Time shifts and phase changes

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

$$Y[n] = X[n-d]$$

so that the $d$th sample of $Y$ is the 0th sample of $X$ and so on. If the integer $d$ is positive (or zero), then $Y$ is a delayed copy of $X$. If $d$ is negative, then $Y$ anticipates $X$; 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 $X$); if you time shift a sum ${X_1}+{X_2}$ 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 $\omega$, 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

$$X[n] = A {Z^n}$$

then

$$Y[n] = X[n-d] = A {Z^{(n-d)}} = {Z^{-d}} A {Z^n} = {Z^{-d}} X[n]$$

so time shifting a complex sinusoid by $d$ samples is the same thing as scaling it by ${Z^{-d}}$--it's just an amplitude change by a particular complex number. Since $\vert Z\vert=1$ for a sinusoid, the amplitude change does not change the magnitude of the sinusoid, only its phase.

The phase change is equal to $- d \omega$, where $\omega = \mathrm{arg}(Z)$ is the angular frequency of the sinusoid. This is exactly what we should expect since the sinusoid advances $\omega$ radians per sample and it is offset (i.e., delayed) by $d$ samples.