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We suspect the LIR is a complementary crossover where one channel is IIR filtered (e.g. HPF) and the other channel (with a LPF response) is created from the difference between the original input (with some small delay) and the filtered signal. When the two channels are added back together, only bulk delay is present and there isn’t any frequency varying phase shift.
We’ve been asked previously about implementing complementary crossovers in FIR Designer M and it’s on the todo list. To match LR’s LIR, we need to check with LR as to exactly what IIR filters they are using. Their paper suggests a 4th order IIR (for the HPF) and in our limited experience with this method, it results in approximately a 2nd order response on the complementary channel.