OFDM Visual Lab: 16-QAM
High Spectral Efficiency (4 bits/symbol) using Gray-Coded Mapping
Step 1: Parallel Bit Groups
In 16-QAM, each subcarrier $k$ carries 4 bits. Two bits determine the In-phase (Real) amplitude, and two determine the Quadrature (Imaginary) amplitude.
$$\text{Bits per Subcarrier: } \{b_0, b_1, b_2, b_3\}$$
Step 2: Gray-Coded Amplitude Mapping
We use Gray Coding so that adjacent points differ by only one bit. The levels are $\{-3, -1, 1, 3\}$. To normalize the average power to 1, we multiply by $1/\sqrt{10}$.
$$X_k = \frac{1}{\sqrt{10}} (A_I + j A_Q), \quad A \in \{-3, -1, 1, 3\}$$
$$\text{Gray Map: } 00 \to -3, \quad 01 \to -1, \quad 11 \to 1, \quad 10 \to 3$$
Step 3: OFDM Multiplexing (IFFT)
The complex symbols are converted to a time-domain wave. Each subcarrier is orthogonal to the others.
$$x[n] = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k e^{j \frac{2\pi nk}{N}}$$
Step 4: Adding the Cyclic Prefix (CP)
The last $N_{cp}$ samples are prepended to the start to protect against channel delays.
Step 5: Receiver 16-Point Grid
The FFT recovers the symbols. In 16-QAM, the points are closer together than BPSK, making the system more sensitive to noise.