The theoretical Bit Error Rate (BER) for Minimum Shift Keying (MSK) is identical to that of BPSK (Binary Phase Shift Keying) and QPSK (Quadrature Phase Shift Keying), provided that coherent detection is used.
The Formula
The probability of bit error (\(P_b\)) for MSK is given by:
Alternatively, expressed using the complementary error function (\(\text{erfc}\)):
Where:
- \(E_b\): Energy per bit.
- \(N_0\): Noise power spectral density (one-sided).
- \(E_b/N_0\): The signal-to-noise ratio per bit.
- \(Q(x)\): The Q-function, which represents the tail probability of the standard normal distribution.
BER Results
| SNR (dB) | BER |
|---|
Why is it the same as BPSK?
Even though MSK is technically a form of Frequency Shift Keying (specifically CPFSK with a modulation index \(h=0.5\)), it can be mathematically viewed as Offset QPSK (OQPSK) with half-sine pulse shaping.
Because the two orthogonal carrier frequencies in MSK are separated such that they can be detected coherently without interfering with one another, MSK achieves the same power efficiency as phase-shift keyed systems.
Key Performance Characteristics
- Coherent vs. Non-coherent: The formula above assumes coherent detection (the receiver is perfectly synchronized in phase and frequency). If non-coherent detection (like a frequency discriminator) is used, the performance degrades significantly, following a curve closer to BFSK: \(P_b = \frac{1}{2} e^{-E_b/2N_0}\).
- Spectral Efficiency: While the BER is the same as BPSK, MSK has a much "narrower" main lobe and faster spectral roll-off. This makes it more bandwidth-efficient and less prone to adjacent channel interference.
- Constant Envelope: MSK has a constant envelope, meaning it is highly resistant to non-linear distortion from power amplifiers, which is why it was chosen for standards like GSM.
BER vs. \(E_b/N_0\) Reference Table
To help visualize the "waterfall" curve for coherent MSK:
| \(E_b/N_0\) (dB) | Approximate BER (\(P_b\)) |
|---|---|
| 0 dB | \(7.8 \times 10^{-2}\) |
| 4 dB | \(1.2 \times 10^{-2}\) |
| 7 dB | \(7.7 \times 10^{-4}\) |
| 8.4 dB | \(1.0 \times 10^{-4}\) |
| 10.5 dB | \(1.0 \times 10^{-6}\) |
| 12.6 dB | \(1.0 \times 10^{-9}\) |
Practical Performance: MSK vs. PSK
In theory (AWGN), they are equal. In reality, MSK usually wins the hardware battle while PSK wins the software/complexity battle.
MSK Advantages
- 0 dB PAPR: Constant envelope allows amplifiers to run at 100% saturation without distortion.
- Spectral Roll-off: Side-lobes drop at \(1/f^4\), preventing "splatter" into adjacent channels.
- Power Efficiency: Higher actual \(E_b/N_0\) delivered for the same battery pull.
PSK Advantages
- Receiver Simplicity: Easier carrier recovery; less sensitive to oscillator phase noise.
- Stability: BPSK is highly forgiving of frequency drift and Doppler shifts.
- High Order Scaling: Easier to move to higher-order modulation (16-QAM, etc.) in modern hardware.
| Metric | Practical PSK | Practical MSK | Winner |
|---|---|---|---|
| Amp Efficiency | Poor (Requires linear PA) | Superb (Saturated PA) | MSK |
| Battery Life | Lower | Higher | MSK |
| Rx Complexity | Simple / Stable | Complex / Finicky | PSK |
| Cheap Oscillators | High Tolerance | Low Tolerance | PSK |
Note: 2G GSM used GMSK because early phone batteries couldn't support the linear amplifiers needed for PSK. Modern 4G/5G returned to PSK/QAM as silicon efficiency improved.