This site uses cookies to deliver our services and to show you relevant ads and job listings. By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service . Your use of the Related Sites, including DSPRelated.com, FPGARelated.com, EmbeddedRelated.com and Electronics-Related.com, is subject to these policies and terms.

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(2πf o t)



[Analog] f o in cycles/second (Hz)



[–F s /2 ≤ f o ≤ F s /2] f o t is \(\frac{cycles}{second} \cdot seconds \)

= cycles. sin(Ω o t)



[Analog] Ω o in radians/second



[–πF s ≤ Ω o ≤ πF s ] Ω o = 2πf o .

f o in cycles/second.

Ω o t is \(\frac{radians}{second} \cdot seconds \)

= radians. sin(2πf o nt s )



[Digital] f o in cycles/second



[–F s /2 ≤ f o ≤ F s /2] n in samples.

f o nt s is

\(\frac{cycles}{second} \cdot samples \cdot \frac{seconds}{sample} \)

= cycles. sin(2πnf o /f s )



[Digital] f o /f s in cycles/sample



[–1/2 ≤ f o /f s ≤ 1/2] n in samples.

nf o /f s is

\(samples \cdot \frac{cycles}{second} \cdot \frac{seconds}{sample} \)

= cycles. sin(2πf o n)



[Digital] f o in cycles/sample



[–1/2 ≤ f o ≤ 1/2] n in samples.

f o n is \( \frac{cycles}{sample} \cdot samples \)

= cycles. sin(ω o n)

(See row below)



[Digital] ω o in radians/sample



[–π ≤ ω o ≤ π] n in samples.

ω o n is \( \frac{radians}{sample} \cdot samples \)

= radians. sin(ω o n)



[Digital] ω o in radians



[–π ≤ ω o ≤ π] n is dimensionless.

ω o n is = radians.