# 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(2πf_{o}t)[Analog] |
f_{o} in cycles/second (Hz)[– F/2 ≤ _{s}f_{o} ≤ F/2]_{s} |
f_{o}t is \(\frac{cycles}{second} \cdot seconds \)= cycles. |

sin(Ω_{o}t)[Analog] |
Ω_{o} in radians/second[–π F ≤ Ω_{s}_{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/2 ≤ _{s}f_{o} ≤ F/2]_{s} |
n in samples.f_{o}nt is_{s}\(\frac{cycles}{second} \cdot samples \cdot \frac{seconds}{sample} \) = cycles. |

sin(2π nf_{o}/f)_{s}[Digital] |
f_{o}/f in cycles/sample_{s}[–1/2 ≤ f_{o}/f ≤ 1/2]_{s} |
n in samples.nf_{o}/f is_{s}\(samples \cdot \frac{cycles}{second} \cdot \frac{seconds}{sample} \) = cycles. |

sin(2π f_{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. |

t = continuous time in seconds

*f _{s}* = sample rate in samples/second

ts = 1/

*f*in seconds/sample

_{s}*F*= sample rate in cycles/second (Hz)

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