東京大学 情報理工学系研究科 電子情報学専攻 2024年8月実施 専門 第5問

Author

adj-matrix

Description

Answer the following questions about discrete-time signal processing.

(1) Describe briefly the functionality and role of an anti-aliasing filter.

(2) For a linear time-invariant and causal discrete-time signal processing system , let the transfer function be where is a complex variable, are real constant numbers, and . In this case, find the difference equation for the relationship between the input and the output of the discrete-time signal for , where is an integer.

(3) In (2), find the range of such that is stable when and .

(4) In (2), when and , find the magnitude response and the phase response in , where is the imaginary unit and is the angular frequency.

(5) In (2), when , and , find the impulse response of using the following discrete-time unit step signal .

Kai

(1)

An anti-aliasing filter is a filter used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist-Shannon sampling theorem over the band of interest.

(2)

Given

i.e.,

Since :

Divide by (assuming non-zero) :

(3)

Substitute parameters into the denominator characteristic equation:

Roots:

For stability, poles must be inside the unit circle: i.e.,

If ,

If ,

i.e.,

(4)

Given

Since ,

Therefore,

Similarly,

In summary:

• If ,
• If ,

(5)

Given

i.e.,

Inverse Z-transform using :