Normalized frequency (signal processing)
In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency () and a constant frequency associated with a system (such as a
Examples of normalization
A typical choice of characteristic frequency is the
Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency as the frequency reference, which changes the numeric range that represents frequencies of interest from cycle/sample to 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 for some arbitrary integer (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by [2]: p.56 eq.(16) [3] The normalized Nyquist frequency is with the unit 1/Nth cycle/sample.
Angular frequency, denoted by and with the unit radians per second, can be similarly normalized. When is normalized with reference to the sampling rate as the normalized Nyquist angular frequency is π radians/sample.
The following table shows examples of normalized frequency for kHz, samples/second (often denoted by
Quantity | Numeric range | Calculation | Reverse |
---|---|---|---|
[0, 1/2] cycle/sample | 1000 / 44100 = 0.02268 | ||
[0, 1] half-cycle/sample | 1000 / 22050 = 0.04535 | ||
[0, N/2] bins | 1000 × N / 44100 = 0.02268 N | ||
[0, π] radians/sample | 1000 × 2π / 44100 = 0.14250 |
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" (PDF). Proceedings of the IEEE. 66 (1): 51–83. S2CID 426548.
- ^ Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies.