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| == Alternative normalizations == | == Alternative normalizations == | ||
| Some programs (such as ]) that design filters with real-valued coefficients use the Nyquist frequency ({{math|''f''<sub>s</sub>/2}}) as the ]. |
Some programs (such as ] toolboxes) that design filters with real-valued coefficients use the ] ({{math|1=''f''<sub>Ny</sub> = ''f''<sub>s</sub> / 2}}) as the ] – 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 {{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}} 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. | ||
| ], 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'' 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'' radians/sample. | ||
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| {| class="wikitable" | {| class="wikitable" | ||
| |- | |- | ||
| | '''Quantity''' | |||
| | '''Unit''' | | '''Unit''' | ||
| | ''' |
| '''Numeric range''' | ||
| | '''Computation''' | | '''Computation''' | ||
| | '''Value''' | | '''Value''' | ||
| |- | |- | ||
| | {{math|''f'' / ''f''<sub>s</sub>}} | |||
| | cycle |
| cycle per sample | ||
| | or | | or | ||
| | 1000 / 44100 | | 1000 / 44100 | ||
| | 0.02268 | | 0.02268 | ||
| |- | |- | ||
| | {{math|''f'' / ''f''<sub>s</sub>}} | |||
| | half-cycle |
| half-cycle per sample | ||
| | or | | or | ||
| | 1000 / 22050 | | 1000 / 22050 | ||
| | 0.04535 | | 0.04535 | ||
| |- | |- | ||
| | {{math|''f'' / ''f''<sub>Ny</sub>}} | |||
| ⚫ | | radian |
||
| | hertz per hertz | |||
| | or | |||
| | 1000 / 22050 | |||
| | 0.04535 | |||
| |- | |||
| | {{math|''ω''}} | |||
| ⚫ | | radian per sample | ||
| | or | | or | ||
| | 2''π'' 1000 / 44100 | | 2''π'' 1000 / 44100 | ||
Revision as of 13:14, 8 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.
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 toolboxes) that design filters with real-valued coefficients use the Nyquist frequency (fNy = fs / 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 / fs 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 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.
| Quantity | Unit | Numeric range | Computation | Value |
| f / fs | cycle per sample | or | 1000 / 44100 | 0.02268 |
| f / fs | half-cycle per sample | or | 1000 / 22050 | 0.04535 |
| f / fNy | hertz per hertz | or | 1000 / 22050 | 0.04535 |
| ω | radian per sample | or | 2π 1000 / 44100 | 0.1425 |
See also
Notes and citations
- Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston,MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.