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Revision as of 12:57, 11 January 2023 editBob K (talk | contribs)Extended confirmed users6,614 edits partially undid my own changesTag: Visual edit← Previous edit Revision as of 17:46, 11 January 2023 edit undoBob K (talk | contribs)Extended confirmed users6,614 edits →Alternative normalizations: made the changes you just "talked" aboutTag: Visual editNext edit →
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|+ |+
!'''Quantity''' !'''Quantity'''
!'''Normalization'''
!'''Numeric range''' !'''Numeric range'''
!'''Computation''' !'''Computation'''
!'''Value''' !'''Value'''
|- |-
|{{math|''f''{{′}}}} |{{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}
⚫ |&nbsp;&nbsp;{{math||size=150%}}&nbsp;
|{{math|''f''<sub>s</sub>}}
⚫ |&nbsp;{{math|size=150%|}}&nbsp; or &nbsp;&nbsp;
|1000&nbsp;cycles/second / 44100&nbsp;samples/second |1000&nbsp;cycles/second / 44100&nbsp;samples/second
|0.02268 cycle/sample |0.02268 cycle/sample
|- |-
|{{math|''f''{{′}}}} |{{math|1=''f''{{′}} = ''f'' / (''f''<sub>s</sub>/2)}}
⚫ |&nbsp;&nbsp;&nbsp;
|{{math|''f''<sub>s</sub> / 2}}
|&nbsp;&nbsp; or &nbsp;&nbsp; |1000&nbsp;cycles/second / 22050&nbsp;samples/half-second
⚫ |1000&nbsp;cycles/second / 22050&nbsp;samples/second
|0.04535&nbsp;half{{nbh}}cycle/sample |0.04535&nbsp;half{{nbh}}cycle/sample
|- |-
|{{math|''ω''{{′}}}} |{{math|''ω''{{′}}}} = {{math|''ω'' / ''f''<sub>s</sub>}}
|&nbsp;&nbsp;&nbsp;
|{{math|''f''<sub>s</sub> / 2π}}
⚫ |1000&nbsp;×2π radians/second / 44100&nbsp;samples/second
⚫ |&nbsp;&nbsp; or &nbsp;&nbsp;
|1000&nbsp;cycles/second / 44100&nbsp;samples/second × 2π
|0.14250&nbsp;radian/sample |0.14250&nbsp;radian/sample
|} |}

Revision as of 17:46, 11 January 2023

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.

A typical choice of characteristic frequency is the sampling rate (fs) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = f / fs, 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), fs 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 fs (or Ts ≡ 1 / fs) 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 toolboxes) that design filters with real-valued coefficients prefer a characteristic frequency of fs / 2 (the Nyquist frequency), which expands the upper limit of useful frequencies from 1/2 to 1.

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 = 44,100 samples/second (often denoted by 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.

Quantity Numeric range Computation Value
f′ = f / fs     1000 cycles/second / 44100 samples/second 0.02268 cycle/sample
f′ = f / (fs/2)     1000 cycles/second / 22050 samples/half-second 0.04535 half‑cycle/sample
ω′ = ω / fs     1000 ×2π radians/second / 44100 samples/second 0.14250 radian/sample

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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