京都大学 情報学研究科 知能情報学専攻 2024年8月実施 専門科目 S-4

Author

itsuitsuki

Description (English)

Let us define the Fourier transform of a real function and the inverse Fourier transform of a function with the following formulas, where and denote real numbers, and .

Answer the following questions, where , and denote positive constants.

Q.1

Prove that the following equation holds for real functions and , where denotes convolution.

Q.2

Compute the Fourier transform of the functions given below. (1) ,

(2)

Q.3

Let be a signal sampled from in Q.2 using a comb function , where denotes the Dirac delta function. Answer the following questions. You may use that holds.

(1) Draw the graph of in the range of .

(2) Show the condition for to satisfy in the range of .

(3) Draw the graph of in the range of .

(4) Draw the graph of in the range of . is given below.