課程名稱︰系統生物數學 課程性質︰選修 課程教師︰歐陽彥正 開課學院： 開課系所︰ 考試日期（年月日）︰104/11/09 考試時限（分鐘）：3小時 試題 : 1.(20%)The family doctor of Mr. A suggested that Mr. A must reduce his body weight. The physician told Mr. A that the calorie intake each day should not exceed 2500 kcal. Assume that the current weight of Mr. A is 90kg and Mr. A needs 35kcal per kilogram of weight to maintain his normal life style. It is known that each gram of fat can generate 9kcal of energy. Can you figure out how the body weight of Mr. A will decrease as time elapses, if Mr. A keeps the calorie intake exactly at 2500kcal everyday? For this problem, you can use 24hours as the unit of time. 2.(10%)Is the system defined in Problem 1 linear? 3.(15%)Assume that the system shown in Figure 1, which is composed of 3 subsystems, is linear and time-invariant. Let g1(t) and g3(t) denote the impulse response functions of subsystems 1 and 3, respectively. Can you figure out the Fourier transform of the impulse response function of subsystem 2 in terms of the Fourier transforms of s(t), r(t), g1(t), and g3(t)? Figure 1. ┌──┐ ┌──-┐ ┌─────-┐ ┌──-┐ ┌──┐ │s(t)│→│g1(t)│→│Subsystem 2│→│g3(t)│→│r(t)│ └──┘ └-──┘ └-─────┘ └-──┘ └──┘ 4.(10%)The linear and time-invariant system shown in Figure 2 has an impulse response function h(t) with h(t)=0 for t<0 and t>=15 seconds. Let Wh(w) be the Fourier transform of h(t) and Wh(w)=0 for │w│>=10Hz. Let h0, h1, ..., hn-1 denote the samples of h(t) taken at the minimum sampling rate to record h(t) without loss of information, Then, what is n? Figure 2. ┌──┐ ┌────────┐ ┌──┐ │s(t)│→│ h(t) │→│r(t)│ └──┘ └────────┘ └──┘ 5.(15%)Now, signal s(t) with s(t)=0 for t<0 and t>=5 second arrives at the input of the system shown in Figure 2. Assume that it has been determined that the sampling rate of 10ms per sample is sufficiently high to record s(t) without loss of information. Let s0, s1,..., s499 denote the 500 samples of s(t) taken during 0<=t<5 second. In order to record the output of the system r(t) without loss of information, we want to obtain the 500-point discrete Fourier transform of the samples of h(t). How can you obtain the 500-point discrete Fourier transform of h(t) based on h0, h1,..., hn-1? 6.(15%)Let s*0, s*1,..., s*499 and h*0, h*1,..., h*499 denote the 500-point discrete Fourier transforms of the samples of s(t) and h(t). Can you compute the 500-point samples of r(t) bases on these two sequences? 7.(15%)Now, assume that signal s(t) is present at the input of another linear and time-invariant system with impulse response g(t). Furthermore, assume that g(t)=0 for t<0 and t>=15 seconds and Wg(w)=0 for│w│>=20Hz, where Wg(w) is the Fourier transform of g(t). In order to figure out the output of the system z(t) efficiently and without loss of information, we want to obtain the discrete Fourier transforms s*0, s*1,...,s*m and g*0, g*1,... g*m of the samples of s(t) and g(t), respectively, and then invoke the convolution theorem to compute z0, z1,..., zm. In this case, what is the minimum value of m? --

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