Hi again,
Steve:
Yes when the questions are not clear or when there is some question about what the OP is asking for i like to post a brief message to try to provoke more questions and possibly get more background information. Luckily you have the book so you can help more here. I'll just make a note of how easy it is to find the difference equation, although i do have to ask the OP how the instructor expected him/her to do this on their own. Did the instructor just hand them the book for the first time and say "do this" or what, cause that is what it almost sounds like ha ha.
For the solution to the difference equation we have a very simple flow graph (from the first message and redrawn in my first reply) where there are no inner feedback loops, so the solution for the output relies solely on the sum of the path transmittances. We have only three paths:
1. The "right' side from x[n] to y[n].
2. The 'left' side from x[n] to y[n].
3. The 'bottom' from y[n] back to y[n].
This makes it quite easy to get the equation for y[n] as we just follow these paths and write down whatever we encounter in the proper terms.
1. For the right side, we see the path start at x[n] and go through one delay T before it reaches the output y[n], so we get simply x[n-1].
2. For the left side, we see the path start at x[n] again and go through one gain -4 so we get -4*x[n], then go through one delay T so we get -4*x[n-1], then through another delay so we get -4*x[n-2].
3. For the bottom, we see the path start at y[n] and go through a gain of 1/4, so we get y[n]/4, then through one delay so we get y[n-1]/4, then through another delay so we get y[n-2]/4.
Summing all these, we get:
y[n]=x[n-1]-4*x*[n-2]+y[n-2]/4
This method might be called "The Path Transmittance Summation Method" but it's really like Mason's Flow Graph Gain Formula except that the method relies on there being no inner loops so before we start we have to reduce all the inner loops to single gain paths using the feedback rule for flow graphs. This usually isnt hard to do and many times we dont even have any inner loops.
In spite of this simplicity however, i still would like to refer the OP to Mason's Flow Graph Gain Formula which is a very ordered way of doing things.
From there to get the transform we simply replace every [n-k] with a multiplication of z^-k and then solve for Y/X as Steve pointed out. We then simplify the equation.
From the limited text i would think that the phrase "z space" simply means any space found in the z plane. This might include a rectangular shaped sub space, triangular space, circular space, or just about any shape; the unit circle would enclose a circular shaped sub space for example.