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Colorado Northwestern Community College

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TOPIC: Insurance in Health Care

You are asked to choose an organization or one you are familiar with and select a method for identifying and mitigating risks. You may choose a method discussed in the articles from the weekly lectures or choose one you are familiar with.

Be sure to include the following sections:

- Introduction
- Background of Organization
- Types of Risks
- Method to Identify & Mitigate Risks
- Implementation Plan
- Benefits and Challenges
- Conclusion

Your writeup:

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Please check the file uploaded for this assignment, MATLAB is used for this assignmentProblem 21:(20 points)1. Simplify the following expressions:(a) (4 points) [δ(t−2)δ(t+ 3) +u(t−2)δ(t−3)u(t+ 3)]ejωt(b) (4 points)u(t+ 1)δ(1−t)e3t−3+sin(π3)+e−(3−jπ)tδ(t−1)2. Evaluate the following integrals:(a) (4 points)∫∞−∞g(τ)δ(t−τ)dτ(b) (4 points)∫∞−∞δ(2τ−3)δ(t+ 1−τ)dτ(c) (4 points)∫t0e−τδ(τ−1)dτProblem 22:(30 points) We use the symbol∗to represent the convolution operation on two signals. Mathematicallythe convolution of two signals,x(t) andy(t) is given as:x(t)∗y(t)≡∫∞−∞x(τ)y(t−τ)dτ1. (16 points) A system with impulse responseh(t) has inputf(t) and zero state responseyZSR(t). The system’sinput-output relation is:yZSR(t) =f(t)∗h(t)(a) (8 points) Using the definition of the convolution given above, prove that the system is linear.(b) (8 points) Using the definition of the convolution given above, prove that the system is time invariant.2. (4 points) Show thatf(t)∗δ(t−T) =f(t−T).3. (6 points) Iff(t)∗g(t) =c(t), show the derivative property of convolution ̇f(t)∗g(t) =f(t)∗ ̇g(t) = ̇c(t).4. (4 points) Derive the identitydu(t)dt=δ(t),whereu(t) is the unit-step function. In order to obtain this identity, you need to show that the functionalsg(t) =du/dtandδ(t) have the same effect on an arbitrary functionf(t), that is∫∞−∞f(t)g(t−T)dt=f(T)whereTis a real-valued constant parameter.Problem 23:(30 points)Systems can be represented by both an ODE and an impulse response function. Given either representation, youcan find the zero-state response for a given input. For example, consider the RC circuit in Figure 1.f (t)y(t)R2R1CFigure 1: RC circuit with input voltagef(t) and output voltagey(t).1. (5 points) Derive the ODE representation of the system andshow that it can be expressed as ̇y+1τy=Kτf.Express the time constantτand parameterKin terms ofR1,R2, andC. What is the physical significance ofthe parameterK?2. (5 points) Solve the ODE in part 1 to determine the zero-state unit-step response.3. (10 points) Determine the impulse response functionh(t) of the circuit by:(a) (5 points) Take the derivative of the zero-state unit-step response.(b) (5 points) Using the method described in Section 2.3 of Lathi.Note that your results should be identical.4. (10 points) Determine the zero-state unit-step responseusing the convolution integral. Check your answeragainst the result obtained in part 2. They should be identical.Problem 24:(20 points)Using the graphical convolution method discussed in section 2.4-2 of the text and lecture, find and sketchy(t) =f(t)∗h(t) for the following signals.1. (10 points)f(t) =u(t+ 1)−u(t−2)h(t) =e−t/5u(t)2. (10 points)f(t) = (t+ 2) (u(t+ 2)−u(t)) + (−t+ 2) (u(t)−u(t−2))h(t) =e−tu(t)Problem 25:(30 points) A future lecture demonstrates that any real-valued periodic signalf(t) with fundamentalperiodTomay be expresses as a superposition of an infinite number of sinusoids,f(t) =ao+∞∑n=1ancos(n ωot) +∞∑n=1bnsin(n ωot),wherea0, a1, a2, . . . , b1, b2. . .are real-valued constant coefficients given byao=1To∫Tof(t)dtan=2To∫Tof(t) cosnωotdtbn=2To∫Tof(t) sinnωotdt,andωo= 2π/To. As an example, the coefficients for the periodic waveform shown in Figure 3.10(a) of Lathi are:ao=13an= (−1)n(2nπ)2bn= 0ωo= 2π/To=π. As it not possible to numerically determinef(t) for an infinite number of terms, consider anapproximation that utilizes the first N terms of the summation,fN(t) =ao+N∑n=1ancos(n ωot) +N∑n=1bnsin(n ωot),IfN <∞, thenf(t) =fN(t) +e(t),wheree(t) is the approximation error.1. (12 points) Write a MATLAB primary function findfN that determinesfN(t) given an integer value ofNanda time vector. The syntax for the calling the function must befN = findfN(t,N);Implement the computation offNusing a For-Loop.Here is a basic example of a Matlab primary function using a for loop:function y = f_N(N)y = 0;for(i = 1:N)y = y+i;endreturnIn Matlab, one way to create the function above is to save the function as an m-file with the same name as thefunction; in this case we would save it asf_N.m. This function can then be called either from the commandline or from other scripts. For additional information on primary functions see:http://matlab.izmiran.ru/help/techdoc/matlab_prog/...h_func8.html2. (18 points) Write a Matlab script m-file that:(a) (9 points) Computesf(t) over the interval−1≤t≤1 using the equationf(t) =t2.The time vector must consist of 10,001 points equally spacedbetween -1 and 1. Plotf(t) using a dot-dashblack line.(b) (3 points) Call the function findfN, using the time vector generated in part (a) andN= 1. Using subplots,plotf(t), using a dashed red line, andf1(t), using a solid blue line, in the upper subplot. Plot the errore(t), in the lower subplot using a dash-dot black line.(c) (6 points)Repeat (b) usingN= 10 andN= 100. Notice that, asNincreases, the approximation erroris reduced. It is possible to reduce the approximation errorto any acceptable level by including enoughterms in the approximation.To earn full credit for Problem 25:•Include your three figures, your script m-file and your function file with your homework solutions.•Include a comment containing your name, your section numberand the due date at the top of both the scriptm-file and function file.•Appropriately label each graph; no credit is given for MATLAB plots whose axes are unlabeled! An exampleis provided in Figure 2.•Use the MATLAB commandgtextto place your name and section name within the figure.Time-1-0.500.51Amplitude-0.500.51The function f(t) compared to its approximation f1(t).f(t)f1(t)Time-1-0.500.51Amplitude-0.100.10.20.3The approximation error, e(t).Figure 2: Example of the plot comparingf(t) tof1(t)Problem 26:(25 points)Using the relationshipf(t)∗δ(t−T) =f(t−T)(1)derived in lecture, evaluation of the convolution integraly(t) =f(t)∗h(t) =∫∞−∞f(τ)h(t−τ)dτis simple when eitherf(t) orh(t) is a sum of weighted impulses. This problem extends this result to the case whereeither the derivative of eitherf(t) orh(t) yields a sum of weighted impulses.1. (10 points) As as an example of the utility of equation (1),suppose thaty(t) =f(t)∗h(t) wheref(t) = 2 (δ(t+ 3) +δ(t−3))h(t) =11 +t2.Determiney(t), and sketchf(t),h(t), andy(t) on a single plot.2. (15 points) Now suppose we apply the inputf(t) =u(t+ 2)−u(t−1)to a LTI system that has the impulse response functionh(t) =u(t−2).Neitherf(t) orh(t) is expressed directly as a sum of weighted impulses.(a) (3 points) Find an expression fordh/dtin terms of an impulse and sketchdh/dt.(b) (3 points) Letg(t) denote the response of a system with impulse responsedh/dtto the inputf, that isg(t) =f(t)∗dhdt.Calculate and sketchg(t).(c) (3 points) Use the derivative property of covolution to show the zero-state response of the system withimpulse responseh(t) to the inputf(t) can be expressed asy(t) =∫t−∞g(τ)dτ.(d) (6 points) Using the last two results, calculate and sketchy(t).Problem 27:(20 points)When the impulse response functionh(t) is a causal signal, then the system is causal. Conversely, if the systemimpulse response is noncausal then the system is noncausal.To illustrate this important concept, consider two LTIsystems that are represented by the impulse response functionsSystem 1:h1(t) =e−tu(t)System 2:h2(t) =etu(−t).1. (6 points) Sketchh1(t) andh2(t), and specify whether or not each impulse response functionis a causal ornoncausal signal.2. (14 points) Let the input to each system bef(t) =e−t/2u(t)(a) (5 points) Use convolution to determine the zero state responsey1(t), for the system,h1(t).(b) (5 points) Use convolution to determine the zero state responsey2(t), for the system,h2(t).(c) (4 points) Compare the sketch of bothy1(t) andy2(t) to the inputf(t). Determine if the systemsh1(t)andh2(t) are causal or noncausal. Explain your answers in a clearly written sentence.Problem 28:(20 points)1. (10 points) Consider two linear time-invariant systems whose impulse responsesh(t) are specified as•h1(t) =u(t)•h2(t) =2t(t2+1)2Classify each system, corresponding to the impulse functions considered above, as either BIBO stable or notBIBO stable. In order to receive credit, justify your answer.2. (10 points) The system shown in Figure 3 is composed of fourLTI systems whose impulse response functionsareh1(t),h2(t),h3(t), andh4(t). The input to the overall system isf(t), and the output isy(t). Using thedistributive, commutative, and associative properties ofconvolution, represent the composite system by a singleblock with an impulse response functionh(t) so thaty(t) =f(t)∗h(t),and expressh(t) in terms ofh1(t),h2(t),h3(t), andh4(t).f (t)h1(t)h2(t)h3(t)h4(t)ΣΣy(t)Figure 3: Block diagram for a system containing subsystems.

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