Normalized frequency (signal processing)

In digital signal processing (DSP), a normalized frequency (f′) 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, typically has the unit cycle per sample regardless of whether the original signal is a function of time, space, or something else. 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. Formulas expressed in terms of f_s (or T_s ≡ 1 / f_s) are readily converted to normalized frequency by setting those parameters to 1. The inverse operation is usually accomplished by replacing instances of the frequency parameter, f, with f / f_s or f T_s.

Alternative normalizations

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients use the Nyquist frequency (f_s / 2) as the normalization constant.

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

The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate f_s = 44,100 samples/second, and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the discrete-time Fourier transform, which is always a periodic function.

Quantity	Unit	Numeric range	Computation	Value
f′	cycle per sample	[−⁠1/2⁠, ⁠1/2⁠]  or  [0, 1]	1000 / 44100	0.02268
f′	half-cycle per sample	[−1, 1]  or  [0, 2]	1000 / 22050	0.04535
ω′	radian per sample	[−π, π]  or  [0, 2π]	2π 1000 / 44100	0.1425

See also

• Prototype filter