CPM Demodulator Baseband

Demodulate signal using CPM method and Viterbi algorithm

• Library:

• Communications Toolbox / Modulation / Digital Baseband Modulation / CPM

• 

Description

The CPM Demodulator Baseband block demodulates an input signal that was modulated using the continuous phase modulation (CPM) method.

CPM is a modulation method with memory. The block processing includes a correlator followed by a maximum-likelihood sequence detector (MLSD) that searches the paths through the state trellis for the minimum Euclidean distance path. The block uses the Viterbi algorithm to perform MLSD.

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

Ports

Input

In — Input signal

scalar | column vector

Input signal, specified as a scalar or column vector. The length of the input signal must be an integer multiple of the number of samples per symbol specified in the Samples per symbol parameter. For more information, see Integer-Valued and Binary-Valued Output Signals.

Data Types: double | single

Output

Out — Output signal

scalar | column vector

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

Supported Data Types

• Double-precision floating point

• Boolean (when Output type is set to Bit)

• 8-, 16-, and 32-bit signed integers (when Output type is set to Integer)

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

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.

Output type — Determines whether output consists of integers or groups of bits

Integer (default) | Bit

Determines whether the output 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.

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

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

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

Dependencies

To enable this parameter, set Output 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 Demodulation.

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 demodulator 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) | scalar in the range [0, 1]

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

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

Frequency pulse shape length, specified as a positive scalar. 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 [– (M – 1), (M – 1)]. M represents the modulation order, which is specified by the M-ary number parameter. 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 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 port sample time. The block implements the rate change by making a size change at the output when compared to the input. The output width is the number of symbols (which is given by dividing the input length by the Samples per symbol parameter value when the Output type parameter is set to Integer).

• Allow multirate processing — The input and output signals have different port sample times. The output period is the same as the symbol period and equals the product of the input period and the Samples per symbol parameter value.

Traceback depth — Traceback depth for Viterbi algorithm

16 (default) | positive integer

Traceback depth for the Viterbi algorithm, specified as a positive integer representing the number of trellis branches that the Viterbi algorithm uses to construct each traceback path. The value of this parameter is also the output delay and the number of zero symbols that precede the first meaningful demodulated symbol in the output. For more information, see Traceback Depth and Output Delays.

Output data type — Output data type

double (default) | boolean | int8 | int16 | int32

Output data type, specified as double, boolean, int8, int16, or int32. For more information, see Supported Data Types in Out.

Block Characteristics

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

More About

CPM Demodulation

The CPM demodulation method process consists of a correlator followed by a maximum-likelihood sequence detector (MLSD) that searches the paths through the state trellis for the minimum Euclidean distance path. When the modulation index is rational (h = m / p), a finite number of phase states exist in the symbol. The implementation uses the Viterbi algorithm to perform MLSD.

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

• h_i = m_i / p_i is the modulation index in proper rational form.

• m_i is the numerator of the modulation index.

• p_i is the denominator of the modulation index.

• m_i and p_i are relatively prime positive numbers.

• The least common multiple (LCM) of {p₀, p₁, p₂, …,p_H-1} is denoted as p.

• h_i= m'_i / p.

{h_i} determines the number of phase states,

$$numPhaseStates=\left\{\begin{array}{l}p,\text{for}\text{all}\text{even}m{\text{'}}_i\\ 2p,\text{for}\text{any}\text{odd}m{\text{'}}_i\end{array}\right\}\text{,}$$

and affects the number of trellis states,

numStates = numPhaseStates×M^(L-1),

• L is the pulse length.

• M is the modulation order.

CPM Method

The input to the demodulator 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.

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]

Integer-Valued and Binary-Valued Output Signals

When the Output type parameter is set to Integer:

• The block produces odd integers between –(M–1) and M–1. The modulation order, M, is specified by the M-ary number parameter.

• The Output datatype parameter cannot be set to boolean.

When the Output type parameter is set to Bit:

• The block produces groupings of k-length binary words. The binary word mapping options are natural binary-coded ordering or Gray-coded ordering.

• The Output datatype can only be double or boolean.

• In binary output mode, the block processing follows this procedure:

  1. Maps each input symbol to an intermediate value, as in the integer output mode.

  2. Maps the odd integer L to the nonnegative integer (L+M–1)/2.

  3. Maps each nonnegative integer to a binary word, using Binary or Gray mapping, as specified by the Symbol set ordering parameter.

Single-Rate Processing

In single-rate processing mode, the input and output signals have the same port sample time. The block implicitly implements the rate change by making a size change at the output when compared to the input. The input width must be an integer multiple of the Samples per symbol parameter value, and the input can be a column vector.

• When you set Output type to Bit, the output width is K times the number of input symbols.

• When you set Output type to Integer, the output width is the number of input symbols.

Multirate Processing

In multirate processing mode, the input and output signals have different port sample times. The input must be a scalar. The output symbol time is the product of the input sample time and the Samples per symbol parameter value.

• When you set Output type to Bit, the output width equals the number of bits per symbol.

• When you set Output type to Integer, the output is a scalar.

Traceback Depth and Output Delays

The traceback depth is the number of trellis branches used to construct each traceback path. Traceback depth influences the output delay, which is the number of zero symbols that precede the first meaningful demodulated value in the output.

The optimal traceback depth setting depends on minimum squared Euclidean distance calculations. Alternatively, you can choose a typical value, dependent on the number of states, using the five-times-the-constraint-length rule, which corresponds to 5log₂(numStates).

For a binary raised cosine pulse shape with a pulse length of 3 and h=2/3, applying this rule (5log₂(3×2²) = 18 gives a result that is close to the optimum value of 20.

• When the Rate options parameter is set to Allow multirate processing, and the model uses a variable-step solver or a fixed-step solver with the Tasking Mode parameter set to SingleTasking, then the delay vector consists of (Traceback depth+1) zero-value symbols.

• When the Rate options parameter is set to Enforce single-rate processing, the delay vector consists of Traceback depth zero-value symbols.

Pair Block

CPM Modulator Baseband — Modulates data using continuous phase modulation.

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
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced before R2006a