CPM Modulator Baseband

Modulate using continuous phase modulation

• Library:

• Communications Toolbox / Modulation / Digital Baseband Modulation / CPM

• 

Description

The CPM Modulator Baseband block modulates an input signal using continuous phase modulation (CPM). The output of the modulator is a baseband representation of the modulated signal:

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

where:

• {α_i} represents 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} represents a sequence of modulation indices and h_i moves cyclically through a set of indices {h₀, h₁, h₂, ..., h_H-1}. When H=1, there is only one modulation index, h₀, which is denoted as h.

For more information about this modulation and the filtering applied, see CPM Modulation and Pulse Shape Filtering.

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 between –(M–1) and M–1. M represents the M-ary number parameter.

When the Input type parameter is set to Bit, the block accepts binary-valued inputs that represent integers. The block collects binary-valued signals into groups of K = log₂(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

• Double-precision floating point

• Boolean (when Input type is set to Bit)

• 8-, 16-, and 32-bit signed integers (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 Single-Rate Processing, and Multirate Processing.

Parameters

M-ary number — Modulation order

4 (default) | positive integer

Modulation order indicating the alphabet size, specified as a positive integer that is a nonzero power of two. M must have the form 2^K for some positive integer K, where K is 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 — Bit mapping

Binary (default) | Gray

Bit mapping, specified as Binary or Gray. 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) — Number of symbol intervals of largest lobe of spectral raised cosine pulse

1 (default) | positive scalar

Number of symbol intervals of the largest lobe of the spectral raised cosine pulse, specified as a positive scalar.

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 time

0.3 (default) | nonnegative scalar

Product of bandwidth and time, specified as a nonnegative 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) — Frequency pulse length

1 (default) | positive scalar

Frequency pulse length, specified as a positive scalar. Refer to LT in Pulse Shape Filtering for more information on the frequency pulse length.

Symbol prehistory — Data symbols used before the start of simulation

1 (default) | scalar | vector

Data symbols used before the start of simulation, specified as a scalar or vector in reverse chronological order. If Symbol prehistory is a vector, then its length must be one less than the Pulse length (symbol intervals) parameter value.

Phase offset (rad) — Initial phase offset

0 (default) | scalar

Initial phase offset of output in radians, 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 Upsample Signals and Rate Changes.

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 — Output data type

double (default) | single

Output data type, specified as double or single.

Block Characteristics

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

More About

CPM Modulation

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

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

where:

• {α_i} represents 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} represents a sequence of modulation indices and h_i moves cyclically through a set of indices {h₀, h₁, h₂, ..., h_H-1}. When H=1, there is only one modulation index, h₀, which is denoted as h.

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

Pulse Shape Filtering

Continuous phase modulation 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	$$g(t)=\frac{1}{L_{main}T}\frac{\sin \left(\frac{2\pi t}{L_{main}T}\right)}{\frac{2\pi t}{L_{main}T}}\frac{\cos \left(\beta \frac{2\pi t}{L_{main}T}\right)}{1-{\left(\frac{4\beta }{L_{main}T}t\right)}^2},0\le \beta \le 1$$
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 rolloff factor of the spectral raised cosine.

• B_bis 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.

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

Symbol Sets

In binary input mode, the block processing follows this procedure:

1. Maps each binary word to k, an integer from 0 to M–1. The binary word mapping options are Binary or Gray, as specified by the Symbol set ordering parameter.

2. Maps k to the intermediate value 2k–(M–1)

3. Proceeds with block processing as in the integer input mode.

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.

Pair Block

CPM Demodulator Baseband — Demodulates continuous phase modulated data.

References

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

Extended Capabilities

C/C++ Code Generation
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