A Table of Digital Frequency Notation

Rick LyonsAugust 5, 2013

When we read the literature of digital signal processing (DSP) we encounter a number of different, and equally valid, ways to algebraically represent the notion of frequency for discrete-time signals. (By frequency I mean a measure of angular repetitions per unit of time.)

The various mathematical expressions for sinusoidal signals use a number of different forms of a frequency variable and the units of measure (dimensions) of those variables are different. It's sometimes a nuisance to keep track of those different algebraic frequency variables. Add to this the fact that the time-index variable n is sometimes dimensionless, and sometimes n is measured in samples.

The following table presents a list of algebraic expressions that I have seen in the literature of DSP. I keep a copy of that table pinned to the wall next to my desk. Perhaps some of you visiting this dsprelated.com web site would like to download a copy of the table.

For simplicity I show no initial phase term in the sinusoidal algebraic expressions in bold font in the left column of the table. For reference, I've included two sinusoidal expressions for continuous-time (analog) sine waves at the top of the table.

Notation Frequency variable
[frequency range]
Units (Dimensions)
sin(2πfot)

[Analog]
fo in cycles/second (Hz)

[–Fs/2 ≤ foFs/2]
fot is \(\frac{cycles}{second} \cdot seconds \)
= cycles.
sin(Ωot)

[Analog]
Ωo in radians/second

[–πFs ≤ Ωo ≤ πFs]
Ωo = 2πfo.
fo in cycles/second.
Ωot is \(\frac{radians}{second} \cdot seconds \)
= radians.
sin(2πfonts)

[Digital]
fo in cycles/second

[–Fs/2 ≤ foFs/2]
n in samples.
fonts is
\(\frac{cycles}{second} \cdot samples \cdot \frac{seconds}{sample} \)
= cycles.
sin(2πnfo/fs)

[Digital]
fo/fs in cycles/sample

[–1/2 ≤ fo/fs ≤ 1/2]
n in samples.
nfo/fs is
\(samples \cdot \frac{cycles}{second} \cdot \frac{seconds}{sample} \)
= cycles.
sin(2πfon)

[Digital]
fo in cycles/sample

[–1/2 ≤ fo ≤ 1/2]
n in samples.
fon is \( \frac{cycles}{sample} \cdot samples \)
= cycles.
sin(ωon)
(See row below)

[Digital]
ωo in radians/sample

[–π ≤ ωo ≤ π]
n in samples.
ωon is \( \frac{radians}{sample} \cdot samples \)
= radians.
sin(ωon)

[Digital]
ωo in radians

[–π ≤ ωo ≤ π]
n is dimensionless.
ωon is = radians.

t = continuous time in seconds
fs = sample rate in samples/second
ts = 1/fs in seconds/sample
Fs = sample rate in cycles/second (Hz)

This table is also available in pdf


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