EE5355 - Project INTDCT’s used in AVS China, H.264 and WMV-9 are given1. Check for the orthogonality of the NxN Integer cosine transform matrices. Display the orthogonalised matrices.Programming:clc;close all;clear all;%initializing the variables% X(8,4,6)=0;I_DCT(8,8,3)=0;DOT(8,8,3)=0;M(8,8)=0;C(8,8,3)=0;S(3,2)=0;p=1;Gain_tc(3)=0;I(8,8,10)=0;k(10)=0;t=0;n(8,3)=0;%inserting the values in the matrices% X(:,:,1)=[8 8 8 8;10 9 6 2;10 4 -4 -10;9 -2 -10 -6;8 -8 -8 8;6 -10 2 9;4 -10 10-4;2 -6 9 -10];X(:,:,2)=[8 8 8 8;12 10 6 3;8 4 -4 -8;10 -3 -12 -6;8 -8 -8 8;6 -12 3 10;4 -8 8 -4;3 -6 10 -12];X(:,:,3)=[12 12 12 12;16 15 9 4;16 6 -6 -16;15 -4 -16 -9;12 -12 -12 12;9 -16 4 15;6 -16 16 -6;4 -9 15 -16];%creating loops to take the diagonization% for j=1:3for i=1:8 X(i,:,j+3)=(((-1)^(i))*X(i,:,j)); %IDCT matrix as described% I_DCT(:,:,j)=cat(2,X(:,:,j),fliplr(-X(:,:,j+3))); %to check the orthogonality ,multiply matrix with its transpose as described by problem% DOT(:,:,j)=I_DCT(:,:,j)*(I_DCT(:,:,j)'); for k=1:8 M(i,k)=(0.95)^(abs(i-k)); endend%diagonilized the matrices and storing the values in matrix% C(:,:,j)=DOT(:,:,j)*M*DOT(:,:,j)';S(j,1)=(sum(diag(C(:,:,j)))/.8); for k=1:8p=p*C(k,k,j);endS(j,2)=(p)^(1./8);Gain_tc(j)=S(j,1)./S(j,2);end disp('Orthogonal Matrices');disp('AVS China');disp(DOT(:,:,1));disp('H.264');disp(DOT(:,:,2));disp('WMV9');disp(DOT(:,:,3));Visualization:Orthogonal Matrices:AVS China 512 0 0 0 0 0 0 0 0 442 0 0 0 0 0 0 0 0 464 0 0 0 0 0 0 0 0 442 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 442 0 0 0 0 0 0 0 0 464 0 0 0 0 0 0 0 0 442H.264 512 0 0 0 0 0 0 0 0 578 0 0 0 0 0 0 0 0 320 0 0 0 0 0 0 0 0 578 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 578 0 0 0 0 0 0 0 0 320 0 0 0 0 0 0 0 0 578WMV9 Columns 1 through 8 1152 0 0 0 0 0 0 0 0 1156 0 0 0 0 0 0 0 0 1168 0 0 0 0 0 0 0 0 1156 0 0 0 0 0 0 0 0 1152 0 0 0 0 0 0 0 0 1156 0 0 0 0 0 0 0 0 1168 0 0 0 0 0 0 0 0 1156Observation:After performing the diagonalization , I get the orthogonal matrices. From the output it can bevisible that all matrices are dimension with 8*8 .and all the diagonalized element are notequal to each other. As the diagonal elements of all the matrices are not equal, i can conclude that those are perfectly orthogonalized.2. Perform comparison with the following properties of these transforms: a. Transform coding gain. b. Residual correlation: Plot the residual correlation in % versus ρ. c. Variance distribution in the transform domain d. Normalized basis restriction error (Jm). e. Normalized MSE. f. Rate versus distortion. Plot rate distortion function RD versus fixed average distortion D.Programming:%diagonilized the matrices and storing the values in matrix% C(:,:,j)=DOT(:,:,j)*M*DOT(:,:,j)';S(j,1)=(sum(diag(C(:,:,j)))/.8); for k=1:8p=p*C(k,k,j);endS(j,2)=(p)^(1./8);%calculating the gain factor for different matrices%Gain_tc(j)=S(j,1)./S(j,2);end disp('Orthogonal Matrices');disp('AVS China');disp(DOT(:,:,1));disp('H.264');disp(DOT(:,:,2));disp('WMV9');disp(DOT(:,:,3)); for l=1:10for i=1:8 for j=1:8 I(i,j,l)=(0.05+((l-1)*0.1))^(abs(i-j)); endendk(l)=100*sum(sum(I(:,:,l)-diag(diag(I(:,:,l)))));end%plotting of residual correlation%semilogy(0.05:0.1:0.95,k);title('Residual Correlation vs row');xlabel('row');ylabel('Residual Correlation in %');disp('Transform gain coding Gain_tc values are');disp('AVS China');disp(Gain_tc(1));disp('H.264');disp(Gain_tc(2));disp('WMV9');disp(Gain_tc(3)); for l=1:3 for i=1:8 for j=i:8 t=t+sum(diag(C(j,j,l))); end n(i,l)=t; endend J(1,:)=n(:,1)./(S(1,1)*8);J(2,:)=n(:,2)./(S(2,1)*8);J(3,:)=n(:,3)./(S(3,1)*8);figure;semilogy(J');legend('AVS China','H.264','WMV9');title('Jm');xlabel('Samples m');ylabel('% Jm');OUTPUT :2.a> Transform gain coding Gain_tc values areAVS China 10.0746H.264 5.1223e-005WMV9 1.9841e-0102.b>Residual correlation:2.c> Variance distribution in the transform domain and plot Variance versus N. Output are:AVS China 512 0 0 0 0 0 0 0 0 442 0 0 0 0 0 0 0 0 464 0 0 0 0 0 0 0 0 442 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 442 0 0 0 0 0 0 0 0 464 0 0 0 0 0 0 0 0 442H.264 512 0 0 0 0 0 0 00 578 0 0 0 0 0 0 0 0 320 0 0 0 0 0 0 0 0 578 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 578 0 0 0 0 0 0 0 0 320 0 0 0 0 0 0 0 0 578WMV9 Columns 1 through 8 1152 0 0 0 0 0 0 0 0 1156 0 0 0 0 0 0 0 0 1168 0 0 0 0 0 0 0 0 1156 0 0 0 0 0 0 0 0 1152 0 0 0 0 0 0 0 0 1156 0 0 0 0 0 0 0 0 1168 0 0