Of course you can implement a DSP processor into an FPGA and make it work, but DSP processor on FPGA performance will be lower that the performance of a similar DSP chip.
DSP on FPGA is preferred when you have to do the same fixed process all the time. In that case you do not implement a DSP soft processor; intead of that, you implement a DSP algorithm using the FPGA DSP structures (take a look at DSP Design Tools from Xilinx, for example). In this case, FPGA performance will be higher than the performance of a DSP processor doing the same process.