One good way to figure out what a bird is is by going step by step with its characteristics. The Cornell Lab of Ornithology has good web pages for this for each species. Note: It is not just the look of a bird that matters. Read the behavior section, as well.
Another thing I want to caution you about with bird ID is that, it is sometimes easy to want to see a more exotic or rarer species. It is probably human nature.
I think this is no doubt a Cooper's Hawk. They are exciting birds. They are very smart. They used to be hawks that lived in the woods. But, they have adapted very well to suburban life where people put out feeders that attract smaller birds...which, in turn, feeds predators like these hawks.
If you really want to see a Peregrine Falcon, post something here:
Or, look online for a local chapter of the Audubon Society to find out how to contact some birders. Most birders are happy to share information like where to find these falcons. I would bet they are nesting on the tall buildings in the downtown area. You can also check Facebook for some local birding groups.
As far as this specific photo goes, look at the photos on the two Cornell websites. Compare the tail of your bird and the other ones. That long striped tail points to a Coop. And, compare the head of the bird in your photo with the other ones. Your hawk has a "cap" of gray feathers. The falcons don't have that. The adult Coops do. Enjoy this hawk if you see it again. They are really cool birds.