The basic concept here is probably "resonance", which is related to the ringing of a bell or the swinging of a pendulum at its natural frequency.
Strike a bell and it will ring. But if you play back the sound of its ringing, it will also start to ring. This is because the resonant frequency is optimal for it accepting energy.
You can keep a swing going over a large range of motion with a small push, correctly timed. Same phenomenon.
LC circuits also "ring". In the case of radios, this is used for tuning: it resonates most strongly near the desired radio frequency. So it absorbs radio waves at that frequency and turns them into an electrical signal, while absorbing waves at different frequency less strongly.
In the case of the heater and welder, it's being used to optimise the power transfer through a transformer to produce a large current.
A full theoretical understanding of AC involves calculus and complex numbers.
Funny, I use both at work :)
The Cortex-M3 (we use STM32s) is a general purpose MCU that is fast and big (flash storage) enough for most complex embedded applications.
However, the R4 is a different beast entirely - at least the Texas Instruments version I use: the RM42, similar to the TMS570. The RM42 is a Cortex-R4 with two cores running in "lock-step" for redundancy, which means that one core is 2 instructions ahead of the other and is used for some error checking and correction.
Also, one of the cores are (physically) mirrored/flipped and turned 90 degrees to improve radiation/noise resilience :)
The RM42 runs at a higher clock speed than the STM32 (100MHz vs 72MHz) and has a slightly different instruction set and performs some of the instructions faster than the M3 (e.g. division instructions execute in one cycle on the R4, not sure they do on M3).
HW timers are VERY precise compared to Cortex-M3. Usually we need a static offset to correct for drift on the M3s - not so with the R4 :)
Where I'd call a Cortex-M3 a general purpose MCU, I'd call the Cortex-R4 a complex real-time/safety MCU. If I am not mistaken, the RM42 is SIL3-compliant...
IMO the R4 is a big step up in complexity even if you're not planning to actually use the real-time/safety features.
A really nice example of the complexity difference: The SPI peripheral has 9 control and status registers on the STM32 whereas the RM42 has 42. It's like this with all the peripherals :)
EDIT:
For what it's worth, in my use cases the Cortex-R4 @ 100MHz is usually 50-100% faster than the Cortex-M3 @ 72MHz when performing the exact same tasks. Maybe because the R4 has data and instruction caches?
Another comparison, a few 1000 lines of C and ASM code are executed on reset before reaching the call to main() with the subset of the safety features I currently use :D and not peripheral initialization or anything, just startup and self test (CPU, RAM, Flash ECC etc.).
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The difference, if there is one, is manufacturer specific. A baseband processor and a baseband controller often perform the same functions - that of controlling every aspect of the RF hardware.
Some manufacturers like to use "baseband processor" in order to indicate that their controller performs functionality you might find higher on a communications stack, such as message decoding. For instance a WiFi baseband controller might still need to be attached to a processor to implement a full WiFi link, whereas a baseband processor might handle more of the communications stack and may attach directly to another network interface, USB, etc.