Normalized frequency (signal processing)

In digital signal processing (DSP), a normalized frequency is a quantity that is equal to the ratio of a frequency and a characteristic frequency of a system.

A typical choice of characteristic frequency is the sampling rate (f_s) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = f / f_s, has the unit cycle per sample regardless of whether the original signal is a function of time or space. For example, when f is expressed in Hz (cycles per second), f_s is expressed in samples per second. This allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. Such a concept is a digital filter design whose bandwidth is specified not in hertz, but as a percentage of the sample rate of the data passing through it. The resultant set of filter coefficients provides that bandwidth ratio for any sample-rate.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (f_s/2) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from [0, 1/2] cycle/sample to [0, 1] half-cycle/sample.

Angular frequency, denoted by ω and with the unit radians per second, can be similarly normalized. When ω is normalized with reference to the sampling rate as ω′ = ω / f_s, the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequencies for a 1 kHz signal (or filter bandwidth), a sampling rate f_s = 44100 samples/second (often denoted by 44.1 kHz), and 3 normalization options.

Quantity	Numeric range	Computation	Value
f / fs	[0, ⁠1/2⁠]	1000 cycles/second / 44100 samples/second	0.02268 cycle/sample
f / (fs/2) = 2f / fs	[0, 1]	2000 half-cycles/second / 44100 samples/second	0.04535 half-cycle/sample
ω / fs	[0, π]	(1000 cycles/second × 2π radians/cycle) / 44100 samples/second	0.14250 radian/sample

See also

• Prototype filter