Normalized frequency (signal processing): Difference between revisions

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	== Alternative normalizations ==		== Alternative normalizations ==

	Some programs (such as ] toolboxes) that design filters with real-valued coefficients ~~use~~ ~~the~~ ] ({{math\|''f''<sub>s</sub> / 2}}~~) as~~ the ].		Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer a characteristic frequency of {{math\|''f''<sub>s</sub>/2}} (the ]), which expands the upper limit of useful frequencies from {{math\|1/2}} to {{math\|1.}} The corresponding unit of frequency, shown in the table below, is ''half-cycles per sample''.

	], denoted by {{math\|''ω''}} and with the unit ], can be similarly normalized. When {{math\|''ω''}} is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π'' radians/sample.		], denoted by {{math\|''ω''}} and with the unit ], can be similarly normalized. When {{math\|''ω''}} is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π'' radians/sample.
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	{\| class="wikitable"		{\| class="wikitable"
		⚫	\|+
		⚫	!'''Quantity'''
			!'''Normalization'''
		⚫	!'''Numeric range'''
		⚫	!'''Computation'''
		⚫	!'''Value'''
	\|-		\|-
		⚫	\|{{math\|''f''{{′}}}}
⚫	\| '''Quantity'''
			\|{{math\|''f''<sub>s</sub>}}
	\| '''Unit'''
			\| {{math\|size=150%\|}}  or
⚫	\| '''Numeric range'''
			\|1000 cycles/sec ÷ 44100 samples/sec
⚫	\| '''Computation'''
			\|= 0.02268 cycles/sample
⚫	\| '''Value'''
	\|-		\|-
	\| {{math\|''f''{{′}}}}		\|{{math\|''f''{{′}}}}
			\|{{math\|''f''<sub>s</sub>/2}}
	\| cycle per sample
	\|    or		\|   or
	\| 1000 / 44100		\|1000 cycles/sec × 2 half-cycles/cycle ÷ 44100 samples/sec
		⚫	\|= 0.04535 half-cycles/sample
	\| 0.02268
	\|-		\|-
	\| {{math\|''f''{{′}}}}		\|{{math\|''ω''{{′}}}}
			\|{{math\|''f''<sub>s</sub>/2π}}
⚫	\| half-~~cycle per~~ sample
	\|    or		\|   or
			\|1000 cycles/sec × 2π radians/cycle ÷ 44100 samples/sec
	\| 1000 / 22050
			\|= 0.14250 radians/sample
	\| 0.04535
⚫	\|-
⚫	\| {{math\|''ω''{{′}}}}
	\| radian per sample
	\|    or
	\| 2''π'' 1000 / 44100
	\| 0.1425
	\|}		\|}

Revision as of 17:45, 10 January 2023

Frequency divided by a characteristic frequency

In digital signal processing (DSP), a normalized frequency (f′) is a quantity that is equal to the ratio of a frequency and a characteristic frequency of a system.

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, typically has the unit cycle per sample regardless of whether the original signal is a function of time, space, or something else. For example, when f is expressed in Hz (cycles per second), f_s is expressed in samples per second.

This allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. Such a concept is a digital filter design whose bandwidth is specified not in hertz, but as a percentage of the sample rate of the data passing through it. Formulas expressed in terms of f_s (or T_s ≡ 1 / f_s) are readily converted to normalized frequency by setting those parameters to 1. The inverse operation is usually accomplished by replacing instances of the frequency parameter, f, with f / f_s or f T_s.

Alternative normalizations

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer a characteristic frequency of f_s/2 (the Nyquist frequency), which expands the upper limit of useful frequencies from 1/2 to 1. The corresponding unit of frequency, shown in the table below, is half-cycles per sample.

Angular frequency, denoted by ω and with the unit radian per second, can be similarly normalized. When ω is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate f_s = 44,100 samples/second (often denoted by 44.1 kHz), and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the discrete-time Fourier transform, which is always a periodic function.


Quantity	Normalization	Numeric range	Computation	Value
f′	f_s	or	1000 cycles/sec ÷ 44100 samples/sec	= 0.02268 cycles/sample
f′	f_s/2	or	1000 cycles/sec × 2 half-cycles/cycle ÷ 44100 samples/sec	= 0.04535 half-cycles/sample
ω′	f_s/2π	or	1000 cycles/sec × 2π radians/cycle ÷ 44100 samples/sec	= 0.14250 radians/sample

Notes and citations

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

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