Continuous-Phase Modulation

Continuous-phase modulation (CPM) is a linear baseband modulation technique in which the message modulates the frequency of a continuous-phase signal. The signal has memory because the phase of the carrier is constrained to be continuous. Communications Toolbox™ software includes these modulation and demodulation functions, System objects, and blocks to model continuous-phase frequency shift keying (CPFSK), continuous-phase modulation (CPM), Gaussian minimum shift keying (GMSK), and minimum shift keying (MSK).

Functions	System objects	Blocks
mskmod, mskdemod	comm.CPMModulator, comm.CPMDemodulator comm.CPFSKModulator, comm.CPFSKDemodulator comm.GMSKModulator, comm.GMSKDemodulator comm.MSKModulator, comm.MSKDemodulator comm.SOQPSKModulator, comm.SOQPSKDemodulator	CPM Modulator Baseband, CPM Demodulator Baseband CPFSK Modulator Baseband, CPFSK Demodulator Baseband GMSK Modulator Baseband, GMSK Demodulator Baseband MSK Modulator Baseband, MSK Demodulator Baseband

CPM

In CPM, the baseband representation of the modulated signal is

Continuous phase modulation includes a convolutional encoder, a symbol mapper, and a modulator.

The output of the modulator is a baseband representation of the modulated signal:

$$\begin{array}{l}s(t)=\exp \left[j2\pi {\displaystyle \sum _{i=0}^n{\alpha }_i{h}_{i}q(t-iT)}\right]\text{ and}\\ nT<t<(n+1)T,\end{array}$$

where:

• {α_i} is a sequence of M-ary data symbols selected from the alphabet ±1, ±3, ±(M–1).

• M must have the form 2^k for some positive integer k, where M is the modulation order and specifies the size of the symbol alphabet.

• {h_i} is a sequence of modulation indices. h_i moves cyclically through a set of indices {h₀, h₁, h₂, ..., h_H-1}.

  • When H=1, only one modulation index exists, h₀, which is denoted as h. The phase shift over a symbol is π × h.

  • When h_i varies from interval to interval, the modulator operates in multi-h. To ensure a finite number of phase states, h_i must be a rational number.

CPM Pulse Shape Filtering

The CPM method uses pulse shaping to smooth the phase transitions of the modulated signal. The function q(t) is the phase response obtained from the frequency pulse, g(t), through this relation: $$q(t)={\displaystyle {\int }_{-\infty }^{t}g(t)dt}$$.

The specified frequency pulse shape corresponds to these pulse shape expressions for g(t).

Pulse Shape	Expression
Rectangular	$$g(t)=\{\begin{array}{cc}\frac{1}{2LT},& 0\le t\le LT\\ 0& \text{otherwise}\end{array}$$
Raised cosine	$$g(t)=\{\begin{array}{cc}\frac{1}{2LT}\left[1-\cos \left(\frac{2\pi t}{LT}\right)\right],& 0\le t\le LT\\ 0& \text{otherwise}\end{array}$$
Spectral raised cosine	$$g(t)=\frac{1}{L_{\text{main}}T}\frac{\sin \left(\frac{2\pi t}{L_{\text{main}}T}\right)}{\frac{2\pi t}{L_{\text{main}}T}}\frac{\cos \left(\beta \frac{2\pi t}{L_{\text{main}}T}\right)}{1-{\left(\frac{4\beta }{L_{\text{main}}T}t\right)}^2},0\le \beta \le 1$$
Gaussian	$$\begin{array}{c}g(t)=\frac{1}{2T}\left\{Q\left[2\pi B_b\frac{t-\frac{T}{2}}{\sqrt{\ln 2}}\right]-Q\left[2\pi B_b\frac{t+\frac{T}{2}}{\sqrt{\ln 2}}\right]\right\},\text{where}\\ Q(t)={\displaystyle {\int }_t^{\infty }\frac{1}{\sqrt{2\pi }}{e}^{-{\tau }^2/2}d\tau }\end{array}$$
Tamed FM (tamed frequency modulation)	$$\begin{array}{l}g(t)=\frac{1}{8}\left[g_0(t-T)+2g_0(t)+g_0(t+T)\right],\text{where}\\ {\text{g}}_0(t)\approx \frac{1}{T}\left[\frac{\sin (\frac{\pi t}{T})}{\frac{\pi t}{T}}-\frac{{\pi }^2}{24}\frac{2\sin \left(\frac{\pi t}{T}\right)-\frac{2\pi t}{T}\cos \left(\frac{\pi t}{T}\right)-{\left(\frac{\pi t}{T}\right)}^2\sin \left(\frac{\pi t}{T}\right)}{{\left(\frac{\pi t}{T}\right)}^3}\right]\end{array}$$

• L_main is the main lobe pulse duration in symbol intervals.

• β is the roll-off factor of the spectral raised cosine.

• B_b is the product of the bandwidth and the Gaussian pulse.

• The duration of the pulse, LT, is the pulse length in symbol intervals. As defined by the expressions, the spectral raised cosine, Gaussian, and tamed FM pulse shapes have infinite length. For all practical purposes, LT specifies the truncated finite length.

• T is the symbol durations.

• Q(t) is the complementary cumulative distribution function.

CPFSK

CPFSK modulation is a specific form of CPM in which the pulse shaping filter, g(t), is a rectangular pulse of duration, LT=1.

GMSK

GMSK is a continuous phase (CPM) scheme with no phase discontinuities because the frequency changes occur at the carrier zero-crossing points. For GMSK (and MSK), the frequency difference between the logical one and logical zero states is always equal to half the data rate. This difference can be expressed in terms of the modulation index. Specifically, an input symbol of 1 causes a phase shift of π/2 radians, which corresponds to a modulation index of 0.5.

GMSK applies a Gaussian pulse shaping filter, as described by the Gaussian filter equation in CPM Pulse Shape Filtering.

MSK

MSK is a specific form of CPM (and CPFSK) in which the modulation index h = 1/2 and the pulse shaping filter, g(t), is a rectangular pulse of duration, LT=1.

SOQPSK

The SOQPSK method is a family of constant-envelope CPM waveforms. This block diagram from Section 2.3.3.2 of IRIG Standard 106-17 shows the input signal flow through a precoder, a pulse shaping filter, and a modulator:

Precoding performed on binary input data generates ternary symbols {–1,0,1}. The Q branch is offset from the I branch by T_b, where T_b is the bit duration in seconds. For each new bit the precoder processes, the modulator outputs a new ternary symbol.

The ternary symbol mapping follows these rules:

$$\begin{array}{l}{\text{a}}_k=\{\begin{array}{c}-1,k^{th}bit=0\\ +1,k^{th}bit=1\end{array}\\ {\alpha }_k={(}^{-}\frac{a_{k-1}(a_k-a_{k-2})}{2}\end{array}$$

The SOQPSK precoder uses this mapping for input data:

SOQPSK Precoding Table for IRIG-106 Compatibility
Map αK from IK					Map αK+1 from QK+1
Ik	Qk–1	Ik–2	ΔΦ	αk	Qk+1	Ik	Qk–1	ΔΦ	αk+1
–1	+1 or –1	–1	0	0	–1	+1 or –1	–1	0	0
+1	+1 or –1	+1	0	0	+1	+1 or –1	+1	0	0
–1	–1	+1	–π/2	–1	–1	–1	+1	+π/2	+1
–1	+1	+1	+π/2	+1	–1	+1	+1	–π/2	–1
+1	–1	–1	+π/2	+1	+1	–1	–1	–π/2	–1
+1	+1	–1	–π/2	–1	+1	+1	–1	+π/2	+1
Included from Table 2-5 in IRIG Standard 106-17

Since I and Q do not vary at the same instant, the ternary symbols generated are constrained so that in a given bit interval, a precoded symbol can be mapped from the set {0,–1} or {0,+1}. Specifically, +1 cannot be followed by –1 and vice versa.

For SOQPSK, the modulation index is ½. The output of the modulator is a baseband representation of the modulated signal:

$$\begin{array}{l}s(t)=\exp \left[j\pi {\displaystyle \sum _{i=0}^n{\alpha }_{i}q(t-iT)}\right],\text{ and}\\ nT<t<(n+1)T.\end{array}$$

References

[1] Proakis, John G. Digital Communications. 5th ed. New York: McGraw Hill, 2007.

[2] Pasupathy, S. “Minimum Shift Keying: A Spectrally Efficient Modulation.”IEEE^® Communications Magazine (July, 1979): 14–22.

[3] Anderson, John B., Tor Aulin, and Carl-Erik Sundberg. Digital Phase Modulation. New York: Plenum Press, 1986.

[4] Inter-Range Instrumentation Group (IRIG) Telemetry Standards, IRIG Standard 106-17, Chapter 2, July 2017.