Normalized frequency (signal processing)
In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency (f) and a constant frequency associated with a system (such as a sampling rate, fs). 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 (fs) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = f / fs, has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when f is expressed in Hz (cycles per second), fs is expressed in samples per second.[1]
Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (fs/2) as the frequency reference, which changes the numeric range that represents frequencies of interest from [0, 1/2] cycle/sample to [0, 1] half-cycle/sample. Therefore, the normalized frequency unit is obviously 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 fs/N, for some arbitrary integer N (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by fs/N. The normalized Nyquist frequency is N/2 with the unit 1/Nth cycle/sample.
Angular frequency, denoted by ω and with the unit radians per second, can be similarly normalized. When ω is normalized with reference to the sampling rate as ω′ = ω / fs, the normalized Nyquist angular frequency is π radians/sample.
The following table shows examples of normalized frequencies for a 1 kHz signal (or filter bandwidth), a sampling rate fs = 44100 samples/second (often denoted by 44.1 kHz), and 4 normalization options.
| Quantity | Numeric range | Calculation |
|---|---|---|
| f / fs | [0, 1/2] cycle/sample | 1000 / 44100 = 0.02268 |
| f / (fs/2) | [0, 1] half-cycle/sample | 1000 / 22050 = 0.04535 |
| f / (fs/N) | [0, N/2] bins | 1000 × N / 44100 = 0.02268 N |
| ω / fs | [0, π] radians/sample | 1000 × 2π / 44100 = 0.14250 |
See also
Citations
- ^ Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.