(Solved): 20-4 The block diagram of a downlink CDMA system operating at baseband in AWGN channel with 2 paths ...

The block diagram of a downlink CDMA system operating at baseband in AWGN channel with 2 paths (one direct and one delayed of one $T_c$ paths) of equal amplitude of 1 is shown in Figure 4 on page 5 , where $b_i$ is the binary antipodal data of $i^{\text{th }}$ user with values $\{\pm 1\},c_i(t)$ is the spreading code of $i^{\text{th }}$ user, $M$ is the system capacity, $T_b$ is the data bit duration of $10^{?3}$ seconds, and $T_c$ is the spreading chip duration. The AWGN power spectral density of $n(t),N_0=10^{?4}\mathrm{W}/\mathrm{Hz}$. The received signal $r(t)=b_1{c}_1(t)+$ ${?}_{i=2}^M{b}_i{c}_i(t)+{?}_{i=1}^M{b}_i{c}_i\left(t?T_c\right)+n(t)$. a) Show that the expression of the output $Y$ can be reduced to: $Y=b_1{T}_b+T_c{?}_{i=2}^M{b}_i{?}_{i1}(0)+T_c{?}_{i=1}^M{b}_i{?}_{i1}(1)+N$ and specify the meanings of the terms ${?}_{i1}(0),{?}_{i1}(1)$ and $N$ in the expression. (11 Marks) b) If the spreading codes are obtained from the full period of Walsh Hadamard codes of length 32 and assuming that the code cross-correlation has an average variance of 25 , determine the system capacity to maintain the bit-error-rate (BER) per user of at least $10^{?3}$. You may assume that the inter-symbol interference (ISI) can be approximated with Gaussian distribution with zero mean. Note that $Q(3.09)?10^{?3}$.