DSPs are designed with processing _digital_ (i.e. time-discrete and value-discret) signals, not analog ones. The connections (input and output) with the analog world are done by ADCs and DACs, respectively, which may or may not be integrated into the DSP.

DSPs used to be "pure" MAC machines with little other hardware beyond what's required to get data into and out of the chip.

Todays DSPs still focus on being fast at DSP-related calculations and offer much more specialized instructions than plain vanilla MACs (the one true-blood DSP I worked with had specific processor instructions for FIR filters, polynomial evaluation, etc), and also things like automatic bit-reversed addressing for FFTs. They come with much more integrated hardware than they used to (UARTs, other types of serial ports, ADCs, DACs).

Todays microcontrollers (especially ARM-based parts) are rapidly invading territory once held by DSPs. They usually have enough CPU power for many signal processing tasks, even things like chewing through a couple of hundred thousand ADC samples per second.

/edit: To answer the question: I wouldn't consider a '51 with added-on hardware filters a DSP - it lacks the flexibility of an actual DSP. Now, if you add a MAC unit, you'd get closer to a "real" DSP, but the '51 architecture still has severe limitations (memory, etc) that a true DSP wouldn't have.

I have done some simple digital signal processing on a '51 and it's definitely feasible, but I wouldn't want to process 10 channels of ADC data coming in at 32000 samples per second and channel with a '51. ;)