The Relation between Time Shift and Phase

Understand the crucial idea of the wave phase as an indicator of its time shift.

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We now study the time shift of a waveform and how it’s related to the phase.

A sinusoidal waveform is given by the expression below:

x(t)=Acos(ωt+θ)x(t) = A \cos (\omega t + \theta)

This signal has these three parameters:

  • Amplitude AA
  • Frequency ω\omega
  • Phase θ\theta

In general, people find it quite straightforward to understand how the amplitude AA represents the signal level while the frequency ω\omega is related to the number of rotations in a second.

ω=2πf=2πT\omega = 2\pi f = \frac{2\pi}{T}

where TT is wave time period. Phase, however, is the tricky part.

Phase as a shadow of frequency

We know that a time shift to the right by τ\tau units of a sinusoidal wave is written as:

x(t)=Acos{ω(tτ)}=Acos(ωtωτ)=Acos(ωt+θ)\begin{align*} x(t)&=A\cos \left\{\omega (t-\tau)\right\}\\ &=A \cos (\omega t - \omega \tau)\\ &=A\cos (\omega t + \theta) \end{align*}

In words, the phase in terms of a time shift is given by:

θ=ωτ\theta = -\omega \tau

This is why we say that the phase is a shadow of frequency.

Note: For the same time shift, the phase is different for each frequency!

And the time shift for a given wave phase can be written as:

τ=θω\tau = -\frac{\theta}{\omega}

Let’s verify this with the help of an example.


Consider a sinusoidal wave, like so:

x(t)=3cos(0.4πt+0.4π)=3cos{0.4π[t(1)]}x(t) = 3\cos(0.4\pi t+0.4\pi)=3\cos\left\{0.4\pi [t-(-1)]\right\}

  • Amplitude: Clearly, the amplitude AA is equal to 33.
  • Frequency: The frequency is:

2πf=0.4πf=0.22\pi f = 0.4\pi \quad \rightarrow \quad f = 0.2

The time period is then:

T=1f=5T = \frac{1}{f}=5

  • Time shift: Since θ=0.4π\theta=0.4\pi, we have:

τ=0.4π0.4π=1\tau = -\frac{0.4\pi}{0.4\pi} = -1

Another way to look at this is that cosine has a peak value when its argument is zero. Putting 0.4πt+0.4π=00.4\pi t+0.4\pi=0, we get time shift as 1-1.

We can now run through different time shifts and phases in a coding environment.

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