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In Forney’s MLSE paper (link: https://ieeexplore.ieee.org/document/1054829), Forney mentioned that the matched filter output is still sufficient statistics even with ISI (second column on page 365).

I understand that a matched filter output is sufficient statistics when the channel has no ISI but only AWGN at the receiver's input. But I am having a difficult time proving & understanding this when there is ISI on the channel.

I found a few articles that explained this problem:

These two papers argue that a matched filter followed by a ZFE provides an output that is sufficient statistics. However, I am not convinced since the ZFE will shape the white noise into colored noise. Therefore, the detector needs more information to distinguish the signal from the colored noise portion (please correct me if I am wrong in this statement).

If ZFE is not used, then the received signal is corrupted by ISI. Even worse, the matched filter will amplify the ISI further! Then we need a sequence of symbols with a length twice the ISI vector. x2 length is because a matched filter convolved with a channel ISI will effectively make ISI longer. But I am not sure if my understanding is correct.

I would greatly appreciate it if anyone could clarify which point my understanding is incorrect.

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