# Todai Entrance Exam: Subject 2019 – Problem 5

This is Kronecker delta. The equation can be interpreted as: If there exists a that then we return , otherwise we return .

**(1)** According the given formula at the end of the problem:

**(2)** Ok, this is Discrete Fourier Transform:

**(3)**

By Modulation Property of the Fourier Transform, we have:

I don’t know much calculus but I think I can apply the inverse Fourier Transform formula here:

**(4)** Thanks to this and this, basically, aliasing is the phenomenon when the continuous function is not bandwidth limited to less than the Nyquist critical frequency (), the power spectral density that lies outside of the range is aliased into the range.

We have:

and

It seems like is a discrete version of with being approximated by . So the larger is (the bigger the sampling rate is), the more approximates .

It is not so easy to see that:

So:

For any , there exists a such that . Therefore, if is not bandwith limited to less than , power spectral density that lies outside of the range is aliased into the range.

To avoid aliasing, must be larger than twice of the largest non-zero angular frequency of .