Documentation

Modulation Techniques

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. Communications Toolbox features enable you to modulate and demodulate signals using various digital modulation techniques. Constellation plots enable you to visualize the constellation diagram of the modulation symbols.

The available methods of modulation depend on whether the input signal is analog or digital. The Modulation category lists the digital and analog modulation techniques supported.

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.

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.

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.

Baseband and Passband Simulation

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

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

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

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 Terminology

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.

Representing Digital Signals

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;]

Integer-Valued Signals and Binary-Valued Signals

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 = log₂(M) bits

where

K represents the number of bits per symbol. Since M = 2^K, K is commonly referred to as the modulation order.

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.

Constellation Ordering (or Symbol Set Ordering)

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

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.

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.

Upsample Signals and Rate Changes

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.

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.

DQPSK Signal Constellation Points and Transitions

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:

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.

Rectangular QAM Modulation and Scatter Diagram

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:

Connect the blocks as shown in the preceding figure. On the Simulation tab, in the Simulate section, set Stop time to 250. The Simulate section appears on multiple tabs.

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.

Compute Symbol Error Rate

The example generates a random digital signal, modulates it, adds noise, demodulates the noisy signal, and computes the symbol error rate. The noisy, modulated data is plotted in a constellation diagram. Numerical results and plot may vary due to the random input data.

Create a random digital message and a constellation diagram System object.

M = 16; % Alphabet size, 16-QAM
x = randi([0 M-1],5000,1);

cpts = qammod(0:M-1,M);
constDiag = comm.ConstellationDiagram('ReferenceConstellation',cpts, ...
    'XLimits',[-4 4],'YLimits',[-4 4]);

Apply 16-QAM modulation and transmit signal through an AWGN channel.

y = qammod(x,M);
ynoisy = awgn(y,15,'measured');

Demodulate ynoisy to recover the message and check the symbol error rate.

z = qamdemod(ynoisy,M);
[num,rt] = symerr(x,z)

num = 79

rt = 0.0158

Create constellation diagram from noisy data. The signal reference constellation has 16 precisely located points but the transmitted symbols with the noise added causes the scatter plot to have a small cluster of points scattered around each reference constellation point.

constDiag(ynoisy)

Combine Pulse Shaping and Filtering with Modulation

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. Although it is less realistic than other kinds of pulse shaping, rectangular pulse shaping can be a first step or an exploratory step in algorithm development. If the transmitter upsamples the modulated signal, then the receiver should downsample the received signal before demodulating. The code below uses the rectpulse function for rectangular pulse shaping at the transmitter and the intdump function for downsampling at the receiver. The “integrate and dump” operation is one way to downsample the received signal.

% Create a random digital message and a constellation diagram System
% object.
M = 16;                     % Alphabet size, 16-QAM
x = randi([0 M-1],5000,1);  % Message signal
Nsamp = 4;                  % Oversampling rate

% Apply 16-QAM modulation and rectangular pulse shaping. Transmit signal
% through an AWGN channel.
y = qammod(x,M);
ypulse = rectpulse(y,Nsamp);
ynoisy = awgn(ypulse,15,'measured');

% Downsample at the receiver.
ydownsamp = intdump(ynoisy,Nsamp);

% Demodulate to recover the message.
z = qamdemod(ydownsamp,M);

Phase Tree for Continuous Phase Modulation

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.

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

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.

First Output Sample in DPSK Demodulation

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.

Example: Delays from Demodulation

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:

Connect the blocks as shown in the preceding figure. On the Simulation tab, in the Simulate section, set Stop time to 1000. The Simulate section appears on multiple tabs. 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.