The sinusoidal sequence
Lets consider continuous-time sinusoid x(t)
x(t) = A sin 2πF0t = A sin Ω0t
F0 and Ω0 are analog frequency in Hertz (or cycles per second) and radians per second, respectively. The sampled version can be given as follows
x(nT) = A sin 2πF0nT = A sin Ω0nT
We can drop the T from x(nT) and can obtain the following result
x(n) = A sin 2πF0nT = A sin Ω0nT, for all n
We can write Ω0T = ω0 which is digital frequency in radians (per sample), such that
x(n) = A sin ω0n = A sin 2πf0n, for all n
By setting ω0 = 2πf0 gives f0 = ω0/2π which is digital frequency in cycles per sample. In the analog domain horizontal axis is calibrated in seconds; "second" is 1 unit of the independent variable, because Ω0 and F0 are in "per second". In digital domain the horizontal axis is calibrated in samples; "sample" is 1 unit of independent variable, so ω0 and f0 are in "per sample".
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