Spectrum After the Time Shift I

One of the most important skills to learn in DSP that proves immensely helpful in any kind of application is the impact of a time shift on the signal spectrum. This is what we explore in this chapter.

Phase at a single frequency

The time reference at zero sets the wave phases just like the zero of a measuring tape. Let’s consider a cosine wave that is shifted to the right by $T/4$. Here, $T$ is the wave period.

$$\begin{align*} \cos2\pi f\left(t-\frac{T}{4}\right) &= \cos \left(2\pi ft -2\pi f\frac{T}{4}\right) = \cos \left(2\pi Ft - \frac{\pi}{2}\right) = \sin 2\pi Ft \end{align*}$$

Clearly, this becomes a sine wave. This is because a full period in time, i.e., $T$, is analogous to traversing a full period in phase, i.e., $2\pi$. Therefore, $T/4$ corresponds to $2\pi/4=\pi/2$.

This is drawn in the figure below where the phase shift $90^\circ$ turns the cosine into a sine: