how much of DSP (digital signal processing) is related VLSI?
First, an example: designing FIR filters pops to mind, or for that matter any signal processing module. The FIR filter is a nice example, since it can easily be implemented in VLSI even at the lowest abstraction level - physical mask level (in Cadence for example). 

If we recall an example of the FIR filter design:


We can see that we only need to design:


Delay
Multiplier
Adder


elements, which are quite easy to implement in VLSI, either at the high level (VHDL, Verilog), or lower level (Cadence, Magic IC, etc.). 

So now we have a mixed problem - the DSP part is designing the FIR filter, which can be designed in various ways, and dealing with the transfer function/frequency response. The VLSI part is actually implementing the filter - creating the layout which can be printed on a real integrated circuit. 





In general, the two fields are connected, but in industry usually one deals in only a single layer of abstraction. Either one deals with circuits/elements as black boxes, without caring so much about the physical implementation, or conversely focuses on the optimal design of a component, which would later be produced and used in projects where DSP is employed.