>build a pair of high gain antennas, connect them
>with coax and put one outside and one inside.
>Of course you will have to match impedances

I would not even worry about high gain antennas.

Working on the assumption that you can get satisfactory reception in the
upper floors of your property, I would make a quarter wave whip with four
radial wires for a ground plane on each end of the coax. With the far end up
on the roof (maybe, or in the attic to keep it out of the weather) and the
other end hanging from the ceiling in the basement I suspect you will get
sufficient signal strength.

If you are using 50 ohm coax try a 5/8 wavelength whip and radials as this
is closer to 50 ohm, although with some reactive impedance.

I would not worry about trying to deal with matching networks unless you are
looking for every last dB of signal transfer, and from your description of
the problem it sounds like you just need to get above marginal signal
levels.

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