DIGITAL COMMUNICATION LAB REPORT
Experiment Number 7 QUADRATURE AMPLITUDE MODULATION SOFTWARE IMPLEMENTATION
Submitted by: R. Sak Saketh
B140540EC
R. M. Alag Alagapp ppan an
B140490EC
R.V.V. Ashish
B140614EC
Venk Venkatesh. U
B140709EC
S.V.Krishna Naik Naik
B120820EC
AIM:
To perform QAM for different constellations (rectangular, 1 circular (1 circle), 2 circular (2 circles)) and to compare the measured error rate with the theoretical error probabilities and to compare the error probabilities for each constellation to find which one has the highest performance. THEORY:
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 (modulating) the amplitudes of two carrier waves, using the amplitude-shift keying (ASK) digital modulation scheme or amplitude modulation (AM) analog modulation scheme. The two carrier waves of the same frequency, 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 final waveform is a combination of both phase-shift keying (PSK) and amplitude-shift keying (ASK), or, in the analog case, of phase modulation (PM) and amplitude modulation. Rectangular QAM constellations are, in general, sub -optimal in the sense that they do not maximally space the constellation points for a given energy. However, they have the considerable advantage that they may be easily transmitted as two pulse amplitude modulation (PAM) signals on quadrature carriers, and can be easily demodulated. The circular QAM constellations highlights the relationship between QAM and PSK. Although, in general, there is a non-rectangular constellation that is optimal for a particular M, they are not often used since the rectangular QAMs are much easier to modulate and demodulate.
Rectangular QAM
Circular QAM
CODE:
clc clear close all nbits=3e5; M=8; nBitsPerSym = log2(M); map_1=[-1+1j,-3+1j,-1-1j,-3-1j,1+1j,3+1j,1-1j,3-1j]; map_2=[2,-2,2j,-2j,(1+j)*sqrt(2),(1-j)*sqrt(2),(1+j)*sqrt(2),(-1-j)*sqrt(2)]; map_3=[1+1j,1-1j,-1+1j,-1-1j,3,3j,-3,-3j]; array=[map_1;map_2;map_3]; EsN0dB=0:18; EbN0dB=EsN0dB-10*log10(nBitsPerSym); symbErrors = zeros(3,length(EbN0dB)); for h=1:3 map=array(h,:); figure plot(real(map),imag(map),'r*'); title('Constellation diagram for Transmitted Symbols'); xlabel('Inphase component'); ylabel('Quadrature component'); axis([-5 5 -5 5]); sym_map =sqrt(1/6)*map; refI = real(sym_map); refQ = imag(sym_map); simulatedBER(h,:)= zeros(1,length(EbN0dB)); theoreticalBER(h,:) = zeros(1,length(EbN0dB)); theoreticalSER(h,:) = zeros(1,length(EbN0dB)); count=1; for i=EbN0dB data_bits=double(rand(1,nbits)>=0.5); inputSymBin=reshape(data_bits,nBitsPerSym,[])'; b = inputSymBin*(2.^((nBitsPerSym-1):-1:0)).'; s=sym_map(b+1).'; EbN0 = 10.^(i/10); EsN0 = 10.^(EsN0dB(count)/10); noiseSigma = sqrt(1./(2*nBitsPerSym*EbN0)); n = noiseSigma*(randn(1,length(s))+1i*randn(1,length(s)))'; y = s + n; demodSymbols = zeros(1,length(y));
for j=1:length(y) [minVal,minindex]=min(sqrt((real(y(j))refI(1,:)).^2+... (imag(y(j))-refQ(1,:)).^2)); demodSymbols(j)=minindex-1; end symbErrors_t=b.'-demodSymbols; symbErrors(h,count)=sum(symbErrors_t(:)~=0)/(nbits/nBitsPerSym ); demodBits=dec2bin(demodSymbols)-'0'; outputSymBin=reshape(demodBits.',1,[])'; bitErrors=sum(sum(xor(outputSymBin.',data_bits))); simulatedBER(h,count) = bitErrors/nbits; theoreticalBER(1,count)= 2*erfc(sqrt((9/14)*EbN0)); theoreticalBER(2,count)= erfc(sqrt(3*EbN0/14))*sin(pi/8); theoreticalBER(3,count)= 3.5*erfc(sqrt(EbN0)); count=count+1; end end theoreticalSER=(theoreticalBER)*(nBitsPerSym); figure; semilogy(EbN0dB,simulatedBER(1,:),'k-o');hold on; semilogy(EbN0dB,simulatedBER(2,:),'r-*');hold on; semilogy(EbN0dB,simulatedBER(3,:),'g-+');hold off; title('BER Vs Eb/N0 (dB) for 8-QAM Modulation-simulated'); legend('rectangular','circular(1)','circular(2)','theoretical' ); axis('tight'); grid on; xlabel('Eb/N0 dB'); ylabel('BER - Bit Error Rate'); grid on; figure; semilogy(EbN0dB,theoreticalBER(1,:),'k-o');hold on; semilogy(EbN0dB,theoreticalBER(2,:),'r-*');hold on; semilogy(EbN0dB,theoreticalBER(3,:),'g-+');hold off; title('BER Vs Eb/N0 (dB) for 8-QAM Modulation-theoretical'); legend('rectangular','circular(1)','circular(2)','theoretical' ); axis('tight'); grid on; xlabel('Eb/N0 dB'); ylabel('BER - Bit Error Rate'); grid on;
figure; semilogy(EbN0dB,simulatedBER(1,:),'k-o');hold on; semilogy(EbN0dB,theoreticalBER(1,:),'b-'); title('BER Vs Eb/N0 (dB) for 8-QAM Modulation'); legend('rectangular','theoretical'); xlabel('Eb/N0 dB'); ylabel('BER - Bit Error Rate'); grid on; figure; semilogy(EbN0dB,simulatedBER(2,:),'r-*');hold on; semilogy(EbN0dB,theoreticalBER(2,:),'b-'); title('BER Vs Eb/N0 (dB) for 8-QAM Modulation'); legend('circular(2)','theoretical'); xlabel('Eb/N0 dB'); ylabel('BER - Bit Error Rate'); grid on; figure; semilogy(EbN0dB,simulatedBER(3,:),'g-+');hold on; semilogy(EbN0dB,theoreticalBER(3,:),'b-'); title('BER Vs Eb/N0 (dB) for 8-QAM Modulation'); legend('circular(1)','theoretical'); xlabel('Eb/N0 dB'); ylabel('BER - Bit Error Rate'); grid on; semilogy(EsN0dB,symbErrors,'k-o');hold on; title('SER Vs Es/N0 (dB) for 8-Ary Modulation'); xlabel('Es/N0 dB'); ylabel('SER - Symbol Error Rate'); grid on; figure; semilogy(EsN0dB,symbErrors(1,:),'k-o');hold on; semilogy(EsN0dB,symbErrors(2,:),'b-o');hold on; semilogy(EsN0dB,symbErrors(3,:),'r-o');hold on; semilogy(EsN0dB,theoreticalSER(1,:),'k-*');hold on; semilogy(EsN0dB,theoreticalSER(2,:),'b-*');hold on; semilogy(EsN0dB,theoreticalSER(3,:),'r-*');hold off; title('SER Vs Es/N0 (dB) for 8-Ary Modulation'); legend('Rect','Circ2','Circ1','TheoreticalRect','Theoreticalci rc2','Theoreticalcirc1'); grid on; xlabel('Es/N0 dB'); ylabel('SER - Symbol Error Rate'); grid on;
PLOTS:
1. Constellation diagrams without noise.
1.
Rectangular
2.
Circular (1 Circle)
3.
Circular (2 Circles)
2. Constellation diagrams with noise(AWGN).
1.
Rectangular
2.
Circular (1 Circle)
3.
Circular (2 Circles)
3. BER Comparison with theoretical values.
Here it can be seen that the circular 2 constellation has the best BER performance among the three. This is because the constellation points are the farthest away from each other in comparison, giving the best performance.
4. Comparison of each scheme with theoretical and simulated.
RESULT:
QAM modulation schemes were implemented as a simulation and their error performance is evaluated on the basis of BER.
