Fourier Transforms, Convolution, Cross-correlation: what is their physical unit exactly?
I agree that the units for Fourier transform are V/Hz. Most of my experience with the convolution integral dealt with a convolution of the impulse response of a linear network with the voltage input to the network. The impulse response for, say, a simple RC network will have the product RC in the denominator. Thus, the unit of impulse response is per second. So, the units of a convolution would be volts-seconds * per second = volts. For correlation, either auto or cross-, in the case of power signals (as opposed to energy signals), you should divide the integral by the period, T. If it is not periodic, still divide by T, but take the limit as T goes to infinity. Now the period, T, has units of seconds and you are integrating over time, which has units of seconds. The time in the numerator and denominator cancel, and you are left with volts squared, which gives power when divided by resistance (assuming that the volts are rms).