Normalized frequency (signal processing)

{{Short description|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, $f' = \tfrac{f}{f_s},$ has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when $f$ 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 $\left[0, \tfrac{1}{2}\right]$ cycle/sample to $[0, 1]$ half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.

File:Normalized_frequency_example.svg

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of $\tfrac{f_s}{N},$ for some arbitrary integer $N$ (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 $\tfrac{f_s}{N}.$ {{rp|p.56 eq.(16)}} The normalized Nyquist frequency is $\tfrac{N}{2}$ with the unit {{sfrac|1|N}}^th cycle/sample.

Angular frequency, 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 $\omega' = \tfrac{\omega}{f_s},$ the normalized Nyquist angular frequency is {{nowrap|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:

class="wikitable"

!Quantity

!Numeric range

!Calculation

!Reverse

f' = \tfrac{f}{f_s}

| {{math|[|size=150%}}0, {{sfrac|1|2}}{{math|]|size=150%}} cycle/sample

|1000 / 44100 = 0.02268

| $f = f' \cdot f_s$

f' = \tfrac{f}{f_s / 2}

| [0, 1] half-cycle/sample

|1000 / 22050 = 0.04535

| $f = f' \cdot \tfrac{f_s}{2}$

f' = \tfrac{f}{f_s / N}

| {{math|[|size=150%}}0, {{sfrac|N|2}}{{math|]|size=150%}} bins

|1000 × {{mvar|N}} / 44100 = 0.02268 {{mvar|N}}

| $f = f ' \cdot \tfrac{f_s}{N}$

\omega' = \tfrac{\omega}{f_s}

| [0, π] radians/sample

|1000 × 2π / 44100 = 0.14250

| $\omega = \omega' \cdot f_s$

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.

}}

Category:Digital signal processing

Category:Frequency

Normalized frequency (signal processing)

Examples of normalization

See also

References