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For example, an amp that has the spec of INPUT SENSITIVITY of 1.5V at 1KHz for 60 watt output and the GAIN of 19DB. What do they mean?
FPIII has the gain of 10db and max output of 10V rms (28V pk-pk) before clipping! Does this mean there will be too much out to the amp in example above?
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For more in-depth information, you can look at my paper on this subject - it's on the Bottlehead.com / Community page, near the top, called "Signals, Noise and Signal-to-noise Ratio". It's not simple, but I was not able to make it less complicated.
To address your specific questions, 60 watts into 8 ohms is 22 volts - power equals voltage squared, divided by resistance. A gain of 19 dB means the output voltage is 19dB greater than the input voltage, and 19dB is a voltage ration of 9. 22 volts divided by 9 is 2.43 volts, which is the input voltage required to produce 60 watts into 8 ohms. This is not 1.5 volts, so the specification you quoted is not correct for an 8 ohm load.
I did some calculations and 1.5v with 19dB voltage gain will produce 60 watts into a 3 ohm load. Perhaps the amp was specified for such low impedance speakers, for example in an automobile?
Foreplay has a maximum gain of 10dB, and most of the volume steps are 3dB. So when the level control is set 3 steps down from the maximum, the gain is 1dB, very close to unity gain. The output is then about equal to the input level. Therefor the gain structure is pretty good if your source (CD player or whatever) is about the right loudness when connected directly to your power amp and the amp is connected to your speakers.