Fundamental Time - Frequency Domain Concepts

Consider a continuous pure sine wave. Its waveform is spread throughout the time domain, but its spectrum is only a single line in the frequency domain. Now consider a signal consisting of only a single spike in the time domain. It has a perfectly constant spectrum spread throughout the frequency domain.

You can check this out using the Daqarta Generator to create a single-sample pulse. (See the Impulse Response section for details.) Then toggle the Spectrum display on and boost the display magnification (or toggle the Y-log power spectrum on) to see the constant low-level spectrum. The level is low because the single time-domain spike doesn't have a lot of energy (or power) to begin with, and the spectrum shows the level at each frequency.

Similarly, if we compare a continuous sine wave to a tone burst of that same sine wave, we find that the narrower we make the burst, the broader the spectrum. This holds as a general rule... you can't limit the extent of a signal in one domain without increasing its extent in the other. There is always a balance between the domains.

This is also an uncertainty principle (like Heisenberg's) that limits our ability to resolve features in both the time domain and frequency domain simultaneously. Consider that to get fine frequency resolution, we must use a large number of samples N, which means a large total time to acquire the sample set. If the input signal is rapidly changing, we will get some sort of "average" spectrum of the N-samples time interval... and we can't assign that whole spectrum to any particular event or sample in the input.

Conversely, we can try to narrow down the input event times by using smaller N, but the spectral resolution becomes more coarse. This is a particular issue with spectrograms, where we are specifically interested in how the spectrum changes with time. The Spectrum Window Width control provides a way to adjust the effective N. See the Spectrogram page of the Screen Shots section of the Daqarta Website for examples of wide and narrow Width applied to a speech spectrogram.

See also Spectrum (Fourier Transform) Theory, Burst Rise/Fall vs. Spectral Width