Impulse response convolution and normalization2
I think you're likely forgetting how the anti-derivative of $h(t)$ affects the gain of the convolution operation.  Recall that somewhere in your convolution integral, you'll be taking an integral of the form $\int We^{W\tau} d\tau$.  The Chain Rule requires the  $W$ in the exponent must appear as a $1/W$ factor after integrating.  This factor cancels the $W$ multiplier in $h(t)$ giving unity gain at DC.