Spectrum After the Time Shift I

Explore the effect of shifting a signal in the time domain on its spectrum from a magnitude and phase viewpoint.

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/4T/4. Here, TT is the wave period.

cos2πf(tT4)=cos(2πft2πfT4)=cos(2πFtπ2)=sin2πFt \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., TT, is analogous to traversing a full period in phase, i.e., 2π2\pi. Therefore, T/4T/4 corresponds to 2π/4=π/22\pi/4=\pi/2.

This is drawn in the figure below where the phase shift 9090^\circ turns the cosine into a sine:

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