digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...

(DSP), a normalized frequency is a ratio of a variable

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

(

f

) and a constant frequency associated with a system (such as a ''

sampling rate In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or s ...

'',

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.

Examples of normalization

A typical choice of characteristic frequency is the ''

'' (

f_s

) that is used to create the digital signal from a continuous one. The normalized quantity,

f' = \tfrac,

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''),

f_s

is expressed in ''samples per second''. Some programs (such as

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...

toolboxes) that design filters with real-valued coefficients prefer the

Nyquist frequency In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a Sampling (signal processing), sampler, which converts a continuous function or signal into a discrete sequence. For a given S ...

(f_s/2)

as the frequency reference, which changes the numeric range that represents frequencies of interest from

\left, \tfrac\right /math>  ''cycle/sample'' to

, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

/math> ''half-cycle/sample''. Therefore, the normalized frequency unit is important when converting normalized results into physical units. Normalized_frequency_example

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of

\tfrac,

for some arbitrary integer

N

(see ). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by

\tfrac.

The normalized Nyquist frequency is

\tfrac

with the unit ^th ''cycle/sample''.

Angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...

, denoted by

\omega

and with the unit '' radians per second'', can be similarly normalized. When

\omega

is normalized with reference to the sampling rate as

\omega' = \tfrac,

the normalized Nyquist angular frequency is . The following table shows examples of normalized frequency for

f = 1

''kHz'',

f_s = 44100

''samples/second'' (often denoted by 44.1 kHz), and 4 normalization conventions:

References

{{reflist, 1, refs= {{cite book , last=Carlson , first=Gordon E. , title=Signal and Linear System Analysis , publisher=©Houghton Mifflin Co , year=1992 , isbn=8170232384 , location=Boston, MA , pages=469, 490 {{cite journal , doi=10.1109/PROC.1978.10837 , last=Harris , first=Fredric J. , title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform , journal=Proceedings of the IEEE , volume=66 , issue=1 , pages=51–83 , date=Jan 1978 , url=http://web.mit.edu/xiphmont/Public/windows.pdf, citeseerx=10.1.1.649.9880 , bibcode=1978IEEEP..66...51H , s2cid=426548 Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies. Digital signal processing Frequency

Examples of normalization

See also

References