The "No signal" message from the monitor means it is not getting a signal from the computer... this can mean the video card is dead, that the analog/digital cable is lose, or that the computer is not running, so the obvious question would be; is there any indication the computer is turning on; like sound comming from any cooling fan/s (Power supply, processor or system cooling fan)?, any sound coming from the Hard Drive, any air coming out from the back of the computer (PSU fan)? Is the computer making any sound at all and are the front panel leds turned on?. Have you checked the monitor analog or digital cable screws, are they or the connector snug enough on the video motherboard terminal. Is the video card added (PCI, PCIe) or integrated on the motherboard?.
If the power supply fan is not running, the first thing to check is the I/O PSU switch, (I on, O is off). Next would be to check if the PSU is supplying enough voltage to power motherboard and components.. the PSU motherboard pinout connector can be checked with a voltmeter, the different voltages the pins should deliver can be checked against the 20 or 24 pin connector image and instructions here: http://www.smpspowersupply.com/connectors-pinouts.html
Next check the power supply cooling fan, next listen other cooling fans and hard drive sounds, check if the front panel led lights are flickering or stable, what the HD led color is, and tell us what's your computer type, age, make and model or system specifications if it's homebuilt so we can better help.
I had a monitor that did this. After flickering for about a minute it would come on and stay on until the next time the monitor was put to sleep or turned off. Not much you can do with a dying monitor except to buy a new one.
Another thing that seems to work sometimes is to hit CTRL-ALT-DEL to bring up windows security, then just hit ESC to cancel out of it. I have no idea why this works, but this issue seems to be ATI related as I've never experienced this on an nvida card, or at least not yet.
Another monitor will tell if it is the monitor or PC. If PC check for domed capasitors, if none try another power supply and/or video card. Reseating the CPU and RAM modules will sometime do the trick as well. Loose chipset heat sinks?