Book suggest
Personally I feel, if you have to go about understanding the nuances of DSP, you have to have a very clear understanding of the basics of Linear Algebra and Complex Analysis. So, if I were you, I would start with "Signals and Systems" by either Oppenheim or Haykins (Both are almost the same) with a book on linear algebra, the best one i found is this "Matrix analysis and applied linear algebra" by carl d. meyers Siam press (available in EDAboard) and Visual complex analysis by Needam (i have it in pdf format scanned if u need it).
Then, for solving problems, proakis DSP is good. For understanding, i would prefer "Understanding digital signal processing:" by Richard G Lyons.
good luck,
cedance.