I second Lyons. In addition, "Digital Signal Processing: A Practical Guide" by Steven Smith is also quite approachable -- immersion in calculus not required.

"A Digital Signal Processing Primer" by Ken Steiglitz is a nice but rigorous intro to the subject. Written by an EE academic, it's more mathematically rigorous than Smith or Lyons.

Allen Downey's "Think DSP" is also worth a look, though its focus is more conceptual than practical, IMO.