Interactive BPSK Pulse Shaping Simulator: Visualize Decoding & ISI

Decoding of Pulse-Shaped BPSK Waveform

Roll-off Factor (α) 0.5

Timing Jitter (τ) 0% Offset

Transmit Bits (a_k)

TX:

RX:

Mathematical Framework

In bandwidth-limited channels, we cannot use rectangular pulses because they have infinite spectral width. Instead, we use the Raised Cosine (RC) pulse, defined in the time domain as:

p(t) = sinc(t/T) * [ cos(παt/T) / (1 - (2αt/T)^2) ]

The total transmitted waveform is the superposition (sum) of these pulses shifted by the symbol period T:

x(t) = Σ a_k * p(t - kT)

1. Nyquist's First Criterion: For zero Intersymbol Interference (ISI), the pulse must satisfy p(0)=1 and p(kT)=0 for all k ≠ 0. In the simulator, notice that when the yellow sticks are perfectly centered, the "tails" of all other pulses cross the zero-line at that exact moment.

2. Impact of Jitter (τ): When a timing offset τ is introduced, the receiver samples at t = iT + τ. The sampled value becomes:

y_i = a_i * p(τ) + Σ [ a_k * p(iT + τ - kT) ]

The first term is the desired bit (attenuated by the pulse shape), and the second term is the ISI component. As the yellow sticks move, the ISI component grows, eventually causing bit errors even in the absence of noise.

3. Roll-off Factor (α): This determines the "excess bandwidth." A low α (0.1) creates a sharp spectral cutoff but results in heavy "ringing" in the time domain, making the system extremely sensitive to jitter.

1. The Decoding Process: From Wave to Bit

In a pulse-shaped system, decoding follows a strict 3-step sequence:

• Timing Synchronization: The receiver aligns its clock (the Yellow Sticks) to the peaks of the incoming pulses. This is the only moment where the signal amplitude is maximized.
• Instantaneous Sampling: At the exact moment indicated by the stick, the receiver captures the instantaneous voltage V_sample.
• Hard Decision (The Slicer): The sampled voltage is compared against a zero-threshold: Decoded Bit = 1 (if V_sample > 0) OR 0 (if V_sample < 0)

2. Mathematical Insight (ISI Component)

When the sampling is perfect (τ = 0), the voltage is pure. However, with Jitter, the sampled value y_i is corrupted by Intersymbol Interference (ISI):

y_i = a_i * p(τ) + Σ [ a_k * p(iT + τ - kT) ]

The summation term represents the "tails" of all other bits leaking into the current sample. As you move the Timing Jitter slider, notice the decoded bits turn red because the ISI term becomes large enough to flip the polarity of the sample.