qammod

Quadrature amplitude modulation (QAM)

Syntax

Y = qammod(X,M)

Y = qammod(X,M,symOrder)

Y = qammod(___,Name=Value)

Description

Y = qammod(X,M) modulates input signal X by using QAM with the specified modulation order M.

Y = qammod(X,M,symOrder) specifies the symbol order.

Y = qammod(___,Name=Value) specifies options using name-value arguments in addition to any of the input argument combinations from previous syntaxes. For example, InputType=bit sets the type of input signal to bits.

Examples

Modulate Data Using QAM

Modulate data using QAM and display the result in a scatter plot.

Set the modulation order to 16 and create a data vector containing each of the possible symbols.

M = 16;
x = (0:M-1)';

Modulate the data using the qammod function.

y = qammod(x,M);

Display the modulated signal constellation using the scatterplot function.

scatterplot(y)

Set the modulation order to 256, and display the scatter plot of the modulated signal.

M = 256;
x = (0:M-1)';
y = qammod(x,M);
scatterplot(y)

Normalize QAM Signal by Average Power

Modulate random data symbols using QAM. Normalize the modulator output so that it has an average signal power of 1 W.

Set the modulation order and generate random data.

M = 64;
x = randi([0 M-1],1000,1);

Modulate the data. Use the 'UnitAveragePower' name-value argument to set the output signal to have an average power of 1 W.

y = qammod(x,M,UnitAveragePower=true);

Confirm that the signal has unit average power.

avgPower = mean(abs(y).^2)

avgPower = 
1.0070

Plot the resulting constellation.

scatterplot(y)
title('64-QAM, Average Power = 1 W')

QAM Symbol Ordering

Plot QAM constellations for Gray, binary, and custom symbol mappings.

Set the modulation order, and create a data sequence that includes a complete set of symbols for the modulation scheme.

M = 16;
d = 0:M-1;

Modulate the data, and plot its constellation. The default symbol mapping uses Gray-coded ordering. The ordering of the points is not sequential.

y = qammod(d,M,PlotConstellation=true);

Repeat the modulation process with binary symbol mapping. The symbol mapping follows a binary-coded order and is sequential.

z = qammod(d,M,'bin',PlotConstellation=true);

Create a custom symbol mapping.

smap = randperm(M)-1;

Modulate and plot the constellation.

w = qammod(d,M,smap,PlotConstellation=true);

QAM Constellation Shape Dependency on Modulation Orders

Plot various M-QAM constellations to demonstrate the shape depends on the modulation order.

Define a vector of vlaues for modulation order, $\mathit{M}$, from the number of bits/symbol, $\mathit{k}$.

k = 1:7;  % Number of bits per symbol
M = 2.^k; % Modulation order

Use a for loop to create a data sequence that includes a complete set of symbols for the modulation scheme, modulate the data, and plot the constellation for each modulation order in the range of values in vector k. Note the constellation shape for depends on the number of bits per symbol, k.

• For even values of k, the constellation is a square.

• For odd values of $\mathit{k}>3$, the constellation is a cross.

• For $\mathit{k}=1$ and $\mathit{k}=3$, the constellation is a rectangle.

for ii = 1:length(M)
    disp(['k = ',num2str(k(ii))])
    d = 0:M(ii)-1;
    y = qammod(d,M(ii),PlotConstellation=true);
    
end

k = 1

k = 2

k = 3

k = 4

k = 5

k = 6

k = 7

Quadrature Amplitude Modulation with Bit Inputs

Modulate a sequence of bits using 64-QAM. Pass the signal through a noisy channel. Display the resultant constellation diagram.

Set the modulation order, and determine the number of bits per symbol.

M = 64;
k = log2(M);

Create a binary data sequence. When using binary inputs, the number of rows in the input must be an integer multiple of the number of bits per symbol.

data = randi([0 1],1000*k,1);

Modulate the signal using bit inputs, and set it to have unit average power.

txSig = qammod(data,M, ...
    InputType='bit', ...
    UnitAveragePower=true);

Pass the signal through a noisy channel.

rxSig = awgn(txSig,25);

Plot the constellation diagram.

cd = comm.ConstellationDiagram(ShowReferenceConstellation=false);
cd(rxSig)

Demodulate QAM Fixed-Point Signal

Demodulate a fixed-point QAM signal and verify that the data is recovered correctly.

Set the modulation order as 64, and determine the number of bits per symbol.

M = 64;
bitsPerSym = log2(M);

Generate random bits. When operating in bit mode, the length of the input data must be an integer multiple of the number of bits per symbol.

x = randi([0 1],10*bitsPerSym,1);

Modulate the input data using a binary symbol mapping. Set the modulator to output fixed-point data. The numeric data type is signed with a 16-bit word length and a 10-bit fraction length.

y = qammod(x,M,'bin', ...
    InputType='bit', ...
    OutputDataType=numerictype(1,16,10));

Demodulate the 64-QAM signal. Verify that the demodulated data matches the input data.

z = qamdemod(y,M,'bin',OutputType='bit');
s = isequal(x,double(z))

s = logical
   1

Average Power Normalization for QAM

Apply average power normalization for hard decision output when using qammod and qamdemod functions by using the helperAvgPow2MinD utility function. Scale the constellation to the normalized average power, and then plot the reference and scaled constellations.

Compute the minimum distance for symbols based on specified average power and modulation order.

M = 64;
avgPwr = 2;
minD = helperAvgPow2MinD(avgPwr,M);

Modulate a signal composed of random integers in the range [0, M - 1], scale the modulated symbols.

x = randi([0,M-1],1000,1);
y = qammod(x,M);
yTx = (minD/2) .* y;

Verify signal average power approximately equals the specified average power, avgPow.

sigPwr = mean(abs(yTx).^2)

sigPwr = 
2.0141

avgPwr

avgPwr = 
2

Assign the transmitted signal to the receive signal without applying any RF or channel impairments. With no impairments to distort the received signal, the demodulated matches the original signal. Demodulate symbols using hard decisions, and confirm correct signal demodulation.

yRx = yTx;
z = qamdemod(yRx*2/minD,M);
checkDemodIsEqual = isequal(x,z)

checkDemodIsEqual = logical
   1

refC = qammod([0:M-1]',M);

Show constellation

maxAx = ceil(max(abs(refC)));
cd = comm.ConstellationDiagram(2, ...
    'ShowReferenceConstellation',0, ...
    'ShowLegend',true, ...
    'XLimits',[-(maxAx) maxAx],'YLimits',[-(maxAx) maxAx], ...
    'ChannelNames', ...
    {'y','yTx'});
cd(y,yTx)

Peak Power Normalization for QAM

Apply peak power normalization for hard decision output when using qammod and qamdemod functions by using the helperPeakPow2MinD utility function. Scale the constellation to the normalized peak power, and then plot the reference and scaled constellations.

Compute the minimum distance for symbols based on specified peak power and modulation order.

M = 16;
pkPwr = 30;
minD = helperPeakPow2MinD(pkPwr,M);

Modulate a signal composed of random integers in the range [0, M - 1], scale the modulated symbols.

x = randi([0,M-1],1000,1);
y = qammod(x,M);
yTx = (minD/2) .* y;

Verify signal peak power approximately equals the specified peak power, pkPow.

sigPwr = max(abs(yTx).^2)

sigPwr = 
30

pkPwr

pkPwr = 
30

Assign the transmitted signal to the receive signal without applying any RF or channel impairments. With no impairments to distort the received signal, the demodulated matches the original signal. Demodulate symbols using hard decisions, and confirm correct signal demodulation.

yRx = yTx;
z = qamdemod(yRx*2/minD,M);
checkDemodIsEqual = isequal(x,z)

checkDemodIsEqual = logical
   1

refC = qammod([0:M-1]',M);

Show constellation

maxAx = ceil(max(abs(refC)));
cd = comm.ConstellationDiagram(2, ...
    'ShowReferenceConstellation',0, ...
    'ShowLegend',true, ...
    'XLimits',[-(maxAx) maxAx],'YLimits',[-(maxAx) maxAx], ...
    'ChannelNames', ...
    {'y','yTx'});
cd(y,yTx)

Input Arguments

X — Input signal

scalar | vector | matrix | array

Input signal, specified as a scalar, vector, matrix, or array. The elements of this input signal must be binary values or integers in the range [0, (M – 1)], where M is the modulation order.

Note

To process input signal as binary elements, specify InputType='bit'. For binary inputs, the number of rows must be an integer multiple of log₂(M). Groups of log₂(M) bits are mapped onto a symbol, with the first bit representing the MSB and the last bit representing the LSB.

Data Types: double | single | fi(S,WL,0) | int8 | int16 | uint8 | uint16

M — Modulation order

positive integer power of two

Modulation order, specified as an positive integer power of two. The modulation order specifies the number of points in the signal constellation.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

symOrder — Symbol order

'gray' (default) | 'bin' | vector

Symbol order, specified as 'gray', 'bin', or a vector.

• 'gray' — Use Gray-coded ordering. For more information, see Gray Code.

• 'bin' — Use binary-coded ordering.

• Vector — Use custom symbol ordering

Vectors must use unique elements in the range [0, (M – 1)]. The first element corresponds to the upper-left point of the constellation, with subsequent elements running down column-wise from left to right.

Example: [0 3 1 2]

Data Types: string | char | double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: y = qammod(x,M,symOrder,'InputType','bit')

InputType — Input type

'integer' (default) | 'bit'

Input type, specified as 'integer' or 'bit'.

• If you specify 'integer', the input signal must consist of integers in the range [0, (M – 1)].

• If you specify 'bit', the input signal must contain binary values, and the number of rows must be an integer multiple of log₂(M).

UnitAveragePower — Unit average power flag

false or 0 (default) | true or 1

Unit average power flag, specified as a numeric or logical 0 (false) or 1 (true).

• When this flag is 1 (true), the function scales the constellation to the average power of one watt referenced to 1 ohm.

• When this flag is 0 (false), the function scales the constellation so that the QAM constellation points are separated by a minimum distance of two.

OutputDataType — Output data type

'double' | 'single' | numerictype object

Output data type, specified as 'double', 'single', or a numerictype object. For more information on constructing these objects, see numerictype (Fixed-Point Designer). This argument determines the data type of the output modulated symbols and the data type used for intermediate computations. Output the fixed-point type as a signed, unscaled numerictype object in MATLAB^® simulation, and as a signed, scaled numerictype object during C code or MEX generation.

• If you do not specify this argument and the input is data type double or a built-in integer, the output data type is double.

• If the input is data type single, the output data type is single.

• If the input is fixed-point, this argument must be specified.

PlotConstellation — Option to plot constellation

false or 0 (default) | true or 1

Option to plot constellation, specified as a numeric or logical 0 (false) or 1 (true). To plot the QAM constellation, specify PlotConstellation=true.

Output Arguments

Y — Modulated signal

scalar | vector | matrix | array

Modulated signal, returned as a complex scalar, vector, matrix, or array of numeric values.

• For integer inputs, output Y has the same dimensions as input signal X.

• For bit inputs, the number of rows in Y is the number of rows in X divided by log₂(M).

For more information on the output data type, see OutputDataType.

Data Types: double | single | fi | int8 | int16 | uint8 | uint16

More About

Gray Code

A Gray code, also known as a reflected binary code, is a system where the bit patterns in adjacent constellation points differ by only one bit.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function supports GPU array inputs. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Usage notes and limitations:

• Fixed-point output data types are not supported.

Version History

Introduced before R2006a

R2024a: Enhance output data type support

The qammod function enhances output data type offering by using the OutputDataType argument. Previously, the output data type was inherited from input data type or set to a fixed-point setting. This addition allows you to also specify the output data type as single or double.

R2023b: Add GPU array support

The qammod function adds support for gpuArray (Parallel Computing Toolbox) object processing to run code on a graphics processing unit (GPU).

R2018b: Initial Phase Input Removed

Starting in R2018b, you can no longer offset the initial phase for the QAM constellation using the qammod function.

To adjust the initial phase of the QAM-modulated data, instead use genqammod to offset the initial phase of the data being modulated, or you can multiply the qammod output by the desired initial phase:

y = qammod(x,M) .* exp(1i*initPhase)

.