What is this sinc doing in my LP-turned-HP filter output? I'm trying to HP-filter a signal by LP-filtering it and subtracting the output from the original filter. To obtain the LP-filtered signal, I backward-forward filtered using a third-order Butterworth filter for a 16,000-sample signal. The cutoff frequency was 0.5 Hz @ 250 Hz. No window was used.

In the first 85% or so of samples, the output of this method seems to overlap completely with the actually HP-filtered signal. However, at the end of the output, there is a supsiciously sinc-like phenomenon.


What causes it?
What could I change to make this method feasible? 


Here are the two signals overlaid. The blue one is HP-filtered, and the red one obtained by subtraction.



Here is the absolute difference per sample. They are never exactly equal; the minimum difference is 8e-12. 87% of the values are below 0.01.



(If there's a better way to obtain an HP filter from an LP filter's $H(z)$, I'd be happy to learn it.)
You are doing a backwards-forwards filter for one, and a forwards-only filter for the other.  Thus the two filters will behave differently at one or both edges.

If your signal is strongly stationary and exactly periodic in the data window length, you could try wrapping the data around circularly and start your filters outset by their impulse response lengths.