Starting from any (real or complex) signal , we can make other signals by
time shifting the signal by a (positive or negative) integer :
Time shifting is a linear operation (considered as a function of the input signal ); if you time shift a sum you get the same result as if you time shift them separately and add afterward.
Time shifting has the further property that, if you time shift a sinusoid of frequency , the result is another sinusoid of the same frequency; time shifting never introduces frequencies that weren't present in the signal before it was shifted. This property, called time invariance, makes it easy to analyze the effects of time shifts--and linear combinations of them--by considering separately what the operations do on individual sinusoids.
Furthermore, the effect of a time shift on a sinusoid is simple: it just
changes the phase. If we use a complex sinusoid, the effect is even simpler.
If for instance
The phase change is equal to , where is the angular frequency of the sinusoid. This is exactly what we should expect since the sinusoid advances radians per sample and it is offset (i.e., delayed) by samples.