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.

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