# Why is there a time-frequency precision tradeoff

I understand the science - I think. Looking at a spectrogram, the lower frequencies are fuzzier due to a lower ratio of: frequency/time.

My main question is: why can I as a human hear lower frequencies sharply, but the FT cant?

Is this a psychological thing - perhaps my ears suffer from the same issue, but my brain is filling in the missing information?

Notice how the band gets wider, demonstrating the increase in innacuracy.

• I am not sure I follow your question completely-- in the FT all frequency bins are separated by 1/T where T is the duration of the time capture. The lower frequencies are spaced the same as the higher frequencies so what do you mean by "fuzzier"? Apr 4, 2021 at 22:15
• @DanBoschen lower accuracy Apr 4, 2021 at 22:25
• @DanBoschen i'll add a diagram for clarity Apr 4, 2021 at 22:26
• So, we were talking about Fourier transforms, and now suddenly we're talking about something called a "convex chirp" (which I can't find with a simple web search, so I'm suspecting it's not a common thing). Please fully explain what you are asking about; a fully-developed question has a chance at getting a sensible answer. Apr 4, 2021 at 23:13
• I think Dan's right. There's two things happening in that chirp diagram: it's dwelling longer at the high frequency, which will naturally lead to better resolution, and it appears to follow a rule -- which I'm pretty sure you're misapplying. Try again with a chirp that's convex downward, that instead dwells at the low frequency. Then report back. Apr 5, 2021 at 0:22

why can I as a human hear lower frequencies sharply, but the FT cant?

No you can't. Human pitch discrimination at low frequencies is actually not particularly good. The "sweet spot" for human pitch perception is between 250 Hz and 4kHz and it gets rapidly worse above and below that. When tuning a bass instrument you typically try to reduce the fundamental and the lower harmonics as much as possible and emphasize the higher harmonics a lot since this makes it a lot easier to determine the correct pitch. A violin slightly out of tune makes everyone cringe: a double bass slightly out of tune is hardly noticeable.

So in this regard the human auditory system is no better than the Fourier Transform.

• in fact, sometimes the "wrong" note at the bass can be the basis for some interesting music. Apr 5, 2021 at 3:53
• @robertbristow-johnson: That's why I play a 5-string! If you don't know, just go low :-) Apr 5, 2021 at 12:11

Frequency resolution in the DFT is the inverse of observation time; specifically each bin of a rectangular windowed DFT forms a Sinc function centered on each bin with the first nulls of the main lobe spaced $$1/T$$ from bin center.

This resolution is the same regardless of the frequency being lower or higher. I suspect what the OP is observing is a longer dwell time for the higher frequencies compared to the lower frequencies such that the chirp is passing through more bandwidth in the observation time of the DFT for the lower frequencies. Changing the characteristics of the chirp (for example try with a linear chirp) will help to clarify this.

Humans seem to not determine pitch over a fixed time interval, such as the duration of some FFT window, but over a variable duration that depends on the pitch (and volume, and timbre or harmonic content, and possibly whether or what any preceding or following sounds were).

Lower pitches take longer for a human to detect them as pitched sounds, and even longer to tell pitches (say a whole tone or semitone) apart, than with higher pitches.

• There a lots of textbooks on hearing, audiology, speech therapy, psychoacoustics, psychology of music, etc. that cover experiments on this topic. Apr 6, 2021 at 2:17