Trading Off Time and Frequency Resolution

[s1735009]

The slides detail the different properties of long windows vs. short windows. Please could you explain a little bit more about what time resolution and frequency resolution mean?

I infer frequency resolution to be not having enough frequencies to accurately reproduce a signal, almost like how the trapezium rule with too few trapezoids is not an accurate estimation for an integral (though I’m not sure). But I cannot understand what time resolution might mean.

Thanks you in advance.

[Simon]

The Fourier transform is invertible, which means that you get as many data points out as you put in. This means that using a longer analysis window in the time domain gives you more points (often referred to as “FFT bins”) in the frequency domain magnitude spectrum (we will ignore phase). Those points (“bins”) are equally spaced from 0 Hz up to the Nyquist frequency.

So, a longer analysis frame in the time domain means that the spectrum has more detail: higher frequency resolution.

The Fourier spectrum is effectively an average over the entire analysis frame (formally, we are making an assumption that the spectrum is constant throughout the frame). A longer time window means averaging over a longer duration of the signal and thus being less precise in the time domain: lower time resolution.

There is a trade-off between time and frequency resolution. In practical terms, this means we have to choose an analysis frame length that is appropriate for our signal. For speech, 25 ms is a common choice.

[Author]

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