BPSK vs MSK vs GMSK Simulation | Compare Spectra and Waveforms

BPSK vs MSK vs GMSK Simulation

Bits: Mode: BPSK MSK GMSK BT: Amp Saturation:

Phase Trajectory (Degrees)

RF Waveform

Spectrum (dB)

Technical Analysis: Baseband to Passband Pipeline

This simulator treats MSK and GMSK as Continuous Phase Modulation (CPM) systems, where information is encoded in the accumulation of phase rather than absolute state jumps.

1. The Pulse Shaping (NRZ to Frequency)

Digital bits are mapped to NRZ (+1/-1). In MSK/GMSK, these are treated as Frequency Commands. In GMSK, these pulses are convolved with a Gaussian filter:

h(t) = (1/&sqrt;2πσT) exp(-t²/2σ²T²)

2. Phase Accumulation (The Integral)

The phase φ(t) is the integral of the shaped frequency pulses. For Minimum Shift Keying, the modulation index is exactly 0.5, ensuring a 90° shift per bit.

BPSK

Phase jumps instantly between 0 and π. Amplitude is constant but phase is discontinuous.

MSK

Phase accumulates linearly. Frequency is constant per bit. Phase change is strict 90°.

GMSK

Phase accumulates as an S-Curve (integral of Gaussian). Smoothest possible transitions.

3. Complex Baseband (I/Q) & Upconversion

Hardware generates the Real (I) and Imaginary (Q) components to drive the mixer:

I(t) = cos(φ(t)), Q(t) = sin(φ(t))
s(t) = I(t)cos(2πf_ct) - Q(t)sin(2πf_ct)

4. Non-Linear Amplification

We simulate High Power Amplifier (HPA) saturation using a hyperbolic tangent function. Constant envelope signals (MSK/GMSK) are highly resistant to this distortion.

s_out = tanh( Saturation · s_in )

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1. The Math of I/Q Modulation

In this simulator, GMSK is converted into Real (I) and Imaginary (Q) waveforms. This Cartesian representation allows hardware to drive mixers without sudden phase jumps.

s(t) = I(t) cos(2πf_ct) - Q(t) sin(2πf_ct)

Where I(t) = cos(φ(t)) and Q(t) = sin(φ(t)). This confirms that even as the phase rotates, the total power I² + Q² = 1 remains constant.

For MSK (Rectangular Frequency) φ(t) = ∑ d_k · (π/2) · (t/T)

For GMSK (Gaussian Frequency) h(t) = (1/&sqrt;2πσT) exp(-t²/2σ²T²)

2. Phase Accumulation: Frequency Integration

In Continuous Phase Modulation (CPM), we do not "set" the phase. Instead, the phase is the running integral of frequency.

MSK: Linear Accumulation

MSK treats bits as constant frequency shifts. This results in Strict Linear Ramps where the phase accumulates exactly ±90° per bit period.

GMSK: Gaussian Accumulation

GMSK integrates Gaussian curves. This creates S-curve phase trajectories, which target 90° but reach it through smooth acceleration and deceleration.

Property	MSK	GMSK
Frequency Pulse	Square Wave	Gaussian Bell Curve
Phase Shape	Straight Zig-Zags	Smooth S-Curves
Accumulation	Strict 90° / Bit	Strict 90° / Bit

3. The GMSK Shaping Process

The core secret of GMSK is passing the digital NRZ signal through the Gaussian filter before integration. This allows us to smooth the spectrum without losing the constant envelope.

Bits (1, 0)

→

NRZ (+1, -1)

→

Gaussian Filter
(Smooth Freq)

→

Integrator
(Smooth Phase)

Critical Insight: BPSK is a Phase-Modulated signal where filtering destroys amplitude. GMSK is a Frequency-Modulated signal where filtering only affects the rate of rotation. This keeps the Amplitude 100% Constant, maximizing amplifier efficiency.