Those remote receiver chips like TSOP4838 are made for a relatively complicated modulation scheme: a double modulation with some 8-20 pulses (some 1-20% PWM) at a 38 kHz and than a pause of at least some 0.2 ms, better more. So the obvious way to create the signal is using a small (e.g. 8 pin) µC, at least if one is familiar with µCs. The analog way would be more like 2 timers like TLC555 or similar for the send part and some extra logic for the detection.
The modulation is at some 38 kHz (there are a few for some 50 kHz) and due to the modulation there is a delay of some 20-40 cycles. So one has to expect a delay of some 0.5-1 ms. It could be more for a lower power version with longer pause. The delay is also not fixed, but more like sampling the barrier at some frequency up to about 2 kHz and than a little extra delay.
With just a simple 38 kHz modulation the system could still work, but less sensitive and thus more power needed. The speed could than be increased a little to maybe a 0.3 ms delay, without much sampling effect.
Modulated IR light really helps to reduce the power needed, as this helps a lot to suppress background light.