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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?
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  • $\begingroup$ 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? $\endgroup$ – BlackMath Jan 2 at 13:26
  • $\begingroup$ 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, $D=1/(L_cT_s)$ $\endgroup$ – user39823 Jan 2 at 15:18
  • $\begingroup$ 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. $\endgroup$ – BlackMath Jan 3 at 3:19
  • $\begingroup$ The bit rate is $1/T_s=1/(L_c\,T_c)$, where $T_c$ is the chip duration. Check your references. $\endgroup$ – BlackMath Jan 3 at 3:23
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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.

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