HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
SCHOOL OF ELECTRONICS AND TELECOMMUNICATION
REPORT FOUR LEVEL QUADRATURE AMPLITUDE MODULATION (4-QAM)
-
Bùi Việt Trường Giang
MSSV: 20131088
- Tr ần Mạnh Tuấn
MSSV: 20134338
Ngô Chí Trung
MSSV: 20134156
-
Class: Electrical Electronic Engineering K58
HaNoi, 09/11/2016
Table of Contents I. Introduction ............................................................. ............................................................................................................................. ........................................................................ ........…. ….3 ....................................................................................................................... ..................................................................................... ..................... 3 II. Definition....................................................... ..................................................................................................................................................... 3 1. QAM .................................................................................................................................................... .................................................................................................................... 3 2. Four level QAM (4-QAM) ..................................................................................................................
III. 4-QAM Modulation .............................................................................. ......................................................................................................................... ........................................... 4 IV.
4-QAM Demodulation ....................................................... ..................................................................................................................... ................................................................ 7
V. 4-QAM Modulation/Demodulation simulation (using Matlab) .................................................... 7 ...................................................................................................................................................... ........... 7 1.Code ............................................................................................................................................ .................................................................................................................................................. 10 2. Result ..................................................................................................................................................
VI. Bit error rate (BER) ......................................................... ...................................................................................................................... ............................................................. 11 ...................................................................................................................... ................................................................................... ................... 11 1. BER theory ...................................................... ................................................................. 13 2. BER simulation using Matlab and comparing with theory .................................................................
VII. Signal error rate ........................................................................................................................... ........................................................................................................................... 15 ........................................................................................................................................ ......... 15 1. SER theory. ................................................................................................................................
2. SER simulation by using Matlab and comparing with theory ............................................................ 17 VIII. CONCLUSION ............................................................................................ ........................................................................................................................... ............................... 19
I. Introduction With the fast development of modern communication techniques, the demand for reliable high date rate transmission is increased significantly, which stimulate much interest in modulation techniques. Different modulation techniques allow you to send different bits per symbol and thus achieve different throughputs or efficiencies. QAM is one of widely used modulation techniques because of its efficiency in power and bandwidth. In QAM system, two amplitude-modulated (AM) signals are combined into a single channel, there by doubling the effective bandwidth. The QAM is one of the adaptive modulation techniques that are commonly used for wireless communications. Different order modulations allow sending more bits per symbol and thus achieving higher throughputs or better spectral efficiencies. This method of modulation has the advantage of reducing or eliminating intermodulation interference caused by a continuous carrier near the modulation sidebands.
II. Definition 1. QAM Quadrature amplitude modulation (QAM) is both an analog and a digital modulation scheme. It conveys two analog message signals, or two digital bit streams, by changing the amplitudes of two carrier waves, using the amplitude-shift keying (ASK) digital modulation scheme or amplitude modulation (AM) analog modulation scheme These two waves, usually sinusoids, are out of phase with each other by 90° and are thus called Quadrature carriers or Quadrature components — hence the name of the scheme. The modulated waves are summed, and the resulting waveform is a combination of both phase-shift keying (PSK) and amplitude-shift keying, or in the analog case of phase modulation (PM) and amplitude modulation. In the digital QAM case, a finite number of at least two phases, and at least two amplitudes are used. PSK modulators are often designed using the QAM principle, but are not considered as QAM since the amplitude of the modulated carrier signal is constant. 2. Four level QAM (4-QAM) 4-QAM consists of four unique combinations of phase and amplitude. Each combination is assigned a 2-bit digital pattern. For example, suppose you want to
pattern, these bits are grouped in two’s so that they can be mapped to the corresponding symbols. In our example, the original bit st ream (1,0,0,1,1,1) is grouped into the three symbols (10,01,11). These combinations are shown as the white dots on the constellation plot in Figure 2.1.4 The red lines represent the phase and amplitude transitions from one symbol to another. Labeled on the constellation plot is the digital bit pattern that each symbol represents. Thus, a digital bit pattern can be sent over a carrier signal by generating unique combinations of phase and amplitude.
Figure 2.1 4-QAM symbol map
III. 4-QAM Modulation The QAM modulation scheme encodes data by varying both amplitude and phase of the carrier signal. Thus, it is sometimes viewed as a combination of ASK and PSK modulation. A more fundamental way of viewing QAM thought is that it encodes data by varying the amplitude of two carrier signals that are in-quadrature (phase difference of 90). Mathematically, 4-ary QAM is described by
cos2+ θ
Quadrature amplitude modulation is a modulation scheme that creates a modulation signal from a binary bit stream. The binary data is broken up into bit sets. Each bit set is represented on a constellation. The position of the point on the constellation representing the bit set is mapped to In-phase and Quadrature components
In above equation,
+ represents the in-phase and
represents the quadrature
component. Since the QAM in the software was at baseband frequencies, mixing of the in phase and quadrature parts of the QAM symbol was not needed. However, for transmission of a QAM symbol it must be mixed to higher frequencies for transmission, and can be represented as:
cos2πt cos2πt cos2 sin2 cos sin =
Where
θ)
0 ℎ ℎ Using the complex In the case of a QAM square constellation, the pairs of coordinates from a square matrix as shown by:
+1, 1 +3, 1 …. 1, 1 +1, 3 +3, 3 … 1,3 .. .. .. {,} . . . +1, +1+3, +1 … 1, +1
Where
= √
For 4-QAM, we have:
{, } 1,1,11 1,1,11
Using the complex envelope notation, a 4 level QAM constellation was used to represent the combinational pairs of binary values.
Figure 3.1 4 level QAM constellation The block diagram of 4-QAM demodulation
Figure 3.2 Block diagram of 4-QAM modulation First the flow of bits to be transmitted is split into two equal parts: this process generates two independent signals to be transmitted. They are encoded separately just like they were in an amplitude-shift keying (ASK) modulator. Then one channel (the one "in phase") is multiplied by a cosine, while the other channel (in "Quadrature") is multiplied by a sine. This way there is a phase of 90° between them. They are simply added one to the other and sent through the real channel.
IV. 4-QAM Demodulation
Figure 4.1 Block diagram of 4-QAM demodulation Multiplying by a cosine (or a sine) and by a low-pass filter it is possible to extract the component in phase (or in Quadrature). Then there is only an ASK demodulator and the two flows of data are merged back.
V. 4-QAM Modulation/Demodulation simulation (using Matlab) 1.Code %>>>>>>>>>>>>>>>>>> Matlab code for 4 ary-QAM modulation >>>>>>>>>>>>>>>>>% clc; clear all; close all; M=4; %M=input(' enter the value of M array for QAM modulation : '); fprintf('\n\n\n'); %XXXXXXXXXXXXXXXXXXXXXXXXXXX input chaking loop XXXXXXXXXXXXXXXXXXXXXXXXXX Ld=log2(M); ds=ceil(Ld); dif=ds-Ld; if(dif~=0) error('the value of M is only acceptable if log2(M)is an integer'); end
%XXXXXXXXXXXXXXXXXXX binary Information Generation XXXXXXXXXXXXXXXXXXXXXXXX nbit=16; %number of information bits msg=round(rand(nbit,1)); % information generation as binary form disp(' binary information at transmitter '); disp(msg); fprintf('\n\n');
x=msg; bp=.000001; bit=[]; for n=1:1:length(x) if x(n)==1; se=ones(1,100); else x(n)==0; se=zeros(1,100); end bit=[bit se];
% bit period
end t1=bp/100:bp/100:100*length(x)*(bp/100); figure(1) subplot(3,1,1); plot(t1,bit,'lineWidth',2.5);grid on; axis([ 0 bp*length(x) -.5 1.5]); ylabel('amplitude(volt)'); xlabel(' time(sec)'); title('transmitting information as digital signal');
% binary information convert into symbolic form for 4-array QAM modulation M=M; % order of QAM modulation msg_reshape=reshape(msg,log2(M),nbit/log2(M))'; disp(' information are reshaped for convert symbolic form'); disp(msg_reshape); fprintf('\n\n'); size(msg_reshape); for(j=1:1:nbit/log2(M)) for(i=1:1:log2(M)) a(j,i)=num2str(msg_reshape(j,i)); end end
as=bin2dec(a); ass=as'; figure(1) subplot(3,1,2); stem(ass,'Linewidth',2.0); title('serial symbol for 4-array QAM modulation at transmitter'); xlabel('n(discrete time)'); ylabel(' magnitude'); disp('symbolic form information for 4-array QAM '); disp(ass); fprintf('\n\n'); %XXXXXXXXXXXXXX Mapping for 4-array QAM modulation XXXXXXXXXXXXXXXXXXXXXXXX M=M; %order of QAM modulation x1=[0:M-1]; p=qammod(ass,M) %constalation design for M-array QAM acording to symbol sym=0:1:M-1; % considerable symbol of M-array QAM, just for scatterplot pp=qammod(sym,M); %constalation diagram for M-array QAM scatterplot(pp),grid on; title('constellation diagram for 4-array QAM'); %XXXXXXXXXXXXXXXXXXXXXX RR=real(p) II=imag(p) sp=bp*2;
4-array QAM modulation XXXXXXXXXXXXXXXXXXXXXXXXXXX
%symbol period for 4-array QAM
f=sr*2; t=sp/100:sp/100:sp; ss=length(t); m=[]; for(k=1:1:length(RR)) yr=RR(k)*cos(2*pi*f*t); % inphase or real component yim=II(k)*sin(2*pi*f*t); % Quadrature or imagenary component y=yr+yim; m=[m y]; end tt=sp/100:sp/100:sp*length(RR); figure(1); subplot(3,1,3); plot(tt,m); title('waveform for 4-array QAM modulation acording to symbolic information'); xlabel('time(sec)'); ylabel('amplitude(volt)'); %XXXXXXXXXXXXXXXXXXXX 4-array QAM demodulation XXXXXXXXXXXXXXXXXXXXXXXXXXXX m1=[]; m2=[]; for n=ss:ss:length(m) t=sp/100:sp/100:sp; y1=cos(2*pi*f*t); % inphase component y2=sin(2*pi*f*t); % quadrature component mm1=y1.*m((n-(ss-1)):n); mm2=y2.*m((n-(ss-1)):n); z1=trapz(t,mm1) % integration z2=trapz(t,mm2) % integration zz1=round(2*z1/sp) zz2=round(2*z2/sp) m1=[m1 zz1] m2=[m2 zz2] end %XXXXXXXXXXXXXXXXXXX de-mapping for 4-array QAM modulation XXXXXXXXXXXXXXXX clear i; clear j; for (k=1:1:length(m1)) gt(k)=m1(k)+j*m2(k); end gt ax=qamdemod(gt,M); figure(3); subplot(2,1,1); stem(ax,'linewidth',2); title(' re-obtain symbol after 4-array QAM demodulation '); xlabel('n(discrete time)'); ylabel(' magnitude'); disp('re-obtain symbol after 4-array QAM demodulation '); disp(ax); fprintf('\n\n');
bi_in=dec2bin(ax); [row col]=size(bi_in); p=1; for(i=1:1:row) for(j=1:1:col) re_bi_in(p)=str2num(bi_in(i,j)); p=p+1;
end disp('re-obtain binary information after 4-array QAM demodulation'); disp(re_bi_in') fprintf('\n\n');
%XX representation of receiving binary information as digital signal XXXXXX x=re_bi_in; bp=.000001; % bit period bit=[]; for n=1:1:length(x) if x(n)==1; se=ones(1,100); else x(n)==0; se=zeros(1,100); end bit=[bit se]; end t1=bp/100:bp/100:100*length(x)*(bp/100); figure(3) subplot(2,1,2); plot(t1,bit,'lineWidth',2.5);grid on; axis([ 0 bp*length(x) -.5 1.5]); ylabel('amplitude(volt)'); xlabel(' time(sec)'); title('receiving information as digital signal after 4-array QAM demoduation'); %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> end of program <<<<<<<<<<<<<<<<<<<<<<<<<
2. Result
Figure 5.1 4-QAM modulation
Figure 5.2 4 level QAM Constellation
Figure 5.3 4-QAM Demodulation
VI. Bit error rate (BER) 1. BER theory Bit error rate, BER is a key parameter that is used in assessing systems that transmit digital data from one location to another. Systems, for which bit error rate-BER is applicable, include radio data links as well as fibre optic data systems, Ethernet, or
any system that transmits data over a network of some form where noise, interference, and phase jitter may cause degradation of the digital signal. Although there are some differences in the way these systems work and the way in which bit error rate is affected, the basics of bit error rate itself are still the same. When data is transmitted over a data link, there is a possibility of errors being introduced into the system. If errors are introduced into the data, then the integrity of the system may be compromised. As a result, it is necessary to assess the performance of the system, and bit error rate, BER, provides an ideal way in which this can be achieved. Unlike many other forms of assessment, bit error rate, BER assesses the full end to end performance of a system including the transmitter, receiver and the medium between the two. In this way, bit error rate, BER enables the actual performance of a system in operation to be tested, rather than testing the component parts and hoping that they will operate satisfactorily when in place. As the name implies, a bit error rate is defined as the rate at which errors occur in a transmission system. This can be directly translated into the number of errors that occur in a string of a stated number of bits. The definition of bit error rate can be translated into a simple formula:
,
Expressions for the symbol-error rate of rectangular QAM are not hard to derive but yield rather unpleasant expressions. For an even number of bits per
symbol, , exact expressions are available. They are most easily expressed in a per carrier sense:
So
21 √ 1 13 1 1
The bit-error rate depends on the bit to symbol mapping, but for
≫1
and a
Gray-coded assignment — so that we can assume each symbol error causes onl y one bit error- the bit-error rate is approximately
3 ≈ 12 4 1 √ 1 1
Since the carriers are independent, the overall bit error rate is t he same as the per-carrier error rate, just like BPSK and QPSK.
An exact and general closed-form expression of the Bit Error Rates (BER) for rectangular type of Quadrature Amplitude Modulation (QAM) o ver AWGN and slow, flat, Rician fading channels were derived analytically. Consider a (L×M)-QAM system with
2.
levels and
2.
levels in the I-channel and Q-channel,
respectively and a two-dimensional grey code mapping employed. The generalized
1 | . ∑= | + ∑= |
expression for the conditional BER on SNR over AWGN channel is
Where
)− . (− − 2 ∙2 − | ∑= 1 . 2 + 12∙2+1 +6 2 2. BER simulation using Matlab and comparing with theory Matlab code
% Step 1. Generate message signal of length >= M^L. M = 64; % Alphabet size of modulation L = 1; % Length of impulse response of channel msg = [0:M-1 0]; % M-ary message sequence of length > M^L
% Step 2. Modulate the message signal using baseband modulation. hMod = comm.RectangularQAMModulator(M);
% Use 4-QAM.
modsig = step(hMod,msg'); % Modulate data Nsamp = 4; modsig = rectpulse(modsig,Nsamp); % Use rectangular pulse shaping.
% Step 3. Apply a transmit filter. txsig = modsig; % No filter in this example
% Step 4. Run txsig through a noiseless channel.
rxsig = txsig*exp(1i*pi/180); % Static phase offset of 1 degree % Step 5. Use the semianalytic function. % Specify the receive filter as a pair of input arguments. % In this case, num and den describe an ideal integrator. num = ones(Nsamp,1)/Nsamp; den = 1; EbNo = 0:20; % Range of Eb/No values under study ber = semianalytic(txsig,rxsig,'qam',M,Nsamp,num,den,EbNo);
% For comparison, calculate theoretical BER. bertheory = berawgn(EbNo,'qam',M);
% Plot computed BER and theoretical BER. figure; semilogy(EbNo,ber,'k*'); hold on; semilogy(EbNo,bertheory,'ro'); title('Semianalytic BER Compared with Theoretical BER'); legend('Semianalytic BER with Phase Offset',... 'Theoretical BER Without Phase Offset','Location','SouthWest'); hold off;
Result
Figure 6.1 BER of 4-QAM
From figure, we see that the simulation results and calculation theory are almost same.
VII. Signal error rate 1. SER theory. The scaling factor of
is for normalizing the average energy of the transmitted
symbols to 1, assuming that all the constellation points are equally likely. Q
1+
1+ I
1 Assuming that the additive noise
1 follows the Gaussian probability distribution
function,
√
Consider the symbol of of y given
with
0 and
, the conditional probability distribution function (PDF)
was transmitted is:
−− 1 |2
Figure 7.1 Probability density function for 4QAM modulat ion
|(>|)>|
As can be seen from the above figure, the symbol
is decoded correctly only if y
falls in the area in the hashed region i.e.
− − (>|)1 1 −∞ 1 12 2 √ ∫∞ − − − 1 (>|)1 −∞ 1 12 2 |1 12 2
Probability of real component of y greater than 0, given
was transmitted is (i.e.
area outside the red region):
Where the complementary error function,
Similarly, probability of imaginary component of y greater than 0, given transmitted is (i.e. area outside the blue region).
The probability of
being decoded correctly is,
was
1 + 1 2+ 14 2
=
Hence, the total symbol error probability: the symbol will be in error, it at least one of the symbol is decoded incorrectly. The probability of symbol error is,
1| 1 + = 1-
=
2. SER simulation by using Matlab and comparing with theory
Matlab code
% symbol error rate for 4-QAM modulation
clear N = 10^5; % number of symbols Es_N0_dB = [-3:20]; % multiple Eb/N0 values ipHat = zeros(1,N); for ii = 1:length(Es_N0_dB) ip = (2*(rand(1,N)>0.5)-1) + j*(2*(rand(1,N)>0.5)-1); % s = (1/sqrt(2))*ip; % normalization of energy to 1 n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % white guassian noise, 0dB variance
y = s + 10^(-Es_N0_dB(ii)/20)*n; % additive white gaussian noise
% demodulation y_re = real(y); % real y_im = imag(y); % imaginary ipHat(find(y_re < 0 & y_im < 0)) = -1 + -1*j; ipHat(find(y_re >= 0 & y_im > 0)) = 1 + 1*j; ipHat(find(y_re < 0 & y_im >= 0)) = -1 + 1*j;
nErr(ii) = size(find([ip- ipHat]),2); % couting the number of errors end
simSer_4QAM = nErr/N; theorySer_4QAM = erfc(sqrt(0.5*(10.^(Es_N0_dB/10)))) (1/4)*(erfc(sqrt(0.5*(10.^(Es_N0_dB/10))))).^2;
close all figure semilogy(Es_N0_dB,theorySer_4QAM,'b.-'); hold on semilogy(Es_N0_dB,simSer_4QAM,'mx-'); axis([-3 15 10^-5 1]) grid on legend('theory-4QAM', 'simulation-4QAM'); xlabel('Es/No, dB') ylabel('Symbol Error Rate') title('Symbol error probability curve for 4-QAM')
Result
Figure 7.2 SER of 4-QAM
From figure, we see that the simulation results and calculation theory are almost same.
VIII. CONCLUSION This report has detailed our implementation of the complete Four Level Quadrature Amplitude Modulation. We simulated 4-QAM modulation/demodulation techniques by using Matlab. Also, Matlab is used to evaluate the performance of 4- QAM technique through plotting the Bit Error Rate (BER) vs Signal Error Rate (SER). Our results match the theoretical values exactly.
REFERENCES
Xiaolong Li, “Simulink -based Simulation of Quadrature Amplitude Modulation (QAM) System”, Proceedings of the 2008 IAJC-IJME International Conference.
https://en.wikipedia.org/wiki/Quadrature_amplitude_modulation
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