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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.

An example of a normalized frequency is the sampling frequency in a system in which a signal is sampled at periodically, in which it equals f′ = f / f_s (with the unit cycle per sample), where f is a frequency and f_s is the sampling rate. For regularly spaced sampling, the continuous time variable, t (with unit second), is replaced by a discrete sampling count variable, n = t / T_s (with the unit sample), upon division by the sampling interval, T_s = 1 / f_s (with the unit second per sample).

The use of normalized frequency allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. An example of 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 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_Ny = f_s / 2) as the normalization constant – that is, they use the (dimensionless) ratio of a frequency to the Nyquist frequency in place of the sampling rate. An alternative may to think of this is as expressing the normalized frequency f′ = f / f_s in units of half-cycles per sample (or equivalently cycles per 2 samples), which produces the same numeric result when the units (hertz per hertz and half-cycle per sample, respectively) are omitted.

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 = 44.1 kHz, 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.

See also

• Prototype filter

Notes and citations

1. Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston,MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.

• Digital signal processing
• Frequency