The concept of normalized frequency

This question has already been asked and answered, but the motivation behind the use of normalized frequency units still evades me.

The Discrete Time Fourier Transform $$X(\tilde{ \omega }) = \sum_{n=-\infty}^{\infty} x_n e^{-i\tilde{\omega}n}$$ in my text is given in terms of the normalized (dimensionless) angular frequency $\tilde{\omega} = \omega \Delta t$ where $\omega$ and $\Delta t$ are the physical frequency and time interval between measurements, respectively.

I want to understand the merits of using normalized frequency units.

1 Answer

Consider a continuous signal $$x(t) = \sin ( 2\pi f t) \$$

of which we have the following measurent points: $$x_n = x(n T_s) = \sin \bigg(2\pi \frac{f}{f_s} n \bigg) \ .$$

We note that $\frac{f}{f_s} \equiv f_n$ arises naturally as the frequency of our discrete set of measurements, which is called the normalized frequency.

Its units are $$\frac{[f]}{[f_s]} = \frac{\text{cycles/second}}{\text{samples/second}} = \text{cycles/sample} \ .$$

Hence, if we consider the (inverse) Discrete Time Fourier Transform, it is natural to use the normalized frequency $\tilde{\omega} = \omega T_s = 2\pi f_n$.

Supplement.

Note that the Nyquist sampling theorem requires $f_d < \frac{1}{2}$ to avoid aliasing. This puts a lower bound on our sampling rate $f_s$.