What is the effect of highly correlated data on a Convolutional Neural Network?
I actually don't believe it really matters.  When applying the same idea to visual problems, you'll find that the raw RGB vs a DCT compressed version lead to similar results as seen here.  If there exist a negative, one may argue that the correlated data may have a slower learning rate.  Nevertheless, the best way to figure out the effect is to give both a try.