Interactive OFDM: The "CP" Effect
Compare how Cyclic Prefix converts linear convolution into circular convolution for perfect channel estimation.
The Mathematical Secret: Why the Simulator Works
The simulator demonstrates a fundamental conflict in digital communications: FFT properties vs. Physical reality. Here is the step-by-step math behind the visualization.
1. The Linear Convolution Problem
In a real-world wireless channel, the transmitted signal x[n] is convolved with the channel impulse response h[n]. This is linear convolution:
If the signal has N samples and the channel has L taps, the output has N + L - 1 samples. The signal "smears" into the next symbol (Inter-Symbol Interference).
2. The "Circular" Requirement of FFT
The Fast Fourier Transform (FFT) at the receiver assumes the signal is periodic. Mathematically, the Discrete Fourier Transform (DFT) only diagonalizes Circular Convolution (⊛), not linear:
Without the Cyclic Prefix, DFT{x[n] * h[n]} ≠ X[k] · H[k]. This is why the "Estimated Channel" in the simulator looks jagged and incorrect when the CP is turned off.
3. How CP Forces Circularity
By copying the last L samples of the OFDM symbol to the front, the receiver's FFT window "sees" a signal that appears to be periodic. Even though the channel performs linear convolution, the math inside the FFT window becomes identical to circular convolution:
- Linear Convolution + CP = Circular Convolution (as seen by the FFT).
4. Channel Estimation Logic
In the simulator, we use Least Squares (LS) Estimation. Because the CP has made the channel response "flat" across each subcarrier, we can estimate the channel gain H[k] for every subcarrier k by simply dividing:
Where:
- Y[k]: Received symbol in frequency domain.
- X[k]: Known Pilot/DMRS symbol (the simulator uses a constant '1' for clarity).
- Ĥ[k]: The estimated complex frequency response of the channel.