a tricky question regarding CDMA with 4 stations

i have encountered this tricky questions that left me really confused, would really appreciate your insight regarding it:

given a cellular system with four stations that utilizes CDMA.

given the following codes(3 out of 4, the forth is not known):

$$x_1: -1 -1 +1 +1 -1 -1 +1 +1$$

$$x_2: -1 -1 -1 -1 +1 +1 +1 +1$$

$$x_3: -1 +1 -1 +1 -1 +1 -1 +1$$

also known that the following code arrived to the base station:

Z = (-2 0 0 +2 0 +2 +2 +4)(i omitted the double dollar sign because i didn't want the zeros to be concatenated, so there are spaces for readability)

the questions:

a)is it possible to know what x_1, x_2,x_3 were transmitting(broadcasting) without knowing what is the code for x_4?

b)knowing that x_4 transmitted(broadcasted) 1, can we find her code?

c)forget what you have done in b) and the result you've obtained, can we know what station x_4 transmitted(broadcasted) without knowing its code? if so, what did it transmit? if not, why?

my attempt:

a)because of the added information about the code that was obtained in the base station, it is possible using statistics to infer to some degree the content of x_1,..,x_3(using collisions). however, i don't think we can know for certain since we cannot know if each station transmitted 0 or 1, so we don't know whether to calculate x_i or x_i'(its complementary).

b)i believe that we can find the code for x_4 using collisions based on the additional information(that it), however, i am not sure it would be easy since it is only told that it transmitted "1", and there are a lot of possibilites and it seems quite impractical in usual methods to try and find(or more accurately, as far as i understand it, guess) the content(code) of x_4. moreover, since we know that x_4 transmitted 1, then we have to take it into the account in the calculation of Z, but i don't seem to find a definitive pattern so that i can say for sure that the content(code) of x_4 is this or that.

c)if we were not told that x_4 transmitted "1" like in the previous question, even though we know that a message was obtained in Z with its code, we cannot know for sure that it came from x_4, thus we can not deduct the code or it contents. we don't know if x_4 is to be calculated using "1" or "0"(complimentary or not) and it seems impossible to solve it using the given information.

please explain to me what i've done wrong if i got to the wrong answers here. i am new here to this is my first post in electronics.

thanks a lot for helping me!

• I've never had any CDMA training in my life, so I can guarantee you I don't have the textbook-correct answer. But if we can presume that x_4 must only consist of the symbols [-1, +1] (that is a safe assumption, correct?) then this problem is straightforward to solve, and no statistics are needed. How I went about it (once again this will be "not the way you're supposed to do it"): aX_1 + bX_2 + cX_3 can only generate 8 possible patterns. For each of those 8 patterns, calculate a hypothetical dX_4 required to generate Z. Eliminate all choices that have symbols outside the [-1, +1] set. – Mr. Snrub Mar 14 at 21:19