# Does the instantaneous frequency contradict the uncertainty principle?

The uncertainty principle states that there is a trade off between time and frequency. So, finding frequency components at specific time is impossible. However, the instantaneous frequency measure the frequency as a function of time. Which means using the instantaneous frequency, the frequency components could be found for a signal at a specific time. How can you interpret this? Why don't we use the instantaneous frequency for time-frequency analysis?

• the premise of your final question is wrong. instantaneous frequency is well defined in multiple ways (one is the derivative of the instantaneous angle of the analytic signal. the time-frequency issue is about resolution of either the instantaneous-amplitude or instantaneous-angle curves. the shorter the window, the more rapidly these signals can change, but the more noisy they are. – robert bristow-johnson Apr 5 '17 at 1:36
• @robertbristow-johnson I think that my misconception was that I didn't understand that we used the instantaneous frequency (IF) in case we have a mono-component signal. So, my explanation that the IF measures the "most probable" frequency components between 3 or 4 others frequencies. Am I correct? – hbak May 14 '17 at 0:49
• i think, what you mean by "IF" (intermediate frequency) is what i mean by the analytic signal bumped up to the IF frequency. – robert bristow-johnson May 14 '17 at 2:04
• @robertbristow-johnson Sorry, I didn't understand. The phrase "bumped up" confused me – hbak May 14 '17 at 13:47
• in this context, "bumped up" means heterodyne. input signal is $x(t)$ the analytic signal is $$x_\text{a}(t) \triangleq x(t) + j \hat{x}(t)$$ (where $\hat{x}(t)$ is the Hilbert transform of $x(t)$) and has no negative frequency components but can go all the way down to DC. we might call those "AF" for "audio frequencies". the IF is heterodyned and "bumped up" from zero to some intermediate frequency, $\omega_0$: $$x_\text{if}(t) = e^{j \omega_0} \cdot x_\text{a}(t)$$ so now $x_\text{IF}$ has the upper sideband starting at intermediate frequency $\omega_0$ and going no lower. – robert bristow-johnson May 14 '17 at 18:48

## 1 Answer

The uncertainty principle works in the presence of (an uncertain amount of) noise or other signals (including possible harmonics), corrupting the exact phase, and thus the rate of change of phase of the signal of interest. If the phase is corrupt, or mixed with the phase of other signals, then deriving an instantaneous frequency from the 1st derivative of that phase might produce nonsense. Time-frequency analysis might be one way to (statistically?) separate information about the signal of interest out of these potentially existing "corrupting" influences.

Whereas the phase of a perfectly analytic signal including zero additive noise is better defined.