Tech Forum

The two solutions above are good but I think they are both overly complex. I assume that the reason for the time delay is that the wireless system produces a pulse whose duration is too short to properly trigger (and latch) the garage door relay. I know that my garage door will stop if the button push is of too short a duration. So we need to stretch the signal from the wireless system and we probably need to make the voltage of the wireless system compatible with the 24V relay in the garage door opener.

Looking back a couple months in Nuts & Volts there is an article about a circuit that turns on a laser to help park a car at the proper place in a garage. The circuit uses the sound of the garage door opener's operation to turn on the lasers for a minute or so. The article is Nuts and Volts - May 2012 - Page 40 - "Build a sound-activated laser parking system." I think the second op-amp stage from that circuit might be all that you need.

The op-amp is setup as a comparator with the reference voltage level set by Potentiometer P1. The other input is from an amplifier stage but I think you could just directly couple the wireless device to that op-amp input (perhaps via a largish value resistor to give some isolation). You would set P1 so that the reference voltage is a bit less than whatever the wireless system puts out.

When the wireless signal is high the comparator swings to the voltage rail charging cap C2. When the wireless signal goes away the op-amp goes off but C2 is kept from discharging back through the op-amp by diode D1. The voltage on the cap turns on the mosfet T1 which you can use to control the current to the relay's coil (replace the laser diodes with the relay's coil in the circuit). The combination of R5 and C2 controls how long the mosfet is turned on.

In the publication the author was going for a couple minutes. If you only want a couple seconds then R5 and C2 can both be made smaller to decrease the on time. As a rough estimate the on time is about 2/(R5*C2) seconds. So R5=1M and C5=1uF should give an on time of about 2 seconds.

I assume that the need is to have a delay before opening or closing the garage door. The circuit diagram below will do this.
 

The circuit shown here, uses a 1Hz oscillator (ICL555) which clocks a 4015 shift register as a sequencer. When the output of pin 2 goes high, the 4N35 optocoupler conducts. The output shorts across a garage door pushbutton.
 

The "Desa Int’l wireless doorbell" will need to close the switch shown as "Enable PB." This can be done with either another optocoupler or suitable transistor logic to simulate the switch closure. In operation, the Enable signal (pin 15) must be active high long enough to be captured on the next clock edge. When the yellow LED goes on, the input has been captured. For the circuit shown, the delay is about four seconds. Increase the clock frequency by decreasing the 560K and/or 330K resistors in the clock section.
 

Note the series wiring of the RLED resistor, optocoupler (LED), and red LED. The 47 ohm resistor must be changed if the circuit is to operate on higher voltages, up to the 15 volt limit of CMOS. Use Ohm's Law to get 20 mA of current through the optocoupler / RLED combo. For the RLED value shown (47 ohm), a six volt supply is required. In practice, I took the circuit board from a cell phone car adapter and "floated" it with a diode to get the six volt regulated output. From a 24 volt source, I use a 12 volt series zener to drop the voltage to 12 volts to the cell phone car adapter. A one watt zener is sufficient.
 

Originally, I tried using a CD4060 for this, but found the circuit to be more predictable. The approximate component placement is shown below the circuit. Actual wiring may vary! The circuit has been in operation for over six months now and works great. If an actual pushbutton is used, the circuit may have potential electrostatic discharge susceptibility through the pushbutton to the D input. While I have not had issues with this, I have a small grounded loop near the pushbutton to reduce static discharge through it. Using a pushbutton with a grounded static shield is recommended.
 

Most parts (including the solderable breadboard) are from AllElectonics.com, though the 4015 is from Digi-Key.

This sounds like a classic application for the ubiquitous 555 timer IC.

The datasheet for the 555 includes many examples of using it, including the one-shot circuit that I think will meet your needs nicely. You can use a common transistor like the 2N2222 to drive the relay coil. Don't forget to put a reversed diode across the relay coil to protect the transistor.