京都大学 情報学研究科 知能情報学専攻 2021年8月実施 専門科目 S-5

Author

祭音Myyura

Description

Let be a discrete-time index. The unit impulse signal and the unit step signal are defined as follows:

Q.1 Compute the -transform of a discrete-time signal given below.

• (1)
• (2)

Q.2 Judge the stability of a system whose transfer function is given below and draw the correponding circuit. In addition, compute the inverse -transform .

• (1)
• (2)

Q.3 Compute the discrete-time Fourier transform of a discrete-time signal given below, where represents a normalized angular frequency.

• (1)
• (2)

Kai

Q.1

(1)

(2)

Note that

hence

Q.2

(1)

The system is stable since the pole of the transfer function is at , which lies within the unit circle.

The corresponding circuit diagram is as follows:

and

(2)

The system is stable since the pole of the transfer function is at , which lies within the unit circle.

The corresponding circuit diagram is as follows:

Note that

hence

Q.3

Note that

and

(1)

By (*) we have

(2)

By (**) we have