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I am learning about digital filters, for example a lowpass butterworth filter.

In many languages, the implmentation requires specification of the order of the filter, which I believe is the number of terms in the closed form equation that generates each filtered sample, given the original samples and previously generated samples.

What I haven't been able to find (at a sufficiently ELI5 level) is how the order affects the frequency and phase response of the filter.

Does a higher order simply mean a better approximation to the optimal filter with (cutoff, stopband) ? Do higher orders mean different or less severe phase differences?

Finally, I'd assume that it affects performance somewhat as well.

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    $\begingroup$ This may help you: dsp.stackexchange.com/questions/31066/… and consider how the moving average of stock market curves look as you go over longer days (it takes more memory, or taps in an FIR filter, or order of the filter which is number of taps -1, to generate a sharper cut-off filter). With more memory means longer delay means a steeper possible phase slope vs frequency can be obtained. How that phase behaves however is specific to the filter implementation. $\endgroup$ – Dan Boschen Apr 10 at 1:44
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If you have Matlab or Octave or similar software there you could find answers quite easily.

Here's one way to look this matter, which shows how filter order affects to the approximation of the optimal filter (example of 1st order IIR LPF, fc = 15000Hz):

enter image description here

(green = analog model, magneta = 1st order, red = 2nd order, black = 4th order, ___ = magnitude, ----- = phase response)

Closer view:

enter image description here

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Does a higher order simply mean a better approximation to the optimal filter with (cutoff, stopband) ?

That depends on how you define "optimal". In general a high order will give you a steeper rolloff

Do higher orders mean different or less severe phase differences?

Higher order will result in more phase distortion

Finally, I'd assume that it affects performance somewhat as well.

That depends on how you define "performance". The "best" filter always depends on the application.

In general higher IIR filter order will give you more steepness in the frequency but also more time smear and phase distortion. Any beginner class on digital filter should cover this. For example, take a look at https://ccrma.stanford.edu/~jos/filters/

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