# How does a shift in time domain result in phase shift in frequency spectrum?

I am aware of the fact that a time shift of say $$t_0$$, results in a phase shift in the frequency spectrum. What confuses me is how this scales the rotational part of the transform by $$t_0$$ and doesn’t add a factor of $$t_0$$.

For example, $$FT(x(t-t_0))=X(j\omega)e^{-j\omega t_0}$$

I understand the magnitude isn’t changed, but how is this a phase shift? This results in $$cos(\omega t_0)+jsin(\omega t_0)$$, but to my understanding this is not a shift, but a scaling of the frequency. A shift would be $$cos(\omega+t_0)+jsin(\omega+t_0)$$.

I probably have a misunderstanding with complex exponential, but the lack of info on this is keeping me up at night.

• Thank you for the edit, jithin. I don’t know how to use Latex yet. – ModularMan Mar 28 '20 at 15:06
• Wishing you a good rest now, see my answer below and let me know if you can now sleep. – Dan Boschen Mar 28 '20 at 15:48
• Yes! Thank you, Dan. That last part of your answer is especially insightful. You’re a DSP hero. – ModularMan Mar 28 '20 at 16:44
• It's the little things that can frustrate us the most... – Dan Boschen Mar 28 '20 at 16:48

Consider first that in the time domain that a phase rotation or shift in phase or phase shift specifically is done by multiplying the waveform with $$e^{j\phi}$$ which rotates or shifts the complex phase for each sample in time by $$\phi$$. Realize that $$e^{j\phi}$$ is a complex phasor of magnitude $$1$$ and angle $$\phi$$. In contrast a time delay is given as $$x(t-t_o)$$ as the OP has indicated.

This is no different in the frequency domain where we have a complex waveform just the same but with a different independent variable (frequency instead of time). Similarly we can multiply the frequency domain waveform by the $$e^{jt_o \omega}$$ which for a given sample at frequency $$\omega$$ will cause a constant phase rotation of that sample.

In the frequency domain $$e^{jt_o \omega} = cos(t_o \omega) + jsin(t_o \omega)$$ is a constant phase shift of $$t_o \omega$$ for each sample given by $$\omega$$ and is not a scaling of frequency as the OP suspects since we are in the frequency domain (in the time domain a scaling of frequency would be of similar form given as $$y(t) = cos((\omega t_o) t)$$ but that is not what is occuring here since we have $$Y(j\omega) = X(j\omega)e^{-j\omega t_o}$$.

Consider a very simple case of $$X(j\omega_1) = Ae^{j\phi_1}$$, which means that the magnitude of frequency $$\omega_1$$ is $$A$$ and the phase of this frequency is $$\phi_1$$. After the time delay given by $$x(t-t_o)$$ the result would be:

$$Ae^{j\phi_1} e^{-j\omega_1 t_o} = Ae^{j(\phi_1-\omega_1 t_o)}$$

Where $$\omega_1 t_o$$ represents a constant phase shift or rotation.

• Dan, I have one more question I am stuck on that I think is keeping me from totally getting this: As in the case of your last example and most I've seen online, you use $\phi_1$. My question is, if it is not $\phi_1$ and is the same variable that is being subtracted, say $\omega_1t$, how would this be a phase shift? It makes perfect sense to me how they would be a phase shift if they are different variables, but in the case of the FT, they are usually both $\omega$ and then you've just changed the $t$ coefficient. – ModularMan Mar 28 '20 at 19:55
• @ModularMan because then again you are in the time domain not the frequency domain. When you are in the time domain then $t$ is changing from sample to sample and $\omega$ represents the frequency which would be constant if the frequency itself is not changing with time. When you are in the frequency domain then $\omega$ is changing from sample to sample and $t$ as used here represents the fixed delay in time. – Dan Boschen Mar 28 '20 at 20:16
• $e^{j\omega_o t}$ is a time domain waveform representing a phasor rotating on the complex plane counter-clockwise at a constant rate of $\omega_o$ radians per second. In the frequency domain for a generic signal each sample is complex value representing the magnitude and phase of that particular frequency. – Dan Boschen Mar 28 '20 at 20:18
• Remember that $Ae^{j\phi}$ is just the magnitude and phase for a complex number; and you can have complex numbers in both the time or the frequency domain. So you can shift (rotate) phase in either the time or frequency domain through that simple complex multiplication. – Dan Boschen Mar 28 '20 at 20:22
• Okay, I think I’m understanding it now. I haven’t dealt with the complex exponential much, so this has been a bit mind bending, but I’m starting to see what you’re saying by the phase rotation at a given frequency. Thanks again! – ModularMan Mar 28 '20 at 21:23