Here is the Federal Antenna Law that says Yes you can install a antenna on a private balcony.
Owners/managers of apartment buildings are aware of this Federal Antenna Law and know that the law provides legal rights for you.
Hoa's/Aho's , housing coven-nuts , compartment owners , and other people con-trolers , all know about the Federal Antenna Law.
And keep their mouth shut about it.
The people that do not know about the Federal Antenna Law are the ones the law is for , You .
http://www.fcc.gov/mb/facts/otard.html.
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The Main Group of Local Digital Broadcast Tv stations/channels are to the , North East .
WSAW-TV Real Digital VHF channel 07.1-2 , virtual number (07.1-2) , CBS and MyNetwork.
WAOW Real Digital VHF channel 09 , virtual number (x) , ABC and The CW and This Tv.
WHRM-DT Real Digital UHF channel 24.1-2-3 , virtual number (20.1-2-3) , PBS channels.
WFXS-DT Real Digital UHF channel 31.1-2-3 , virtual number (55.1-2-3) , FOX and Me-Tv and This Tv.
WTPX-DT Real Digital UHF channel 46 , virtual number (x) , ION and Qubo and ION Life.
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And 1 Local channel to the West.
WEAU-TV Real Digital UHF channel 38 , virtual number (x) , NBC.
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Your apartment windows and balcony are West.
The Main group of Tv stations are to the North East.
If your west facing apartment is on the north end of the building , will make reception of north east group easier.
Install a Antennas Direct , Clear Stream 2V Complete Uhf/Vhf antenna , out on the balcony away from the building the best you can on the north part of the balcony and aim the CS2V antenna the best you can to the north east at about 58 degree magnetic compass direction.
Here is how to aim antennas ,
http://www.kyes.com/antenna/pointing/pointing.html.
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If the apartment is some where in the middle or south end of the building , then you will do the best you can with antenna aiming.
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A Winegard LNA-100 'Boost' antenna system amplifier will most likely help with reception.
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Here are some places to buy antennas and etc. .
http://www.solidsignal.com.
www.amazon.com.
http://www.winegarddirect.com.
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The Tv Must Channel Scan for , OTA=Over The Air , Digital Broadscat Tv stations/channels , often named the 'Air Channels' or 'Antenna Channels' in the Tv Setup Menu because the Tv transmissions travel through the Air from the transmitting antenna to the receiving antenna.
Some Digital Tv's will Automatic channel scan for cable tv channels.
DO NOT channel scan for cable tv channels.
Go into the Tv Setup Menu and select , 'Air Channels' ~ 'Antenna Channels'.
Scan for channels.
Digital Tuners can develop - Digital Glitches - that are not cleared out with simple channel scans.
Do a Double Rescan ,
http://www.dtv.gov/rescan.html