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In most media for communication, only a fixed range of frequencies is available for
transmission. One way to communicate a message signal whose frequency spectrum does not
fall within that fixed frequency range, or one that is otherwise unsuitable for the
channel, is to alter a transmittable signal according to the information in your message
signal. This alteration is called *modulation*, and it is the
modulated signal that you transmit. The receiver then recovers the original signal
through a process called *demodulation*.

The Communications Toolbox™ supports these modulation techniques for digital data. All the methods at the far right are implemented in library blocks.

Like analog modulation, digital modulation alters a transmittable signal according to the information in a message signal. However, for digital modulation the message signal is restricted to a finite set. Modulation functions output the complex envelope of the modulated signal. Using the Communications Toolbox, you can modulate or demodulate signals using various digital modulation techniques, and plot signal constellations.

Unless otherwise indicated, the modulation and demodulation functions do not perform pulse shaping or filtering. See Combining Pulse Shaping and Filtering with Modulation for more information about filtering.

The available methods of modulation depend on whether the input signal is analog or digital. For a list of the modulation techniques the Communications Toolbox supports see Digital Baseband Modulation, Analog Baseband Modulation, and Analog Passband Modulation.

**Accessing Digital Modulation Blocks. **Open the Modulation library by double-clicking the icon in the main block
library. Then open the Digital Baseband sublibrary by double-clicking its
icon in the Modulation library.

The Digital Baseband library has sublibraries of its own. Open each of these sublibraries by double-clicking the icon listed in the table below.

Kind of Modulation | Icon in Digital Baseband Library |
---|---|

Amplitude modulation | AM |

Phase modulation | PM |

Frequency modulation | FM |

Continuous phase modulation | CPM |

Trellis-coded modulation | TCM |

Some digital modulation sublibraries contain blocks that implement specific modulation techniques. These specific-case modulation blocks use the same computational code that their general counterparts use, but provide an interface that is either simpler or more suitable for the specific case. This table lists general modulators along with the conditions under which is general modulator is equivalent to a specific modulator. The situation is analogous for demodulators.

**General and Specific Blocks**

General Modulator | General Modulator Conditions | Specific Modulator |
---|---|---|

General QAM Modulator Baseband |
Predefined constellation containing
2
| Rectangular QAM Modulator Baseband |

M-PSK Modulator Baseband | M-ary number parameter is
`2` . | BPSK Modulator Baseband |

M-ary number parameter is
`4` . | QPSK Modulator Baseband | |

M-DPSK Modulator Baseband | M-ary number parameter is
`2` . | DBPSK Modulator Baseband |

M-ary number parameter is
`4` . | DQPSK Modulator Baseband | |

CPM Modulator Baseband | M-ary number parameter is
`2` , Frequency pulse
shape parameter is
`Gaussian` . | GMSK Modulator Baseband |

M-ary number parameter is
2, Frequency pulse
shape parameter is
`Rectangular` ,
Pulse length parameter is
`1` . | MSK Modulator Baseband | |

Frequency pulse shape parameter
is `Rectangular` ,
Pulse length parameter is
`1` . | CPFSK Modulator Baseband | |

General TCM Encoder | Predefined signal constellation containing
2^{K}
points on a rectangular lattice. | Rectangular QAM TCM Encoder |

Predefined signal constellation containing
2^{K}
points on a circle. | M-PSK TCM Encoder |

Furthermore, the CPFSK Modulator Baseband block is similar to the M-FSK Modulator Baseband block, when the M-FSK block uses continuous phase transitions. However, the M-FSK features of this product differ from the CPFSK features in their mask interfaces and in the demodulator implementations.

For a given modulation technique, two ways to simulate modulation techniques
are called *baseband* and *passband*.
Baseband simulation, also known as the *lowpass equivalent
method*, requires less computation. The Communications Toolbox supports baseband simulation for digital modulation and passband
simulation for analog modulation.

**Baseband Modulated Signals Defined**

If you use baseband modulation to produce the complex envelope
`y`

of the modulation of a message signal
`x`

, then `y`

is a
*complex-valued* signal that is related to the output of
a passband modulator. If the modulated signal has the waveform

$${Y}_{1}(t)\mathrm{cos}(2\pi {f}_{c}t+\theta )-{Y}_{2}(t)\mathrm{sin}(2\pi {f}_{c}t+\theta )\text{\hspace{0.17em}},$$

where *f*_{c} is the carrier frequency
and θ is the carrier signal's initial phase, then a baseband simulation
recognizes that this equals the real part of

$$[({Y}_{1}(t)+j{Y}_{2}(t)){e}^{j\theta}]\mathrm{exp}(j2\pi {f}_{c}t)\text{\hspace{0.17em}}.$$

and models only the part inside the square brackets. Here *j*
is the square root of -1. The complex vector `y`

is a sampling
of the complex signal

$$({Y}_{1}(t)+j{Y}_{2}(t)){e}^{j\theta}\text{\hspace{0.17em}}.$$

If you prefer to work with passband signals instead of baseband signals, then you can build functions that convert between the two. Be aware that passband modulation tends to be more computationally intensive than baseband modulation because the carrier signal typically needs to be sampled at a high rate.

Modulation is a process by which a *carrier signal* is
altered according to information in a *message signal*. The
*carrier frequency*,
*F*_{c}, is the frequency of the
carrier signal. The *sampling rate* is the rate at which the
message signal is sampled during the simulation.

The frequency of the carrier signal is usually much greater than the highest
frequency of the input message signal. The Nyquist sampling theorem requires
that the simulation sampling rate, *F*_{s},
be greater than two times the sum of the carrier frequency and the highest
frequency of the modulated signal in order for the demodulator to recover the
message correctly.

To modulate a signal using digital modulation with an alphabet having M
symbols, start with a real message signal whose values are integers from 0 to
M-1. Represent the signal by listing its values in a vector,
`x`

. Alternatively, you can use a matrix to represent a
multichannel signal, where each column of the matrix represents one
channel.

For example, if the modulation uses an alphabet with eight symbols, then the
vector `[2 3 7 1 0 5 5 2 6]'`

is a valid single-channel input
to the modulator. As a multichannel example, the two-column
matrix

[2 3; 3 3; 7 3; 0 3;]

`3`

.All digital modulation blocks process only discrete-time signals and use the baseband representation. The data types of inputs and outputs are depicted in the following figure.

If you want to separate the in-phase and quadrature components of the complex modulated signal, use the Complex to Real-Imag block in the Simulink Math Operations library.

Some digital modulation blocks can accept either integer-valued or
binary–valued signals. The corresponding demodulation blocks can output either
integers or groups of individual bits that represent integers. This section
describes how modulation blocks process integer or binary inputs; the case for
demodulation blocks is the reverse. You should note that modulation blocks have
an **Input type** parameter and that demodulation blocks have
an **Output type** parameter.

When you set the **Input type** parameter to
`Integer`

, the block accepts integer values from
`0`

to *M*-`1`

.
*M* represents the **M-ary number** block
parameter.

When you set the **Input type** parameter to
`Bit`

, the block accepts binary-valued inputs
that represent integers. The block collects binary-valued signals into
groups of *K* =
log_{2}(*M*) bits

where

*K* represents the number of bits per symbol. Since
*M* = 2* ^{K}*,

The input vector length must be an integer multiple of
*K*. In this configuration, the block accepts a group
of *K* bits and maps that group onto a symbol at the block
output. The block outputs one modulated symbol for each group of
*K* bits.

Depending on the modulation scheme, the **Constellation
ordering** or **Symbol set ordering** parameter
indicates how the block maps a group of *K* input bits to a
corresponding symbol. When you set the parameter to
`Binary`

, the block maps
[*u*(1) *u*(2) ... *u*(K)]
to the integer

$$\sum _{i=1}^{K}u(i){2}^{K-i}$$

and assumes that this integer is the input value. *u*(1) is
the most significant bit.

If you set *M* = 8, **Constellation
ordering** (or **Symbol set ordering**) to
`Binary`

, and the binary input word is
[1 1 0], the block converts [1 1 0] to the integer 6. The
block produces the same output when the input is 6 and the **Input
type** parameter is `Integer`

.

When you set **Constellation ordering** (or **Symbol
set ordering** or **Symbol mapping**) to
`Gray`

, the block uses a Gray-coded arrangement and
assigns binary inputs to points of a predefined Gray-coded signal constellation.
The predefined M-ary Gray-coded signal constellation assigns the binary
representation

M = 8; P = [0:M-1]'; de2bi(bitxor(P,floor(P/2)), log2(M),'left-msb')

to the *P*^{th}
integer.

The following tables show the typical Binary to Gray mapping for
*M* = 8.

**Binary to Gray Mapping for Bits**

Binary Code | Gray Code |
---|---|

000 | 000 |

001 | 001 |

010 | 011 |

011 | 010 |

100 | 110 |

101 | 111 |

110 | 101 |

111 | 100 |

**Gray to Binary Mapping for Integers**

Binary Code | Gray Code |
---|---|

0 | 0 |

1 | 1 |

2 | 3 |

3 | 2 |

4 | 6 |

5 | 7 |

6 | 5 |

7 | 4 |

**Gray Encoding a Modulated Signal**

For the PSK, DPSK, FSK, QAM, and PAM modulation types, Gray constellations are obtained by setting the symbol mapping to Gray-encoding in the corresponding modulation function or System object®.

For modulation functions, you can specify 'gray' for the symbol order input argument to obtain Gray-encoded modulation.

The following example uses the `qammod`

function with Gray-encoded symbol mapping.

y = [0:15]; y = de2bi(y); M = 16; symorder = 'gray'; xmap = qammod(y,M,symorder,'InputType','bit','PlotConstellation',true);

Checking the constellation plot, you can see the modulated symbols are Gray-encoded because all adjacent elements differ by only one bit.

Some digital modulation blocks can output an upsampled version of the
modulated signal, while their corresponding digital demodulation blocks can
accept an upsampled version of the modulated signal as input. In both cases, the
**Rate options** parameter represents the upsampling
factor, which must be a positive integer. Depending on whether the input signal
is single-rate mode or multirate mode, the block either changes the signal's
vector size or its sample time, as the following table indicates. Only the OQPSK
blocks deviate from the information in the table, in that S is replaced by 2S in
the scaling factors.

**Process Upsampled Modulated Data (Except OQPSK Method)**

Computation Type | Input Status | Result |
---|---|---|

Modulation | Single-rate processing | Output vector length is S times the number of integers or binary words in the input vector. Output sample time equals the input sample time. |

Modulation | Multirate processing | Output vector is a scalar. Output sample time is 1/S times the input sample time. |

Demodulation | Single-rate processing | Number of integers or binary words in the output vector is 1/S times the number of samples in the input vector. Output sample time equals the input sample time. |

Demodulation | Multirate processing |
Output signal contains one integer or one binary word. Output sample time is S times the input sample time. Furthermore, if S > 1 and the demodulator is from the AM, PM, or FM sublibrary, the demodulated signal is delayed by one output sample period. There is no delay if S = 1 or if the demodulator is from the CPM sublibrary. |

**Illustrations of Size or Rate Changes. **The following schematics illustrate how a modulator (other than OQPSK)
upsamples a triplet of frame-based and sample-based integers. In both cases,
the **Samples per symbol** parameter is
`2`

.

The following schematics illustrate how a demodulator (other than OQPSK or
one from the CPM sublibrary) processes three doubly sampled symbols using
both frame-based and sample-based inputs. In both cases, the
**Samples per symbol** parameter is
`2`

. The sample-based schematic includes an output
delay of one sample period.

For more information on delays, see Delays in Digital Modulation.

The model below plots the output of the DQPSK Modulator Baseband block. The image shows the possible transitions from each symbol in the DQPSK signal constellation to the next symbol.

To open this model enter
`doc_dqpsk_plot`

at the MATLAB command line. To build
the model, gather and configure these blocks:

Random Integer Generator, in the Random Data Sources sublibrary of the Comm Sources library

Set

**M-ary number**to`4`

.Set

**Initial seed**to any positive integer scalar, preferably the output of the`randn`

function.Set

**Sample time**to`.01`

.

DQPSK Modulator Baseband, in the PM sublibrary of the Digital Baseband sublibrary of Modulation

Complex to Real-Imag, in the Simulink Math Operations library

XY Graph, in the Simulink Sinks library

Use the blocks' default parameters unless otherwise instructed. Connect the blocks as in the figure above. Running the model produces the following plot. The plot reflects the transitions among the eight DQPSK constellation points.

This plot illustrates π/4-DQPSK modulation, because the default
**Phase offset** parameter in the DQPSK Modulator Baseband
block is `pi/4`

. To see how the phase offset influences the
signal constellation, change the **Phase offset** parameter in
the DQPSK Modulator Baseband block to `pi/8`

or another value.
Run the model again and observe how the plot changes.

The model below uses the M-QAM Modulator Baseband block to modulate random data. After passing the symbols through a noisy channel, the model produces a scatter diagram of the noisy data. The diagram suggests what the underlying signal constellation looks like and shows that the noise distorts the modulated signal from the constellation.

To open this model, enter
`doc_qam_scatter`

at the MATLAB command line. To build
the model, gather and configure these blocks:

Random Integer Generator, in the Random Data Sources sublibrary of the Comm Sources library

Set

**M-ary number**to`16`

.Set

**Initial seed**to any positive integer scalar, preferably the output of the`randn`

function.Set

**Sample time**to`.1`

.

Rectangular QAM Modulator Baseband, in the AM sublibrary of the Digital Baseband sublibrary of Modulation

Set

**Normalization method**to`Peak Power`

.

AWGN Channel, in the Channels library

Set

**Es/No**to`20`

.Set

**Symbol period**to`.1`

.

Constellation Diagram, in the Comm Sinks library

Set

**Symbols to display**to`160`

.

Connect the blocks as in the figure. From the model window's
**Simulation** menu, select **Model
Configuration parameters**. In the Configuration Parameters
dialog box, set **Stop time** to `250`

.
Running the model produces a scatter diagram like the following one. Your plot
might look somewhat different, depending on your **Initial
seed** value in the Random Integer Generator block. Because the
modulation technique is 16-QAM, the plot shows 16 clusters of points. If there
were no noise, the plot would show the 16 exact constellation points instead of
clusters around the constellation points.

The example generates a random digital signal, modulates it, and adds noise. Then it creates a scatter plot, demodulates the noisy signal, and computes the symbol error rate.

The output and scatter plot follow. Your numerical results and plot might vary, because the example uses random numbers.

num = 83 rt = 0.0166

The scatter plot does not look exactly like a signal constellation. Where the signal constellation has 16 precisely located points, the noise causes the scatter plot to have a small cluster of points approximately where each constellation point would be.

Modulation is often followed by pulse shaping, and demodulation is often preceded by a filtering or an integrate-and-dump operation. This section presents an example involving rectangular pulse shaping. For an example that uses raised cosine pulse shaping, see Pulse Shaping Using a Raised Cosine Filter.

**Rectangular Pulse Shaping. **Rectangular pulse shaping repeats each output from the modulator a fixed
number of times to create an upsampled signal. Rectangular pulse shaping can
be a first step or an exploratory step in algorithm development, though it
is less realistic than other kinds of pulse shaping. If the transmitter
upsamples the modulated signal, then the receiver should downsample the
received signal before demodulating. The “integrate and dump”
operation is one way to downsample the received signal.

The code below uses the `rectpulse`

function for
rectangular pulse shaping at the transmitter and the
`intdump`

function for downsampling at the
receiver.

M = 16; % Alphabet size, 16-QAM x = randi([0 M-1],5000,1); % Message signal Nsamp = 4; % Oversampling rate % Modulate y = qammod(x,M); % Follow with rectangular pulse shaping. ypulse = rectpulse(y,Nsamp); % Transmit signal through an AWGN channel. ynoisy = awgn(ypulse,15,'measured'); % Downsample at the receiver. ydownsamp = intdump(ynoisy,Nsamp); % Demodulate to recover the message. z = qamdemod(ydownsamp,M);

This example plots a phase tree associated with a continuous phase modulation scheme. A phase tree is a diagram that superimposes many curves, each of which plots the phase of a modulated signal over time. The distinct curves result from different inputs to the modulator.

This example uses the CPM Modulator Baseband block for its numerical computations. The block is configured using a raised cosine filter pulse shape. The example also illustrates how you can use Simulink and MATLAB together. The example uses MATLAB commands to run a series of simulations with different input signals, to collect the simulation results, and to plot the full data set.

In contrast to this example's approach using both MATLAB and Simulink, the
`commcpmphasetree`

example produces a phase tree using
a Simulink model without additional lines of MATLAB code.

Open the model by typing
`doc_phasetree`

at the MATLAB command line. To build the
model, gather and configure these blocks:

Constant, in the Simulink Commonly Used Blocks library

Set

**Constant value**to`s`

(which will appear in the MATLAB workspace).Set

**Sampling mode**to`Frame-based`

.Set

**Frame period**to`1`

.

Set

**M-ary number**to`2`

.Set

**Modulation index**to`2/3`

.Set

**Frequency pulse shape**to`Raised Cosine`

.Set

**Pulse length**to`2`

.

To Workspace, in the Simulink Sinks library

Set

**Variable name**to`x`

.Set

**Save format**to`Array`

.

Do not run the model, because the variable `s`

is not yet
defined in the MATLAB workspace. Instead, save the model to a folder on your
MATLAB path, using the filename `doc_phasetree`

.

The second step of this example is to execute the following MATLAB code:

% Parameters from the CPM Modulator Baseband block M_ary_number = 2; modulation_index = 2/3; pulse_length = 2; samples_per_symbol = 8; L = 5; % Symbols to display pmat = []; for ip_sig = 0:(M_ary_number^L)-1 s = de2bi(ip_sig,L,M_ary_number,'left-msb'); % Apply the mapping of the input symbol to the CPM % symbol 0 -> -(M-1), 1 -> -(M-2), etc. s = 2*s'+1-M_ary_number; sim('doc_phasetree', .9); % Run model to generate x. % Next column of pmat pmat(:,ip_sig+1) = unwrap(angle(x(:))); end; pmat = pmat/(pi*modulation_index); t = (0:L*samples_per_symbol-1)'/samples_per_symbol; plot(t,pmat); figure(gcf); % Plot phase tree.

This code defines the parameters for the CPM Modulator, applies symbol mapping, and plots the results. Each curve represents a different instance of simulating the CPM Modulator Baseband block with a distinct (constant) input signal.

The log-likelihood ratio (LLR) is the logarithm of the ratio of probabilities of a
0 bit being transmitted versus a 1 bit being transmitted for a received signal. The
LLR for a bit *b* is defined as:

$$L(b)=\mathrm{log}\left(\frac{\mathrm{Pr}(b=0|r=(x,y))}{\mathrm{Pr}(b=1|r=(x,y))}\right)$$

Assuming equal probability for all symbols, the LLR for an AWGN channel can be expressed as:

$$L(b)=\mathrm{log}\left(\frac{{\displaystyle \sum _{s\in {S}_{0}}{e}^{-\frac{1}{{\sigma}^{2}}\left({(x-{s}_{x})}^{2}+{(y-{s}_{y})}^{2}\right)}}}{{\displaystyle \sum _{s\in {S}_{1}}{e}^{-\frac{1}{{\sigma}^{2}}\left({(x-{s}_{x})}^{2}+{(y-{s}_{y})}^{2}\right)}}}\right)$$

where the variables represent the values shown in the following table.

Variable | What the Variable Represents |
---|---|

$$r$$
| Received signal with coordinates (x,
y). |

$$b$$
| Transmitted bit (one of the K bits in an M-ary symbol, assuming all M symbols are equally probable. |

$${S}_{0}$$
| Ideal symbols or constellation points with bit 0, at the given bit position. |

$${S}_{1}$$
| Ideal symbols or constellation points with bit 1, at the given bit position. |

$${s}_{x}$$
| In-phase coordinate of ideal symbol or constellation point. |

$${s}_{y}$$
| Quadrature coordinate of ideal symbol or constellation point. |

$${\sigma}^{2}$$
| Noise variance of baseband signal. |

$${\sigma}_{x}^{2}$$
| Noise variance along in-phase axis. |

$${\sigma}_{y}^{2}$$
| Noise variance along quadrature axis. |

Noise components along the in-phase and quadrature axes are assumed to be independent and of equal power (i.e., $${\sigma}_{x}^{2}={\sigma}_{y}^{2}={\sigma}^{2}/2$$).

Approximate LLR is computed by taking into consideration only the nearest constellation point to the received signal with a 0 (or 1) at that bit position, rather than all the constellation points as done in exact LLR. It is defined as [8]:

$$L(b)=-\frac{1}{{\sigma}^{2}}\left(\underset{s\in {S}_{0}}{\mathrm{min}}\left({(x-{s}_{x})}^{2}+\text{}\text{}{(y-{s}_{y})}^{2}\right)-\underset{s\in {S}_{1}}{\mathrm{min}}\left({(x-{s}_{x})}^{2}+\text{}\text{}{(y-{s}_{y})}^{2}\right)\right)$$

Digital modulation and demodulation blocks sometimes incur delays between their inputs and outputs, depending on their configuration and on properties of their signals. The following table lists sources of delay and the situations in which they occur.

**Delays Resulting from Digital Modulation or Demodulation**

Modulation or Demodulation Type | Situation in Which Delay Occurs | Amount of Delay |
---|---|---|

FM demodulator | Sample-based processing | One output period |

All demodulators in CPM sublibrary | Multirate processing, and the model uses a variable-step
solver or a fixed-step solver with the Tasking
Mode parameter set to
`SingleTasking`
D = Traceback
length parameter | D+1 output periods |

Single-rate processing, D = Traceback
depth parameter | D output periods | |

OQPSK demodulator | Single-rate processing |
For more information, see OQPSK Demodulator Baseband. |

Multirate processing, and the model uses a fixed-step solver
with Tasking Mode parameter set to
`Auto` or
`MultiTasking` . | ||

Multirate processing processing, and the model uses a
variable-step solver or the Tasking Mode
parameter is set to
Single`Tasking` . | ||

All decoders in TCM sublibrary | Operation mode set to
`Continuous` , Tr =
Traceback depth parameter, and code
rate k/n | Tr*k output bits |

As a result of delays, data that enters a modulation or demodulation block at time T appears in the output at time T+delay. In particular, if your simulation computes error statistics or compares transmitted with received data, it must take the delay into account when performing such computations or comparisons.

In addition to the delays mentioned above, the M-DPSK, DQPSK, and DBPSK demodulators produce output whose first sample is unrelated to the input. This is related to the differential modulation technique, not the particular implementation of it.

Demodulation in the model below causes the demodulated signal to lag, compared
to the unmodulated signal. When computing error statistics, the model accounts
for the delay by setting the Error Rate Calculation block's
**Receive delay** parameter to `0`

. If the
**Receive delay** parameter had a different value, then the
error rate showing at the top of the Display block would be close to 1/2.

To open this model
, enter `doc_oqpsk_modulation_delay`

at the MATLAB
command line. To build the model, gather and configure these blocks:

Random Integer Generator, in the Random Data Sources sublibrary of the Comm Sources library

Set

**M-ary number**to`4`

.Set

**Initial seed**to any positive integer scalar.

OQPSK Modulator Baseband, in the PM sublibrary of the Digital Baseband sublibrary of Modulation

AWGN Channel, in the Channels library

Set

**Es/No**to`6`

.

OQPSK Demodulator Baseband, in the PM sublibrary of the Digital Baseband sublibrary of Modulation

Error Rate Calculation, in the Comm Sinks library

Set

**Receive delay**to`1`

.Set

**Computation delay**to`0`

.Set

**Output data**to`Port`

.

Display, in the Simulink Sinks library

Drag the bottom edge of the icon to make the display big enough for three entries.

Connect the blocks as shown above. From the model window's
**Simulation**, select **Model
Configuration parameters**. In the **Configuration
Parameters** dialog box, set **Stop time** to
`1000`

. Then run the model and observe the error rate at
the top of the Display block's icon. Your error rate will vary depending on your
**Initial seed** value in the Random Integer Generator
block.

[1] Jeruchim, M. C., P. Balaban, and K. S. Shanmugan,
*Simulation of Communication Systems*, New York, Plenum
Press, 1992.

[2] Proakis, J. G., *Digital
Communications*, 3rd ed., New York, McGraw-Hill, 1995.

[3] Sklar, B., *Digital Communications:
Fundamentals and Applications*, Englewood Cliffs, NJ,
Prentice-Hall, 1988.

[4] Anderson, J. B., T. Aulin, and C.-E. Sundberg,
*Digital Phase Modulation*, New York, Plenum Press,
1986.

[5] Biglieri, E., D. Divsalar, P.J. McLane, and M.K.
Simon, *Introduction to Trellis-Coded Modulation with
Applications*, New York, Macmillan, 1991.

[6] Pawula, R.F., “On M-ary DPSK Transmission
Over Terrestrial and Satellite Channels,” *IEEE Transactions on
Communications*, Vol. COM-32, July 1984, pp. 752–761.

[7] Smith, J. G., “Odd-Bit Quadrature
Amplitude-Shift Keying,” *IEEE Transactions on
Communications*, Vol. COM-23, March 1975, pp. 385–389.

[8] Viterbi, A. J., “An Intuitive Justification and a Simplified
Implementation of the MAP Decoder for Convolutional Codes,”
*IEEE Journal on Selected Areas in Communications*,
vol. 16, No. 2, pp 260–264, Feb. 1998