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1. What is the reason most developers use NIOS II? Is it because it can handles more complex math functions?
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Nios is a microprocessor, with all its advantages: code execution in the form of sequential instructions, high level language programming (C/C++), memory, peripherals, timers and so on.
Then, complex math functions, control tasks or communication protocols are far easier to implement than with HDL.
Clearly, you pay this with lower speed.
If you decide to use Nios (or other uP) in a fpga system, usually is convenient to keep some functions in hardware, to improve performance. The great advantage of a soft uP core implemented in fpga, compared to a physical uP, is the fact you can somehow choose the separation line between hw and sw, because you are able to define the architecture of the system.
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2. What means or software do I need to use the NIOS II package?
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All you need is included in the Quartus II web edition intallation, which can be freely downloaded from Altera website. The Nios dev tools are included here.
A licence is required for using Nios in a production device. You don't need it if you simply use the core for testing purposes: an unlicensed Nios will keep on working as long as your development system remains connected.
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3. Are there basic tutorials available?
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There's plenty of documentation and tutorials on the Altera website.
Since you already have a DE2 board, I'd suggest you start with the Nios Architect Design Tutorial. This and much other helpful documentation can be found here:
http://www.altera.com/devices/processor/nios2/literature/lit-nio2.jsp