qamdemod

Quadrature amplitude demodulation

Syntax

z = qamdemod(y,M)

z = qamdemod(y,M,symOrder)

z = qamdemod(___,Name,Value)

Description

z = qamdemod(y,M) returns a demodulated signal, z, given quadrature amplitude modulated (QAM) signal y of modulation order M.

z = qamdemod(y,M,symOrder) returns a demodulated signal and specifies the symbol order.

z = qamdemod(___,Name,Value) specifies demodulation behavior using Name,Value pairs and any of the previous syntaxes.

Input Arguments

y — Input signal

scalar | vector | matrix | 3-D array

Input signal resulting from quadrature amplitude modulation, specified as a complex scalar, vector, matrix, or 3-D array. Each column is treated as an independent channel.

Data Types: double | single | fi
Complex Number Support: Yes

M — Modulation order

scalar integer

Modulation order, specified as a power-of-two scalar integer. The modulation order specifies the number of points in the signal constellation.

Example: 16

Data Types: double

symOrder — Symbol order

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

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

• 'gray' — Use Gray Code ordering

• 'bin' — Use natural binary-coded ordering

• Vector — Use custom symbol ordering

Vectors must use unique elements whose values range from 0 to 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: char | double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'UnitAveragePower' — Unit average power flag

false (default) | true

Unit average power flag, specified as the comma-separated pair consisting of UnitAveragePower and a logical scalar. When this flag is true, the function scales the constellation to an average power of 1 watt referenced to 1 ohm. When this flag is false, the function scales the constellation so that the QAM constellation points are separated by a minimum distance of 2.

Data Types: logical

'OutputType' — Output type

'integer' (default) | 'bit' | 'llr' | 'approxllr'

Output type, specified as the comma-separated pair consisting of OutputType and one of the following: 'integer', 'bit', 'llr', or 'approxllr'.

Data Types: char

'NoiseVariance' — Noise variance

1 (default) | positive scalar | vector of positive values

Noise variance, specified as a positive scalar or vector of positive values.

• When specified as a scalar, the same noise variance value is used on all input elements.

• When specified as a vector, the vector length must be equal to the number of elements in the last dimension of the input signal. Each element of the vector specifies noise variance for all the elements of the input along the corresponding last dimension.

Tip

When OutputType = ‘llr’, if the demodulation computation outputs Inf or -Inf values, it is likely because the specified noise variance values are significantly smaller than the SNR. Since the LLR algorithm computes exponentials using finite precision arithmetic, computation of exponentials with very large or very small numbers can yield positive or negative infinity.

Try using OutputType = ‘approxllr’ instead, because the approximate LLR algorithm does not compute exponentials.

Dependencies

This input argument applies only when 'OutputType' is set to 'llr' or 'approxllr'.

Data Types: double

'PlotConstellation' — Option to plot constellation

false (default) | true

Option to plot constellation, specified as the comma-separated pair consisting of 'PlotConstellation' and a logical scalar. To plot the QAM constellation, set PlotConstellation to true.

Data Types: logical

Output Arguments

z — Demodulated output signal

scalar | vector | matrix | 3-D array

Demodulated output signal, returned as a scalar, vector, matrix, or 3-D array. The data type is the same as that of the input signal, y. The dimensions of the output vary depending on the specified OutputType value.

'OutputType'	Return Value of qamdemod	Dimensions of Output
'integer'	Demodulated integer values from 0 to M – 1	z has the same dimensions as input y.
'bit'	Demodulated bits	The number of rows in z is log2(M) times the number of rows in y. Each demodulated symbol is mapped to a group of log2(M) bits, where the first bit represents the MSB and the last bit represents the LSB.
'llr'	Log-likelihood ratio value for each bit
'approxllr'	Approximate log-likelihood ratio value for each bit

Examples

Demodulate 8-QAM Signal

Demodulate an 8-QAM signal and plot the points corresponding to symbols 0 and 3.

Generate random 8-ary data symbols.

data = randi([0 7],1000,1);

Apply 8-QAM.

txSig = qammod(data,8);

Pass the modulated signal through an AWGN channel.

rxSig = awgn(txSig,18,'measured');

Demodulate the received signal using an initial phase of $$\pi$$/8.

rxData = qamdemod(rxSig.*exp(-1i*pi/8),8);

Generate the reference constellation points.

refpts = qammod((0:7)',8) .* exp(1i*pi/8);

Plot the received signal points corresponding to symbols 0 and 3 and overlay the reference constellation. Only the received data corresponding to those symbols is displayed.

plot(rxSig(rxData==0),'g.');
hold on
plot(rxSig(rxData==3),'c.');
plot(refpts,'r*')
text(real(refpts)+0.1,imag(refpts),num2str((0:7)'))
xlabel('In-Phase')
ylabel('Quadrature')
legend('Points corresponding to 0','Points corresponding to 3', ...
    'Reference constellation','location','nw');

QAM Demodulation with WLAN Symbol Mapping

Modulate and demodulate random data by using 16-QAM with WLAN symbol mapping. Verify that the input data symbols match the demodulated symbols.

Generate a 3-D array of random symbols.

x = randi([0,15],20,4,2);

Create a custom symbol mapping for the 16-QAM constellation based on WLAN standards.

wlanSymMap = [2 3 1 0 6 7 5 4 14 15 13 12 10 11 9 8];

Modulate the data, and set the constellation to have unit average signal power. Plot the constellation.

y = qammod(x,16,wlanSymMap,'UnitAveragePower', true,'PlotConstellation',true);

Demodulate the received signal.

z = qamdemod(y,16,wlanSymMap,'UnitAveragePower',true);

Verify that the demodulated signal is equal to the original data.

isequal(x,z)

ans = logical
   1

Demodulate QAM Fixed-Point Signal

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

Set the modulation order, 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

Estimate BER for Hard and Soft Decision Viterbi Decoding

Estimate bit error rate (BER) performance for hard-decision and soft-decision Viterbi decoders in AWGN. Compare the performance to that of an uncoded 64-QAM link.

Set the simulation parameters.

clear; close all
rng default
M = 64;                 % Modulation order
k = log2(M);            % Bits per symbol
EbNoVec = (4:10)';       % Eb/No values (dB)
numSymPerFrame = 1000;   % Number of QAM symbols per frame

Initialize the BER results vectors.

berEstSoft = zeros(size(EbNoVec)); 
berEstHard = zeros(size(EbNoVec));

Set the trellis structure and traceback length for a rate 1/2, constraint length 7, convolutional code.

trellis = poly2trellis(7,[171 133]);
tbl = 32;
rate = 1/2;

The main processing loops performs these steps:

• Generate binary data.

• Convolutionally encode the data.

• Apply QAM modulation to the data symbols. Specify unit average power for the transmitted signal.

• Pass the modulated signal through an AWGN channel.

• Demodulate the received signal using hard decision and approximate LLR methods. Specify unit average power for the received signal.

• Viterbi decode the signals using hard and unquantized methods.

• Calculate the number of bit errors.

The while loop continues to process data until either 100 errors are encountered or 1e7 bits are transmitted.

for n = 1:length(EbNoVec)
    % Convert Eb/No to SNR
    snrdB = EbNoVec(n) + 10*log10(k*rate);
    % Noise variance calculation for unity average signal power.
    noiseVar = 10.^(-snrdB/10);
    % Reset the error and bit counters
    [numErrsSoft,numErrsHard,numBits] = deal(0);
    
    while numErrsSoft < 100 && numBits < 1e7
        % Generate binary data and convert to symbols
        dataIn = randi([0 1],numSymPerFrame*k,1);
        
        % Convolutionally encode the data
        dataEnc = convenc(dataIn,trellis);
        
        % QAM modulate
        txSig = qammod(dataEnc,M,'InputType','bit','UnitAveragePower',true);
        
        % Pass through AWGN channel
        rxSig = awgn(txSig,snrdB,'measured');
        
        % Demodulate the noisy signal using hard decision (bit) and
        % soft decision (approximate LLR) approaches.
        rxDataHard = qamdemod(rxSig,M,'OutputType','bit','UnitAveragePower',true);
        rxDataSoft = qamdemod(rxSig,M,'OutputType','approxllr', ...
            'UnitAveragePower',true,'NoiseVariance',noiseVar);
        
        % Viterbi decode the demodulated data
        dataHard = vitdec(rxDataHard,trellis,tbl,'cont','hard');
        dataSoft = vitdec(rxDataSoft,trellis,tbl,'cont','unquant');
        
        % Calculate the number of bit errors in the frame. Adjust for the
        % decoding delay, which is equal to the traceback depth.
        numErrsInFrameHard = biterr(dataIn(1:end-tbl),dataHard(tbl+1:end));
        numErrsInFrameSoft = biterr(dataIn(1:end-tbl),dataSoft(tbl+1:end));
        
        % Increment the error and bit counters
        numErrsHard = numErrsHard + numErrsInFrameHard;
        numErrsSoft = numErrsSoft + numErrsInFrameSoft;
        numBits = numBits + numSymPerFrame*k;

    end
    
    % Estimate the BER for both methods
    berEstSoft(n) = numErrsSoft/numBits;
    berEstHard(n) = numErrsHard/numBits;
end

Plot the estimated hard and soft BER data. Plot the theoretical performance for an uncoded 64-QAM channel.

semilogy(EbNoVec,[berEstSoft berEstHard],'-*')
hold on
semilogy(EbNoVec,berawgn(EbNoVec,'qam',M))
legend('Soft','Hard','Uncoded','location','best')
grid
xlabel('Eb/No (dB)')
ylabel('Bit Error Rate')

As expected, the soft decision decoding produces the best results.

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.

Compatibility Considerations

Initial Phase Input Removed

Errors starting in R2018b

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

Instead use genqamdemod to offset the initial phase of the QAM data being demodulated or you can multiply the modulated input to qamdemod by the desired initial phase:

z = qamdemod(y .* exp(-1i*initPhase,M))

Extended Capabilities

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