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I am working on a CDMA system with only 4 chips/symbol. I want to use $R=\frac{1}{2}log_2(1+SNR)$ chips/channel use to calculate what $SNR$ I require to successfully receive a given transmission. In the equation listed, I set $R=4$ because I want to send 4 chips/symbol and one symbol per channel use.

Is this correct thinking?

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What you cite above is the Shannon limit, which gives you an upper bound for achievable data rate at arbitrarily little error rate.

It is, correctly stated (note: $\le$, not $=$),

$$ R \le B \log_2(1+\text{SNR})\,\text{.} $$

Now, Shannon says something about the data rate, not the chip rate: A single chip only contains only a fraction of the info of a symbol that you've spread!

So, you need to think like this:

A given transmission has data rate $R$ and uses bandwidth $B$. What's the $\text{SNR}$ I need to have to even theoretically be able to receive that with arbitrarily low probability of error?

and insert $R$ and $B$ in above inequality and solve for $\text{SNR}$.

You need two of the three variables to solve for the third!

In communications theory, bandwidth, however, is often arbitrarily set to $B=1$, as one can always argue that you can just scale your capacity by scaling your bandwidth (subject to SNR staying constant).

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Your query is mixing up ideas and formulas from very different systems to arrive at very questionable answers.

  • The expression $\frac 12 \log_2(1+\text{SNR})$ is the capacity (measured in bits per use) of a discrete-time Gaussian channel. The model for this channel is that the $i$-th use of the channel consists of the transmission of a single real number value $X_i$ and receipt of $Y_i=X_i+N_i$ where $N_i$ are independent zero-mean Gaussian random variables with variance $\sigma^2$. The $X_i$ are also modeled as random variables with $E[X_i^2]=A^2$, and SNR is the ratio $\frac{A^2}{\sigma^2}$. The claim then is that no matter how we choose the distributions of the $X_i$'s, it is not possible to convey information across this channel at rates exceeding $\frac 12 \log_2(1+\text{SNR})$ bits per use. Getting even close to this capacity requires the joint distribution of the $X_i$ to effectively be close to that of correlated zero-mean Gaussian random variables with variance $A^2$.
  • A CDMA system with 4 chips per symbol can be idealized into a Gaussian channel but the information is carried in the symbols and not the chips. Setting $R=4$ because there are 4 chips per symbol is very muddled thinking.
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