The Significance of a Linear Phase

We know that a phase shift is directly proportional to a time shift in a sinusoid.

$$\begin{align*} \cos \left(\omega t +\phi\right)&=\cos \left[\omega \left(t +\frac{\phi}{\omega}\right)\right]\\ &=\cos \left(\omega \left(t -\tau\right)\right) \end{align*}$$

Here, $\tau$ is a time shift. From this, we observe the exact relationship below:

$$\phi = -\omega\tau$$

Clearly, the time shift of each sinusoid depends on its phase shift normalized by its own frequency. This is a linear expression of the form $y=mx$ in which the slope $m$ is given by $-\tau$.

Signal distortion

Almost all practical signals consist of multiple sinusoids. If there is a delay in the signal processing chain through a system, e.g., an amplifier or a filter, then this delay affects all the constituent sinusoids according to their own frequencies.

Let’s see an example in the code below. A square wave is again generated by a sum of sinusoids, but their phases can be altered to observe their impact.