# data rate in digital communications

I am a beginner in digital communications and I have question for the data rate with CDMA. Let, the length of the spreading sequence is $$L_c$$, The modulation is BPSK and the symbol time is $$T_s$$.

To calculate the data rate, I make the following reasoning: I have $$1$$ bit for $$1$$ symbol. After, the spreading, the length is $$L_c$$.

Here my questions:

1. What is the duration of the spreading sequence?
2. On the internet, I found that the data rate for CDMA is $$D=\frac{1}{L_cT_s}$$. Why $$L_cT_s$$?
3. Many people calculate the data rate with the length of a frame, can you explain to me how to do that?
• 1- The sequence duration equals the bit duration. The chip duration is $T_s/L_c$. 2- The data rate should be $1/T_s$ b/s. Where did you find your answer? Can you reference it? 3- Can you explain more what you mean? – BlackMath Jan 2 '19 at 13:26
• For me, the duration of the spreading sequence is $L_cT_s$ because a spreading sequence is set of bits and the duration for $1$ bit is $T_s$. So we have, $D=1/(L_cT_s)$ – user39823 Jan 2 '19 at 15:18
• This isn't true. In a symbol time $T_s$ we transmit $L_c$ chips. Refer to any digital communication textbook. The chip rate is $1/T_c=1/(T_s/L_c)=L_c/T_s$ chip/second, but this isn't the bit rate. – BlackMath Jan 3 '19 at 3:19
• The bit rate is $1/T_s=1/(L_c\,T_c)$, where $T_c$ is the chip duration. Check your references. – BlackMath Jan 3 '19 at 3:23

Spreading refers to spreading of the frequency spectrum of the signal, and is achieved by dividing up the symbol interval $$T_s$$ into many shorter intervals called chip intervals ($$L_c$$ chip intervals in your notation) and
• transmitting successive bits of the spreading sequence ($$L_c$$ bits long, remember?) in these $$L_c$$ chip intervals if the data bit you want to transmit in the symbol interval under consideration happens to be a $$0$$
• transmitting the complements of the successive bits of the spreading sequence in these $$L_c$$ chip intervals if the data bit you want to transmit in the symbol interval under consideration happens to be a $$1$$
Symbolically, if $$L_c = 3$$ and the spreading sequence is $$011$$, then in a $$T_s$$-second symbol interval, you get to transmit either $$011$$ or $$100$$ in that $$T_s$$-second interval which we have mentally divided into $$3$$ subintervals of length $$T_s/3$$ each. So, the bits transmitted during each of these chip intervals are of shorter duration (by a factor of $$3$$) and thus require more bandwidth (by a factor of 3 too!) to transmit than we would need if we were transmitting just one data bit every $$T_s$$ seconds.
"But, but, but,..." you splutter, "I don't have data bits, but rather these beautifully shaped RRC pulses that are of duration $$T_s$$ seconds. How do I create a CDMA signal from these?" Well, you have to go back to the purely digital domain where bits are bits, use a frequency multiplier on your clock to get clock pulses at a rate of $$L_c/T_s$$ Hz so that you can replace each data bit at the $$1/T_s$$ Hz clock rate with $$L_c$$ chip bits at the $$L_c/T_s$$ Hz clock rate, and tell the analog guys to design their circuits to produce RRC pulses of duration $$T_s/L_c$$ to accommodate your higher rate digital input signal that is going to send them bits at rate $$L_c/T_s$$ Hz.