CPM Modulator Baseband
Modulate signal using CPM method
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 CPM Modulation and Pulse Shape Filtering.
Examples
View CPM Phase Tree Using Simulink
The doc_cpm_phase_tree
model uses the Eye Diagram block to view the inphase 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 inphase and quadrature signals. Doubleclick 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. Doubleclick 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 zeroorder 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
orGaussian
.
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.
When the Input type parameter is set to
Integer
, the block accepts odd integers
in the range [ –(M–1), (M–1)].
M is the modulation order which is specified by
the Mary number parameter.
When the Input type parameter is set to Bit
,
the block accepts binaryvalued inputs that represent integers. The block collects
binaryvalued signals into groups of k =
log_{2}(M) bits. k
is the number of bits per symbol and M is the modulation order.
The input vector length must be an integer multiple of k. The
block maps each group of k bits onto a symbol, as specified by
the Symbol set ordering parameter. For each group of
k bits, the block outputs one modulated symbol,
oversampled by the Samples per symbol parameter value.
Supported Data Types
Doubleprecision floating point
Boolean is permitted when Input type is set to
Bit
8, 16, and 32bit signed integers are permitted when Input type is set to
Integer
Data Types: double
 Boolean
 int8
 int16
 int32
Output
Out — Output signal
scalar  column vector
Output signal, returned as a scalar or column vector.
When the Input type parameter is set to
Integer
, the block outputs one modulated symbol for each input symbol.When the Input type parameter is set to
Bit
, the block outputs one modulated symbol for each group of k bits.
In both cases, the modulated symbols are oversampled by the Samples per symbol parameter value.
Data Types: double
 single
For more information on the processing rates, see SingleRate Processing, and Multirate Processing.
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.
Mary number — Modulation order
4
(default)  power of two scalar
Modulation order, specified as a poweroftwo scalar. The modulation order, M = 2^{k} specifies the number of points in the signal constellation, where k is a positive integer indicating the number of bits per symbol.
Input type — Integer or group of bits input indicator
Integer
(default)  Bit
Indicates whether the input consists of integers or groups of bits,
specified as Integer
or
Bit
.
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 binarycoded ordering.Set this parameter to
Gray
to map symbols using Graycoded ordering.
For more information, see Symbol Sets.
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.
{h} represents a sequence of modulation indices. For more information, see CPM Modulation.
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 on the
filtering options, 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 by the modulator to pulseshape 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 from 0 to 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
Data symbols used before the start of simulation, specified as scalar or vector with odd integer elements in the range [– (Mary number – 1), (Mary number – 1)]. The Symbol prehistory parameter defines the data symbols used by the modulator prior to 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 of the modulated waveform, specified as a scalar.
Samples per symbol — Symbol sampling rate
8
(default)  positive scalar
Symbol sampling rate, specified as a positive scalar. This parameter represents the number of samples output for each integer or binary word input. For all nonbinary schemes, as defined by the pulse shapes, this value must be greater than 1.
For more information, see Signal Upsampling and Rate Changes.
Rate options — Block processing rate
Enforce singlerate processing
(default)  Allow multirate processing
Block processing rate, specified as one of these options:
Enforce singlerate 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 — Output data type
double
(default)  single
Output data type, specified as double
or
single
.
Block Characteristics
Data Types 

Multidimensional Signals 

VariableSize Signals 

More About
CPM Modulation
The output of the modulator is a baseband representation of the modulated signal:
$$\begin{array}{l}s(t)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{exp}\left[j\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}^{n}{\alpha}_{i}{h}_{i}q(tiT)}\right],\text{and}\\ nT\text{\hspace{0.17em}}t(n+1)T.\end{array}$$
where:
{α_{i}} is a sequence of Mary 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 {h_{0}, h_{1}, h_{2}, ..., h_{H1}}. When H=1, only one modulation index exists, h_{0}, which is denoted as h.
h_{i} specifies the modulation index. When h_{i} varies from interval to interval, the block operates in multih. 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: $$q(t)={\displaystyle {\int}_{\text{\hspace{0.17em}}\infty}^{t}g(t)dt}$$.
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  $$g(t)=\{\begin{array}{cc}\frac{1}{2LT}\left[1\mathrm{cos}\left(\frac{2\pi t}{LT}\right)\right],& 0\le t\le LT\\ 0& \text{otherwise}\end{array}$$ 
Spectral raised cosine  $$g(t)=\frac{1}{{L}_{\text{main}}T}\frac{\mathrm{sin}\left(\frac{2\pi t}{{L}_{\text{main}}T}\right)}{\frac{2\pi t}{{L}_{\text{main}}T}}\frac{\mathrm{cos}\left(\beta \frac{2\pi t}{{L}_{\text{main}}T}\right)}{1{\left(\frac{4\beta}{{L}_{\text{main}}T}t\right)}^{2}},\text{\hspace{1em}}0\le \beta \le 1$$ 
Gaussian  $$\begin{array}{c}g(t)=\frac{1}{2T}\left\{Q\left[2\pi {B}_{b}\frac{t{\scriptscriptstyle \frac{T}{2}}}{\sqrt{\mathrm{ln}2}}\right]Q\left[2\pi {B}_{b}\frac{t+{\scriptscriptstyle \frac{T}{2}}}{\sqrt{\mathrm{ln}2}}\right]\right\},\text{\hspace{0.17em}}\text{where}\\ Q(t)={\displaystyle {\int}_{t}^{\infty}\frac{1}{\sqrt{2\pi}}{e}^{{\tau}^{2}/2}d\tau}\end{array}$$ 
Tamed FM (tamed frequency modulation)  $$\begin{array}{l}g(t)={\scriptscriptstyle \frac{1}{8}}\left[{g}_{0}(tT)+2{g}_{0}(t)+{g}_{0}(t+T)\right],\text{\hspace{0.17em}}\text{where}\\ {\text{g}}_{0}(t)\approx \frac{1}{T}\left[\frac{\mathrm{sin}({\scriptscriptstyle \frac{\pi t}{T}})}{{\scriptscriptstyle \frac{\pi t}{T}}}\frac{{\pi}^{2}}{24}\frac{2\mathrm{sin}\left({\scriptscriptstyle \frac{\pi t}{T}}\right){\scriptscriptstyle \frac{2\pi t}{T}}\mathrm{cos}\left({\scriptscriptstyle \frac{\pi t}{T}}\right){\left({\scriptscriptstyle \frac{\pi t}{T}}\right)}^{2}\mathrm{sin}\left({\scriptscriptstyle \frac{\pi t}{T}}\right)}{{\left({\scriptscriptstyle \frac{\pi t}{T}}\right)}^{3}}\right]\end{array}$$ 
L_{main} is the main lobe pulse duration in symbol intervals.
β is the rolloff 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].
Symbol Sets
In binary input mode, the block processing follows this procedure.
Divide the input bits into klength 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 theMary number
parameter. The binary word mapping options are binarycoded ordering or Graycoded ordering, as specified by theSymbol set ordering
parameter.Map each integer L to signed integers, as 2L–(M–1).
Proceed with modulation processing as in the integer input mode.
SingleRate Processing
In singlerate 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.
References
[1] Anderson, John B., Tor Aulin, and CarlErik Sundberg. Digital Phase Modulation. New York: Plenum Press, 1986.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced before R2006a
See Also
Blocks
 CPM Demodulator Baseband  CPFSK Modulator Baseband  GMSK Modulator Baseband  MSK Modulator Baseband
Objects
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