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Normalized frequency (signal processing)

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Revision as of 18:05, 7 January 2023 by Quondum (talk | contribs) (Alternative normalizations: rm confused statement that gets it wrong)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff) Frequency divided by a characteristic frequency

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 / fs (with the unit cycle per sample), where f is a frequency and fs 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 / Ts (with the unit sample), upon division by the sampling interval, Ts = 1 / fs (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 fs or Ts 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 / fs or f Ts.

Alternative normalizations

Some programs (such as MATLAB) that design filters with real-valued coefficients use the Nyquist frequency (fs/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 fs = 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.

Unit Domain Computation Value
cycle/sample    or    1000 / 44100 0.02268
half-cycle/sample    or    1000 / 22050 0.04535
radian/sample    or    2π 1000 / 44100 0.1425

See also

Notes and citations

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