CPM Demodulator Baseband

Demodulate signal using CPM method and Viterbi algorithm

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Libraries:
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. The input to this block is a baseband representation of the modulated signal. For more information about this demodulation and the filtering applied, see Algorithms.

Examples

CPM Modulation and Demodulation

Compute the bit error rate for a continuous phase modulated signal passed through an AWGN channel.

The cm_cpm_demod model passes a CPM-modulated signal through AWGN, demodulates the signal, and then computes the error rate of the received bits.

Run the model to compute the bit rate error.

BER = 0.002033
Number of errors = 2
Number of samples = 984

Ports

Input

In — CPM-modulated baseband signal

scalar | column vector

CPM-modulated baseband 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 information on the processing rates, see Single-Rate Processing and Multirate Processing.

This port is unnamed on the block.

Data Types: double | single

Output

Out — Demodulated output signal

scalar | column vector

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

This port is unnamed on the block.

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

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.

Output type — Integer or group of bits output indicator

Integer (default) | Bit

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

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

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

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

Symbol set ordering — Symbol mapping

Binary (default) | Gray

Symbol mapping of bit outputs, specified as Binary or Gray. This parameter determines how each integer maps to a group of output bits.

• 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 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. The modulator operates in multi-h. 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 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, 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 scalar

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 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. The traceback depth specifies 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.

• When you set the Output type parameter to false, you can set the output to double-precision or signed-integer data types.

• When you set the Output type parameter to true, you can set the output to double-precision, signed-integer, or logical data types.

Block Characteristics

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

More About

Integer-Valued and Binary-Valued Output Signals

When you set the Output type parameter to Integer:

• The block outputs an integer column vector of length equal to N/N_SPS. The output values are odd integers in the range [–(M–1), (M–1)].

• You cannot set the Output datatype parameter to boolean.

When you set the Output type parameter to Bit:

• The block outputs a binary column vector of length equal to k×(N/N_SPS).

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

  1. Map each demodulated symbol to an odd integer L in the range [–(M–1), (M–1)].

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

  3. Map each nonnegative integer to a k-length binary word using binary-coded ordering or Gray-coded ordering, as specified by the Symbol set ordering parameter.

• You can set the Output datatype parameter to only double, boolean, int8, in16, or int32.

N is the number of input baseband modulated symbols in the input signal (specifically, the length of the input signal). N_SPS represents the value of the Samples per symbol parameter. M represents the value of the M-ary number parameter. k = log2(M) and k is the number of bits per symbol.

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, D, is the number of trellis branches used to construct each traceback path in the trellis description of the modulation scheme for demodulation using the Viterbi algorithm.

Determine the optimal setting for traceback depth by calculating the minimum squared Euclidean distance. Alternatively, you can use a common heuristic based on the number of states. Specifically, the five-times-the-constraint-length rule, which approximates the traceback depth as D=5log₂(numStates).

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

The Viterbi algorithm processing results in a delay preceding the first meaningful demodulated value in the output.

The traceback depth equals the value of the Traceback depth parameter. The length of the delay vector is:

• (D+1) zero-value symbols, 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.

• D zero-value symbols, when the Rate options parameter is set to Enforce single-rate processing.

Algorithms

CPM Demodulation

CPM demodulation processing consists of a correlator followed by a maximum-likelihood sequence estimation (MLSE) detector 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 MLSE detection. The input to the demodulator is a baseband representation of the modulated signal.

{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

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
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Version History

Introduced before R2006a