Time shifts and phase changes

Starting from any (real or complex) signal , we can make other signals by
time shifting the signal by a (positive or negative) integer :

so that the th sample of is the 0th sample of and so on. If the integer is positive, then is a delayed copy of . If is negative, then anticipates ; this can be done to a recorded sound but isn't practical as a real-time operation.

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

then

so time shifting a complex sinusoid by samples is the same thing as scaling it by --it's just an amplitude change by a particular complex number. Since for a sinusoid, the amplitude change does not change the magnitude of the sinusoid, only its phase.

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.