MSD and GMSD Algorithms
1. Why MSD and GMSD Are Used
In Wireless Communication, signals suffer from:
- Scattering by surrounding objects
- Reflections
- The above two leads non-line-of-sight (NLOS) propagation
As a result, the received signal is composed of multiple delayed replicas of previously transmitted symbols, causing inter-symbol interference (ISI).
Discrete-time channel model
Let binary symbols \( b[k] \in \{0,1\} \) be transmitted using IM/DD modulation. The received signal is:
\[ r[k] = \sum_{i=0}^{L} h[i]\; b[k-i] + n[k] \]- \(h[i]\): channel impulse response taps
- \(L\): ISI memory length
- \(n[k]\): noise (shot + thermal)
Hence, detecting a symbol independently is suboptimal.
2. Maximum Likelihood Sequence Detection (MLSD)
\[ \hat{\mathbf{b}} = \arg\max_{\mathbf{b}} p(\mathbf{r}|\mathbf{b}) \]However, MLSD has exponential complexity:
\[ \mathcal{O}(2^K) \]where \(K\) is the packet length, making it impractical for real systems.
3. Multiple-Symbol Detection (MSD)
Core Idea
\[ \mathbf{b} = [b[k], b[k-1], \dots, b[k-N+1]] \]Observation model
\[ \mathbf{r} = [r[k], r[k-1], \dots, r[k-N+1]]^T \] \[ \mathbf{r} = \mathbf{H}\mathbf{b} + \mathbf{n} \]\(\mathbf{H}\) is a Toeplitz convolution matrix.
MSD Decision Rule
\[ \hat{\mathbf{b}} = \arg\min_{\mathbf{b} \in \{0,1\}^N} \left\| \mathbf{r} - \mathbf{H}\mathbf{b} \right\|^2 \]MSD optimally mitigates ISI but has complexity:
\[ \mathcal{O}(2^N) \]4. Generalized Multiple-Symbol Detection (GMSD)
Motivation
Wireless channels exhibit long delay spreads, but ISI power decays rapidly. GMSD exploits this by considering only dominant ISI components.
Truncated channel model
\[ r[k] = \sum_{i=0}^{M} h[i] b[k-i] + \sum_{i=M+1}^{L} h[i] b[k-i] + n[k] \]The second summation is treated as additional noise.
GMSD Decision Rule
\[ \hat{\mathbf{b}}_M = \arg\min_{\mathbf{b}_M} \left\| \mathbf{r} - \mathbf{H}_M \mathbf{b}_M \right\|^2 \] \[ \mathcal{O}(2^M), \quad M \ll N \]5. MSD vs GMSD Comparison
| Feature | MSD | GMSD |
|---|---|---|
| Optimality | Optimal | Near-optimal |
| ISI Handling | Full | Truncated |
| Complexity | High | Reduced |
| Real-time feasibility | No | Yes |
Summary
MSD performs joint maximum-likelihood detection over multiple symbols to combat ISI, while GMSD reduces complexity by considering only dominant ISI components, making it practical for wireless systems with long scattering-induced delay spreads.