# A Table of Digital Frequency Notation

August 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 ≤ fo ≤ Fs/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 ≤ fo ≤ Fs/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 Previous post by Rick Lyons:
A Quadrature Signals Tutorial: Complex, But Not Complicated
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Is It True That j is Equal to the Square Root of -1 ?

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