The actual technology is not all that complex. Transient detection and triangulation by decorrelation from a microphone (edit: not microwave!) array. You could probably prototype it in a week if you leaned on an established FFT library. Obviously they would not want to reveal their source code and so on, but if you know digital audio you know this isn't that hard.
I think what they really want to conceal is not the technology but its use in operation. I don't know what Chicago pays for the service, but Sacramento CA spends about $1m a year on it and Sac is a fairly small city. https://www.sacbee.com/news/local/article231997777.html