will the CC2540 wake up when a paired device sends it a communication?

Hi Matt,

  No, the CC2540 will not wake-up, as in SLEEP mode the radio is not active, therefore, cannot receive and process any signals.  The SLEEPTIMER is required to wake-up the CC2540, it will then go in to active mode and listen for any communication (see datasheet page19).   The Analog Comparator can also be used for this purpose, which when activated, will generate an interrupt to get out of PM2 (see page 20).  Also see page 25 discussion on the XTAL accuracy for WAKEUP timing, as the network typically will have specified time intervals for each receiver to wake and listen for the controller, and to minimize the listen time (when the radio is in RX mode consuming the most power), the XTAL accuracy and synchronization is important.

-Leonard

BTW,  TI's BLE stack will automatically wake up the cc2540 before the next scheduled RF interval.   Before going into PM2 the halSleep() code checks for the next osal timer and the next rf time.  It will wake for whichever comes sooner.   As long as the connection is maintained you can use PM2 power saving feature.  However if the connection is lost you will need to go through advertising to re-establish the connection.

For more details you can look at Components/hal/target/CC2540EB/hal_sleep.c .