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)
- Comments
- Write a Comment Select to add a comment
To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.
Please login (on the right) if you already have an account on this platform.
Otherwise, please use this form to register (free) an join one of the largest online community for Electrical/Embedded/DSP/FPGA/ML engineers:
