# Rate in CDMA setup

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?

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).

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.