Normalized frequency (signal processing)
In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency ([math]\displaystyle{ f }[/math]) and a constant frequency associated with a system (such as a sampling rate, [math]\displaystyle{ f_s }[/math]). 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.
Examples of normalization
A typical choice of characteristic frequency is the sampling rate ([math]\displaystyle{ f_s }[/math]) that is used to create the digital signal from a continuous one. The normalized quantity, [math]\displaystyle{ f' = \tfrac{f}{f_s}, }[/math] has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when [math]\displaystyle{ f }[/math] is expressed in Hz (cycles per second), [math]\displaystyle{ f_s }[/math] is expressed in samples per second.[1]
Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency [math]\displaystyle{ (f_s/2) }[/math] as the frequency reference, which changes the numeric range that represents frequencies of interest from [math]\displaystyle{ \left[0, \tfrac{1}{2}\right] }[/math] cycle/sample to [math]\displaystyle{ [0, 1] }[/math] 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 [math]\displaystyle{ \tfrac{f_s}{N}, }[/math] for some arbitrary integer [math]\displaystyle{ N }[/math] (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by [math]\displaystyle{ \tfrac{f_s}{N}. }[/math][2]:p.56 eq.(16) The normalized Nyquist frequency is [math]\displaystyle{ \tfrac{N}{2} }[/math] with the unit 1/Nth cycle/sample.
Angular frequency, denoted by [math]\displaystyle{ \omega }[/math] and with the unit radians per second, can be similarly normalized. When [math]\displaystyle{ \omega }[/math] is normalized with reference to the sampling rate as [math]\displaystyle{ \omega' = \tfrac{\omega}{f_s}, }[/math] the normalized Nyquist angular frequency is π radians/sample.
The following table shows examples of normalized frequency for [math]\displaystyle{ f = 1 }[/math] kHz, [math]\displaystyle{ f_s = 44100 }[/math] samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:
| Quantity | Numeric range | Calculation | Reverse |
|---|---|---|---|
| [math]\displaystyle{ f' = \tfrac{f}{f_s} }[/math] | [0, 1/2] cycle/sample | 1000 / 44100 = 0.02268 | [math]\displaystyle{ f = f' \cdot f_s }[/math] |
| [math]\displaystyle{ f' = \tfrac{f}{f_s / 2} }[/math] | [0, 1] half-cycle/sample | 1000 / 22050 = 0.04535 | [math]\displaystyle{ f = f' \cdot \tfrac{f_s}{2} }[/math] |
| [math]\displaystyle{ f' = \tfrac{f}{f_s / N} }[/math] | [0, N/2] bins | 1000 × N / 44100 = 0.02268 N | [math]\displaystyle{ f = f ' \cdot \tfrac{f_s}{N} }[/math] |
| [math]\displaystyle{ \omega' = \tfrac{\omega}{f_s} }[/math] | [0, π] radians/sample | 1000 × 2π / 44100 = 0.14250 | [math]\displaystyle{ \omega = \omega' \cdot f_s }[/math] |
See also
References
- ↑ Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.
- ↑ Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform". Proceedings of the IEEE 66 (1): 51–83. doi:10.1109/PROC.1978.10837. Bibcode: 1978IEEEP..66...51H. http://web.mit.edu/xiphmont/Public/windows.pdf.
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