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| == Alternative normalizations == | == Alternative normalizations == | ||
| Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math|''f''<sub>s</sub>/2}}) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from {{math|}} to {{math|}}. | Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math|''f''<sub>s</sub>/2}}) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from {{math|}} ''cycles/sample'' to {{math|}} ''half-cycles/sample''. | ||
| ], denoted by {{math|''ω''}} and with the unit '']'', can be similarly normalized. When {{math|''ω''}} is normalized with reference to the sampling rate as {{math|1=''ω''′ = ''ω'' / ''f''<sub>s</sub>}}, the normalized Nyquist angular frequency is ''π 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 as {{math|1=''ω''′ = ''ω'' / ''f''<sub>s</sub>}}, the normalized Nyquist angular frequency is ''π radians/sample''. | ||
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| |{{math|1=''ν''′ = ''f'' / (''f''<sub>s</sub>/2) = 2''f'' / ''f''<sub>s</sub>}} | |{{math|1=''ν''′ = ''f'' / (''f''<sub>s</sub>/2) = 2''f'' / ''f''<sub>s</sub>}} | ||
| | | | | ||
| |2000 |
|2000 half-cycles/second / 44100 samples/second | ||
| |0.04535 cycle/sample | |0.04535 half-cycle/sample | ||
| |- | |- | ||
| |{{math|''ω''′}} = {{math|''ω'' / ''f''<sub>s</sub>}} | |{{math|''ω''′}} = {{math|''ω'' / ''f''<sub>s</sub>}} | ||
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| |0.14250 radian/sample | |0.14250 radian/sample | ||
| |} | |} | ||
| ==See also== | ==See also== | ||
Revision as of 15:44, 14 January 2023
Frequency divided by a characteristic frequencyIn 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 or space. 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 the Nyquist frequency (fs/2) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from cycles/sample to half-cycles/sample.
Angular frequency, denoted by ω and with the unit radians per second, can be similarly normalized. When ω is normalized with reference to the sampling rate as ω′ = ω / fs, 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 = 44100 samples/second (often denoted by 44.1 kHz), and 3 normalization options.
| Quantity | Numeric range | Computation | Value |
|---|---|---|---|
| f′ = f / fs | 1000 cycles/second / 44100 samples/second | 0.02268 cycle/sample | |
| ν′ = f / (fs/2) = 2f / fs | 2000 half-cycles/second / 44100 samples/second | 0.04535 half-cycle/sample | |
| ω′ = ω / fs | (1000 cycles/second × 2π radians/cycle) / 44100 samples/second | 0.14250 radian/sample |
See also
Citations
- Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.