A Table of Digital Frequency Notation
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)
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