Can the (point-wise) multiplication of two discrete band-limited functions be inherently aliased?

(1) is it possible that the point-wise multiplication of two discrete, band-limited functions is aliased itself? Yes.

1. How can a band limited signal be sampled without aliasing?
2. Why convolution is multiplication in frequency domain?
3. When DFT is real?
4. What is DFT used for?

How can a band limited signal be sampled without aliasing?

Thus, band-limited signals can be sampled and fully recovered only when observing the Nyquist criterion. For bandpass signals the Nyquist criterion will ensure no aliasing only when the recovery of the signal is done with a bandpass filter; otherwise a higher sampling frequency will be required.

Why convolution is multiplication in frequency domain?

We know that a convolution in the time domain equals a multiplication in the frequency domain. In order to multiply one frequency signal by another, (in polar form) the magnitude components are multiplied by one another and the phase components are added.

When DFT is real?

The real DFT transforms an N point time domain signal into two N/2 + 1 point frequency domain signals. The time domain signal is called just that: the time domain signal.

What is DFT used for?

The Discrete Fourier Transform (DFT) is of paramount importance in all areas of digital signal processing. It is used to derive a frequency-domain (spectral) representation of the signal.