[AtomicOrbital]

https://m.youtube.com/watch?v=D1WfID6kk90

take a time series dataset like an audio file or stock ticker price over time ... give your self a healthy period of time ... for example a second of broadcast quality audio gives you 44,100 data points spread across that time period stored as information ... importantly this time series audio curve wobbles up and down as it's recorded over time ... in order to justify taking 44,100 audio samples per second (on the X axis) you must balance that by breaking up the granularity of your measurement of the up and down wobble (Y axis) by devoting two bytes (a bit depth of 16 bits) of memory storage per data point which gives you 2 raised to the 16 power distinct gradations of resolution

above defines the time domain representation of the one second of audio data ... now feed this dataset into a Fourier transform which will output the same information you started with but now in the frequency domain ... it will give you not 44,100 points in time but instead 44,100 distinct frequencies ... super cool side note you can feed this new frequency domain representation of the dataset into an inverse Fourier transform to rescue back the original time series audio

If instead of a second of audio we start with a fraction of that number this reduction of recording duration will compromise the frequency resolution of the data in the frequency domain giving it less granularity hence larger increments to the next frequency

[TheOtherHobbes]

No, you're confusing two things. The uncertainty in a Fourier transform applies whether it's continuous or discrete. It does not require sampling. It still appears in the sampled DFT, but that's an extra wrinkle. It's a feature of the transform itself, not the sampling process.

You're also confusing horizontal and vertical resolution. Sampling bit depth sets the maximum possible dynamic range resolution of both pre-transformed samples and post-transformed frequency components.

The number of samples defines how many frequency components there are. The number of bits define how accurate their levels are.

The uncertainty trade off is in the number of samples. You can do an FFT on multi-second chunks of music. You get superb frequency resolution, and it will transform back to the original. But you can't use the spectrum to see fine detail in individual notes, because the frequency domain view is just a bar graph with the same number of samples, and shorter features - like individual notes - are smeared out across the entire frame.

[quantadev]

First paragraph is a description of single channel redbook CD-AUDIO format right? Second paragraph is basically describing FFT and inverse FFT. Third paragraph is basically how MP3s work right? Wavlets that are essentially FFTs (list of frequencies) over a finite range of time, with enough of them removed to compress the data.

[kragen]

'wavelets' means something different