Minimum Shift Keying (MSK) is one of the most spectrally efficient modulation schemes available. Due to its constant envelope, it is resilient to nonlinear distortion and was therefore chosen as the modulation technique for the GSM cell phone standard.
MSK is a special case of ContinuousPhase Frequency Shift Keying (CPFSK) which is a special case of a general class of modulation schemes known as ContinuousPhase Modulation (CPM). It is worth noting that CPM (and hence CPFSK) is a nonlinear modulation and hence by extension MSK is a nonlinear modulation as well. Nevertheless, it can also be cast as a linear modulation scheme, namely Offset Quadrature Phase Shift Keying (OQPSK), which is a special case of Phase Shift Keying (PSK). As a borderline case, these relationships are illustrated in Figure below.
Figure 1: MSK as a special case of both nonlinear and linear modulation schemes
At this point, you would be thinking about the following question: How can a modulation be both nonlinear and linear? As we see later in this tutorial, originally MSK is a nonlinear modulation but a certain depiction of its actual digital symbols known as pseudosymbols turns it into an OQPSK representation.
Modulation is a simple topic to understand but owing to the above description, MSK can sometimes be an intimidating concept. Here, our purpose is to present it in an uncomplicated manner by building it through the fundamentals.
The starting point is one of the simplest digital modulations possible: FSK.
Binary Frequency Shift Keying (BFSK)
In Frequency Shift Keying (FSK), digital information is transmitted by changing the frequency of a carrier signal. Naturally, Binary FSK (BFSK) is the simplest form of FSK where the two bits 0 and 1 correspond to two distinct carrier frequencies and to be sent over the air. The bits can be translated into symbols through the relations
This enables us to write the frequencies with as
where is the nominal carrier frequency and is the peak frequency deviation from this carrier frequency. Consequently,
(1)
where
Figure 2 below displays a BFSK waveform for a random stream of data at a rate of . Note that we are not distinguishing between a bit period and a symbol period because both are the same for a binary modulation technique.
Figure 2: A Binary Frequency Shift Keying (BFSK) waveform
As is evident from Figure 2 above, the phase transitions at the boundaries of bit transitions are — in general — discontinuous.
Minimum Frequency Spacing
Although any two distinct frequencies and can be used for communication purpose, it greatly helps in receiver design if the two distinct signals are orthogonal to each other, i.e.,
A question that arises at this stage is the following: how close can the two frequencies and be? Or in other words, what is the smallest possible value of ? The reason for asking this question is spectral efficiency. The closer the two frequencies, the more the number of channels available for other users in the same spectrum.
For a BFSK case,
or
Since while , the first term goes to zero and we can write
which is true for (remember for integer )
From here, the orthogonality condition can be concluded as
This also yields the minimum frequency separation for .
for orthogonal signaling. Thus, the peak frequency deviation can be computed as
MSK as ContinuousPhase FSK
From the above information, a BFSK signal with minimum tone spacing can be constructed by replacing by in Eq (1) as
This is a CPBFSK signal with minimum tone spacing defined over a single bit interval . There are two more steps to construct an actual MSK waveform.
 In a real communication system, the signal is constructed by transmitting a sequence of bits in succession, where is the bit index. As we have seen earlier, bits are converted to symbols .
Then, in the interval , the above signal can be written as
(2)
where is the bit index within a long bit stream and the second term indicates the underlying baseband message.

Observe in the above equation that the phase continuity is not necessarily maintained from one symbol to the next. To ensure phase continuity, we must add a phase component for each symbol as
For this purpose, the phase at both sides of must be equal, as illustrated in Figure 3.
Figure 3: Phase on both sides of must be equal to maintain continuity
Thus, at the instant , the following equation must be satisfied.
which can be written as
Another way to write the above recursive relation is
Assume that . Then,
In general,
When the phase follows this rule during each bit/symbol interval, the phase continuity is ensured and the resulting waveform is shown for an example sequence in Figure 4.
Figure 4: MSK as ContinuousPhase Binary FSK
Notice how Figure 4 is different than Figure 2 in phase continuity.Below, we plot as a function of time in Figure 5 to see how it evolves. One can observe that it indeed changes values in steps of depending on the last data bit.
Figure 5: evolving with time
MSK as Offset QPSK
First, write Eq (2) assuming as
Using Eq (3), we can further manipulate as
Notice that when
 because the second term in Eq (6) becomes zero.
 is even because the second term in Eq (6) is a multiple of (remember that is when not zero).
We conclude that can only change when and is odd. In that case, it will always change by an odd multiple of . In summary, considering modulo operations,
Next, we use the following identities
to open Eq (5).
Now we process both terms one by one.
because implies , see Eq (7). Furthermore, and . We have also defined
Similarly, Term 2 can be expanded as
since as before and . Here, we have defined as
Finally, plugging both Term 1 and 2 into for the th bit period,
Using the identity , the above equation can be revised as
where is a halfsinusoidal pulse shape of period .
The above expression resembles an OQPSK waveform if the bit rate for and is or bit period is , since the time offset in the term must be half the bit period for OQPSK.
So we have to check if and change values every other symbol.
From the definition, and Eq (7) tells that can only change values for odd . Hence, is indeed an rate stream.
On the other hand, . Again, Eq (7) says that can only change when changes but that means that stays the same. Therefore, can only change for even and when changes values. Consequently, is also an rate stream.
Since changes values for odd while does the same for even , is offset with respect to by seconds, the same amount as in Eq (8). Summing up everything so far, MSK can indeed be represented as an OQPSK waveform. The data rate is the same as in CPBFSK format since two bits are being transmitted in two bit periods here as well.
This representation is illustrated in Figure 6.
Figure 6: MSK as Offset QPSK
Compare Figure 6 with Figure 4. I did not choose separate blue and red colors in this figure so as not to confuse s with s (that can raise a misunderstanding that is odd s and is even , which is not correct).
Observe from Figure 6 that is changing values every two s at odd multiples of , while is changing values every two s at even multiples of . Due to this offset behavior, at every , either or waveform is zero at while the other reaches its maximum value. This is how phase remains continuous during symbol transitions.
Figure 7 draws as a supplement to above findings.
Figure 7: evolving with time
On a final note, observe from some equations (e.g., (2) and (5)) that the continuous phase has two parts, one of which arises due to the delay of the th symbol. This information can be used to refine a phase estimate.
We are also clear now why MSK can act both like a linear and a nonlinear modulation. In reality, MSK is a nonlinear modulation scheme (see Eq(2)) for . Pseudosymbols themselves are nonlinear functions of information bits. So it is only from viewpoint that MSK can be seen as a linear modulation scheme.
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A Generalization: From MSK to ContinuousPhase FSK (CPFSK)
After understanding MSK, it can be expanded into a general modulation scheme known as ContinuousPhase Frequency Shift Keying (CPFSK). As we see in the next section, CPFSK is a special case of ContinuousPhase Modulation (CPM), which is a class of nonlinear digital modulation schemes in which the phase of the signal is constrained to be continuous from one symbol to the next. As with MSK, the most attractive feature of such a signal is that it has a constant envelope as a result of the amplitude being independent of the modulating information. Consequently, a CPM signal can be amplified without distortion by a nonlinear amplifier operating near the saturation point allowing low cost and more efficiency as compared to a linear amplifier.
Usually CPFSK (and hence CPM) is not a straightforward concept to master. However, starting from MSK, its basics can easily be understood by building on the same expressions. For this purpose,
 Remember that .
 Modulation symbols can carry more than 1 information bit. For example, 00, 01, 11 and 10 can be sent through four symbols . In general, is a sequence from the alphabet . Thus, we call as symbols and replace with (a symbol time) from here onwards. Symbol rate is then .
Now let us start with rewriting Eq (2) and Eq (4) and then substituting the former into the latter.
where we have defined the following terms.
It turns out that is a general concept known as
modulation index, which in case of MSK is equal to . In general, is defined as
which describes the peak frequency deviation in terms of a percentage of the symbol rate. With the definition of , we can write
The above equation is true for the interval . Considering from its definition that is after a symbol interval, we can also write as
where is defined as the derivative of as
Notice that in the present case with defined as above, its derivative is a rectangular pulse shape. Consequently, is the standard baseband Pulse Amplitude Modulated (PAM) waveform with rectangular pulse shape and this is how its discontinuities are transformed into a continuousphase signal.
It is refreshing to conclude that the starting point for a continuousphase modulated signal is a standard pulse amplitude waveform, in which the discontinuities are smoothed out by the integral operation.
More Generalization: From ContinuousPhase FSK (CPFSK) to ContinuousPhase Modulation (CPM)
Now referring to Eq (9), the expression for a CPM signal can be written as
Here, is a sequence of modulation indices: a ratio of two relatively prime integers.
This is the case of multih CPM. When all , the modulation index is the same for all symbols. This is the category we saw in CPFSK and MSK above.
Finally, is a waveform shape known as the phase response of the modulator which is normalized as
where is the length of a pulse in symbols. Since angular frequency is the rate of change of phase, this is the derivative of and known as the frequency response of the modulator.
For any pulse shape , results in full response CPM, while the other case is the partial response CPM. Commonly used pulse shapes are rectangular (as used in the case of MSK), raised cosine and Gaussian.
We can see that CPM is in fact a very large class of modulation schemes owing to different pulse shapes , modulation indices and the modulation alphabet size . That is both a blessing and a curse: blessing due to the remarkable variety of signals as its offsprings all yielding excellent spectral and power efficiencies, and curse due to the high receiver complexity. By virtue of Moore’s law, this is becoming less of an issue — thanks to powerful baseband processors in the modern age.
References
[1] Mengali and D’Andrea, Synchronization Techniques for Digital Receivers, 1997.
[2] John Proakis, Digital Communications, 4th ed, 2001.
Hello. I read your article, Ihave some questions.
1)There are a lot of sources, e.g. “http://www.dsplog.com/2009/06/16/msktransmitterreceiver/”, “https://en.wikipedia.org/wiki/Minimumshift_keying”,the article “Minimum shift keying: A spectrally efficient modulation Subbarayan Pasupathy, IEEE Communications Magazine, July 1979” where the coefficients Di,n and Dq,n are suggested to be the odd and even bits of the input bit stream An. You told that it is incorrect. Why?
Thanks in advance. Your site is very interesting.
Thanks for your appreciation. You can have even and odd input bit streams to generate a halfsinusoidal shaped OQPSK waveform. However, in that case, the 0s and 1s that generate two different carrier frequencies will not be directly coming from the input, but a function of the input bits. Hope that clarifies the question.