Normalization means to have a measure for a signal in the same, fixed, easy to use range such as [0,..,1] and this measure should not have physical units.
For normalizing the frequency f to a value fn in range [0,...,1], the most obvious way is to divide f by the sampling frequency fs:
fn = f / fs (1)
Formula (1) is a number (no units) showing how many sampling periods with frequency fs are in the sampled signal f.
However, according to Nyquist-Shannon theorem, the sampling frequency is always at least two times the frequency f. Thus (1) is never larger than 1/2. To have fn in the range [0, 1], we multiply (1) with a factor 2:
fn = 2 * f / fs (2)
Now if we replace fs to be the Nyquist frequency, fs=2*f then we obtain fn=1. For higher sampling frequencies than Nyquist frequency, the value fn will be less than one since we divide 2*f by an increasingly large number, thus we obtain the range [0,..,1] for fn as we wanted. If one would not use the multiplier 2 in (2), then the range would be always [0,...,1/2] which is not so nice.
In graphical representations in Matlab, the normalized frequency is shown as “(x pi rad/sample)”. This comes from the following calculation. It is known that omega in [rad/sec] and f in [Hz] are related by the formula:
omega = 2 * pi * f (3)It means
f = omega / (2*pi) (4)
Replacing f above in (2) gives:
fn = 2*omega/(2*pi) / fs = (omega / pi) / fs (5)
We multiply (5) by pi and obtain:
fn * pi = omega / fs (6)
Now (omega / fs) has units [rad/sec] / [sample / sec] = [rad/sample]
Therefore we have fn in units [pi * rad / sample] as shown in graphics. However, creating a unit in this way is somewhat misleading since the relative frequency has no units, as seen in formula (1).
