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The capacitor works as in a power supply filtering out any noise and bounces from the switch and keeping the voltage at a certain level for a brief moment.
Therefore the supply voltage will have to be higher than the desired ON voltage to compensate for the slow voltage drop and guarantee a solid ON state, and keep in mind the diode voltage drops.
The diode's purpose is to eliminate the 10K resistor from the circuit in the charging phase so that the cap charges immediately and the ON voltage is instantaneous, however maybe there should still be some resistance to avoid excessive inrush current through the cap, thus the 500R resistor after the diode in the NO circuit. In the NC circuit the cap will charge through the top resistor so its value will have to be large enough to avoid excessive inrush current through the cap but small enough to still charge it instantaneously.
In the discharging phase the diode no longer is part of the circuit and the capacitor slowly discharges through the 10K resistor next to it along with the bottom 1K resistor in the NO circuit, so its value will be chosen to keep the discharge from being too much slower than the 10K alone.
In resume, the cap keeps the voltage above a certain level, which should be higher than the minimum ON voltage, and for a certain amount of time wich should be slower than the slowest bounce in the switch, otherwise 2 fast pushes will count as 1 single push. When the button is pressed the diode will make the cap charge almost immediately, in the NC circuit through the top resistor and in the NO circuit through the added resistor after the diode. When the button is released the cap will discharge slowly through the 10K resistor in the NC circuit and through the 10K plus the bottom resistor in the NO circuit. Press the buttons very quickly to simulate the switch bounce and see how the original traces go from 0 to 10V immediately but the de-bounced signal drops only a little. If you press and release the button slowly the caps have enough time to discharge and guarantee clean ON and OFF states.
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