots for EMBED Equation.3 . What good is this filter? Now re-do this experiment for  EMBED Equation.3 , which incidentally is the matched filter for  EMBED Equation.3 . (You will study these in communications). The function plotLTP.m thinks EMBED Equation.3  is causal and unstable. How can an unstable system be all-pass? Explain what is going on, and what you would do to recognize  EMBED Equation.3  as a stable, anti-causal transfer function. If you are ambitious you might try to re-write plotLTP.m to account for causal and anti-causal transfer functions. "_`)4LTQ^ iyz . > ? a b c d l m  }  " # 6 ؿjEHUVjp? UV jUjEHUVjd? UVjOJQJU OJQJ\6] OJQJ]OJQJ5 6\]ji+> EHUVEH jp@EHU 5>*CJ\:"_PQ]^ !5AENW`ij$`a$$^a$$^a$$a$$a$:J;F   ! . /   G Q W ` r $]`a$$`a$$^a$$a$  - ? Q c u ~  ; < T U xy+,?@$^a$$a$$`a$6 7 8 9 ; S Z ^  !NOhiy +>CDIJyz{|}  +,0nopqrtuvw B*phj{EHUVj? UVEH6]OJQJ6 jU jGEHUjz? UVN@*+CT$^a$$`a$$a$wxyz{|}~AH'()-.2378@AEF=>QRSTvwj(? 5UV\j5U\5\ jUjC)@ UV j EHUjD&? UVCJ j8 EHUj? UV jU6OJQJ6]?v$^a$$a$$<^<a$$`a$!"#$BCDEXYlmno<DHX¸魣͙|jEHUVj? UVEH jU6] jw56\]j5EHU\j? 5UV\jg5EHU\j? 5UV\ 56\]jH5EHU\j1? 5UV\5\j5U\jO5EHU\0#$%&nopq%>ghijs*+12TUVWdeABdȿзj"EHUVj? UVjEHUVj? UVEHjtEHUVj? UVEH jU\ 6\] jw6]66]B,-_`@hi+,?@3 4 ` a t!u!!!$a$defg !hoty !&25*,?@VWXYZ\]^_abcghijpqrs9 : \ ] ^ _ !!!!b!c!d!e!j (EHUVj? UVEH65OJQJ6] jUj%EHUVj? UVOe!t!!!!!!!!h"i"j"k"l"n"o"p"q"s"t"u""""""#Z#[#_#b#c##############$$$$$$%$&$'$$$$$$$$$$$%6%9%@%E%%%%%%&'&8&k&r&&&2'G'V'b'}'~'''' jp6] 6CJ] jw6]OJQJ6]56X!"""""##&&'&8&9&'''' * ******++++$^a$$`a$$^a$$a$''''''''''''((( (.(/(F(K(O(P(Q(R(~(((((((((((())V)W)j)k)l)m))))))) * *r*w****+9+a+b+c+++++++++++++\] j,U jw6] js6]OJQJ B*ph j*EHUj'M? UV jUOJQJ6] 6\]]6H++++++++++++++Q,R,~,,---F.G.s.t....$^a$$a$+++++++++<,=,>,?,@,D,E,F,G,H,I,V,W,y,z,{,|,,,,,,,,,- -M-N-O-P-V-W-X-Y-----------..K.L.n.o.p.q..1/B/C/D/E/l/m//ǿEHjFEHUVj? UVEHOJQJ j DEHUjS@ UV6j@EHUVjS@ UVEH jUCJ6]F......///!/*/1/2/j/k///K0L0M0N0O0P0Q0R0S0T0U0V0$a$$`a$//////////////X00 111111111111111111222%3&3+343]3^3c3d3f333333*4.4e4i44455556666&6,6f6o66757;7žŸŸ OJQJ\ B*\ph\5 6CJ ] jw6] B*ph 6\]6]OJQJ jUjIEHUVj=? UVHV0W0X000 222252M2222%3&3f3g33366777888$`a$$^a$$a$;7^7g777777#8'888J9K9|9}9~999::::::::::::::::;; ; ;;;;;; ;";#;*;+;1;2;3;4;8;9;C;D;F;G;K;W;a; 56] mHnHu0JmHnHu0J j0JUjUmHnHuj5UmHnHu5jLEHUVj? UVEH jUOJQJ jp6]6] B*ph=888:::::::::::::::::::::::::::$a$::::::::::::::::::::::::::::::$a$:::::::::::::::::::::::::::::$a$:::::::::::::::::::::::::::: $)@&#$a$$a$::::;$;%;H;I;J;K;W;X;;;<<= =5=6=y=z=>>>$ & Fa$$ & Fa$$a$$a$$a$a;m;;;;;;;#<,<{<<<<<<<<<<<<<<<======1=2=3=4=6=7=C=D=E=F=b=c=d=e===============Ի jbYEHUj](? UV jVEHUj"(? UVjSEHUVj4? UVjPEHUVj1? UVEH jw6]6] B*ph jNEHUj1T? UV jUOJQJ:=>> > >[>e>>>>>#?$?U?V?W?X????????????@@@@@@@@@@@@AABBgBmBqBwB~BBBBBBBBBBBB;C?CqCĻ粫粫 56\] B*ph 6CJH*] 6CJ ]CJ j_EHUVjr? UVj]EHUVjj? UVEHOJQJ66] jU jn[EHUjx(? UV>>>????AAAABBBBCCCCCCCCC$a$$ a$$Y^Ya$$^a$$^a$ & F$^a$$a$qCrCCCCCCCCCCC DDD-D3D=DADBDUDVDWDXDDDDDDDDDDDDDDD)E2E:E;ENEOEPEQEEEEEEECFLFdFgFF» jEHU jEHUj)? UV jEHUjg*? UV jhEHUj? UV jfEHUj^*? UV6] B*ph jEHUjzV? UV jU j,bUOJQJ8CCCCCCFFFFF$ & Fa$$^a$$^a$$^a$$a$$ & Fa$  0/ =!"#$%DdTJ  C A? "2@n.yi' D`!n.yi' l 0XJxڍK+Ag7?N/ќ`!h+ET^FR`%(boc)(Z vuD;wn9'$F-Wg̷#50>78]H0rVqٙ]+}Wa>u{7"i+1.O\l]X 5)OպLkn||HyXs>FQ{5!gs>׷6;p3J'A9Ho= A<@yQ.mє7VF]t,8{\\w\(݇ geq@eiDdTJ  C A? "2C0b̡0"`!C0b̡0 8>XJmxcdd``.cd``baV d,FYzP1C&,7\ A?d\ jx|K2B* Rt1 @001d++&1k[~!}6>.:]j mҪY ۱dz5V_ 5 +'~kfy;>0\ga/FG\F$˂_/󯲡_gAw`7 W&007)MHۿ Ѕ3׆A|[8#D )д 1F&&\r[v.F{@4f`}A4DdlTJ  C A? "2ovw<򮚇.Cr`!jovw<򮚇.Cl XJ8xcdd``6cd``baV d,FYzP1C&,7\f! KA?H n5rC0&dT20]I?1 @001d++&1k[~!}y+ P.P5`ոbށĚwa#7(Ϛ0o1eBܤ9D'? 2-DŽ.ӏ?-8 v0o8+KRs13t106WoDdTJ  C A? "2sKϥ;`!sKϥ; F)XJxڍ;OAgxȝ(bT:,5F01ho-_@ 1ZswneF@a  7,i!:Q֪Gx/~b ;N@% 50# >^XȕՃ<7>Y2pF5]ȗ|E[.rP?|mU4IFƈ Z4tjq3HSOZcxEI#^1 7iOv_,jqN㜃$PF _y[/bsn@69XޜW<2:`ja@yErCF^>D_"g"gY=f9_K#lp.}~50]%~N_d]G- I:,#c;*8ݰDr9iCq zl5X?VDdTJ  C A? "2nprjrgБf| `!nprjrgБf@ PVXJZxcdd``Ncd``baV d,FYzP1C&,7\ A?dVj䆪aM,,He`7Ӂ`0LY ZZǰml\ \ ,@u@@ڈհwd|5|AO5*!+'26`JX b W&00ah&̐1u #aE^F p,sɈa+N`Ƶ;+KRsʫҜA06z0 zDd,J  C A? "2~OQD>rY~Z `!ROQD>rY~h xcdd``fbd``baV d,FYzP1C&,7\&! KA?H @Sjx|K2B* Rt1 @001d++&1k[~!} j.:]j mx rJ/Qn>#Y3HM(f. W&00ݹf)L$wAv7\F`TrAC ;LLJ% ,iAh`ZDd@ yw\  s *rA? "21 j=ҥY`!1 j=ҥY)xڝTkA}df7IZDz"hR1OS-i{6b4Db'z<"E BCZ  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEGHIJKLMHPShTUVWXaYZ[\]^_`bcdgefikjlmntopqrsuxvwyz|{}Root Entry  F QmdIR*Data FْWordDocument !ObjectPool \pldI QmdI_1049635689F`AldI`ldIOle PIC TMETA   !"#$%&'()*+,-./0123456789:;<=>?@ABCDFIJKLMNOPQRSTUVWXYZ\]_`adfghijklmnopqrstuvwxyz|Trr L    .  & & ' dxpr  " currentpoint "30 dict begin currentpoint 3 -1 PICT   CompObj cObjInfo OlePres000 $roll sub neg 3 1 roll sub 256 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate end dMATH re FMicrosoft Equation 3.0 Equation Equation.39q Ole10Native _1057485412$F`ldIPmldIOle PIC TT\@ /  + .  & Times" ǥwwgw" -!A Times wwgw -!( TimesMETA f CompObjEfObjInfoGOlePres000H" ȥwwgw" -!s Times wwgw  -!) PSymbol" ɥwwgw" -!= Times wwgw  -!s #Times" ʥwwgw" -!4)PSymbol wwgw  -!+ 0Times" ˥wwgw" -!10 8Times wwgw  -!s DTimes" ̥wwgw" -!3IPSymbol wwgw  -!+ PTimes" ͥwwgw" -!35 YTimes wwgw  -!s eTimes" Υwwgw" -!2kPSymbol wwgw  -!+ rTimes" ϥwwgw" -!50 {Times wwgw  -!s PSymbol" Хwwgw" -!+ Times wwgw -!24 PSymbol" ѥwwgw" -!= Times wwgw -!( Times" ҥwwgw" -!s PSymbol wwgw -!+ Times" ӥwwgw" -!1 !) !( Times wwgw -!s PSymbol" ԥwwgw" -!+ Times wwgw -!2 !) !( Times" եwwgw" -!s PSymbol wwgw  -!+ Times" ֥wwgw" -!3 !) !( Times wwgw  -!s PSymbol" ץwwgw" -!+ Times wwgw  -!4 !) & & ' FMicrosoft Equation 3.0 DS Equation Equation.39q"Wl 6 .1   &@ & MathTypePTimes New Romanwgw -2 )y2 <4y2 )(2  3(2 ~)(2 2(2 E)(2 1(2 ((2 *242 ( 502  352 102 )02 (0 Times New RomanwgwA  -2 m 202 302 e40Symbol ¥wwgw -2 G+02 "+02 +02 +02 =02 5+02 = +02 +02 5+02 =0Times New RomanwgwA  -2 s02 _s02 &s02 )s02 rs02  s02 Qs02 s02 s02 TA0 & "Systemwf   - A(s)=s 4 +10s 3 +35s 2 +50s+24=(s+1)(s+2)(s+3)(s+4)V_ A(s)=s 4 +10s 3 +35s Ole10Native[Equation Native ^_1057485424FPmldIPldIOle b2 +50s+24=(s+1)(s+2)(s+3)(s+4)T @    f .  & Times fwwgw f -PIC cTMETA eZCompObj{fObjInfo}!s Times =wwgw = -!2PSymbol gwwgw g -!+ Times >wwgw > -!1 PSymbol hwwgw h -!= Times ?wwgw ? -!( 'Times iwwgw i -!s ,PSymbol @wwgw @ -!+ 3Times jwwgw j -!j >Times Awwgw A -!) B!( FTimes kwwgw k -!s JPSymbol Bwwgw B -!- QTimes lwwgw l -!j \Times Cwwgw C -!) ` & ' FMicrosoft Equation 3.0 DS Equation Equation.39qC W v .1    OlePres000~&Ole10NativeDEquation Native k_1057480826?!FPOldIPOldI&  & MathTypePTimes New Romanwgw -2  )y2 >)(2 _((2 1( Times New Romanwgw -2 2(Times New Romanwgw -2  j(2 s(2 j(2 s(2 @s(Symbol wwgw -2 -(2 +(2 ^=(2 +( & "Systemwf  -@ s 2 +1=(s+j)(s-j)OT s 2 +1=(s+j)(s"j)Ole CompObj "fObjInfo#Equation Native F FMicrosoft Equation 3.0 DS Equation Equation.39q*` (s"1) 3Tm8@_1057485499S(F@ldI@ ldIOle PIC %'TMETA , m    .  & Timesu wwgwu  -!( Times" ,wwgw" , -!s Timesu wwgwu  -!2 PSymbol" -wwgw" - -!+ Timesu wwgwu  -!2 Times" .wwgw" . -!s !PSymbolu wwgwu  -!+ )Times" /wwgw" / -!1 1!) 6!( :Timesu wwgwu  -!s >Times" 0wwgw" 0 -!2DPSymbolu wwgwu  -!+ KTimes" 1wwgw" 1 -!4 TTimesu wwgwu  -!s [PSymbol" 2wwgw" 2 -!+ bTimesu wwgwu  -!3 k!) qPSymbol" 3wwgw" 3 -!= wTimesu wwgwu  -!s Times" 4wwgw" 4 -!4PSymbolu wwgwu  -!+ Times" 5wwgw" 5 -!6 Timesu wwgwu  -!s Times" 6wwgw" 6 -!3PSymbolu wwgwu  -!+ Times" 7wwgw" 7 -!12 Timesu wwgwu  -!s Times" 8wwgw" 8 -!2PSymbolu wwgwu  -!+ Times" 9wwgw" 9 -!10 Timesu wwgwu  -!s PSymbol" :wwgw" : -!+ Timesu wwgwu -!3 & ' FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj&*fObjInfoOlePres000)+VOle10Native,ZW  .1   &` & MathTypePTimes New Romanwgw= c -2 3y2 102 6122 U622 ' )22  322 / 422 )(2 _1(2 2(2 6(( Times New Romanwgw  -2 2(2 3(2 4(2 ^2(2 E2(Symbol= dwwgw= d -2 +(2 +(2 _+(2 `+(2  =(2  +(2 5+(2 +(2 +(Times New Romanwgw  -2 s(2 s(2 s(2  s(2  s(2 s(2 s(2 s( & "Systemw#fT  - (s 2 +2s+1)(s 2 +4s+3)=s 4 +6s 3 +12s 2 +10s+3V_ (s 2 +2s+1)(s 2 +4s+3)=s 4 +6s 3 +12sEquation Native _1057480887/F@ ldI0TldIOle CompObj.0f 2 +10s+3 FMicrosoft Equation 3.0 DS Equation Equation.39qCk* A(s)=s 2 +2s+1ObjInfo1Equation Native __10586128044F0TldI0TldIOle CompObj35fObjInfo6Equation Native 5_10764908199 F0ldI ldI FMicrosoft Equation 3.0 DS Equation Equation.39q9  (t)  FMicrosoft Word Picture MSWordDocWord.Picture.89q1TableCompObj8;hObjInfoObjectPool:=0ldI0ldI  i8@8 NormalCJ_HaJmH sH tH <A@< Default Paragraph Font    @V 00008@j(  NB  S DH  # 0e NB  S Df  s *(k f  s * B S  ?XtX(t(hP 8tP  t xt :: @@UnknownGz Times New Roman5Symbol3& z Arial"h c c!!x20G2ENS UserENS UserWordDocumentSummaryInformation(<>DocumentSummaryInformation8_1057481256-IAF ldI ldI[ bjbj " jjl@@@t  t66666      $> ^ J-@-66Bl6@6,@6* W)Wt(X0 L Tf H h ( t ) ( ( t )       jd6CJ]CJ CJ 6CJ ] 6CJ]56CJ$\]jCJUmHnHu $a$  1hN N!"#!$!%Oh+'0p  , 8 DPX`hss ENS User NS NS  Normal.dot ENS Usert2S Microsoft Word 9.0@@W@W՜.+,0 hp  Colorado State Universityi2  TitleOle CompObj@BfObjInfoCEquation Native : FMicrosoft Equation 3.0 DS Equation Equation.39q` e st FMicrosoft Equation 3.0 DS Equation Equation.39q_1057481265FF ldI ldIOle CompObjEGfObjInfoHEquation Native J_1057481349DNKF bldI bldIOle CompObjJLf.h5U H(s)e st FMicrosoft Equation 3.0 DS Equation Equation.39q&V_ e jtObjInfoMEquation Native B_1057481436PFldIldIOle       !"#$%&'()+./0123456789:;<=?@BCDGIJKLMNOPQRSTUVWXYZ[\^abcdefghijknqstuvwxyz{|}~ FMicrosoft Equation 3.0 DS Equation Equation.39q!8l H(j)TS0CompObjOQfObjInfoREquation Native =_1057485524WFldIgldIOle PIC TVTMETA fCompObjUY*fS /   .  & Times fwwgw f -!hTimes" ᥇wwgw" -!(Times gwwgw g -!t Times" ⥇wwgw" -!)PSymbol hwwgw h -!=Times" 㥇wwgw" -!e Times iwwgw i -!at &Times" 䥇wwgw" -!u,Times jwwgw j -!(2Times" 奇wwgw" -!t6Times kwwgw k -!):Times" 楇wwgw" - !Laplace `PSymbol lwwgw l -! T! w! ^! h! nTimes" 祇wwgw" -!HTimes mwwgw m -!(Times" 襇wwgw" -!sTimes nwwgw n -!)PSymbol" 饇wwgw" -!=Times owwgw o -!1 Times" ꥇwwgw" -!sPSymbol pwwgw p -!-Times" 륇wwgw" -!a-  & ' FMicrosoft Equation 3.0 DS Equation Equation.39qR  .1  @& & MathTypeObjInfo,OlePres000XZ-Ole10Native[>Equation Native A- Times New Romanwgw_ -2 ay2 +sy2 sy2 GHy2 ty2 uy2 _ey2 Ity2 ;hy Times New Romanwgw f - 2 b Laplace2 patSymbol_ wwgw_ -2 -t2  =t2 N t2  t2 ] t2 v t2 D t2 k=tTimes New Romanwgw g -2 71t2 Y)t2 Z(t2 O)t2 x(t2 )t2 (t & "Systemw f  - h(t)=e at u(t) Laplace EH(s)=1s-a h(t)=e at u(t) Laplace !H(s)=1s"aThX_1057485766 `FgldIldIOle EPIC ]_FTMETA H"h   > .  & Timesk wwgwk > -!=TimesV ᥇wwgwV -!B 'Timesk ?wwgwk ? -!( .TimesV ⥇wwgwV -!s 2Timesk @wwgwk @ -!) 7TimesV 㥇wwgwV -!A'Timesk Awwgwk A -!(.TimesV 䥇wwgwV -!s2Timesk Bwwgwk B -!)7- % ; & ' FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj^b]fObjInfo_OlePres000ac`Ole10Nativedl=q ` .1  &> & MathType- xTimes New Romanwgw4 -2 )y2 (y2 7)y2 7(y2 W)y2 X(yTimes New Romanwgw K -2 rsy2 =Ay2 7rsy2 7.By2 sy2 EHySymbol4 Υwwgw4 -2  =y & "Systemwf>  -9 H(s)=B(s)A(s)FV H(s)=B(s)A(s)Equation Native mb_1057485769iFIldIldIOle oPIC fhpTTf Tf R   .  & PMT Extra wwgw  -! Times 楇wwgw -!B Times wwgw  -!( META r CompObjgkfObjInfoOlePres000jlTimes 祇wwgw -!s Times wwgw  -!) PSymbol 襇wwgw -!= Times wwgw  -!b #Times 饇wwgw -!1(Times wwgw  -!s +Times ꥇwwgw -!m1PSymbol wwgw  -!-6Times 륇wwgw -!1:PSymbol wwgw  -!+ ATimes 쥇wwgw -!b JTimes wwgw  -!2PTimes wwgw -!s TTimes wwgw  -!mYPSymbol wwgw -!-_Times wwgw  -!2cPSymbol 磊wwgw -!+ iPMT Extra wwgw  -!K oPSymbol wwgw -!+ {Times wwgw  -!b Times wwgw -!mPSymbol wwgw  -!-Times wwgw -!1Times wwgw  -!s PSymbol wwgw -!+ Times wwgw  -!b Times wwgw -!m & ' FMicrosoft Equation 3.0 DS Equation Equation.39q W Z .1   & & MathTypeP Times New Romanwgw B -2 my2 Smy2 my2 EmyTimes New RomanwgwM f -2 {by2 sy2 by2  sy2 by2 sy2 by2 sy2 EBySymbol Cwwgw C -2 +y2  +y2  +y2 +y2 =y SymbolM gwwgwM g -2 -y2 Z -y2 -y Times New Romanwgw D -2 e1y2 2y2 2y2 W1y2 41yTimes New RomanwgwM h -2 )y2 (yMT Extra Ewwgw E -2 w Ky & "Systemw4f  - B(s)=b 1 s m-1 +b 2 s m-2 +K+b m-1 s+b m( B(s)=b Ole10NativemEquation Native _1057485772ewrFmdItmdIOle 1 s m"1 +b 2 s m"2 +& +b m"1 s+b mT T R   .  & PIC oqTMETA  CompObjptfObjInfoPMT Extra `wwgw ` -! Timesk Hwwgwk H -!A Times awwgw a -!( Timesk Iwwgwk I -!s Times bwwgw b -!) PSymbolk Jwwgwk J -!= Times cwwgw c -!a #Timesk Kwwgwk K -!1)Times dwwgw d -!s ,Timesk Lwwgwk L -!n1PSymbol ewwgw e -!-6Timesk Mwwgwk M -!19PSymbol fwwgw f -!+ @Timesk Nwwgwk N -!a ITimes gwwgw g -!2OTimesk Owwgwk O -!s STimes hwwgw h -!nYPSymbolk Pwwgwk P -!-]Times iwwgw i -!2aPSymbolk Qwwgwk Q -!+ gPMT Extra jwwgw j -!K mPSymbolk Rwwgwk R -!+ yTimes kwwgw k -!a Timesk Swwgwk S -!nPSymbol lwwgw l -!-Timesk Twwgwk T -!1Times mwwgw m -!s PSymbolk Uwwgwk U -!+ Times nwwgw n -!a Timesk Vwwgwk V -!n & ' FMicrosoft Equation 3.0 DS Equation Equation.39qW Z .1   &` & MathTypeP Times New RomanwgwC  -2 OlePres000suOle10NativevEquation Native _1057485791{FtmdI/mdIny2 <ny2 ny2 _nyTimes New Romanwgw o -2 <ay2 sy2 ay2  sy2 ay2 sy2 ay2 sy2 TAySymbolC wwgwC -2 P+y2  +y2 y +y2 +y2 =y Symbol pwwgw p -2 -y2 + -y2 -y Times New RomanwgwC -2 1y2 2y2 2y2 @1y2 N1yTimes New Romanwgw q -2 )y2 (yMT Extra wwgwC -2 G Ky & "SystemwMfx  -    !"#$%&'()*+,-.023689:;<=>?@ABCDEFGHIJKLMNOPQRSTUWXYZ[\]^_`abcdefghijklmnopqsvwxyz{|}~ A(s)=a 1 s n-1 +a 2 s n-2 +K+a n-1 s+a nAB A(s)=a 1 s n"1 +a 2 s n"2 +& +a n"1 s+a nOle PIC xzTMETA  CompObjy}fTh Fh F L  [ .  & PMT Extra wwgw -! Times wwgw -!bPSymbol wwgw -!= Times wwgw -![Times wwgw -!bTimes wwgw -!1Times wwgw -!b(Times wwgw -!2-PMT Extra wwgw -!K7Times wwgw -!bJTimes wwgw -!mOTimes wwgw -!]V & ' FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoOlePres000|~ Ole10Native/pEquation Native 1p +L  .1  ` &  & MathTypePTimes New Romanwgw  -2 @ ]y2 @[y Times New Romanwgw3 " -2 2y2 1y Times New Romanwgw -2 myTimes New Romanwgw3 # -2 @Wby2 @]by2 @xby2 @1byMT Extra wwgw -2 @XKySymbol3 $wwgw3 $ -2 @&=y & "Systemw f6  -l b=[b 1 '  b 2 '  K'  b m ]w8v y b=[b 1 '  b 2 '  & '  b m ]T  X    | ._1057485799nF/mdI mdIOle 4PIC 5TMETA 7  & Timese wwgwe -!HTimes xwwgw x -!( Timese wwgwe -!sTimes ywwgw y -!)PSymbole wwgwe -!=Times zwwgw z -!s CTimese wwgwe  -!2HPSymbol {wwgw { -!+ OTimese wwgwe  -!1 W!4&Times |wwgw | -!s-Timese wwgwe  -!32PSymbol }wwgw } -!+9Timese wwgwe  -!4BTimes ~wwgw ~ -!sHTimese wwgwe  -!2NPSymbol wwgw  -!+UTimese wwgwe  -!2^Times wwgw -!sdPSymbole wwgwe  -!+lTimes wwgw -!1t-%y & '|dxpr  |"| currentpoint ",Times .+H) (PICT VCompObjrfObjInfotOlePres000u)s)), Symbol)=( Cs (H2 ++)1(&4)s (23 ++) 4)s (N2 ++) 2)s)+)1"%T30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3968 div 960 3 -1 roll exch div scale currentpoint translate 64 46 translate 9 562 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 307 562 moveto 384 /Times-Roman f1 (\() show 449 562 moveto 384 /Times-Italic f1 (s) show 610 562 moveto 384 /Times-Roman f1 (\)) show 827 562 moveto 384 /Symbol f1 (=) show 2086 323 moveto 384 /Times-Italic f1 (s) show 2252 152 moveto 224 /Times-Roman f1 (2) show 2483 323 moveto 384 /Symbol f1 (+) show 2743 323 moveto 384 /Times-Roman f1 (1) show 1173 855 moveto (4) show 1380 855 moveto 384 /Times-Italic f1 (s) show 1543 684 moveto 224 /Times-Roman f1 (3) show 1764 855 moveto 384 /Symbol f1 (+) show 2062 855 moveto 384 /Times-Roman f1 (4) show 2269 855 moveto 384 /Times-Italic f1 (s) show 2435 684 moveto 224 /Times-Roman f1 (2) show 2666 855 moveto 384 /Symbol f1 (+) show 2957 855 moveto 384 /Times-Roman f1 (2) show 3165 855 moveto 384 /Times-Italic f1 (s) show 3398 855 moveto 384 /Symbol f1 (+) show 3658 855 moveto 384 /Times-Roman f1 (1) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1145 463 moveto 2696 0 rlineto stroke end d{MATHo H(s)=s 2 +14s 3 +4s 2 +2s+1 4 FMicrosoft Equation 3.0 DS Equation Equation.39q"  .1   &@ c & MathType-!3 Times New Romanwgw -2 v 1y2 vR 2y2 v@4y2 v<4y2  1y2 `V)y2 `W(y Times New Romanwgw -2 2y2 v3y2 2ySymbol wwgw -2 v +y2 vX +y2 vF+y2 +y2 `=yTimes New Romanwgw -2 v sy2 vsy2 vsy2 Lsy2 `sy2 `EHy & "Systemwf  -o H(s)=s 2 +14s 3 +4s 2 +2s+1А H(s)=sOle10NativesEquation Native _10580984712F mdI[mdIOle  2 +14s 3 +4s 2 +2s+1 FMicrosoft Equation 3.0 DS Equation Equation.39qs-X   h(t)!H(s)CompObjfObjInfoEquation Native I_10871325567F[mdImdIOle PIC TMETA 6 CompObjfTi,i   ! .  & Times xwwgw x -!hTimes. {wwgw. { -!(Times ywwgw y -!t Times. |wwgw. | -!)PSymbol zwwgw z -!=PSymbol. }wwgw. } -!g0Times {wwgw { -!k6!k"PSymbol. ~wwgw. ~ -!=&Times |wwgw | -!1*Times. wwgw.  -!n&PSymbol }wwgw } -!!Times. wwgw. -!e:PSymbol ~wwgw ~ -!a @Times. wwgw. -!kETimes wwgw  -!t HTimes. wwgw. -!uKTimes wwgw -!(QTimes. wwgw. -!tUTimes wwgw -!)YTimes. wwgw. - !Laplace PSymbol wwgw -! u! ! ! ! Times. wwgw. -!HTimes wwgw -!(Times. wwgw. -!sTimes wwgw -!)PSymbol. wwgw. -!=PSymbol wwgw -!g Times. wwgw. -!kTimes wwgw -!sPSymbol. wwgw. -!-PSymbol wwgw -!aTimes. wwgw. -!k- !kPSymbol wwgw -!=Times. wwgw. -!1Times wwgw -!nPSymbol. wwgw. -! & ' FMicrosoft Equation 3.0 DS Equation Equation.39q# ^ .  ` &@  & MathType-00% Symbolw@ ww0-2 @y2 tyObjInfoOlePres000Ole10NativeEquation Native F Symbolw@ ww0-2 =y2 =ySymbolw@ ww0-2 z-y2 @=y2 ky2 y2 y2 y2 hy2 t=y Times New Romanww0-2 ny2 kky 2 Laplace2  ta2 ka2  na2 kaTimes New Romanww0-2 z ka2 z(sa2 ka2 sa2 pHa2  ta2  ua2 6ea2 Fka2 Hta2 ;ha Times New Romanww0-2 S1a2 ? ]a2 i[a2 1aTimes New Romanww0-2 z]a2 z[a2 ]a2 [a2 )a2 (a2 g )a2  (a2 ]a2 [a2 )a2 (aSymbolw@ ww0-2 zaa2 ga2 $ga Symbolw@ ww0-2 aa & "System 0-NANI h(t)=g kk=1n  e a k t u(t) Laplace EH(s)=g k s-a kk=1n *x O h(t)=[k] k=1n " e [k]t u(t) Laplace !H(s)=[k]s"[k] k=1n " FMicrosoft Equation 3.0 DS Equation Equation.39qN7 [k]=B([k])' elim s!_1087132666FmdImdIOle CompObjfObjInfoEquation Native _1057485776FmdIBmdIOle PIC T[k] (s"[k])A(s)TvOvO H    .  & Times wwgw -META  CompObj0fObjInfo2OlePres0003      !"#$%&'()*+,-./1456789:;<=>?@ABCDEFHIJLMNORTUVWXYZ[\]^_`abcdefghijklmnopruvwxyz{|}~    * '!"#$%&()+/F,-.0412365798:;<=?>@CABDEGIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~!HTimes pwwgw p -!( Times wwgw -!sTimes qwwgw q -!)PSymbol wwgw -!=Times rwwgw r -!s FTimes wwgw -!2KPSymbol swwgw s -!+ STimes wwgw -!1 [!(&Times twwgw t -!s*PSymbol wwgw -!+1Times uwwgw u -!1:!)?!(CTimes wwgw -!sGPSymbol vwwgw v -!+OTimes wwgw -!2X!)^!(bTimes wwwgw w -!sfPSymbol wwgw -!+nTimes xwwgw x -!3v!)|-%PSymbol wwgw -!=Times ywwgw y -!s Times wwgw -!2PSymbol zwwgw z -!+ Times wwgw -!1 Times {wwgw { -!sTimes wwgw -!3PSymbol |wwgw | -!+Times wwgw -!6Times }wwgw } -!sTimes wwgw -!2PSymbol ~wwgw ~ -!+Times wwgw -!11Times wwgw  -!sPSymbol wwgw -!+Times wwgw -!6 & ' FMicrosoft Equation 3.0 DS Equation Equation.39qE V .1  `&  & MathType-   Times New Romanwgw{ -2 }Y6y2 }u112 }612 s112 Z& )12 Z 312 Z )(2 Za 2(2 Z)(2 ZA1(2 Z4((2  1(2 `W)(2 `X(( Times New Romanwgw 2 -2 2(2 3(2 2(2 2(Symbol{ 륇wwgw{ -2 }i+(2 }+(2 }+(2 +(2 `=(2 Z +(2 Zl+(2 Zo+(2 +(2 `=(Times New Romanwgw 3 -2 }s(2 }.s(2 }-s(2 ,s(2 Z s(2 Zs(2 Zs(2 s(2 `s(2 `EH( & "Systemwf~  - H(s)=s 2 +1(s+Ole10NativeGEquation Native K_1057485885FBmdImdIOle P1)(s+2)(s+3)=s 2 +1s 3 +6s 2 +11s+6@C H(s)=s 2 +1(s+1)(s+2)(s+3)=s 2 +1s 3 +6s 2 +11s+6T d 0     .  & Times awwgw a -!HTimesH 2wwgwH 2 -!( TimesPIC QTMETA S\CompObjqfObjInfos bwwgw b -!sTimesH 3wwgwH 3 -!)PSymbol cwwgw c -!=TimesH 4wwgwH 4 -!5 .Times dwwgw d -!s&PSymbolH 5wwgwH 5 -!+-Times ewwgw e -!36- % ;PSymbolH 6wwgwH 6 -!+?!- OTimes fwwgw f -!5 UTimesH 7wwgwH 7 -!sJPSymbol gwwgw g -!+QTimesH 8wwgwH 8 -!2Z I `PSymbol hwwgw h -!+dTimesH 9wwgwH 9 -!1 uTimes iwwgw i -!snPSymbolH :wwgwH : -!+uTimes jwwgw j -!1} m & ' FMicrosoft Equation 3.0 DS Equation Equation.39qX  .1  @@& & MathType-U OlePres000tOle10NativedEquation Native _1057490395FmdIn!mdI  Times New Romanwgw4 -2 O 1y2 8p 1y2  2y2 8 5y2 3y2 85y2 U)y2 W(ySymbol  gwwgw  g -2 } +y2  +y2 +y2 8q-y2 +y2 +y2 =yTimes New Romanwgw4 -2  sy2 sy2 3sy2 sy2 EHy & "Systemwf3  -` H(s)=5s+3+-5s+2+1s+1tГx H(s)=5s+3+"5s+2+1s+1Ole PIC TMETA CompObjfTO>O>     .  & PSymbol wwgw -!w PSymbolu wwgwu -!= Times wwgw -!0  & ' FMicrosoft Equation 3.0 DS Equation Equation.39q  .1  @&U & MathType0ObjInfoOlePres000.Ole10NativeEquation Native 1Times New Romanwgw -2 U0ySymbol wwgw -2 a=ySymbol ֥wwgw -2 "wy & "SystemwfT  - w=0  =0_1058100273F#mdI)&mdIOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qs*X   (s"1) 3TXEquation Native F_1057486385F)&mdI(mdIOle PIC T G   .  & Times 磊wwgw -!HTimes. 奇wwgw. -!( Times wwgw -!sTimes. 楇wwgw. -META CompObjfObjInfoOlePres000!)PSymbol wwgw -!=Times. 祇wwgw. -!s cTimes wwgw -!( h!2 mTimes. 襇wwgw. -!s sPSymbol wwgw -!+ zTimes. 饇wwgw. -!1 !) Times wwgw -!s'Times. ꥇwwgw. -!3,PSymbol wwgw -!+3Times. 륇wwgw. -!(<!3@!/H!2O!)UTimes wwgw -!sYTimes. 쥇wwgw. -!2^PSymbol wwgw -!+fTimes. wwgw. -!(n!13r!/!16!)Times wwgw -!sPSymbol. wwgw. -!+Times wwgw -!(!5!/!16!)- & & ' FMicrosoft Equation 3.0 DS Equation Equation.39q&h 4 .1  &] & MathType-Times New Romanwgw -2 *)y2 162 x/62 562 C(62 )62  162 f /62  132  (32 )32 B232 /32 332 }(32 8)32 8 132 8J 232 8 (32 W)32 X(3 Times New Romanwgw/ J -2  232 33Symbol wwgw -2 S+32  +32 +32 8 +32 =3Times New Romanwgw/ K -2 s32 Z s32 6s32 8 s32 8P s32 s32 EH3 & "Systemwf  - H(s)=s(2s+1)s 3 +(3/2)s 2 +(13/16)s+(5/16)V H(s)=s(2s+1)s 3 +(3/2)s 2 +(13/16)s+Ole10NativeEquation Native _1057486388F(mdIU-mdIOle (5/16)TX    .  & Times wwwgw w -!GTimes wwgw PIC TMETA @CompObj fObjInfo     !"$%'(),/27:=?@ABCEHIJKLMNSUVWXYZ[\]^_`abcdefghijlopqrstuvwxyz{|~ -!( Times xwwgw x -!sTimes wwgw -!)PSymbol ywwgw y -!=Times wwgw -!( e!2 iTimes zwwgw z -!s oPSymbol wwgw -!+ wTimes {wwgw { -!1 !) Times wwgw -!s&Times |wwgw | -!3+PSymbol wwgw -!+2Times }wwgw } -!(;!3?!/G!2N!)TTimes wwgw  -!sXTimes ~wwgw ~ -!2]PSymbol wwgw  -!+eTimes wwgw  -!(m!13q!/!16!)Times wwgw  -!sPSymbol wwgw -!+Times wwgw  -!(!5!/!16!)- % & ' FMicrosoft Equation 3.0 DS Equation Equation.39qOlePres000Ole10Native#Equation Native &_1058613282FU-mdIp/mdIX , .1  &] & MathType-nTimes New Romanwgw -2 )y2 162 >/62 r562  (62 )62  162 . /62  132 x (32 )32  232 x/32 332 I(32 8 )32 8 132 8 232 8X (32 $)32 %(3 Times New Romanwgw" -2  232 33Symbol wwgw -2 +32  +32 Z+32 87 +32 =3Times New Romanwgw" -2 Ws32 $ s32 s32 8t s32 s32 1G3 & "Systemwf  - G(s)=(2s+1)s 3 +(3/2)s 2 +(13/16)s+(5/16)l G(s)=(2s+1)s 3 +(3/2)s 2 +(13/16)s+(5/16) FMicrosoft Equation 3.0 DS Equation Equation.39q9OX   F(s)=G)s 0 ()Ole *CompObj+fObjInfo-Equation Native .k_1058613341Fp/mdIp2mdIOle 0CompObj1fObjInfo3 FMicrosoft Equation 3.0 DS Equation Equation.39q96DQ  0 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native 46_1058613368Fp2mdIp4mdIOle 5CompObj6fObjInfo8Equation Native 9N_1057491050Fp4mdI`6mdIOle ;92@ 210 6TX,X     .  & PIC <TMETA >DCompObjDfObjInfoFTimes ͥwwgw -!f Times wwgw -!c  & ' FMicrosoft Equation 3.0 DS Equation Equation.39q  OlePres000GOle10NativeOEquation Native P6_1057491058F`6mdI`;mdI.1  &` & MathTypeP Times New Romanwgw  -2 cyTimes New Romanwgw $ -2 @wfy & "Systemw{f  - f cGh f cTy ,y    o .  & Times wwgw -!h Times ͥwwgw Ole QPIC RTMETA TCompObjkf -!( Times wwgw -!t Times Υwwgw -!) PSymbol wwgw -!= Times ϥwwgw -!2 !Times wwgw -!f )Times Хwwgw -!c .PSymbol wwgw -! 4Times ѥwwgw -!sinc 9!( L!2 PPSymbol wwgw -!p WTimes ҥwwgw -!f ^Times wwgw -!c cTimes ӥwwgw -!t fTimes wwgw -!) j & ' FMicrosoft Equation 3.0 DS Equation Equation.39q f  .1   ObjInfomOlePres000nOle10Native}]Equation Native &  & MathTypePTimes New Romanwgw  -2 @ )y2 @|2y2 @ (y 2 @sinc2 @u2i2 @)i2 @(iTimes New Romanwgw: -2 @ ti2 @ fi2 @Vfi2 @Hti2 @;hi Times New Romanwgw  -2 ) ci2 ciSymbol: ۥwwgw: -2 @ piSymbol wwgw  -2 @i2 @r=i & "Systemwf  -Y h(t)=2f c sinc(2pf c t)mȇD h(t)=2f c "sinc(2f c t) FMicrosoft Equation 3.0 DS Equation Equation.39qsRX   F(s)="s"1/2s+1/23lZMvgǛBH}O9LC=gNƔ˱mv*1vhVj/׫5mLzWqFnWZJ`_{H3eT|6.?׳:(r,&Jt.o9%U"Ǵw&q3]!#˚).tHE lO;'K)(ϊԡu `aDVHGzQ[' F㟨 \8pCn!g@|$sGwdU?LXa{Q̬4R+oz=WE$O(̋4-d+U7J Q%Doh'B5SkcA z+4fU=5ghpG;i'(8?dSעǸi Bf1Z]Yı[3ț؟Jb/ < "`V {Üv=]Q;}LU}X{ּ}Kmy˶hsNnupBn٪p:xa!93Z_KVDd,J   C A ? "2[@&a-mY,+7`!/@&a-mY,+Hxcdd``> @c112BYL%bL0Yn,56~) m @ k775bnĒʂT+~3);a~&brdKF&&\li1 ] A4 s;.Ff~IDdTJ   C A ? " 2ٮ]_AR7he]`!Uٮ]_AR7he$` PXJ#xcdd``bd``baV d,FYzP1C&,7\S A?db@P5< %! `WfRv ,L ! ~ Ay ߖ_H_>kP]b m!yk"6 ~N-ρM,]@| {;)? 2 ۫ 'lF 4!  `p021)W2`KD~f` Dd,J   C A ? " 2+'#hSK4H^`!V+'#hSK4Hh$xcdd``fbd``baV d,FYzP1C&,7\&! KA?H @jx|K2B* R ͤ `YB2sSRsֵ-5+R.:]b m/XgI%gUB?Xװqy#g401LHn& Ma`sRH^np;]V^ %.pH2M%=#RpeqIj.CR ] A4f~ؗ^"Dd,J   C A ? " 2܆eo lWY:|``!X܆eo lWY:|@2&xcdd``bd``baV d,FYzP1C&,7\F! KA?H 5rC0&dT20]?c&,``abM-VK-WMcXBul\sXt9c<3Hf%F%`Mq_ 4f>0L@'%y7ln G%4Ը! h 0y{qĤ\Y\P5d ^DdJ   C A ? " 2-1v ج 6 `!1v ج 6@P% xڍ=,A߼s'(. !BBBGDd>⎋[F+!*^C%ӈhQ5ofwX{3 8j(oq 2+ ªV H`v*^/ /:m?Ǜ X'[3+ʹNъCk $Gv>=hYk sυMݘ0LfUyZnMyoSJܫ!J(*Ԉm},<]Oo>q 2E)gĸrk1fYv79f3Z3|o՛^=9*>5;AVĝ0Oa^+&>V'>R2/yUߪ&㶳WGgQ8 }=Ͽ!X6\t&nCq󎝅Ϯ`{ O`;fDd80J  C A? " 2:Nkp`S`!:Nkp`Sv kjxڅRJA=#fpQ `0M &FL`.`O؊ *`-&e"> 9ss3<arb9FL! |ƱZH2z@es dl{5 MD1E6;g 1c3\?Q^hVS:wyO*a7]]s~m^~zo5&֭ |K*e*euu!?_ٴZ)G;YS>6_YJo%ې|,>ǭ @m <8L1Ɉ4Y[ [AxbRowޫqd}x'ksDd TJ  C A? "2AVʗ{e `!Vʗ{e XJxڍTK@wMZVCp(TD[*0*袣 82%9s~{{{+(@h`byŽ0kuz2xDkړJ 2[n%1oO L,ޱd`Kq&m>ܳy)+/ Hd_ln"b6iX>%Qge$@IzŒ{"ǧ-3*(B? |Ԝe,/>VIMV}uBVzD?iALɈ/vyE{̋msuZ~Hgsp;; hHrq9>2}6kٵjdNo#F-Ż ,}NaW {rV`³F|v}M! ^tDdD TJ  C A? "2=VXMV}u"~D9nq9(ΠY;ڼXyqs-rssd|:_:ĹM5q P\gN7ꣻ8x(g@ժ5z(?b[;C%WJRfA1 ճkk `'vKuDd|J  C A? "2c7br]<:g%`!c7br]<:gL` xtyxcdd``^ @c112BYL%bL0YnL \B@?6 :깡jx|K2B* Rt1 @001d++&1k[~!}i6.i.:]j mt55bb͟5a>#gF0?=0 ? 02!=| n;0MS, * oc'&;|C\+Jظd݀?Ys``\ Ma`g! >>Cȶ ڸ)д&aF&&\ligAh`DdpXJ  C A? "2&CG4+M(`!&CG4+ xڍ=H@KŠ"" AQhEqPcVhiɦt+M\8t讳AUw4.>^ 5xRKLj{oajե`'Sþθ j5XH6n BVoڻtpT4RZyb=fL)e;pyXx* f⬈V^6Z8Gɜ'򸹂'>EK"QE#7dO~%1>;}6Y{ٚ!'b0mKʩ:~ŘNJγ>7WJqsz$|yGQz-x۹qsSt< ;ⷁ]YеH40LJw G򱥿f*<DdLJ  C A? "2- w![kz*`!r- w![kT H@xcdd``bd``baV d,FYzP1C&,7\lB@?6 u10Lj䆪aM,,He`7Ӂ`0LY ZZǰmظ Xt91Hf%o kQ3@g$`+3~F0kQd=| %p5X +ss6Z tR8!pwju%4Թ1^h 0y{qĤ\Y\P-d> bbDd`i<  C A, \b"?Jm}V @~//-n?Jm}V @~/PNG  IHDRgMXsRGBIDATx^DL |'hCJ?bb-J+#Q6 dȠLŮˁ 0 e o=.lIx9 7nqS%w9ߍ-'gpU3g+)z0F0˙/gsՁf;ӻAgpWΌ ]9[ RՄF]",$YϘ3FF?iᓏI|Nξ\.P,w#TlH3, Qz\r,d?G>[U;t֪ '@-7tneEͧ, oZ <ˀmVD Ofuw@2`.Kli0@-*s,Vp--`p@0˖ 08 pYeK˅As,^(W C@#mdVVpQ.,糩+1;Lq=9$8 .Ts~m36vIp@@G dAk=w?H)Xn:"u\QRq&}Myd&J&+L3/3?/6[&ߐK>ӍFO94BY0 \]ݷPDWv(|fk K1f$@ZĽqg5\że X=)DB Fq}k+nX[\&<FE  p6*^@ L 6lv e+kybap1NDA'$hHb0(&bmq WH\`pgz!0 .*hF5 8w¢!bjl\je+_m#apaTBg#b00pmVXm@[\º'6D pB d@ Fqmr /~ Qw¨M.[aa]mq Z`\Q `p',! .Ʃ&ʶe+nX[\&<FE  p6*^@ L 6lv e+kybap1NDA'$hHb0(&bmq WH\`pgz!0 .*hF5 8w¢!bjl\je+_m#apaTVP~LC8>'}Ύ3 ?=z[sPF9y4r4 o O"哑7z*"6{eKGr0opD@'%nuOqߨk0"U~[6OgXfsIosJap95BF7r٢f5Pde+LA]% CJ 1 N 4mr @0uq N t Χ(081Дζe+L_EF[\9081PA>08Z@h۬6P[bpEA !i80$b"-.[a ( O-P `pbtQ@emV T(ODӃmr @0uq N t Χ(081Pz26+m *]TCH' xƌ "6lkhV8'J:@S @b(]T=Am.!x$A`w0Ĵk(7Ǐ_s()SUB%{$b NS4}rnqsReS̤ŴUuڙN{U;I`pB56ngp3fX[zPݷ%˗˧$nWWʍZ\T` xokapf [{st_"-.[a ( O-P `pb>"0sAsF ! Wke4.6lkhV8'J:@S @bhJg"-.[a ( O-P `pb)mVl)s`pb|`p>@  &EѶYi+lP $@@Cp$  `HŶe+L_EF[\9081PA>08Z@h۬6PtQ # 8+3 0o"-.[a ( O-P `pbtQ@emV T(ODӃmr @0uq N t Χ(081Pz26+m *]TCH' xƌ "6lkhV8'J:@S @b!Nyt}R H]Cjryn"V57O+IF^-.\~=cc2exJ0w3-j/ n6˷ӤV~ FD6J;cbp$ A0PC5!H 4G)@ =WҮa8g7y{8r8)Zzfz&=X19i9Zઊ'@0 0b_}ѷln <{nH! MOWy)3Cx]08Nm[n e"bpEmWC n@q08=[۶}ayj+VXP0 'kڙ㾫`WZ UÐ䔟=ԡ+`p2>/"-.[a ( O-P `pb)mVl)s`pb|`p>@  &tM.[a(2.΁!%'JUtPFfA3,   NÑ,! N_&0} mq SP@I|j@@L:(mVؠBE5$@xg mVl)s`pb|`p>@  &EѶYi+lPG p<Vp3f@` N޶e+L_EF[\9081PA>08Z@h۬6PtQ # 8+3 0o"-.[a ( O-P `pbtQ@emV T(OD#ѶM.[aclq _1]DAw#n"p=_O.A9 .4'L-y>  TmQX(?rl# ;e>I^ky　R-^?}S^pp=z+\B e֋#Xؚz[s[eV˳vnL'apN=SFW1u`pZ>vz'#l3bpI1{ރ?r@@G`ɎVϴBRj%M&oO>u6>yrU}NyzShFV9owl=0zMFKx{uJX{u뮣A:A2$ r9(;w&o:6Jxt:qɝYsCx5£.jwEq*Խ%|Ig]ɭCaN_鶴>"I~}?q7L5>}VsaSDY#o 4|:2BOqjv)-3V:KH)J/O Dd<lJ  C A? "2lpe&Q's<H8G`!@pe&Q's<ݬ`%xڍAkA{$Ĭie*>_za`QS ?9+cJosFe 8^-2ʱ`|y9;ޤ[75˨Tg&l7[1})=OEg>8'w8ȾlR}JIlVq3S. H֊Ut"E![+EzQ=vExN`$"DBҚQۗ$'RazanM` " tn] |!{OrV72LU{ӄ֡_9{SӪx.^خ_VJpv569/`(5XV""5iZZ=jT<.y^*^w(9xMvC?Oϛr-/[?u}{FdnQ; IN`G/߯}t}ܮӚnqA,:%]Dy(Wi-?'VcBDdJ  C A? "2|/_a/miXL`!P/_a/mi@ Rxcdd`` @c112BYL%bL0YnFf! KA?HȺ 1305rC0&dT20]?c&,``abM-VK-WMcXB\Xt9c(mXcBܤC\Kf1H&C\F4sl.Nps@WrAC r`CDkF&&\iAh`[Y5DdlTJ  C A? "2[[mO''$sO`!k[[mO''$l XJ9xcdd``6cd``baV d,FYzP1C&,7\f! KA?H n5rC0&dT20]I?1 @001d++&1k[~!}<& \ ,@u@@ڈ3Hq%ļ5a#7(Ϛ0o1eBܤ,ډ92O?> \4.=8fdbR ,.Ie(?Œ2D~`pDd DJ  C A? "2X@΃ H!ZK>84MQ`!,@΃ H!ZK>8hp xڍKA{fDz(ҖVEDPBl F zPZ('CyQ? AD(Zɀۙlkiwٙ7AHXH>-AAD~.GO73Rقڠ\二eg-Iګڛ@Zk21zYΈ22U7~@O`O.*R57Z^X߫գ&\"ceUZi`oӪo튟ka~4Zz.o_v4a͛q|S\.f9h35'M}4Cf>,z9[?WڗuLua[b }n~s7M߽yH_DޯOQM n).}kc.L0acDmo-6ę|#QoVY+z?tW }-)~Dd DJ  C A? "2SʵY&ٿ{>/CT`!'ʵY&ٿ{>X+ xڍKA{dnDќӞ)et;=4p%wdE2 KDs01|=^AV^RljiO,dTV_QG8sy>;P5*jr}n >l[ "MFD:/Իj;X̏j|X󛛊G5;w4z IBR+Wy5zgYOsyzz2fL= YfO0}f>@0pZ<}YOY:}w;ծ\^>kUym$?Oー@| 'ɚR ]ǀd+e͜@o1( l`3}DSFVV [U8L- L0Dd<,J  C A? "2 2{df4W`! 2{df` xcdd``~$d@9`,&FF(`T̐ & 1CRcgb TFnĒʂT$/&`b]F"L@ 0dƽ fr1L@(\%m: g`ˈ˞ HF%!RXn!#c=2Bj ,@sC2sSRsܩNLPs'0ʂ5{{{ a#f8+"n;0j-f.!~[V p?bŌ pD8m+) K7LLJ% ,iu ] A4gf~M Dd,J  C A? "2n^EKJY`!B^EKHxcdd``> @c112BYL%bL0Yn,56~) m @ k+75MbnĒʂT+~3);a&br-ځO8`ˈf.н>.hsc,а``㔑I)$5<9d>  lDd,J  C A? "2ZZÐ @c112BYL%bL0Yn ,56~) m @ k+'5MbnĒʂT+~3);a~&brXKKZjJ]" jŭE%-y߹w&;g@\r333gT"n jA[>ofgRS_r1X%tvwD7SeA_4;'3"㦎{Kyb?h( &` PCO@&0J `5lja ? oxK > >Ox+ N' q8A N' q"1"dD,#ϴep>m%G0_~ s !8ӑth*ӊ;fjhh4>>_MUG$kMpϮe h)]Msg O+j ]$M9싎-3}3> b]7_ɼJ} !4Kǽ, N2r3r7؆\|sM"Gc6&7Ǵaq)8g?Ih>eʏŇN5eً?B#}QƵ%-wv"67|V̴N::Ygç-NyuֈSRLĩ3VI/d8Vg8jƹ^t3VUqV9| N:cŹF35x[,֌Sq+IqNr4Ƀ8ՙH{lǹiO!Seyp\24k+O_6bdA?H`{#əه)'Ӭ[OAGN :4߲޾i;M7@~>`9&IQEu'c86 B93_og-o3)o)wkMyO7t2=[1{1VR^i[Aho| <(ilC,ù9CGsc٧od{L[t|tTz0F<:K]큻>Gy]u:6}\L떷OJJd94w]G]G'ﺭ;泑3fZj^GykwRyt]U](oqkBƵ b:uQ7t_ g otR׃Fݨv 3ϫ6FicfqO ;ț6uQw:̋צMwH7}{g=RS Sçz|Z;fL̴O4}sTORMv;xO;4y)O^K뙽v%8. g+p3^?7JPG;3Fc~"Z@Fc{Ov?bL`bPL,\5N"#^1vS܌<ʹ1ѸYx>rʃ8utFZ3jOĩ3Zq:| NzqJyOfZ'N׌3Sqt{VI[}Z.t3V4iq+48>y[zqtK;=8>:ꟇNUZz9Jk xqi<(ʅݿrs@pf=ԵB {j=m+AE>$}&+fZ6R isX|z7+w\uن.٥4t^,̴iR׍"Vf^:+׶nsfZ e|›R(ϋ|y gRn4n4Msp?Ըɋ7mmۯE>rǣ{9>= f^sk&^Gi؟;ٻ:^GW+72\D"pXG׉B`JzZ 7_+qU7Sg|? ǿC;`G #?Jm ox ;y+q?My,]=~j!Hx;Kt _ρp X !̺!)2AIb'5]8S QOv~J={P/R7؇zu5Q/NN,&ěP>>6;4^GD>MDEMnŇaԑ##ѥ81MPg>>C8N3D=oԡ'[ . W?̀d gT9Ђi@K %Z6<[g8t9@'\<3_-@W+݁@O }y@`_#Y@60 ."F(`4<` p!08uB7w"1&"HdAyA]m.h!]\A4:hgjW\LKG۟9jh虃ӚM c_Y^#gjܷ|VdzPwhXawWhyS+'W|'}-a&%8>cv}ZCo~77>s>͡R*h惣kmcε>w$%tCo9/̈%^ rn0i̋Co5*p6z{cff^sETruqXO}j1Z;z2S~Kf:%?<ۼ9]͚|) inNؖdiwG NַYڃ8utZwgçZk=N<{ttZvSqt{V)ñ6Ã5nuƊwp_Ay+qnw݃8z8;|ZAnuƊX3bOVg8GW|{: s5D}gL=Zg< c3>]].Wxĥ_u{.<u*R3c]>zA#sٻ]O3T\XzWjWajWvMNyfZwM5yq.T%5Ճ\ޮv=u'y6W(mfu&׍Uu ^5y-vZ:uI^):4yUwLX^CƒwתiC:D^+x1\+0¯k3׼C$ﻐ϶s܇lXsioBkg"+Y~60 &\&Nk(Eׂ4ԊEt!pQX,e w~&`9d ]+6~{+a6 1| ]X bX ~Ǿ?~ /k#t 6{aװ|x~>x>= x0G!t<]O@矠6 {aI<lm_ y>!"tۡ%zضy;ag']4 Kon* U(=()nu"7߃ً^0L{~0Ѿh_Go 7/t:Mޡس`~S9Ҁ3ԖY@{:GOsS:?9p=zJwewc;7 {ps|& h 8g큳@Gp.pЙ{q 8 t=@z@/ 2c0!P`p F#\`0f:o|1X`0ޏߺT~.胆^\P3UܵNsp^/9 aO[8>\P7Q+rzT7`ܘy^GN72l!3˲u2rFN|֣|ag1Η8_,!_͓|'`l6h W ,ra'Y/$ `[s5CE B\ &FR3( OD1mDNLgɢ=p6~w$Ĺb,p8OD÷alv7`ؿ|1@З `cl , À  >#o.dr!; :Fbx99K}AP2L¶;W8T'rD=7czɐLb+ep) 7` r[@pm~Wlρj`.t<`>P\ \,bb`/wS }e`)/nn |>ST7Av9t:W#V`%{pϼ7dy._ ݾ[Ǽ:W !8kֳƷn{G~f O ! t*q8A N' $;C q8A v6"1"dD#'-9k&?8Ɋw}=Y53ne<1 K2儘'qx?戃[ŅN7Ď`id6nӹr8?k~ႥeEWG+3$dȷ j's. J F!f#zaTrG07p.?lrdb尉F幣/)LL;u"L1- 4'ሼ')VOO|ڋ^MRc*xۉhÏz8~#Ǡ7Dk15l>ѩG#_oR\wu<2VsT=]la͂rU+[YZ.ܳZe 6 ^˅{֑U-td=Wo൜ʱقɂzDrN&({6fagrNҔ=KYHSLUN3N[}٩\T圞%r.?GVM5\S%i[*VKBJz(G W@*(\yJ y|Epc/rQmZ*WW}ϊ{6k}?c]z^.rz=]a7að+o=_\+ i:Ƀv,=g)֟Jqs#Dp iyX}q!%>4ʔ7z-ki8!|H Ly'PyHc2H;g횮zvj1?`i|b"^֑m B/OU~f=8[E}WpJĻ/BaUĻ#q7Gc}GtM=O2c~܌w ^* -%/dJK_TFpM-{YVjT/ՈӠY(&)b-sGw@Z..49Ȯ$a(h:{w+Tkŀs.YPVT"lPJQ|yculğMmyEkg$,ߌ"Lͤ&*e l ZB͂r*[YZ.ܳw v1\gmS9VZPHUY_o^3U9gە=n]3Ugm+{*RTճbbeTPʜc4̒cOzMҶah䰛~ofI(a73K8̒z"`[P^U|?ցׁAWǟYRol+E+ErR??~:NYlj,%ⱎg(p5ޕm$"z.Sjdm 9Hvew5DZWV+ѕkD`V/uF?,[DJD6v΍E9{zp_w: ͦ}C9/r&G.O8\8⢲%syF[g"|M6cd75Yg&iKjWOz?߬x~W1no<>qq_owOWîx]ͧlbd_--e.HƟl/Sw]jo!Yg=ꗾo+x+ ]iE .Z8Ңk,>WY^E K,(-LS^6234xQYzEˊ{[{-na7u濐sOInl$7qT#G~ھsxoig7kVnOnRiͯWIӓ*SKNb' E |7>ͅ:'qeڮ\Ƶ܍x~v\ȇ> oJ_/gG7&+C]cWA'BSÿA`!9o8bNV>xtHxcdd`` @c112BYL%bL0Yn,56~) m @ */ UXRYvoaA $37X/\!(?71a] YXt9m06'303|F0ɞ`2n\PqC =0cdbR ,.Ie(O`Iegbal v320 GI%Dd0,J ' C A'? " 224a wdc`![24a wdk)xcdd``bd``baV d,FYzP1C&,7\f! KA?H  Sjx|K2B* Rt1 @001d++&1k[~!}:O.:]j m6J_Jt ŕ0L@' lO1W߭?P 27)?_Kv ? 0 vn%4Ը! h 0y{qĤ\Y\P̕f2D~b;]\%Dd0,J ( C A'? "224a wdc`![24a wdk)xcdd``bd``baV d,FYzP1C&,7\f! KA?H  Sjx|K2B* Rt1 @001d++&1k[~!}:O.:]j m6J_Jt ŕ0L@' lO1W߭?P 27)?_Kv ? 0 vn%4Ը! h 0y{qĤ\Y\P̕f2D~b;]\INE %username%EuseuseNormalm ENS User%E67 Microsoft Word 9.0R@`I@x=dI@ (@nadIM-՜.+,0 hp|  ENS`72 LINEAR SYSTEMS LABORATORY 2: Title  FMicrosoft Word Document MSWordDocWord.Document.89q i0@0 Normal_HmH sH tH 8@8 Heading 1$$@&a$5\<@< Heading 2$$@&a$ 6\]<< Heading 3$$@&a$ 5CJ \8@8 Heading 4$@& 56\]:@: Heading 5$$@&a$CJ\<A@< Default Paragraph Font&)@& Page Number4@4 Header  !OJQJ, @, Footer  !,B@", Body Text$a$."@. Caption$a$5\S@ BS@ C B"_PQ]^ !5AENW`ij!./GQW`r -?Qcu~; < T U x y + , ? @ * + C T v,-_`@hi+,?@34`atu&"'"8"9"#### & &&&&&&'''''''''''''''''Q(R(~(()))F*G*s*t*********+++!+*+1+2+j+k+++K,L,M,N,O,P,Q,R,S,T,U,V,W,X,,, ....5.M....%/&/f/g///22333444446666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666667$7%7H7I7J7K7W7X777889 95969y9z9::::;;;;====>>>>??????????????BBBB0H00"0"0"00"0"0"0"0"0"0"00"0"00000"0"0"0"0"00"0"0"000"0"0"00"0"0"0"0000"0"0"0"00000000000"0"0"0"0"00"0"0"0"00"0"0"0"0"00"0"0"00"0"00"0"0"0"0"0"0"0"0"00"0"0"0"0"0"00"00"0"0"0"00"00"0"0"0"0"00"0"00n0n000P000000000000000000000000000000000000000000000000000000000000000000000000000{(0{(0{(00{(000{(00{(0{(0{(0{(0{(0{(0{(0{(0{(0000{(0000{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(00{(00{(0{(0{(0{(0{(0{(0{(0{(0{(0{(00{(0{(0{(00{(0{(0{(0{(00{(0{(0{(0{(00{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0{(0@0@0@0@0@0@0@0@0@0 0000 00 00000000 00 0000000000 000000000000800S 0S0Z0Z0Z00Z 0Z 4XXX[6 wde!'+/;7a;=qCF%*,./12468:@AC @!+.V08::::>CF&()+-03579;<=>?BDF'_>ac" 6 8 =QSvXln1TVdAdf9\^V%j%l%V(y({()))K*n*p*l+++J5|5~5B:::::::::::::::::::::::: #.19@BGRU[!!!tt=QSSgi  @rt H \ ^   C ::::::::::::::f 0e0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S"@d(    60e?"   p0e?. ``TT`TT"   60e?" B S  ?666BQ"!tQ-$P,T@-$ t _1076235054 _1076235346 _1076490809B@@@B>dfklmQT" 9 < G H S Z ^ 6 = > A G N O R ,0AH4=TVZ[]vXoqruv}1WY^`cdAgiklr9_adeo  !!*"/"0"5"k"r"""V#[#\#a#V%m%o%t%u%x%%%r&w&&&&&&&V(|((((())))))K*q*t*v*w*y*****l++++++ ----.....%.&.3.....+/4///11f2o222 33^3g3#4'44444J5555556666777#7*74787G7I7J777#8,8{8888899499999999 ::::;#;X;Y;\;];^;;;;;;;<<<<<<<<>>>>??????>@A@X@Z@^@_@c@@@@@@@@@@@@@)A2A4A:AQARATAUA[AAAAAAACBLBBB"#67AB/>dHMQT" 9 : N S z {   + 3 E F V Y  =TVvXonp-1W^`d@Aghik49_`ad;= *"0"9"="U#\#%V%m%%&&&&b'd'R(V(|(}((())))G*K*q*r*********+ +*+++k+l+++--...%..... / /&/e/22 3344444J5556666777#7*74787G7I7J7a7f7778889949C9E999999 :#;X;;;<<\>e>>>>>>>??A@X@@@@@:AQAAABB333333333333333333333333333333333333333333333333333333333333333333333333333333333>d" 9 +,01=TvXo1WdAg9_``aa ##$$$$%%%%V%m%V(|((())))K*q*l++333333J55666667777#7*74787G7I7J7J7K7BBENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS UsergC:\Documents and Settings\mbuehner\Application Data\Microsoft\Word\AutoRecovery save of Lab_Notes_4.asdENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docENS User=U:\work\Signals_and_Systems_I\lab04\Lab_Notes\Lab_Notes_4.docA. g $< pQMBPs!>\se `hh^h`o(.^`.pLp^p`L.@ @ ^@ `o(.^`.L^`L.^`.^`.PLP^P`L.hh^h`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.hh^h`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.hh^h`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.hh^h`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.A. e `BP pQMg !>\  NÎB     Hj        'j                t        xi        6J7=?BB@`=B@@UnknownGz Times New Roman5Symbol3& z ArialA"GenevaArial71CourierY New YorkTimes New Roman"1hWF-hf,hfCM-`0d7 2QLINEAR SYSTEMS LABORATORY 2: %username%ENS UserCompObjj