I have tried a restore 3 times now.
Did you restore your iPhone from your iPhone's backup, or as a new iPhone or not from your iPhone's backup?
Wi-Fi settings are included with your iPhone's backup. If there is a problem with your iPhone's backup or with the wi-fi settings included with your iPhone's backup, restoring from your iPhone's backup will also restore the problem.
If the most recent restore was not as a new iPhone or not from your iPhone's backup, give this a try. Before doing so, you can try selecting forget this network with your iPhone - Settings > wi-fi. Select your wi-fi network to view the settings for the network. At the top of the settings for your wi-fi network, select Forget this Network. This will require re-joinging the network.
If no change after doing so, go to Settings > General > Reset and select Reset Network Settings. This will require rejoining your network along with any other wi-fi network you have access to and have previously joining with your iPhone.
If no change after reset network settings on your iPhone, try restoring your iPhone with iTunes as a new iPhone or not from your iPhone's backup.
If no change after this, more than likely your iPhone has a hardware problem. The iPhone includes a one year warranty. You can call AppleCare or make an appointment at an Apple store if there is one nearby.