If you want to build a very simple cheap antenna here is one and the story behind it.
This is a simple broadband TV antenna that works.
A breakfast buddy kept telling us about his antenna and
we kept po poing it because it wasn't standard.
My daughter needed an antenna so I quickly built one using
aluminum foil spray glued to cardboard, I simply did a physical
contact connection for the coax connections.
My daughter is still using this antenna in the window, over 5 years later.
1-1/2 years ago after a hurricane knocked out cable,
my neighbor tried to buy a TV antenna and found there was a local
run on antennas and he couldn't get one.
I spent less than an hour building one with aluminum flashing
screwed to a piece of wood. I used screws to make the connections.
we put it on a 10 ft mast (chain link fence top rail) and it received all but the local PBS channel,
so I added another 10ft section of mast, it got that last station.
A few months later he bought an antenna and put the one I built up at his mothers house.
I suspect the wood was a little lossy and I could have did better, but it was
just a quick build and worked .
This link has a success story for someone that built it.
Long wire antenna for DTV? - The RadioBoard Forums
Read the last 3 posts in the thread. Note: I think he made an error, because he talks about a transformer, there is no transformer, the two coax wires connect directly to the individual aluminum sheets. In the pictures of his build note the center support is fiberglas not metal, (which would be electrically conductive)
I haven't modeled this, but my breakfast buddy did, and I think he optimized the dimensions. He has his behind a picture in his living room, I'm sure it works better outdoors up in the air. But, my daughter has had her's propped up in a window two feet off the ground.