Minimum Shift Keying (MSK)

• 📘 Overview
• 🧮 Mathematical Representation
• 🧮 Steps for minimum shift keying
• 🧮 Why Use MSK?
• 🧮 Similarities between FSK and MSK
• 🧮 Differences between FSK and MSK
• 🧮 MATLAB Codes
• 🧮 Simulator for MSK
• 📚 Further Reading

Minimum Shift Keying (MSK)

In digital modulation, Minimum-Shift Keying (MSK) is a type of continuous-phase frequency-shift keying. Similar to OQPSK, MSK encodes bits alternating between quadrature components, with the Q component delayed by half the symbol period.

However, instead of using square pulses like OQPSK, MSK encodes each bit as a half sinusoid. This results in a constant-modulus signal, which reduces issues caused by non-linear distortion. In addition to being viewed as a variant of OQPSK, MSK is also seen as a Continuous Phase Frequency Shift Keyed (CPFSK) signal with a frequency separation of one-half the bit rate. 

 

Mathematical Representation

The resulting signal can be expressed as:

s(t) = a_I(t)  cos(πt / 2T)  cos(2πf_ct) − a_Q(t)  sin(πt / 2T)  sin(2πf_ct)

where aI(t) and aQ(t) encode even and odd bits respectively with square pulses of duration 2T.

Using trigonometric identities, the expression can be rewritten to show phase and frequency modulation more clearly:

s(t) = cos(2πf_ct + φ_k + πt / 2T  ·  b_k)

where:

• bk = +1 if aI(t) = aQ(t), and -1 otherwise
• φk = 0 if aI(t) = 1, and π otherwise

This shows that the signal is modulated in both frequency and phase, and the phase changes in a continuous and linear manner. 

 

Minimum shift keying (MSK) is a type of continuous frequency shift keying (CFSK). Like continuous frequency shift keying, we can also define bit "1" and "0" by two distinct frequency carriers, but the frequency shift will be half of the baud rate, i.e., f2 - f1 will be Rb/2.

where f2 is the carrier frequency for binary bit "1"

f1 is the carrier frequency for binary bit "0"

Rb = Baud Rate  

Here, the frequency shift between two carrier frequencies is minimum (Baud Rate) / 2. So, it is called minimum shift keying.

Steps for minimum shift keying

1. Firstly, generate a bitstream of binary bits "1" and "0"

2. Then Convert it to NRZ Signal

3. Then perform MSK modulation on the NRZ Signal

4. If the carrier frequency is fc, then bit "1" will be represented by slightly higher than the carrier frequency, fc + (baud rate)/4, and bit "0" by slightly lower than the carrier frequency, fc - (baud rate)/4.

Gaussian minimum shift keying (GMSK) is quite similar to the MSK. Here, MSK modulation is followed by the Gaussian filtering.

 

Why Use MSK?

Binary data with sharp transitions (like from "1" to "0") can create wide sidebands around the carrier frequency, which causes interference with adjacent channels. MSK, with its constant envelope and smooth phase transitions, minimizes these sidebands and fits better within limited bandwidth, making it ideal for wireless communications.

 

Similarities between FSK and MSK

Frequency Shift Keying (FSK)	Minimum Shift Keying (MSK)
1. Bits "1" and "0" are represented by two different carrier frequencies 2. Here, you can define bits "1" and  "0" by carrier frequencies f2 and f1 , respectively.	1. Here also, bits "1" and "0" are represented by two different carrier frequencies, but the frequency  difference between them will be half of the baud rate 2. Here, you can define bits "1" and  "0" by fc + (baud rate)/4 and fc - (baud rate)/4 , respectively. Where fc is the carrier frequency.

Differences Between MSK and FSK

Phase Continuity in FSK vs MSK

1. Frequency‑Shift Keying (FSK)

In FSK, when the data changes from 0 to 1 (and f₀ ≠ f₁), the carrier hops between the two symbol frequencies. Unless we deliberately use continuous‑phase FSK (CPFSK), the phase is not preserved across bit boundaries, so abrupt phase jumps occur at every transition.

2. Minimum‑Shift Keying (MSK)

MSK guarantees a continuous phase trajectory. The phase evolves linearly within each bit interval according to its value; it never jumps. The required frequency deviation is Δf = ±1 ⁄ 4T_b, where T_b is the bit duration. MSK “remembers” the carrier phase—the phase at the end of one symbol becomes the starting phase of the next.

3. Worked Example (Bitstring 0110, 100 bps, f_c = 1 kHz)

Parameters

• Bit rate: 100 bps → T_b = 0.01 s
• Carrier: f_c = 1 kHz

3.1 MSK Symbol Frequencies & Phase Ranges

MSK frequencies are offset by ±1 ⁄ 4Tb = ±25 Hz:

• Bit 0 → f = 975 Hz
• Bit 1 → f = 1025 Hz

Bit #	Value	Phase Equation (within its interval)	Phase Range
1	0	φ(t) = −πt ⁄ (2Tb)	0 → −π ⁄ 2
2	1	φ(t) = −π ⁄ 2 + π(t−0.01) ⁄ (2Tb)	−π ⁄ 2 → 0
3	1	φ(t) = π(t−0.02) ⁄ (2Tb)	0 → π ⁄ 2
4	0	φ(t) = π ⁄ 2 − π(t−0.03) ⁄ (2Tb)	π ⁄ 2 → 0

3.2 Binary FSK Symbol Frequencies & Phase Ranges

To keep the two tones orthogonal over one bit, choose Δf = 1 ⁄ (2T_b) = 50 Hz:

• Bit 0 → f₀ = 950 Hz
• Bit 1 → f₁ = 1050 Hz

(Standard FSK resets the phase at the start of each symbol.)

Bit #	Value	Phase Equation	Phase Range
1	0	φ(t) = 2π·950·t	0 → 19π ≈ 59.7 rad
2	1	φ(t) = 2π·1050·(t−0.01)	0 → 21π ≈ 66.0 rad
3	1	φ(t) = 2π·1050·(t−0.02)	0 → 21π ≈ 66.0 rad
4	0	φ(t) = 2π·950·(t−0.03)	0 → 19π ≈ 59.7 rad

4. Key Take‑Aways

• MSK = narrow phase swing (±π/2), fully continuous.
• FSK = large phase swing (≈60–66 rad), discontinuous unless CPFSK is used.
• Continuous phase → better spectral efficiency and smoother spectra.

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Read more about

1. Minimum Shift Keying (MSK) Simulator
2. MATLAB Code for MSK
3. Gaussian minimum shift keying (GMSK)
4. MATLAB Code for GMSK