I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to correct me in the probably various places I am wrong)

I am trying to make some basic computations, like for example compute the whole algebra probably, $H_K$ (the left-invariant on $K$ distributions) and the center (which in the rest of literature is what is usually denoted by $H_K$ I think, essentially the bi-$K$-invariant distributions). Now the center can be computed by the Satake isomorphism (and it is isomorphic to $\mathbb{C}[z_1^{\pm},z_2^{\pm}]^{S_2}$). For the rest I really could not come up with an explicit computation.

Now searching around, I found that most sources do not define the Hecke algebra as the locally constant compactly supported distributions, but by a purely algebraic definition with some generators over the Weyl group. I really cannot understand this definition.

Can you provide me with some source that explains this explicit presentation of the Hecke algebra, and why is it the same as Bernstein's one?