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\begin{solution}
\begin{tasks}
+ \task
+ The length of both codes is $N = 4$.
+ \begin{equation*}
+ \begin{split}
+ \left\langle \vect{C}_{4,1}, \vect{C}_{4,2} \right\rangle &= \sum\limits_{i = 0}^{N - 1} C_{4,1}[i] C_{4,2}[i] \\
+ &= 1 \cdot 1 + 1 \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot 1 \\
+ &= 0
+ \end{split}
+ \end{equation*}
+ The inner product of the codes is zero. This means that the codes are orthogonal and are suitable for CDMA.
+
+ \task
+ %TODO Symbol spreading block diagram
+ $\vect{S} = \left[1,1,-1,-1,-1,-1,1,1\right]$
+
+ \task
+ {
+ \tiny
+ $\vect{S}$ is extended by zero. That means, $S[n] = 0 \quad \forall n < 0$ and $S[n] = 0 \quad \forall n \geq 8$.
+ \begin{equation*}
+ S[n] = \left[\ldots,0,0,0,0,\underline{1},1,-1,-1,-1,-1,1,1,0,0,0,0,\ldots\right]
+ \end{equation*}
+ \begin{remark}
+ The underline marks the sample at $n = 0$.
+ \end{remark}
+
+ Same zero-extension is applied to the codes.
+ \begin{equation*}
+ C_{4,1}[n] = \left[\ldots,0,0,0,0,\underline{1},1,-1,-1,0,0,0,0,\ldots\right]
+ \end{equation*}
+
+ Now, the cross-correlation can be applied.
+ \begin{equation*}
+ \mathrm{R}_{XY}[n] = \sum\limits_{i = -\infty}^{+\infty} X[n] \cdot Y[n+i]
+ \end{equation*}
+
+ The cross-correlation effectively slides the code over the chips.
+ %TODO Figure
+
+ \begin{equation*}
+ \begin{split}
+ \mathrm{R}_{SC_{4,1}}[-3] &= S[-3] \cdot C_{4,1}[0] + S[-2] \cdot C_{4,1}[1] + S[-1] \cdot C_{4,1}[2] + S[0] \cdot C_{4,1}[3] \\
+ &= (0) \cdot (1) + (0) \cdot (1) + (0) \cdot (-1) + (1) \cdot (-1) \\
+ &= -1 \\
+ \mathrm{R}_{SC_{4,1}}[-2] &= S[-2] \cdot C_{4,1}[0] + S[-1] \cdot C_{4,1}[1] + S[0] \cdot C_{4,1}[2] + S[1] \cdot C_{4,1}[3] \\
+ &= (0) \cdot (1) + (0) \cdot (1) + (1) \cdot (-1) + (1) \cdot (-1) \\
+ &= -2 \\
+ \mathrm{R}_{SC_{4,1}}[-1] &= S[-1] \cdot C_{4,1}[0] + S[0] \cdot C_{4,1}[1] + S[1] \cdot C_{4,1}[2] + S[2] \cdot C_{4,1}[3] \\
+ &= (0) \cdot (1) + (1) \cdot (1) + (1) \cdot (-1) + (-1) \cdot (-1) \\
+ &= 1 \\
+ \mathrm{R}_{SC_{4,1}}[0] &= S[0] \cdot C_{4,1}[0] + S[1] \cdot C_{4,1}[1] + S[2] \cdot C_{4,1}[2] + S[3] \cdot C_{4,1}[3] \\
+ &= (1) \cdot (1) + (1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) \\
+ &= 4 \\
+ \mathrm{R}_{SC_{4,1}}[1] &= S[1] \cdot C_{4,1}[0] + S[2] \cdot C_{4,1}[1] + S[3] \cdot C_{4,1}[2] + S[4] \cdot C_{4,1}[3] \\
+ &= (1) \cdot (1) + (-1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) \\
+ &= 2 \\
+ \mathrm{R}_{SC_{4,1}}[2] &= S[2] \cdot C_{4,1}[0] + S[3] \cdot C_{4,1}[1] + S[4] \cdot C_{4,1}[2] + S[5] \cdot C_{4,1}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) \\
+ &= 0 \\
+ \mathrm{R}_{SC_{4,1}}[3] &= S[3] \cdot C_{4,1}[0] + S[4] \cdot C_{4,1}[1] + S[5] \cdot C_{4,1}[2] + S[6] \cdot C_{4,1}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (1) + (-1) \cdot (-1) + (1) \cdot (-1) \\
+ &= -2 \\
+ \mathrm{R}_{SC_{4,1}}[4] &= S[4] \cdot C_{4,1}[0] + S[5] \cdot C_{4,1}[1] + S[6] \cdot C_{4,1}[2] + S[7] \cdot C_{4,1}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (1) + (1) \cdot (-1) + (1) \cdot (-1) \\
+ &= -4 \\
+ \mathrm{R}_{SC_{4,1}}[5] &= S[5] \cdot C_{4,1}[0] + S[6] \cdot C_{4,1}[1] + S[7] \cdot C_{4,1}[2] + S[8] \cdot C_{4,1}[3] \\
+ &= (-1) \cdot (1) + (1) \cdot (1) + (1) \cdot (-1) + (0) \cdot (-1) \\
+ &= -1 \\
+ \mathrm{R}_{SC_{4,1}}[6] &= S[6] \cdot C_{4,1}[0] + S[7] \cdot C_{4,1}[1] + S[8] \cdot C_{4,1}[2] + S[9] \cdot C_{4,1}[3] \\
+ &= (1) \cdot (1) + (1) \cdot (1) + (0) \cdot (-1) + (0) \cdot (-1) \\
+ &= 2 \\
+ \mathrm{R}_{SC_{4,1}}[7] &= S[7] \cdot C_{4,1}[0] + S[8] \cdot C_{4,1}[1] + S[9] \cdot C_{4,1}[2] + S[10] \cdot C_{4,1}[3] \\
+ &= (1) \cdot (1) + (0) \cdot (1) + (0) \cdot (-1) + (0) \cdot (-1) \\
+ &= 1 \\
+ \end{split}
+ \end{equation*}
+
+ The cross-correlation peak is at $n = 0$, indicating that $\vect{S}$ is correlated to $\vect{C}_{4,1}$ and is spread by $\vect{C}_{4,1}$.
+ }
+
+ \task
+ {
+ \tiny
+ Zero-extension of $\vect{C}_{4,2}$.
+ \begin{equation*}
+ C_{4,2}[n] = \left[\ldots,0,0,0,0,\underline{1},-1,-1,1,0,0,0,0,\ldots\right]
+ \end{equation*}
+
+ Cross-correlation:
+ \begin{equation*}
+ \begin{split}
+ \mathrm{R}_{SC_{4,2}}[-3] &= S[-3] \cdot C_{4,2}[0] + S[-2] \cdot C_{4,2}[1] + S[-1] \cdot C_{4,2}[2] + S[0] \cdot C_{4,2}[3] \\
+ &= (0) \cdot (1) + (0) \cdot (-1) + (0) \cdot (-1) + (1) \cdot (1) \\
+ &= 1 \\
+ \mathrm{R}_{SC_{4,2}}[-2] &= S[-2] \cdot C_{4,2}[0] + S[-1] \cdot C_{4,2}[1] + S[0] \cdot C_{4,2}[2] + S[1] \cdot C_{4,2}[3] \\
+ &= (0) \cdot (1) + (0) \cdot (-1) + (1) \cdot (-1) + (1) \cdot (1) \\
+ &= 0 \\
+ \mathrm{R}_{SC_{4,2}}[-1] &= S[-1] \cdot C_{4,2}[0] + S[0] \cdot C_{4,2}[1] + S[1] \cdot C_{4,2}[2] + S[2] \cdot C_{4,2}[3] \\
+ &= (0) \cdot (1) + (1) \cdot (-1) + (1) \cdot (-1) + (-1) \cdot (1) \\
+ &= -3 \\
+ \mathrm{R}_{SC_{4,2}}[0] &= S[0] \cdot C_{4,2}[0] + S[1] \cdot C_{4,2}[1] + S[2] \cdot C_{4,2}[2] + S[3] \cdot C_{4,2}[3] \\
+ &= (1) \cdot (1) + (1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (1) \\
+ &= 0 \\
+ \mathrm{R}_{SC_{4,2}}[1] &= S[1] \cdot C_{4,2}[0] + S[2] \cdot C_{4,2}[1] + S[3] \cdot C_{4,2}[2] + S[4] \cdot C_{4,2}[3] \\
+ &= (1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (1) \\
+ &= 2 \\
+ \mathrm{R}_{SC_{4,2}}[2] &= S[2] \cdot C_{4,2}[0] + S[3] \cdot C_{4,2}[1] + S[4] \cdot C_{4,2}[2] + S[5] \cdot C_{4,2}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (1) \\
+ &= 0 \\
+ \mathrm{R}_{SC_{4,2}}[3] &= S[3] \cdot C_{4,2}[0] + S[4] \cdot C_{4,2}[1] + S[5] \cdot C_{4,2}[2] + S[6] \cdot C_{4,2}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) + (1) \cdot (1) \\
+ &= 2 \\
+ \mathrm{R}_{SC_{4,2}}[4] &= S[4] \cdot C_{4,2}[0] + S[5] \cdot C_{4,2}[1] + S[6] \cdot C_{4,2}[2] + S[7] \cdot C_{4,2}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (-1) + (1) \cdot (-1) + (1) \cdot (1) \\
+ &= 0 \\
+ \mathrm{R}_{SC_{4,2}}[5] &= S[5] \cdot C_{4,2}[0] + S[6] \cdot C_{4,2}[1] + S[7] \cdot C_{4,2}[2] + S[8] \cdot C_{4,2}[3] \\
+ &= (-1) \cdot (1) + (1) \cdot (-1) + (1) \cdot (-1) + (0) \cdot (1) \\
+ &= -3 \\
+ \mathrm{R}_{SC_{4,2}}[6] &= S[6] \cdot C_{4,2}[0] + S[7] \cdot C_{4,2}[1] + S[8] \cdot C_{4,2}[2] + S[9] \cdot C_{4,2}[3] \\
+ &= (1) \cdot (1) + (1) \cdot (-1) + (0) \cdot (-1) + (0) \cdot (1) \\
+ &= 0 \\
+ \mathrm{R}_{SC_{4,2}}[7] &= S[7] \cdot C_{4,2}[0] + S[8] \cdot C_{4,2}[1] + S[9] \cdot C_{4,2}[2] + S[10] \cdot C_{4,2}[3] \\
+ &= (1) \cdot (1) + (0) \cdot (-1) + (0) \cdot (-1) + (0) \cdot (1) \\
+ &= 1 \\
+ \end{split}
+ \end{equation*}
+
+ The cross-correlation has no clear peaks, indicating that $\vect{S}$ and $\vect{C}_{4,2}$ are uncorrelated.
+ }
+
+ \task
+ {
+ \tiny
+ Autocorrelation is the cross-correlation of the (zero-extended) code $\vect{C}_{4,2}$ with itself.
+
+ \begin{equation*}
+ \begin{split}
+ \mathrm{R}_{C_{4,2}C_{4,2}}[-3] &= C_{4,2}[-3] \cdot C_{4,2}[0] + C_{4,2}[-2] \cdot C_{4,2}[1] + C_{4,2}[-1] \cdot C_{4,2}[2] + C_{4,2}[0] \cdot C_{4,2}[3] \\
+ &= (0) \cdot (1) + (0) \cdot (-1) + (0) \cdot (-1) + (1) \cdot (1) \\
+ &= 1 \\
+ \mathrm{R}_{C_{4,2}C_{4,2}}[-2] &= C_{4,2}[-2] \cdot C_{4,2}[0] + C_{4,2}[-1] \cdot C_{4,2}[1] + C_{4,2}[0] \cdot C_{4,2}[2] + C_{4,2}[1] \cdot C_{4,2}[3] \\
+ &= (0) \cdot (1) + (0) \cdot (-1) + (1) \cdot (-1) + (-1) \cdot (1) \\
+ &= -2 \\
+ \mathrm{R}_{C_{4,2}C_{4,2}}[-1] &= C_{4,2}[-1] \cdot C_{4,2}[0] + C_{4,2}[0] \cdot C_{4,2}[1] + C_{4,2}[1] \cdot C_{4,2}[2] + C_{4,2}[2] \cdot C_{4,2}[3] \\
+ &= (0) \cdot (1) + (1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (1) \\
+ &= -1 \\
+ \mathrm{R}_{C_{4,2}C_{4,2}}[0] &= C_{4,2}[0] \cdot C_{4,2}[0] + C_{4,2}[1] \cdot C_{4,2}[1] + C_{4,2}[2] \cdot C_{4,2}[2] + C_{4,2}[3] \cdot C_{4,2}[3] \\
+ &= (1) \cdot (1) + (-1) \cdot (-1) + (-1) \cdot (-1) + (1) \cdot (1) \\
+ &= 4 \\
+ \mathrm{R}_{C_{4,2}C_{4,2}}[1] &= C_{4,2}[1] \cdot C_{4,2}[0] + C_{4,2}[2] \cdot C_{4,2}[1] + C_{4,2}[3] \cdot C_{4,2}[2] + C_{4,2}[4] \cdot C_{4,2}[3] \\
+ &= (-1) \cdot (1) + (-1) \cdot (-1) + (1) \cdot (-1) + (0) \cdot (1) \\
+ &= -1 \\
+ \mathrm{R}_{C_{4,2}C_{4,2}}[2] &= C_{4,2}[2] \cdot C_{4,2}[0] + C_{4,2}[3] \cdot C_{4,2}[1] + C_{4,2}[4] \cdot C_{4,2}[2] + C_{4,2}[5] \cdot C_{4,2}[3] \\
+ &= (-1) \cdot (1) + (1) \cdot (-1) + (0) \cdot (-1) + (0) \cdot (1) \\
+ &= -2 \\
+ \mathrm{R}_{C_{4,2}C_{4,2}}[3] &= C_{4,2}[3] \cdot C_{4,2}[0] + C_{4,2}[4] \cdot C_{4,2}[1] + C_{4,2}[5] \cdot C_{4,2}[2] + C_{4,2}[6] \cdot C_{4,2}[3] \\
+ &= (1) \cdot (1) + (0) \cdot (-1) + (0) \cdot (-1) + (0) \cdot (1) \\
+ &= 1 \\
+ \end{split}
+ \end{equation*}
+ }
\end{tasks}
\end{solution}
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