Online Simulator for GMSK (Gaussian Minimum Shift Keying)

GMSK Waveform Generator

Time Scale: Seconds

Input Bitstream

BT (Smoothing)

Carrier Freq (Hz)

Step 1: NRZ Encoding
$m(t) = \sum a_n \cdot rect(t-nT)$

Step 2: Gaussian Pulse Shaping
$g(t) = m(t) * h_{gauss}(t)$

Step 3: Smooth Phase Trajectory
$\phi(t) = 2\pi h \int g(\tau) d\tau$

Step 4: Final GMSK Carrier Waveform
$s(t) = \cos(2\pi f_c t + \phi(t))$

How GMSK is Produced: Step-by-Step

The mathematical journey from raw bits to a smooth radio wave.

Step 1

NRZ Bit Encoding

The digital input (0s and 1s) is converted into a **Non-Return-to-Zero (NRZ)** signal. Binary '1' becomes $+1$ and binary '0' becomes $-1$. This creates the raw "pulses" seen in the first plot of the simulator.

Step 2

Gaussian Pulse Shaping

The NRZ pulses are passed through a **Gaussian Filter**. This is the most critical part. It "smears" the sharp edges of the square pulses.
The amount of smoothing is controlled by the **$BT$ product** (Bandwidth-Time).

$$h(t) = \frac{1}{\sqrt{2\pi}\sigma T} \exp\left(-\frac{t^2}{2\sigma^2 T^2}\right)$$

Step 3

Phase Integration

To keep the phase continuous (no jumps), the filtered signal is **integrated** over time. This transforms frequency shifts into a smooth, wandering phase angle $\theta(t)$.

$$\phi(t) = \sum b_n \pi h \int_{-\infty}^{t-nT} g(\tau) d\tau$$

Step 4

Carrier Modulation (I/Q)

Finally, the phase is applied to a high-frequency carrier wave. In the simulator, this is often done using In-phase (I) and Quadrature (Q) components:

• I Signal: $\cos(\phi(t)) \cdot \cos(2\pi f_c t)$
• Q Signal: $\sin(\phi(t)) \cdot \sin(2\pi f_c t)$

The sum of these two creates the final GMSK wave that has a **constant envelope** and a very narrow bandwidth.

Pro Tip: When using the simulator, try changing the BT Product to 0.3 (Standard for GSM). You will notice that as BT gets smaller, the Filtered Signal becomes smoother, and the Frequency Spectrum becomes narrower.

🎯 The Magic of the BT=0.3 Product

In GMSK modulation, the TBP (referred to as the BT Product) determines the "smoothness" of the phase transitions.

BT = 0.3 (GSM Standard)
Optimal balance. Narrowest main-lobe with manageable Inter-Symbol Interference (ISI).

Higher BT (> 0.5)
Cleans up ISI but creates a "fatter" spectrum that interferes with neighboring radio channels.

Interconnection: By pre-filtering bits with a Gaussian pulse of a specific TBP, GMSK achieves a Constant Envelope, allowing mobile phones to use cheap, high-power amplifiers without distorting the signal.

Return to Simulations Main Page →