CPM Modulator Baseband

Modulate signal using CPM method

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Libraries:
Communications Toolbox / Modulation / Digital Baseband Modulation / CPM

Description

The CPM Modulator Baseband block modulates an input signal using the continuous phase modulation (CPM) method. The output of the modulator is a baseband representation of the modulated signal. For more information about the modulation and the filtering applied, see Algorithms.

Examples

View CPM Phase Tree Using Simulink

The doc_cpm_phase_tree model uses the Eye Diagram block to view the in-phase and quadrature components, phase trajectory, phase tree, and instantaneous frequency of a CPM modulated signal.

Explore Model

A random integer signal is converted to bits and then CPM modulated. The CPM modulated signal values are converted from complex to magnitude, and angle, and then the phase is unwrapped.

Plot Eye Diagrams

Eye Diagram blocks are named to reflect the signal each displays. When you run the example, these Eye Diagram blocks show how the CPM signal changes over time:

• Modulated Signal block — Displays the in-phase and quadrature signals. Double-click the block to open the scope. The modulated signal is easy to see in the eye diagram only when the Modulation index parameter in the CPM Modulator Baseband block is set to 1/2. For a modulation index value of 2/3, the modulation is more complex and the features of the modulated signal are difficult to decipher. Unwrapping the phase and plotting it is another way to illustrate these more complex CPM modulated signals.

• Phase Trajectory block — Displays the CPM phase. Double-click the block to open the scope. The Phase Trajectory block reveals that the signal phase is also difficult to view because it drifts with the data input to the modulator.

• Phase Tree block — Displays the phase tree of the signal. The CPM phase is processed by a few simple blocks to make the CPM pulse shaping easier to view. This processing holds the phase at the beginning of the symbol interval and subtracts it from the signal. This zero-order hold resets the phase to zero every three symbols. The resulting plot shows the many phase trajectories that can be taken by the signal from any given symbol epoch.

• Instantaneous Frequency block — Displays the instantaneous frequency of the signal. The CPM phase is differentiated to produce the frequency deviation of the signal. Viewing the CPM frequency signal enables you to observe the frequency deviation qualitatively, as well as make quantitative observations, such as measuring peak frequency deviation.

Running the doc_cpm_phase_tree model opens and plots the phase tree and instantaneous frequency eye diagram plots.

Further Exploration

To learn more about the example, try changing the following parameters in the CPM Modulator Baseband block:

• Change Pulse length to a value between 1 and 6.

• Change Frequency pulse shape to one of the other settings, such as Rectangular or Gaussian.

You can observe the effect of changing these parameters on the phase tree and instantaneous frequency of the modulated signal.

Ports

Input

In — Input signal

scalar | column vector

Input signal, specified as a scalar or column vector. For more information, see Integer-Valued and Binary-Valued Input Signals.

This port is unnamed on the block.

Data Types: double | Boolean | int8 | int16 | int32

Output

Out — CPM-modulated baseband signal

scalar | column vector

CPM-modulated baseband signal, returned as a scalar or column vector. The modulated output symbols are oversampled by the Samples per symbol parameter value. For information on the processing rates, see Single-Rate Processing and Multirate Processing.

Use the Output data type parameter to specify the output data type.

This port is unnamed on the block.

Data Types: double | single

Parameters

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

M-ary number — Modulation order

4 (default) | power-of-two scalar

Modulation order, specified as a power-of-two scalar. The modulation order M = 2^k specifies the number of points in the symbol alphabet. k is a positive integer indicating the number of bits per symbol.

Input type — Integer or group of bits input indicator

Integer (default) | Bit

Integer or group of bits input indicator, specified as Integer or Bit.

• Set this parameter to Integer to input data as integers.

• Set this parameter to Bit to input data as bits.

For more information, see Integer-Valued and Binary-Valued Input Signals.

Symbol set ordering — Symbol mapping

Binary (default) | Gray

Symbol mapping of bit inputs, specified as Binary or Gray.

• Set this parameter to Binary to map symbols using binary-coded ordering.

• Set this parameter to Gray to map symbols using Gray-coded ordering.

For more information, see Integer-Valued and Binary-Valued Input Signals.

Dependencies

To enable this parameter, set Input type to Bit.

Modulation index — Modulation index {h_i}

0.5 (default) | nonnegative scalar | column vector

Modulation index {h_i}, specified as a nonnegative scalar or column vector. The modulator operates in multi-h. For more information, see CPM Method.

Frequency pulse shape — Type of pulse shaping

Rectangular (default) | Raised Cosine | Spectral Raised Cosine | Gaussian | Tamed FM

Type of pulse shaping used to smooth the phase transitions of the modulated signal, specified as Rectangular, Raised Cosine, Spectral Raised Cosine, Gaussian, or Tamed FM. For more information, see Pulse Shape Filtering.

Main lobe pulse duration (symbol intervals) — Main lobe duration

1 (default) | positive integer

Main lobe duration of the largest lobe in the spectral raised cosine pulse, specified as a positive integer representing the number of symbol intervals used to pulse-shape the modulated signal.

Dependencies

To enable this parameter, set Frequency pulse shape to Spectral Raised Cosine.

Rolloff — Rolloff factor of spectral raised cosine pulse shape

0.2 (default) | nonnegative scalar

Rolloff factor of the spectral raised cosine pulse, specified as a scalar in the range [0, 1].

Dependencies

To enable this parameter, set Frequency pulse shape to Spectral Raised Cosine.

BT product — Product of bandwidth and symbol time of Gaussian pulse shape

0.3 (default) | positive scalar

Product of the bandwidth and symbol time of the Gaussian pulse shape, specified as a positive scalar. Use BT product to reduce the bandwidth, at the expense of increased intersymbol interference.

Dependencies

To enable this parameter, set Frequency pulse shape to Gaussian.

Pulse length (symbol intervals) — Length of frequency pulse shape

1 (default) | positive integer

Length of the frequency pulse shape in symbol intervals, specified as a positive integer. For more information on the frequency pulse length, refer to LT in Pulse Shape Filtering.

Symbol prehistory — Data symbols used before the start of simulation

1 (default) | scalar | vector

Symbol prehistory, specified as a scalar or vector with odd integer elements. Values must be in the range [–(M – 1), (M – 1)]. M is the modulation order specified byM-ary number. The Symbol prehistory parameter defines the data symbols used by the modulator before the first call of the block, in reverse chronological order.

• A scalar value expands to a vector of length L_P – 1. L_P represents the pulse length, which is specified by the Pulse length (symbol intervals) parameter.

• For a vector, the length must be L_P – 1.

Phase offset (rad) — Initial phase offset

0 (default) | scalar

Initial phase offset in radians, specified as a scalar. This property value is the initial phase offset of the modulated waveform.

Samples per symbol — Symbol sampling rate

8 (default) | positive integer

Symbol sampling rate, specified as a positive integer. This parameter specifies the output symbol upsampling factor for each input sample.

For more information, see Signal Upsampling and Rate Changes.

Tip

To accurately model nonbinary pulse shapes, specifically pulse shapes other than rectangular, you should set the symbol sampling rate to values greater than 4.

Rate options — Block processing rate

Enforce single-rate processing (default) | Allow multirate processing

Block processing rate, specified as one of these options:

• Enforce single-rate processing — The input and output signals have the same sample time. The block implements the rate change by making a size change at the output when compared to the input. The output width equals the product of the number of symbols and the Samples per symbol parameter value.

• Allow multirate processing — The input and output signals have different sample times. The output sample time equals the symbol period divided by the Samples per symbol parameter value.

Output data type — Data type of output

double (default) | single

Data type of the output, specified as double or single.

Block Characteristics

Data Types	Boolean | double | integer | single
Multidimensional Signals	no
Variable-Size Signals	no

More About

Integer-Valued and Binary-Valued Input Signals

When you set the Input type parameter to Integer:

• The block inputs odd integers in the range [ –(M–1), (M–1)]. M is the modulation order specified by the M-ary number parameter.

• The input signal must have a double-precision or signed-integer data type.

When you set the Input type parameter to Bit:

• The block inputs groupings of k bits. Each grouping is called a binary word. The input vector length must be an integer multiple of k.

• In bit input mode, the block follows this process:

  1. Divide the input bits into k-length bit words and map each to an integer L in the range [0, M – 1], where k = log2(M) and M is the modulation order specified by the M-ary number parameter. The options for mapping binary words are binary-coded ordering and Gray-coded ordering, as specified by the Symbol set ordering parameter.

  2. Map each integer L to signed integers, as 2L–(M–1).

  3. Proceed with modulation processing as in the integer input mode.

• The input signal must have a double-precision or Boolean data type.

Single-Rate Processing

In single-rate processing mode, the input and output signals have the same port sample time. In this mode, the input to the block can be multiple symbols. The block implicitly implements the rate change by making a size change at the output when compared to the input.

• When you set Input type to Integer, the input can be a scalar or a column vector with the length equal to the number of input symbols.

• When you set Input type to Bit, the input width must be an integer multiple of the number of bits per symbol.

The output width equals N_Sym × N_SPS, where N_Sym is the number of symbols in the frame and N_SPS is the number of samples per symbol.

Multirate Processing

In multirate processing mode, the input and output signals have different port sample times. In this mode, the input to the block must be one symbol.

• When you set Input type to Integer, the input must be a scalar.

• When you set Input type to Bit, the input width must equal the number of bits per symbol.

The output sample time equals T_Sym / N_SPS, where T_Sym is the symbol period and N_SPS is the number of samples per symbol.

Algorithms

CPM Method

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.

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.

For more information on pulse shape filtering, see [1].

References

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

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced before R2006a