# Exact formula for alias of Discrete Fourier transform for periodic sigals

Suppose that $f(t): \mathbb{R} \to \mathbb{C}$ is a $T$-periodic signal, with highest frequency $f_h$. Now suppose that our sampling rate frequency is lower than $f_h$, and is not any multiples of $1/T$. Furthermore, we use discrete Fourier transform, meaning that we will only be taking $N$ samples, which are of finite quantity.

What will be the formula for alias of DFT in such a case?

• The signal to noise ratio is the most critical part . For practical purposes if you could band pass filter your signal before sampling it you will get a better input into your DFT. The greater the number of samples you use will provide a better resolution for your measurement. Basically, Nyquist requires that the sampling frequency will be higher than the bandwidth of the signal that is analyzed. The desired signal is "folded" around the sampling frequency. – Moti Apr 8 '15 at 17:01
• Related: this question and its answers. – Matt L. Apr 8 '15 at 17:23