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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-cycle 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'' 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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| |{{math|''f''<sub>s</sub>}} | |{{math|''f''<sub>s</sub>}} | ||
| | {{math|size=150%|}} or | | {{math|size=150%|}} or | ||
| |1000 |
|1000 cycles/second ÷ 44100 samples/second | ||
| |0.02268 cycle/sample | |0.02268 cycle/sample | ||
| |- | |- | ||
| |{{math|''f''{{′}}}} | |{{math|''f''{{′}}}} | ||
| |{{math|''f''<sub>s</sub>/2}} | |{{math|''f''<sub>s</sub> / 2}} | ||
| | or | | or | ||
| |1000 cycles/second × 2 half{{nbh}}cycles/cycle ÷ 44100 samples/second | |1000 cycles/second × 2 half{{nbh}}cycles/cycle ÷ 44100 samples/second | ||
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| |- | |- | ||
| |{{math|''ω''{{′}}}} | |{{math|''ω''{{′}}}} | ||
| |{{math|''f''<sub>s</sub>/2π}} | |{{math|''f''<sub>s</sub> / 2π}} | ||
| | or | | or | ||
| |1000 cycles/second × 2π radians/cycle ÷ 44100 samples/second | |1000 cycles/second × 2π radians/cycle ÷ 44100 samples/second | ||
Revision as of 01:45, 11 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, 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/second ÷ 44100 samples/second | 0.02268 cycle/sample |
| f′ | fs / 2 | or | 1000 cycles/second × 2 half‑cycles/cycle ÷ 44100 samples/second | 0.04535 half‑cycle/sample |
| ω′ | fs / 2π | or | 1000 cycles/second × 2π radians/cycle ÷ 44100 samples/second | 0.14250 radian/sample |
See also
Notes and citations
- Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.