BPSK vs MSK vs GMSK Simulation
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:
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:
s(t) = I(t)cos(2πfct) - Q(t)sin(2πfct)
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.
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.
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.
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.
(Smooth Freq)
(Smooth Phase)
📡 Why BT = 0.3 Is Important in GMSK
In Gaussian Minimum Shift Keying (GMSK), the BT product (Bandwidth–Time product) controls how strongly the Gaussian filter smooths the transmitted data before modulation. This parameter directly influences the signal's bandwidth and the amount of Inter-Symbol Interference (ISI).
Provides an effective compromise between spectral efficiency and ISI, resulting in a compact spectrum while maintaining reliable communication.
Reduce ISI by applying less smoothing, but increase the occupied bandwidth, making the transmitted spectrum wider and more likely to affect adjacent channels.
Key Insight: The Gaussian pre-filter shapes the bit stream before modulation, allowing GMSK to produce a constant-envelope signal. This enables the use of highly efficient, non-linear power amplifiers in mobile devices without introducing significant signal distortion.