# Time-Frquency Resolution issues

While Im studying wavelet transform, I have this questions in my mind which can't find it's answer:

1. I understand that when we take longer time windows and take the fourier transform we would suffer in the time resolution, But Why if we take a shorter time window we would suffer in frequency resolution?

2. why longer time intervals are needed for high frequency signals? and shorter time intervals are needed for low frequency signals? In another word, Why we want good frequency resolution for high frequencies and good time resoution for low frequencies?

3. If I have a signal contains frequencies from 0 Hz to 50 Hz, How can I define the range of low and high frequencies?

I understand that when we take longer time windows and take the fourier transform we would suffer in the time resolution, But Why if we take a shorter time window we would suffer in frequency resolution?

Imagine that your signal is two different sinusoids that are close in frequency and they both start at phase 0. If the time window is short their end phase will be almost identical because their frequencies are close. Thus, they are difficult to distinguish with a short time window.

If the time window is long the end phase will be quite different- eventually they will be opposite phases (180 degrees out of phase), and thus very easy to distinguish. That is why longer time windows give better frequency resolution.

why longer time intervals are needed for high frequency signals? and shorter time intervals are needed for low frequency signals? In another word, Why we want good frequency resolution for high frequencies and good time resoution for low frequencies?

You do not need longer time intervals for high frequency signals. You'll need to clarify what you're trying to get at on #2 if you want a better response.

If I have a signal contains frequencies from 0 Hz to 50 Hz, How can I define the range of low and high frequencies?

Again, I'm not sure what you mean. What you define as a "high frequency" or "low frequency" is completely arbitrary. What you will need to do is sample the signal at more than 100 Hz to make sure that the Nyquist frequency is above 50 Hz.

• what you mean by "end phase" ? Aug 13, 2015 at 14:58
• The phase of the sinusoid at the end of the time window. Aug 13, 2015 at 16:01