Online Simulator for GMSK (Gaussian Minimum Shift Keying)

Gaussian Minimum Shift Keying Modulation (GMSK)

• Note: Use input fields for bits, frequency, BT, and baud rate.
• In GMSK, the filtered NRZ signal is integrated to produce a continuous phase signal.

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Number of Bits:

Carrier Frequency (Hz):

Time-Bandwidth (BT):

Baud Rate (Hz):

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GMSK Demodulation

Click 'Demodulate' to extract the original bitstream using a low-pass filter and phase estimator.

Perform Full Demodulation Analysis

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.

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