Normalized frequency (signal processing)

<h2 id="examples-of-normalization">Examples of normalization</h2>
A typical choice of characteristic frequency is the <a href="/facts/Sampling_rate/xxck7aLN">sampling rate</a> (
 
 
 
 
 f
 
 s
 
 
 
 
 {\displaystyle f_{s}}
 
) that is used to create the digital signal from a continuous one. The normalized quantity, 
 
 
 
 
 f
 ′
 
 =
 
 
 
 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 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is expressed in <a href="/facts/Hertz/6Wshj5qm">Hz</a> (cycles per second), 
 
 
 
 
 f
 
 s
 
 
 
 
 {\displaystyle f_{s}}
 
 is expressed in samples per second.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
Some programs (such as <a href="/facts/MATLAB/qPjLISCk">MATLAB</a> toolboxes) that design filters with real-valued coefficients prefer the <a href="/facts/Nyquist_frequency/zwFUKdVR">Nyquist frequency</a> 
 
 
 
 (
 
 f
 
 s
 
 
 
 /
 
 2
 )
 
 
 {\displaystyle (f_{s}/2)}
 
 as the frequency reference, which changes the numeric range that represents frequencies of interest from 
 
 
 
 
 [
 
 0
 ,
 
 
 
 1
 2
 
 
 
 
 ]
 
 
 
 {\displaystyle \left[0,{\tfrac {1}{2}}\right]}
 
 cycle/sample to 
 
 
 
 [
 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 
 
 
 
 
 
 
 
 f
 
 s
 
 
 N
 
 
 
 ,
 
 
 {\displaystyle {\tfrac {f_{s}}{N}},}
 
 for some arbitrary integer 
 
 
 
 N
 
 
 {\displaystyle N}
 
 (see <a href="/facts/Discrete-time_Fourier_transform/Em8WprVs">§ Sampling the DTFT</a>). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by 
 
 
 
 
 
 
 
 f
 
 s
 
 
 N
 
 
 
 .
 
 
 {\displaystyle {\tfrac {f_{s}}{N}}.}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>: p.56 eq.(16) <a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The normalized Nyquist frequency is 
 
 
 
 
 
 
 N
 2
 
 
 
 
 
 {\displaystyle {\tfrac {N}{2}}}
 
 with the unit ⁠1/N⁠th cycle/sample.
<a href="/facts/Angular_frequency/VqLm3L3R">Angular frequency</a>, denoted by 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 and with the unit <a href="/facts/Radian_per_second/cG0plKHD">radians per second</a>, can be similarly normalized. When 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 is normalized with reference to the sampling rate as 
 
 
 
 
 ω
 ′
 
 =
 
 
 
 ω
 
 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 
 
 
 
 f
 =
 1
 
 
 {\displaystyle f=1}
 
 kHz, 
 
 
 
 
 f
 
 s
 
 
 =
 44100
 
 
 {\displaystyle f_{s}=44100}
 
 samples/second (often denoted by <a href="/facts/44.1_kHz/7SBI1DYS">44.1 kHz</a>), and 4 normalization conventions:

<table><tbody><tr><th>Quantity</th><th>Numeric range</th><th>Calculation</th><th>Reverse</th></tr><tr><td> f ′ = f f s {\displaystyle f'={\tfrac {f}{f_{s}}}} </td><td>  [0, ⁠1/2⁠] cycle/sample</td><td>1000 / 44100 = 0.02268</td><td> f = f ′ ⋅ f s {\displaystyle f=f'\cdot f_{s}} </td></tr><tr><td> f ′ = f f s / 2 {\displaystyle f'={\tfrac {f}{f_{s}/2}}} </td><td>  [0, 1] half-cycle/sample</td><td>1000 / 22050 = 0.04535</td><td> f = f ′ ⋅ f s 2 {\displaystyle f=f'\cdot {\tfrac {f_{s}}{2}}} </td></tr><tr><td> f ′ = f f s / N {\displaystyle f'={\tfrac {f}{f_{s}/N}}} </td><td>  [0, ⁠N/2⁠] bins</td><td>1000 × N / 44100 = 0.02268 N</td><td> f = f ′ ⋅ f s N {\displaystyle f=f'\cdot {\tfrac {f_{s}}{N}}} </td></tr><tr><td> ω ′ = ω f s {\displaystyle \omega '={\tfrac {\omega }{f_{s}}}} </td><td>  [0, π] radians/sample</td><td>1000 × 2π / 44100 = 0.14250</td><td> ω = ω ′ ⋅ f s {\displaystyle \omega =\omega '\cdot f_{s}} </td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Prototype_filter/D5VTDvao">Prototype filter</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384. <a href="8170232384" target="_blank">8170232384</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837. S2CID 426548. <a href="http://web.mit.edu/xiphmont/Public/windows.pdf" target="_blank">http://web.mit.edu/xiphmont/Public/windows.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies. <a href="https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies" target="_blank">https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Normalized frequency (signal processing) open-in-new

Normalized frequency (signal processing)