東京大学 情報理工学系研究科 電子情報学専攻 2015年8月実施 専門 第5問

Author

Josuke

Description

Let us denote the Fourier transform of the signal as , where and represent a time variable and an angle frequency, respectively.

(1) Show the definition of the Fourier transform of the signal . Also, explain the difference between the Fourier transform and the Fourier series expansion.

(2) Explain why represents the power spectrum, i.e., the power at a certain angle frequency of .

(3) Derive the following Parseval's theorem in the Fourier transform and determine .

You may use , . You may also use the Fourier transform of the following convolution integrals

is the imaginary unit and is a real constant.

You may include when answering .

(4) Explain the physical meaning of the Parseval's theorem in the Fourier transform.

Kai

(1)

Fourier series can be applied to periodic signal. Fourier transform can be applied to non-periodic signal.

(2)

Thus represents the power of certain angle frequency .

(3)

if

So

(4)

The energy in time domain equals to times energy in frequency domain.