Digital Modulation — BPSK / QPSK / 8PSK / BFSK / 16-QAM / 64-QAM

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The six common schemes on one workbench. Edit the modulation, the pulse shape, the Eb/N0, and the bit count — see the constellation cloud, the I/Q + RF waveform, the eye diagram, and the BER vs Eb/N0 curve update live. Built to address the most common student confusions: Eb/N0 vs SNR, BER vs SER, Gray vs natural mapping, and why QAM beats PSK at high M.

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Modulation

Scheme:

Pulse shape:

Roll-off α:

Mapping:

Channel

Eb/N0 (dB):

Symbol rate (Hz):

Carrier (Hz):

Bit source

Source:

Hex (e.g. 1F2A):

N bits to simulate:

Samples/symbol:

⚠ Common confusion: Eb/N0 ≠ SNR. SNR = Eb/N0 · log₂(M) = energy per bit · bits per symbol, divided by noise. The KV panel shows both — keep them straight.

① Constellation (with AWGN noise cloud at current Eb/N0)

Ideal M-point constellation (large open circles) + received points from a Monte-Carlo run at the current Eb/N0 (small filled dots, one color per transmitted symbol). The cloud tightens as Eb/N0 rises.

ideal constellation point received (with noise)

② Time-domain waveforms (first 12 symbols)

I(t) and Q(t) are the baseband in-phase and quadrature signals. RF(t) = I·cos(2πf_ct) − Q·sin(2πf_ct) is the actual transmitted signal at the carrier.

I(t) (in-phase baseband) Q(t) (quadrature baseband) RF(t) (upconverted signal)

Y-axis is log scale (BER, bit error rate). Solid lines = theoretical Q-function formulas. Filled circles = Monte-Carlo simulation at the current Eb/N0 for the selected scheme. Higher M is more bandwidth-efficient (curves shifted right) but needs more Eb/N0 for the same BER (curves shifted up).

BPSK QPSK 8PSK 16-QAM 64-QAM current (● simulation)

Modulation

Scheme

—

M (constellation size)

—

k = log₂(M) bits/sym

—

Mapping

—

BER plot range (X = Eb/N0 dB, Y = log BER)

X_min (dB)

X_max (dB)

Y_min (BER)

Y_max (BER)

Type values directly (e.g. 1e-9 for Y_min = 10⁻⁹). Default = auto-fit on every render: X spans [0, max(30, Eb/N0 + 5)] dB; Y spans [10⁻¹², 1].

Rates + Bandwidth

R_s (sym rate)

—

R_b = R_s·k

—

BW (≈)

—

η = R_b/BW (bits/s/Hz)

—

Eb/N0 ↔ SNR

Eb/N0 (linear)

—

SNR = Eb/N0 · k

—

SNR (dB)

—

⚠ Not the same: Eb/N0 is energy per bit / noise spectral density. SNR is total signal power / noise power. For M-ary: SNR = k · (Eb/N0).

BER (at current Eb/N0)

Theoretical (formula)

—

Simulated (Monte Carlo)

—

Agreement

—

Common student confusions

① Eb/N0 vs SNR: SNR = k · (Eb/N0), where k = log₂(M).
② SER vs BER: for Gray-coded M-ary, BER ≈ SER / k (one bit flips per symbol error).
③ Why BPSK = QPSK BER: same minimum distance 2√E_s between adjacent points.
④ Why QAM beats PSK at high M: 16/64-QAM uses a 2D grid (more packing); PSK is on a circle (constant amplitude, less packing).
⑤ Gray vs natural: Gray makes adjacent symbols differ by 1 bit, so symbol error → 1 bit error. Natural: symbol error → up to k bit errors.

📖 Deep-dive blog (planned)

Hard concepts that don't fit cleanly into a single interactive tool. Each will get a long-form post in the blog with the full derivation, edge cases, and worked GATE problems.

Why Q(√(2·Eb/N0)) for BPSK AND QPSK? The minimum-distance argument: BER depends on the noise projection along the direction between adjacent points, and both schemes have the same 2√E_s distance.
The full 16-QAM exact BER formula — derivation from the noise distribution and the decision boundaries; the textbook approximation (3/4)·Q(√(4·Eb/N0 / 5)) is off by up to 0.5 dB at low BER.
Gray code construction — bit-reversal trick, why it minimizes bit errors, the boundary-effect at the wraparound (00 vs 11).
Spectral efficiency vs power efficiency — the trade-off table for all 5 schemes (bits/s/Hz vs Eb/N0 for BER=10⁻⁵), and where each one is used in practice (BPSK in deep-space, QPSK in satellite, 64-QAM in cable/DSL, 256-QAM in Wi-Fi).
Phase ambiguity and differential coding — DBPSK / DQPSK / π/4-DQPSK, and why some systems (e.g. cell phones) need pilot symbols.
Carrier phase recovery (Costas loop, PLL) — what happens when the receiver doesn't know the carrier phase; the 4-phase ambiguity in QPSK.
Raised-cosine vs root-raised-cosine — matched filtering, why RRC on TX + RRC on RX gives an overall RC response with no ISI.

Posts are planned. Tool ships first; deep-dives follow.