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Revision as of 17:45, 10 January 2023 editBob K (talk | contribs)Extended confirmed users6,614 edits →Alternative normalizations: expand the computation column for clarityTag: Visual edit: Switched← Previous edit Revision as of 17:50, 10 January 2023 edit undoBob K (talk | contribs)Extended confirmed users6,614 editsm →Alternative normalizations: plural to singular where appropriateTag: Visual editNext edit →
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== Alternative normalizations == == Alternative normalizations ==


Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer a characteristic frequency of {{math|''f''<sub>s</sub>/2}} (the ]), which expands the upper limit of useful frequencies from {{math|1/2}} to {{math|1.}} The corresponding unit of frequency, shown in the table below, is ''half-cycles per sample''. Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer a characteristic frequency of {{math|''f''<sub>s</sub>/2}} (the ]), which expands the upper limit of useful frequencies from {{math|1/2}} to {{math|1.}} The corresponding unit of frequency, shown in the table below, is ''half-cycle per sample''.


], denoted by {{math|''ω''}} and with the unit ], can be similarly normalized. When {{math|''ω''}} is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π''&nbsp;radians/sample. ], denoted by {{math|''ω''}} and with the unit ], can be similarly normalized. When {{math|''ω''}} is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π''&nbsp;radians/sample.
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|&nbsp;{{math|size=150%|}}&nbsp; or &nbsp;&nbsp; |&nbsp;{{math|size=150%|}}&nbsp; or &nbsp;&nbsp;
|1000 cycles/sec ÷ 44100 samples/sec |1000 cycles/sec ÷ 44100 samples/sec
|= 0.02268 cycles/sample |= 0.02268 cycle/sample
|- |-
|{{math|''f''{{′}}}} |{{math|''f''{{′}}}}
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|&nbsp;&nbsp; or &nbsp;&nbsp; |&nbsp;&nbsp; or &nbsp;&nbsp;
|1000 cycles/sec × 2 half-cycles/cycle ÷ 44100 samples/sec |1000 cycles/sec × 2 half-cycles/cycle ÷ 44100 samples/sec
|= 0.04535 half-cycles/sample |= 0.04535 half-cycle/sample
|- |-
|{{math|''ω''{{′}}}} |{{math|''ω''{{′}}}}
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|&nbsp;&nbsp; or &nbsp;&nbsp; |&nbsp;&nbsp; or &nbsp;&nbsp;
|1000 cycles/sec × 2π radians/cycle ÷ 44100 samples/sec |1000 cycles/sec × 2π radians/cycle ÷ 44100 samples/sec
|= 0.14250 radians/sample |= 0.14250 radian/sample
|} |}


Revision as of 17:50, 10 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. The corresponding unit of frequency, shown in the table below, is half-cycle per sample.

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 Normalization Numeric range Computation Value
f′ fs    or    1000 cycles/sec ÷ 44100 samples/sec = 0.02268 cycle/sample
f′ fs/2    or    1000 cycles/sec × 2 half-cycles/cycle ÷ 44100 samples/sec = 0.04535 half-cycle/sample
ω′ fs/2π    or    1000 cycles/sec × 2π radians/cycle ÷ 44100 samples/sec = 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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