TIME-FREQUENCY RESOLUTION

A consequence of Fourier transforms being built from $e^{i\omega t}$ is that scaling a function to be narrower in one domain scales it to be wider in the other domain. Scaling $\omega$ implies inverse scaling of t to keep the product $\omega t$ constant. For example, the FT of a rectangle is a sinc. Making the rectangle narrower broadens the sinc in proportion because $\omega t$ is constant. A pure sinusoidal wave has a clearly defined frequency, but it is spread over the infinitely long time axis. At the other extreme is an impulse function (often called a delta function), which is nicely compressed to a point on the time axis but contains a mixture of all frequencies. In this section we examine how the width of a function in one domain relates to that in the other.

By the end of the section, we will formalize this into an inequality:

For any signal, the time duration $\Lambda T$ and the spectral bandwidth $\Lambda F$ are related by  

$$\Lambda F \; \Lambda T \quad \geq \quad 1$$ (1)

This inequality is the uncertainty principle.

Since we are unable to find a precise and convenient analysis for the definitions of $\Lambda F$ and $\Lambda T$,the inequality (1) is not strictly true. What is important is that rough equality in (1) is observed for many simple functions, but for others the inequality can be extremely slack (far from equal). Strong inequality arises from all-pass filters.

An all-pass filter leaves the spectrum unchanged, and hence $\Lambda F$ unchanged, but it can spread out the signal arbitrarily, increasing $\Lambda T$ arbitrarily. Thus the time-bandwidth maximum is unbounded for all-pass filters. Some people say that the Gaussian function has the minimum product in  (1), but that really depends on a particular method of measuring $\Lambda F$ and $\Lambda T$.