CDMA System with DS Spread Spectrum and BPSK Modulation

CDMA (Code Division Multiple Access) is a multiple access technique used in wireless communication systems to allow multiple users to share the same frequency band simultaneously. DS (Direct Sequence) spread spectrum is a technique used in CDMA systems to spread the signal over a wider bandwidth by multiplying it with a pseudo-random sequence. BPSK (Binary Phase Shift Keying) is a modulation scheme used in CDMA systems to represent binary data by shifting the phase of the carrier signal.

Processing Gain and Error Probability

Processing gain is the ratio of the spread bandwidth to the original signal bandwidth. It represents the amount of spreading of the signal and is directly proportional to the resistance to interference and noise. In this question, the processing gain is given as 1000.

Error probability is the probability of incorrect detection or decoding of the transmitted signal. It depends on the modulation scheme, the signal-to-noise ratio (SNR), and the desired level of performance. In this question, the desired level of performance is given as an error probability of 10^-6.

Number of Users

Assuming all users have equal power, the maximum number of users that can be supported by the CDMA system can be calculated as follows:

- The SNR required to achieve the desired error probability for BPSK modulation is given by: SNR = 10 log10(1/2Pe) = 20 log10(1/SQRT(2))/Pe, where Pe is the error probability.
- The SNR required for a processing gain of 1000 is given by: SNR = 10 log10(1000) = 30 dB.
- The total received power is the sum of the powers of all users. Since all users have equal power, the total received power is N times the power of a single user, where N is the number of users.
- The interference power is the power of the signals from other users. Since all users have equal power, the interference power is (N-1) times the power of a single user.
- The SNR at the receiver is the ratio of the received power to the interference power.
- Equating the SNR required for the desired error probability and the SNR at the receiver, we get: N = (1 + SNR)/(1 + SNR/1000).

Substituting the values, we get: N = (1 + 10^(-6))/(1 + 10^(-6)/1000) ≈ 216.

Therefore, the number of users that can be supported by the CDMA system is 216. The correct answer is option D.