Normalized frequency (signal processing)
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In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency ( f {\displaystyle f} ) and a constant frequency associated with a system (such as a sampling rate, f s {\displaystyle f_{s}} ). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.
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β’ See also
β’ References
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Examples of normalization
A typical choice of characteristic frequency is the sampling rate ( f s {\displaystyle f_{s}} ) that is used to create the digital signal from a continuous one. The normalized quantity, f β² = f f s , {\displaystyle f'={\tfrac {f}{f_{s}}},} has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when f {\displaystyle f} is expressed in Hz (cycles per second), f s {\displaystyle f_{s}} is expressed in samples per second.cite-ref-carlson-1-0[1]
Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency ( f s / 2 ) {\displaystyle (f_{s}/2)} as the frequency reference, which changes the numeric range that represents frequencies of interest from [ 0 , 1 2 ] {\displaystyle \left[0,{\tfrac {1}{2}}\right]} cycle/sample to [ 0 , 1 ] {\displaystyle [0,1]} half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.
A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of f s N , {\displaystyle {\tfrac {f_{s}}{N}},} for some arbitrary integer N {\displaystyle N} (see Β§ Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by f s N . {\displaystyle {\tfrac {f_{s}}{N}}.} cite-ref-harris-2-0[2]cite-ref-taboga-3-0[3] The normalized Nyquist frequency is N 2 {\displaystyle {\tfrac {N}{2}}} with the unit β 1/Nβ th cycle/sample.
Angular frequency, denoted by Ο {\displaystyle \omega } and with the unit radians per second, can be similarly normalized. When Ο {\displaystyle \omega } is normalized with reference to the sampling rate as Ο β² = Ο f s , {\displaystyle \omega '={\tfrac {\omega }{f_{s}}},} the normalized Nyquist angular frequency is Ο radians/sample.
The following table shows examples of normalized frequency for f = 1 {\displaystyle f=1} kHz, f s = 44100 {\displaystyle f_{s}=44100} samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:
| Quantity | Numeric range |
|---|---|
| f β² = f f s {\displaystyle f'={\tfracβ¦ | [ 0, β 1 / 2 β ] cycle/sample |
| f Ⲡ= f f s / 2 {\displaystyle f'={\t⦠| [0, 1] half-cycle/sample |
| f β² = f f s / N {\displaystyle f'={\tβ¦ | [ 0, β N / 2 β ] bins |
| Ο β² = Ο f s {\displaystyle \omega '={β¦ | [0, Ο ] radians/sample |
| Quantity | Calculation |
|---|---|
| f Ⲡ= f f s {\displaystyle f'={\tfrac⦠| 1000 / 44100 = 0.02268 |
| f Ⲡ= f f s / 2 {\displaystyle f'={\t⦠| 1000 / 22050 = 0.04535 |
| f Ⲡ= f f s / N {\displaystyle f'={\t⦠| 1000 à N / 44100 = 0.02268 N |
| Ο β² = Ο f s {\displaystyle \omega '={β¦ | 1000 Γ 2Ο / 44100 = 0.14250 |
| Quantity | Reverse |
|---|---|
| f β² = f f s {\displaystyle f'={\tfracβ¦ | f = f β² β
f s {\displaystyle f=f'\cdo⦠|
| f β² = f f s / 2 {\displaystyle f'={\tβ¦ | f = f β² β
f s 2 {\displaystyle f=f'\c⦠|
| f β² = f f s / N {\displaystyle f'={\tβ¦ | f = f β² β
f s N {\displaystyle f=f'\c⦠|
| Ο β² = Ο f s {\displaystyle \omega '={β¦ | Ο = Ο β² β
f s {\displaystyle \omega =β¦ |
See also
References
cite-note-harris-22. β citerefharris1978Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51β83. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837. S2CID 426548.
cite-note-taboga-33. β Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies.