108年 - 108 國立中山大學_碩士班招生考試_資工系(乙組)：工程數學#105772

複選題
All the following questions are multiple-choice questions.

1.You can expand the function defined by f(x)=x²+3, 0＜x＜3 in a Fourier series, a cosine series, or a sine series. Please choose the correct answers. (A)1(6) = 3 for Fourier sine series； (B)(3) = 12 for Fourier cosine series； (C) (0) = 3 for Fourier series； (D) (-1) = 4 for Fourier sine series

2.Please determine which the following matrix is a diagonal matrix. (A) (B) (C) (D)

複選題

3.Please select the right following description (A) = and have the same eigenvalues.(B) If matrix A is diagonalizable, then the column of A are linearly independent.(C) Every orthogonal set of V must be linearly independent.(D) If A is an orthogonal matrix, then A is symmetric

4.Please select the function which is with the largest period (A) f(x) =cos5xcos3x； (B) g(x) = sin2.xsin2x； (C)h(x) = sin5x cos4x；(D) q(x) = cos 2x

複選題

5.There are six vectors shown below. Please determine the right description

(A) ,are linear independent

(B) , are linear independent(C), are linear independent(D) , are linear independent

【已刪除】5.(10%)There are six vectors shown below. Please determine the right description [0 [1] 5= Vz 질 V -1 -1 0 (A), Vs , are linear independent (B) vi, ,V3, are linear independent (C) v., are linear independent (D) vi , V2 , ', are linear independent IL. All the questions in the following group questions are multiple-choice questions. You are a nature scientist dedicated to observing a special particle movement phenomenon. Eventually, you find that the observing particle movement track belong to time x can be formulated as f(x) = cos 3x.

6.Your team leader ask you to analyze the result of the particle movement in the frequency domain. You first decide to apply the Fourier Transform to do this job. Please select the right answer after applying the Fourier Transform. (A)π(δ(ω-3))；(B)δ(ω-3)+δ(ω+3)；(C)π(δ(ω-3)+δ(ω+3)；(D)3π(δ(ω-3)+δ(ω+3))

7.While receiving the signal, you adopt a circuit to process this signal. After your analysis, you
can model the circuit by using the following ordinary differential equation (ODE).
y(x)+3y(x)+2y(x)= [δ(x-3)+δ(x+3)]

Please use the result in the previous problem and the symmetric property to solve the above ODE and

determine the right y(x).(A) π(δ(ω))；(B)π(δ(ω-3)+δ(ω+3)).；(C)π(δ(ω+3)-δ(ω-3)).；(D)π(δ(ω-3)+δ(ω-3)).

8. Now, you decide to transfer one signal s(x) (i.e., s(x) = )and make the f(x) as the carrie wave. Based on the modulation technique, you can create one new signal r(x) containing the s(x) and f(x). Please describe the formulation of the r(x) in the frequency domain. (A)(B) (C) (D)

9.After your analysis, you find that it is not practical to analyze the phenomenon in the
frequency domain in the continuous time. Afterward, you decide to use Discrete Fourier Transform
(DFT) instead. Assume you sample the new created r(x) signal every 2 unit time and collect 4 data
from time zero (ie., x are 0, 2, 4, and 6 sequentially.) If you know that the DFT can be formulated as

where the L is the length of the samples and the n indicates the sample number. Please select the right
following result of your analysis in the frequency domain after applying your derived DFT.(A) ；(B) ；(C)；(D)；

10. To facilitate the DFT analysis, you decide to program a DFT code. Besides, you find that you can adopt a matrix to perform this kind of transform. Please determine the proper Fourier matrix below, which can transform a length-4 data sequence from time domain to frequency domain.

(A)(B)(C)(D)

108年 - 108 國立中山大學_碩士班招生考試_資工系(乙組)：工程數學#105772

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