Normalized frequency (signal processing): Difference between revisions

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Revision as of 12:28, 6 February 2023 editBob K (talk \| contribs)Extended confirmed users6,614 editsm →Examples of normalization: {{mvar}} template← Previous edit		Revision as of 00:12, 7 February 2023 edit undoBob K (talk \| contribs)Extended confirmed users6,614 editsm move image lower downNext edit →
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	In ] (DSP), a '''normalized frequency''' is a ratio of a variable ] ({{math\|''f''}}) and a constant frequency associated with a system (such as a '']'', {{math\|''f''<sub>s</sub>}}). 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.		In ] (DSP), a '''normalized frequency''' is a ratio of a variable ] ({{math\|''f''}}) and a constant frequency associated with a system (such as a '']'', {{math\|''f''<sub>s</sub>}}). 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 ===		=== Examples of normalization ===
	A typical choice of characteristic frequency is the '']'' ({{math\|''f''<sub>s</sub>}}) that is used to create the digital signal from a continuous one. The normalized quantity, {{math\|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when {{mvar\|f}} is expressed in ] (''cycles per second''), {{math\|''f''<sub>s</sub>}} is expressed in ''samples per second''.<ref>{{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}}</ref>		A typical choice of characteristic frequency is the '']'' ({{math\|''f''<sub>s</sub>}}) that is used to create the digital signal from a continuous one. The normalized quantity, {{math\|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when {{mvar\|f}} is expressed in ] (''cycles per second''), {{math\|''f''<sub>s</sub>}} is expressed in ''samples per second''.<ref>{{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}}</ref>
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	Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math\|''f''<sub>s</sub>/2}}) as the frequency reference, which changes the numeric range that represents frequencies of interest from {{math\|}} ''cycle/sample'' to {{math\|}} ''half-cycle/sample''. Therefore, the normalized frequency unit is obviously important when converting normalized results into physical units.		Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math\|''f''<sub>s</sub>/2}}) as the frequency reference, which changes the numeric range that represents frequencies of interest from {{math\|}} ''cycle/sample'' to {{math\|}} ''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 {{math\|''f''<sub>s</sub>/''N''}}, for some arbitrary integer {{mvar\|N}} (see {{slink\|Discrete-time_Fourier_transform\|Sampling_the_DTFT\|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by {{math\|''f''<sub>s</sub>/''N''}}. The normalized Nyquist frequency is {{math\|''N''/2}} with the unit {{sfrac\|1\|N}}<sup>th</sup> ''cycle/sample''.		A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of {{math\|''f''<sub>s</sub>/''N''}}, for some arbitrary integer {{mvar\|N}} (see {{slink\|Discrete-time_Fourier_transform\|Sampling_the_DTFT\|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by {{math\|''f''<sub>s</sub>/''N''}}. The normalized Nyquist frequency is {{math\|''N''/2}} with the unit {{sfrac\|1\|N}}<sup>th</sup> ''cycle/sample''.

Revision as of 00:12, 7 February 2023

Frequency divided by a characteristic frequency

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, 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 sampling rate (f_s) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = 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 is expressed in Hz (cycles per second), f_s is expressed in samples per second.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (f_s/2) 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 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 f_s/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 f_s/N. The normalized Nyquist frequency is N/2 with the unit ⁠1/N⁠ 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 / f_s, the normalized Nyquist angular frequency is π radians/sample.

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:


Quantity	Numeric range	Calculation	Reverse
f′ = f / f_s	cycle/sample	1000 / 44100 = 0.02268	f = f′ × f_s
f′ = f / (f_s/2)	half-cycle/sample	1000 / 22050 = 0.04535	f = f′ × f_s / 2
f′ = f / (f_s/N)	bins	1000 × N / 44100 = 0.02268 N	f = f′ × f_s / N
ω′ = ω / f_s	radians/sample	1000 × 2π / 44100 = 0.14250	ω = ω′ × f_s

Citations

Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.

Categories:

Revision as of 12:28, 6 February 2023 editBob K (talk \| contribs)Extended confirmed users6,614 editsm →Examples of normalization: {{mvar}} template← Previous edit		Revision as of 00:12, 7 February 2023 edit undoBob K (talk \| contribs)Extended confirmed users6,614 editsm move image lower downNext edit →
Line 5:		Line 5:
	In ] (DSP), a '''normalized frequency''' is a ratio of a variable ] ({{math\|''f''}}) and a constant frequency associated with a system (such as a '']'', {{math\|''f''<sub>s</sub>}}). 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.		In ] (DSP), a '''normalized frequency''' is a ratio of a variable ] ({{math\|''f''}}) and a constant frequency associated with a system (such as a '']'', {{math\|''f''<sub>s</sub>}}). 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 ===		=== Examples of normalization ===
	A typical choice of characteristic frequency is the '']'' ({{math\|''f''<sub>s</sub>}}) that is used to create the digital signal from a continuous one. The normalized quantity, {{math\|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when {{mvar\|f}} is expressed in ] (''cycles per second''), {{math\|''f''<sub>s</sub>}} is expressed in ''samples per second''.<ref>{{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}}</ref>		A typical choice of characteristic frequency is the '']'' ({{math\|''f''<sub>s</sub>}}) that is used to create the digital signal from a continuous one. The normalized quantity, {{math\|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when {{mvar\|f}} is expressed in ] (''cycles per second''), {{math\|''f''<sub>s</sub>}} is expressed in ''samples per second''.<ref>{{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}}</ref>
Line 11:		Line 10:
	Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math\|''f''<sub>s</sub>/2}}) as the frequency reference, which changes the numeric range that represents frequencies of interest from {{math\|}} ''cycle/sample'' to {{math\|}} ''half-cycle/sample''. Therefore, the normalized frequency unit is obviously important when converting normalized results into physical units.		Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math\|''f''<sub>s</sub>/2}}) as the frequency reference, which changes the numeric range that represents frequencies of interest from {{math\|}} ''cycle/sample'' to {{math\|}} ''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 {{math\|''f''<sub>s</sub>/''N''}}, for some arbitrary integer {{mvar\|N}} (see {{slink\|Discrete-time_Fourier_transform\|Sampling_the_DTFT\|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by {{math\|''f''<sub>s</sub>/''N''}}. The normalized Nyquist frequency is {{math\|''N''/2}} with the unit {{sfrac\|1\|N}}<sup>th</sup> ''cycle/sample''.		A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of {{math\|''f''<sub>s</sub>/''N''}}, for some arbitrary integer {{mvar\|N}} (see {{slink\|Discrete-time_Fourier_transform\|Sampling_the_DTFT\|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by {{math\|''f''<sub>s</sub>/''N''}}. The normalized Nyquist frequency is {{math\|''N''/2}} with the unit {{sfrac\|1\|N}}<sup>th</sup> ''cycle/sample''.

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