Does anybody know of a type of antenna that will give something like 12 dBi gain in a smaller package, like 5 feet long?
One antenna fits within your dimensional requirements: AntennaCraft's Y5-7-13. (Winegard's competing model, YA-6713, has been discontinued.) The Y5 will only get you to maybe 9 dBi of gain, which is about half of what you seek. Therefore, you'll need to deploy other strategies for getting that last three dBi:
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More height above ground level. The classic rule of thumb is that every additional 3 metres of height is worth about another decibel of gain. This factor can be dependent on the intervening terrain, and occasionally, more height doesn't help at all. (On the bright side, this isn't usually the case in flat areas such as the Toronto-Buffalo lake plain.)
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Different location on the roof. Signals have horizontal-plane "hot spots" whose sizes vary inversely with increasing frequency. In other words, you can move a UHF antenna several centimetres laterally to find (or lose) a hot spot, but it can take a couple of metres, or more, to achieve the same effect in mid-VHF frequencies such as channel 7. This strategy will require quite a bit of experimentation. Translation: You'll likely be up on the ladder and/or roof repeatedly.
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Use a Jointenna or signal injector with the dedicated VHF antenna. The antenna will pick up signals from many other channels besides the one you want in a heavily saturated market like Toronto. Together, they will interfere with the weak signal you seek, even if they're on other channels (and even the FM band!). Think of this device as a "collar" that lets channel 7's signal through while attenuating everything else on both sides of channel 7. Channel Master
Jointennas are inexpensive, but their F-connectors have a bad reputation for snapping off under anything more than the slightest pressure; Tin Lee Electronics right in Toronto makes high-quality
signal injectors, but you'll pay dearly for their custom work.
• Finally, a low-gain pre-amplifier added to the VHF antenna might help to counteract cable losses, but note that you will need to insert the amp and its power injector
before a Jointenna/signal injector to avoid creating other reception headaches.
All of this is a very roundabout way of saying you'd probably be much better off compromising on those dimensions by getting the YA-1713 everyone else recommends and feeding it into a Jointenna/signal injector...