Interactive BPSK Waveform Decoding Simulator using Matched Filter

BPSK Waveform Decoding using Matched Filter (Convolution)

Noise (SNR dB) 5 dB

Carrier Frequency 2 cycles/bit

Binary Data

TX:

RX:

Stage 1: Noisy BPSK Stream x[n] (Observe Phase Flips)

Stage 2: Filter Output y[n] (Convolution Peaks at Yellow Sticks)

Simulation Methodology: BPSK Convolution

In BPSK, the pulse shape is a sine wave. To decode this optimally, we use a Direct Matched Filter whose impulse response is a time-reversed version of the carrier cycle.

1. The Linear Convolution: The green waveform is generated by sliding the BPSK template across the noisy input stream. Mathematically: y[n] = Σ x[k] · h[n - k] Because the BPSK carrier is perfectly "matched" to the filter h[n], the output constructively interferes to create a Positive Peak for bit '1' and a Negative Peak for bit '0'.

2. Noise Suppression: Even at 5dB SNR, where the input (Stage 1) is nearly indistinguishable, the convolution process accumulates signal energy over the entire bit period. Random noise, being zero-mean, does not accumulate effectively, allowing the signal peak to "rise" above the noise floor.

3. Optimal Sampling: The Yellow Sticks represent the symbol clock. We sample the output of the convolution exactly when the current bit is fully contained within the filter's memory. This is the point of Maximum Likelihood.