update module 07

morea/07-antenna-diversity/reading-07-space-time-codes.md

@@ -75,4 +75,89 @@ In conclusion, the Alamouti scheme achieves the diversity gain of 2 while transm
 
 ## General space-time codes
 
+### Definition
 
+Space-time codes modulate transmit signals across space (i.e., antennas) and time. Mathematically, we can represent a space-time code by a set of complex codewords $\\{\mathbf{X}_i\\}$, where each codeword $\{\mathbf{X}_i \in \mathbb{C}^{L \times N}\}$ is a complex-valued $L$ by $N$ matrix. Here, $L$ is the number of antennas, and $N$ is the *block length* of the code (i.e., the number of time slots).
+
+For example, a repetition code over 2 time slots and 2 antennas and with binary phase shift keying (BPSK) modulation can be represented by the two codewords below
+
+$$
+\mathbf{X}_1 = \left[ \begin{array}{cc} +1 & 0 \\ 0 & +1 \end{array} \right], \quad \mathbf{X}_2 = \left[ \begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array} \right].
+$$
+
+The Alamouti scheme over 2 time slots and 2 antennas and with binary phase shift keying (BPSK) modulation can be represented by 16 codewords of the form
+
+$$
+\left[ \begin{array}{lr} u_1 & -u_2^\ast \\ u_2 & u_1^\ast \end{array} \right], 
+$$
+
+where $u_1$ and $u_2$ can be $1+j$, $1-j$, $-1+j$, and $-1-j$.
+
+For fair comparison, we normalize the codewords so that hte average energy per symbol is $1$. Therefore, the signal-to-noise ratio (SNR) is $\textsf{SNR} = 1/N_0$.
+
+### Performance analysis
+
+Assuming that the channel does not change during the block length (i.e., for $N$ time slots), the signal model can be written as
+
+$$
+\mathbf{y}^T = \mathbf{h}^T \cdot \mathbf{X} + \mathbf{w}^T,
+$$
+
+where
+
+$$
+\mathbf{y} = \left[ \begin{array}{c} y[1] \\ \vdots \\ y[N] \end{array} \right], \quad \mathbf{h} = \left[ \begin{array}{c} h_1 \\ \vdots \\ h_N \end{array} \right], \quad \mathbf{w} = \left[ \begin{array}{c} w[1] \\ \vdots \\ w[N] \end{array} \right].
+$$
+
+As in [performance analysis of the rotation code](../06-time-diversity/reading-06-beyond-repetition-coding.html), we focus on pariwise error probability, which is a good approximation of the exact error probability. Conditioned on the channel vector $\mathbf{h}$, the probability of confusing $\mathbf{X}_B$ with $\mathbf{X}_A$ is
+
+$$
+\mathbf{P}\left( \mathbf{X}_A \rightarrow \mathbf{X}_B | \mathbf{h} \right) = Q\left( \frac{\Vert \mathbf{h}^T (\mathbf{X}_A - \mathbf{X}_B) \Vert}{2 \sqrt{N_0/2}} \right),
+$$
+
+which comes from the [vector Gaussian detection problem](../06-time-diversity/reading-06-performance-gain-time-diversity.html#detection-in-a-complex-vector-space) between $\mathbf{h}^T \mathbf{X}_A$ and $\mathbf{h}^T \mathbf{X}_B$.
+
+Therefore, the pairwise error probability is
+
+$$
+\mathbf{P}\left( \mathbf{X}_A \rightarrow \mathbf{X}_B \right) = \mathbf{E}_{\mathbf{h}} \left[ Q\left( \sqrt{ \frac{ \textsf{SNR} \cdot \mathbf{h}^T (\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H \mathbf{h}^\ast }{2} } \right) \right]
+$$
+
+The $L \times L$ matrix $(\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H$ is Hermitian and positive semidefinite, and thus can diagonalized by
+
+$$
+(\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^H,
+$$
+
+where $\mathbf{U}$ is a unitary matrix, namely $\mathbf{U} \mathbf{U}^H = \mathbf{U}^H \mathbf{U} = \mathbf{I}$, and $\mathbf{\Lambda} = \text{diag} \left( \lambda_1^2, \ldots, \lambda_L^2 \right)$. Here, $\lambda_\ell$ is the $\ell$-th singular value of the codeword difference matrix $\mathbf{X}_A - \mathbf{X}_B$.
+
+Then we can rewrite the pairwise error probability as
+
+$$
+\mathbf{P}\left( \mathbf{X}_A \rightarrow \mathbf{X}_B \right) = \mathbf{E}_{\mathbf{h}} \left[ Q\left( \sqrt{ \frac{ \textsf{SNR} \cdot \sum_{\ell=1}^L \vert \tilde{h}_\ell \vert^2 \lambda_\ell^2 }{2} } \right) \right],
+$$
+
+where $\mathbf{\tilde{h}} = \mathbf{U}^T \mathbf{h}$. Since $\mathbf{h}$ is a circularly symmetric complex Gaussian random vector, its linear transformation $$\mathbf{\tilde{h}}$ is also a circularly symmetric complex Gaussian random vector.
+
+Using the property of the Q function $Q(x) \leq e^{-x^2/2}$ and the fact that $\vert \tilde{h}_\ell \vert^2$ are independent exponential random variables, we can bound the average pairwise error probability as
+
+$$
+\mathbf{P}\left( \mathbf{X}_A \rightarrow \mathbf{X}_B \right) \leq \mathbf{E}_{\mathbf{h}} \left[ e^{ -\frac{ \textsf{SNR} \cdot \sum_{\ell=1}^L \vert \tilde{h}_\ell \vert^2 \lambda_\ell^2 }{4} } \right] 
+= \prod_{\ell=1}^L \mathbf{E}_{\tilde{h}_\ell} \left[ e^{ -\frac{ \textsf{SNR} \cdot \vert \tilde{h}_\ell \vert^2 \lambda_\ell^2 }{4} } \right] 
+= \prod_{\ell=1}^L \frac{1}{ 1 + \textsf{SNR} \cdot \lambda_\ell^2 / 4 } .
+$$
+
+From here, we can see that it is important to have all the eeigenvalues $\lambda_\ell^2$ to be strictly positive. In this case, we can bound the error probability by
+
+$$
+\mathbf{P}\left( \mathbf{X}_A \rightarrow \mathbf{X}_B \right) \leq \frac{4^L}{\textsf{SNR}^L \prod_{\ell=1}^L \lambda_\ell^2} = \frac{4^L}{\text{det}\left[ (\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H \right]} \cdot \textsf{SNR}^{-L}.
+$$
+
+## Take-away
+
+For a general space-time code, we need to design the codewords $\\{ \mathbf{X}_i \\}$ with the following rules.
+ - The diversity gain is determined by the number of strictly positive eigenvalues $\lambda_\ell^2$, or equivalently the rank, of the matrix $(\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H$.
+ - To achieve the maximum diversity gain of $L$, there should be $L$ strictly positive eigenvalues $\lambda_\ell^2$ of $(\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H$ over all pairs of codewords.
+ - To minimize the multiplier before $\textsf{SNR}^{-L}$, we should maximize the minimum of the determinant $\text{det}\left[ (\mathbf{X}_A - \mathbf{X}_B) (\mathbf{X}_A - \mathbf{X}_B)^H \right]$ over all pairs of codewords.
+
+Note that to achieve the maximum diversity gain of $L$, we need to have a block length $N$ greater than or equal to the number of transmit antennas $L$.