Normalized frequency (signal processing)

In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency (${\displaystyle f}${\displaystyle f}) and a constant frequency associated with a system (such as a sampling rate , ${\displaystyle f_{s}}${\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.

A typical choice of characteristic frequency is the sampling rate (${\displaystyle f_{s}}${\displaystyle f_{s}}) that is used to create the digital signal from a continuous one. The normalized quantity, ${\displaystyle f'={\tfrac {f}{f_{s}}},}${\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 ${\displaystyle f}${\displaystyle f} is expressed in Hz (cycles per second), ${\displaystyle f_{s}}${\displaystyle f_{s}} is expressed in samples per second.^[1]

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency ${\displaystyle (f_{s}/2)}${\displaystyle (f_{s}/2)} as the frequency reference, which changes the numeric range that represents frequencies of interest from ${\displaystyle \left[0,{\tfrac {1}{2}}\right]}${\displaystyle \left[0,{\tfrac {1}{2}}\right]} cycle/sample to ${\displaystyle [0,1]}${\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 ${\displaystyle {\tfrac {f_{s}}{N}},}${\displaystyle {\tfrac {f_{s}}{N}},} for some arbitrary integer ${\displaystyle N}${\displaystyle N} (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by ${\displaystyle {\tfrac {f_{s}}{N}}.}${\displaystyle {\tfrac {f_{s}}{N}}.}^{[2] : p.56 eq.(16) [3]} The normalized Nyquist frequency is ${\displaystyle {\tfrac {N}{2}}}${\displaystyle {\tfrac {N}{2}}} with the unit ⁠1/N⁠^th cycle/sample.

Angular frequency, denoted by ${\displaystyle \omega }${\displaystyle \omega } and with the unit radians per second , can be similarly normalized. When ${\displaystyle \omega }${\displaystyle \omega } is normalized with reference to the sampling rate as ${\displaystyle \omega '={\tfrac {\omega }{f_{s}}},}${\displaystyle \omega '={\tfrac {\omega }{f_{s}}},} the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequency for ${\displaystyle f=1}${\displaystyle f=1} kHz, ${\displaystyle f_{s}=44100}${\displaystyle f_{s}=44100} samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:

• Prototype filter

• Digital signal processing
• Frequency

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