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Dear StackExchange gurus,

I would like to pose this question from a practical standpoint rather than a theoretical one, although perhaps some context in the latter might come useful.

Real-life devices (say Digital-to-Analog converters for example) have a roll off in their frequency response due to analog front-ends. To find said response (and their inverse) one can use spectral methods (T(f) = out(f)/in(f)), or time-domain methods that will extract the impulse response (Wiener-Hopf equations, LMS, etc).

Now, in time-domain methods we can chose our solution to have the dominant coefficient at the center tap of the filter, or we can choose it to be the first tap (i.e. single sided and with zero delay). Both of these give similar results in some occasions, but there are times when the single sided inverse does not invert the system as well as the two-sided does.

Can someone give me some insight on why this happens? Which of the two approaches is better for equalization (regardless of implementation complexity)?

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The analog front end will always contain some sort of low pass filter. This can be chosen to be either linear phase or minimum phase. Linear phase preserves the phase response of the original signal but the time domain response is non-causal, i.e. the impulse response is symmetric around the center tap.

Minimum phase has a one sided impulse response, with the strongest tap up front, but it incurs some phase distortion. Ideally we want both, but we have to pick one.

The same consideration implies to the inverse: the inverse of a linear phase filter is linear phase and the inverse of a minimum filter is minimum phase. However there is an added complexity: not all filters are invertible: If the magnitude of the original transfer function is very small or zero at some frequencies, the inverse is not well defined. You get "divide by zero" or "divide by noise" problems.

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